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Linear and nonlinear estimators of the O-D matrix Wills, Michael Jeffrey 1978

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LINEAR AND NONLINEAR ESTIMATORS OF THE O-D MATRIX by MICHAEL JEFFREY WILLS M.A.Soc.Sc.(hons), Un i v e r s i t y of St. Andrews, 1968 M.A., U n i v e r s i t y of B r i t i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES i n THE DEPARTMENT OF GEOGRAPHY We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1978 © Michael J e f f r e y W i l l s , 1978 In present ing th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar l y purposes may be granted by the Head of my Department or by h is representat ives . It i s understood that copying or p u b l i c a t i o n of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. Department of gSPfrfeft-PWy The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date CMt>Ur 1Q , t ^ H E i i ABSTRACT The general o b j e c t i v e of t h i s work i s to construct, t e s t and apply a method of estimating a matrix of passenger t r i p s between o r i g i n s and d e s t i n a t i o n s ( 0 - D ) from e x i s t i n g data and without recourse to a survey. This o b j e c t i v e i s attained i n f i v e steps. F i r s t , i t i s shown that e x i s t i n g 0 - D survey methods are expensive, cumbersome and u n r e l i a b l e . Then, three f a m i l i e s of models are hypothesized to estimate the 0 - D matrix from the t r a f f i c volumes observed on highway l i n k s ; these are nonlinear, l i n e a r and sequential models. The t h i r d step s e l e c t s the nonlinear c l a s s of models, which are estimated and systematically tested on a v a r i e t y of data, i n c l u d i n g data f o r Canada as a whole. Here i t i s shown that these estimates give good approximations of the 0 - D matrix together with reasonable parame ters. Given the construction of the f i r s t i n t e r c i t y car and bus 0 - D matrices f o r Canada, a fourth step uses these data to estimate what seems to be the f i r s t complete i n t e r c i t y multimodal passenger t r a v e l demand model f o r the e n t i r e country. This model i s shown to be a s p e c i a l case of a more general and " f l e x i b l e " model, which i s i n turn estimated and analysed from several points of view. In the empirical parts of the work, l i k e l i h o o d r a t i o t e s t s are used throughout and f a m i l i e s of models are h i e r a r c h i c a l l y nested, leading to a natural framework f o r the evaluation of successive r e s t r i c t i o n s on the most general formulations. E f f i c i e n t algorithms are developed which permit the estimation of large and complex models on extensive datasets. Table of Contents Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES x LIST OF FIGURES x i i i ACKNOWLEDGEMENTS XV CHAPTER 1 INTRODUCTION 1 1.1 Objectives and organisation 1 1.2 D e f i n i t i o n s 12 1.3 Notation 14 2 ON THE STRUCTURE OF TRAVEL DEMAND ESTIMATION 18 2.1 Background 18 2.2 Sequential models 20 2.3 Direct models 27 2.3.1 Direct and s p e c i f i c 28 2.3.2 Direct and abstract 29 2.3.3 Quasi-direct and s p e c i f i c 30 2.3.4 Quasi-direct and quasi-abstract 31 3 THEORETICAL SETTING 32 3.1 Introduction 32 3.2 Inadequacy of O-D survey methods 32 3.3 Nature of the O-D matrix and arc volumes 38 3.4 An estimate of the O-D matrix 40 3.4.1 Form of the estimation problem 40 3.4.2 Some sources of s p e c i f i c a t i o n error 44 3.5 Antecedents 48 (iv) (v) Page 4 DIRECT UNCONSTRAINED MODELS. I NONLINEAR MODELS 53 4.1 General structure of the estimation 5 3 problem 4.2 S p e c i f i c a t i o n of the 0-D matrix 54 4.2.1 Some general forms 5 5 4.2.2 Special cases of the general 57 model 4.2.3 S p e c i f i c models 60 4.3 S p e c i f i c a t i o n of assignment 63 4.3.1 Multipath assignment 63 4.3.2 S p e c i f i c models 67 5 DIRECT UNCONSTRAINED MODELS. II LINEAR MODELS 7 0 5.1 Derivation of the basic l i n e a r model 71 5.2 Transformations of the l i n e a r model 72 5.3 Switching strategies f o r a l i n e a r 0-D estimator 75 5.3.1 Dependent v a r i a b l e c r i t e r i o n 77 5.3.2 J o i n t l y determined by independent variables 78 5.4 Switching strategies with i n e q u a l i t y r e s t r i c t i o n s 79 5.4.1 Least absolute errors 8 0 5.4.2 Least squares 8 3 5.5 Interaction polynomials 86 6 SEQUENTIAL AND COMBINED MODELS 89 6.1 Separate generation and d i s t r i b u t i o n models 89 6.1.1 General form 8 9 6.1.2 Singly constrained t r i p d i s t r i b u t i o n 91 6.1.3 Doubly constrained t r i p d i s t r i b u t i o n 95 6.2 Estimating 0-D from t r i p ends 96 6.3 Combined d i s t r i b u t i o n and assignment 100 6.4 Direct demand with constrained assignment 105 6.4.1 Arc capacity constraints 106 (vi) Page 7 ALGORITHMS FOR O-D ESTIMATION 109 7.1 Introductory remarks 109 7.2 A l i n e a r estimate of the constant i n d i r e c t unconstrained models 110 7.3 Assignment algorithms for proportional methods 116 7.4 Elimination of assignment 121 7.5 Maximal interchange s e l e c t i o n procedure f o r c a l i b r a t i o n 12 3 7.6 Response surfaces 128 7.7 An improved shortest path algorithm 132 8 ASPECTS OF ESTIMATION 138 8.1 Structure of the l i k e l i h o o d function 139 8.1.1 The logarithmic concentrated l i k e l i h o o d function 139 8.1.2 Maximum l i k e l i h o o d estimation of the O-D matrix 141 8.2 S p e c i f i c a t i o n of the s t a t i s t i c a l model 142 8.2.1 Estimation of a generalised l i k e l i h o o d function 14 2 8.2.2 A l t e r n a t i v e d i s t r i b u t i o n a l models 144 8.2.3 Truncated models 145 8.3 S p a t i a l l y dependent error covariances 147 8.3.1 Maximum l i k e l i h o o d estimation 147 8.3.2 The structure of s p a t i a l dependence 153 8.3.3 Estimating s p a t i a l error covariances 157 8.4 A pseudo-capacity estimator 161 8.5 Estimation of s t r u c t u r a l change 164 9 EMPIRICAL RESULTS I: B.C. DATA 166 9.1 Two models 166 9.1.1 A basic model 166 9.1.2 An extended model 16 7 9.2 B r i t i s h Columbia data 169 9.2.1 Introduction 16 9 9.2.2 The basic model 169 ( v i i ) Page 9.2.2.1 Hypothesis t e s t i n g 16 9 9.2.2.2 Maximal interchange 17 4 9.2.2.3 St r u c t u r a l change over time 174 9.2.3 Aspects of the extended model 184 9.2.3.1 P a r t i t i o n s of the o r i g i n constant 184 9.2.3.2 Estimation of l o c a l t r a f f i c 187 9.2.3.3 Maximal interchange i n the extended model 189 9.2.3.4 A l t e r n a t i v e extended models and 0-D estimates 194 9.2.4 Estimating the functional form of the 0-D matrix 19 8 10 EMPIRICAL RESULTS I I : CANADIAN DATA 204 10.1 Introduction 204 10.2 The basic model 205 10.2.1 Hypothesis t e s t i n g 205 10.2.2 Maximal interchange 211 10.3 Aspects of the extended model 215 10.3.1 Estimation of l o c a l t r a f f i c 215 10.3.2 Maximal interchange i n the extended model 218 10.4 Estimation of a bus 0-D matrix 223 10.4.1 Avail a b l e sources of data 223 10.4.2 A comparison of the bus and car estimation problems 2 24 10.4.3 D e f i n i t i o n s of terms 225 10.4.4 The basic model 226 10.4.5 Transfers and intervening opportunities 228 10.4.6 Empirical r e s u l t s 228 11 APPLICATION TO INTERCITY TRAVEL DEMAND I METHODS 234 11.1 Objectives 234 11.2 Methodology 235 11.2.1 Cross-sectional models 235 11.2.2 Abstract mode models 2 37 11.3 S p e c i f i c a t i o n of a demand function 238 ( v i i i ) Page 11.3.1 A two-stage model 23 8 11.3.2 U t i l i t y function 239 11.3.3 Mode s p l i t 240 11.3.4 Impedance and t o t a l demand 241 11.3.5 Special cases 242 11.4 Estimation procedures 245 12 APPLICATION TO INTERCITY TRAVEL DEMAND II EMPIRICAL 248 12.1 Nature of the data 248 12.1.1 Aggregation problems 24 8 12.1.2 Transportation variables 251 12.1.3 Socio-economic variables 254 12.1.4 Form of the model estimated 257 12.2 Results, experiments and comparisons 257 12.2.1 Introduction 257 12.2.2 Experiments with a l l modes 259 12.2.3 Experiments with surface modes 263 12.2.4 Experiments with common c a r r i e r s 265 12.2.5 A comparison of e l a s t i c i t i e s with other models and data 272 13 APPLICATION TO INTERCITY TRAVEL DEMAND III EXTENSIONS AND TESTS 277 13.1 Introduction 277 13.1.1 C r i t e r i a for the comparison of models 279 13.2 Theoretical framework 2 81 13.2.1 General form of the mode s p l i t equation 281 13.2.2 Special cases and t h e i r e l a s t i c i t i e s 282 13.2.3 General form of the t o t a l t r a v e l demand equation 284 13.2.4 Special cases and t h e i r e l a s t i c i t i e s 284 13.3 Approach to estimation 287 13.3.1 Rearrangement for estimation 287 13.3.2 Models selected 288 (ix) Page 13.4 Results f o r the mode s p l i t equation 290 13.4.1 Estimation and tests of the func t i o n a l form 13.4.2 Analysis of residuals 13.4.3 Functional form and t - s t a t i s t i c s 13.4.4 E l a s t i c i t i e s 13.5 Results for the t o t a l t r a v e l demand equation 313 14 CONCLUSIONS 317 BIBLIOGRAPHY 321 APPENDICES 329 A A l t e r n a t i v e d e f i n i t i o n s of the objective function 329 B A n a l y t i c a l gradient of the d i r e c t unconstrained O-D model 3 34 C L i s t of c i t i e s and c i t y - p a i r s 336 290 300 302 305 L i s t of Tables Table Page 3.1 Percentage error bounds for the 0-D matrix 35 3.2 Percentage error bounds for t r i p s generated by a zone 35 9.1 The basic model: unconstrained maximum l i k e l i h o o d estimates of a l t e r n a t i v e hypotheses 171-2-9.2 The basic model: unconstrained maximum l i k e l i h o o d estimates with maximal interchange experiments 175.. 9.3 The basic model: unconstrained maximum l i k e l i h o o d estimates of a serie s of annual cross-sections 17.8 . 9.4 The basic model: constrained maximum l i k e l i h o o d estimates of the simple gravity model for a serie s of annual cross-sections 181_. 9.5 The basic model: ex post facto p r e d i c t i o n using 196 3 parameter values 183 9.6 The extended model: p a r t i t i o n s of the o r i g i n constant 185 9.7 The extended model: p a r t i t i o n s of the arc constant 188 9.8 Maximal interchange i n the extended model: estimated parameters 191 9.9 Maximal interchange i n the extended model: sample estimated O-D's 193 9.10 Al t e r n a t i v e extended models 195 9.11 Estimated 0-D from a l t e r n a t i v e extended models 196 9.12 The basic model. Estimating the functional form. Location of the optimum 200 9.13 The basic model. Estimating the functional form. A comparison of a l t e r n a t i v e forms with X>0 201 (x) (xi) Table Page 9.14 The basic model. Estimating the func t i o n a l form. A comparison of a l t e r n a t i v e forms with X<0 202 10.1 Likelihood r a t i o t ests for the basic model 206 10.2 The basic model: unconstrained maximum l i k e l i h o o d estimates of a l t e r n a t i v e hypotheses 207-8 10.3 The basic model. Likelihood r a t i o t e s t r e s u l t s 209 10.4 Maximal interchange i n the basic model: estimated parameters 212 10.5 The extended model: regional p a r t i t i o n s of the arc constant 216 10.6 Maximal interchange i n the extended model: estimated parameters 219 10.7 Maximal interchange i n the extended model: sample estimated O-D 221 10.8 Estimation and te s t s of a bus O-D model (i) 230 10.9 Estimation and tests of a bus O-D model ( i i ) 231 12.1 Estimates of multimodal parameters 1: experiments on a l l modes (i) 267' 12.2 Estimates of multimodal parameters: experiments on a l l modes ( i i ) 26 8 12.3 Estimates of multimodal parameters: experiments with surface modes (i) 269 12.4 Estimates of multimodal parameters: experiments with surface modes ( i i ) 270 12.5 Estimates of multimodal parameters: experiments with common c a r r i e r s 271 12.6 Comparison of e l a s t i c i t i e s with models using data obtained by other methods 274 12.7 Combined mode s p l i t and t o t a l t r a v e l e l a s t i c i t i e s 276 ( x i i ) Table Page 13.1 Likelihood r a t i o tests f o r the functional form 295 13.2 Conditional t e s t for X 296 13.3 Conditional t e s t for \i 297 13.4 Unconditional t e s t for A and u 29 8 13.5 Extended Box-Tukey transformation of the mode s p l i t equation 301 13.6 Market share e l a s t i c i t i e s for u=0 310 13.7 Market share e l a s t i c i t i e s for y=10 311 13.8 Market share e l a s t i c i t i e s f o r A=-0.2 312 13.9 A l t e r n a t i v e models for t o t a l t r a v e l demand 316 L i s t of Figures Figure Page 1.1 Plan of the work 3 3.1 General scheme to estimate the O-D matrix from arc volumes 41 3.2 Minimally connected network 46 3.3 F u l l y connected network 4 6 3.4 F u l l y connected subgraph i n a general network 4 6 7.1 Response surface f o r l i k e l i h o o d function 131 7.2 Demonstration network for a t r e e - b u i l d i n g algorithm 137 9.1 Convergence paths of parameters with increasing maximal interchange s e l e c t i o n 176 size 9.2 The basic model: unconstrained maximum l i k e l i h o o d estimates of a series of annual cross-sections 179 9.3 The basic model: constrained maximum l i k e l i h o o d estimates of a series of annual cross-sections 182 11.1 Geometric i n t e r p r e t a t i o n of the e l a s t i c i t i e s 244 13.1 The mode s p l i t equation: l i k e l i h o o d function for X given y 291 13.2 The mode s p l i t equation: l i k e l i h o o d surface for X and u 293 13.3 The mode s p l i t equation: skewness and kurtosis as a function of X 303 13.4 The mode s p l i t equation: t - s t a t i s t i c s as a function of X, given y=0 304 13.5 The mode s p l i t equation: t - s t a t i s t i c s as a function of X, given y=20 306 ( x i i i ) (xiv) Figure Page 13.6 The mode s p l i t equation: fare e l a s t i c i t i e s 308 13.7 The mode s p l i t equation: time e l a s t i c i t i e s 309 13.8 The t o t a l t r a v e l equation: l i k e l i h o o d surface f o r a two-A model 314 Acknowledgements xv The author would l i k e to acknowledge the a s s i s t a n c e and encouragement he has been g i v e n by h i s c o l l e a g u e s over the past s e v e r a l years i n the completion of t h i s d i s s e r t a t i o n . In p a r t i c u l a r , Dr. Robert North i s i n s t r u m e n t a l i n encouraging the completion and i n c o o r d i n a t i n g communication with the committee. The l a t t e r i s thanked f o r having s t r u g g l e d through a r a t h e r lengthy document. During the p a s t two y e a r s , a d d i t i o n a l encouragement has been forthcoming from many c o l l e a g u e s a t the U n i v e r s i t e de Montreal, T r a n s p o r t Canada and the Canadian T r a n s p o r t Commission. At the U. de M., p r o f e s s o r s Marc Gaudry and Michael F l o r i a n have g i v e n me hours of s t i m u l a t i n g d i s c u s s i o n . At the CTC, Dr. John Rea and John P l a t t s were a b l e to p r o v i d e me w i t h u s e f u l sources of data and an o p e r a t i o n a l a p p l i c a t i o n of the work. But above a l l , I am indebted to Joseph Greer who has proved a most s t i m u l a t i n g c o l l e a g u e i n completing the e m p i r i c a l aspects of the work. I am a l s o indebted to the T r a n s p o r t Development Agency F e l l o w s h i p Program f o r f i n a n c i a l a s s i s t a n c e i n the p e r i o d 1971-74. The t e x t was typed by J i l l Ansten, Diana Box and Monique S i n o t t e and I am g r a t e f u l f o r t h e i r c a r e f u l work. CHAPTER 1 INTRODUCTION 1.1 Objectives and organisation One input into the development and implementation of passenger transportation p o l i c i e s and plans i s the construction and analysis of mathematical models of i n t e r c i t y t r a v e l demand. These models are expected to address the key questions of the e f f e c t on t r a v e l demands of changes i n the le v e l s of service of e x i s t i n g modes of t r a v e l , of p o t e n t i a l new modes and of exogenous changes i n the socio-economic environment. The models achieve t h i s objective for a set of o r i g i n s and destinations, by r e l a t i n g the observed t r a v e l demand (0-D) to a set of socio-economic and l e v e l of service variables with a set of weights or parameters incorporated into a functional form. These parameters i d e n t i f y the r e l a t i v e importance of the factors a f f e c t i n g the demand for t r a v e l , and from them are obtained the e l a s t i c i t i e s , dimensionless numbers which measure the proportional e f f e c t on one var i a b l e of a proportional change i n another. Whereas most of the data for the s p e c i f i c a t i o n and estimation of a multimodal t r a v e l demand model i s quite r e a d i l y a v a i l a b l e , c e r t a i n variables may be nonexistent thereby f r u s t r a t i n g the completion of the model. In t h i s context, the paucity of 0-D t r a v e l demand data f o r the automobile i s undoubtedly the most c r u c i a l d e f i c i e n c y . Very l i t t l e - 1 -- 2 -data at a l l are extant for t h i s mode, and t h i s seems incongruous for a mode which comprises about 85 percent of a l l i n t e r c i t y t r a v e l . This v i r t u a l absence of O-D demand data f o r the automobile i s explained by a number of f a c t o r s . Unlike other modes, there i s no obvious and inexpensive data c o l l e c t i o n mechanism analogous with the t i c k e t analyses possible f o r a i r , r a i l and bus. Questionnaire-type surveys are expensive to administer and process, and as a consequence are r e s t r i c t e d i n scope and coverage, both s p a t i a l l y and temporally. The problem i s thus defined: how to obtain an estimate of the O-D matrix of t r a v e l by the automobile to permit the s p e c i f i c a t i o n of a multimodal t r a v e l demand model. The p r i n c i p a l objective of t h i s d i s s e r t a t i o n i s to demonstrate that such an estimate can indeed be made, using e x i s t i n g and r e a d i l y a v a i l a b l e inexpensive data. A f t e r an introduction to some basic concepts, the discussion begins with a perspective on the structure of t r a v e l demand, followed by an examination of the s e t t i n g for estimators of the O-D matrix. Developing from t h i s basis, several classes of models are presented together with algorithmic procedures f o r t h e i r e f f i c i e n t implementation. Following t h i s , several aspects of s t a t i s t i c a l estimation are examined with p a r t i c u l a r reference to the structure of the error covariances. The l o g i c a l structure of the work i s completed by presenting empirical r e s u l t s obtained from - 3 -Figure 1.1 Plan of the work 5. Direct l i n e a r models 1. Introduction & ou t l i n e 2. Context of transport demand models 3. Theoreti fo r O-D < zal basis estimation 4. Di r e c t non-linear models 6. Sequential & combined models 7. Algorithms 8. Aspects of estimation 10. Empirical r e s u l t s II: Canadian data 9. Empirical r e s u l t s I: B.C. data 11. I n t e r c i t demand: y t r a v e l methods 12. I n t e r c i t y t r a v e l demand: r e s u l t s 13. Extensions & tests of a generalised demand model - 4 -various datasets, including a series of cross-sections over time f o r B.C. and a s i n g l e cross-section pertaining to Canada as a whole. In addition, the method i s extended to the estimation of a bus 0-D matrix derived from head counts at staging points. F i n a l l y , estimates of car and bus 0-D f o r Canada are added to a multiple mode choice abstract-mode type model i n order to compute e l a s t i c i t i e s . The work i s organised i n the following manner. Chapter 1 o u t l i n e s objectives, organisation, d e f i n i t i o n s and notation. Chapter 2 traces the main elements involved i n the structure of t r a v e l demand estimation as i t i s currently achieved. The chapter opens with a perspective on how the present models have evolved through d i f f e r e n t planning problems and research climates. From t h i s o r i g i n d i f f e r i n g methodologies have appeared and these are contrasted, a strand which i s eventually taken up i n the models fo r 0-D estimation. This i s p r i m a r i l y a contrast between sequential separable and d i r e c t demand approaches. Against t h i s background chapter 3 establishes the s e t t i n g for estimators of the 0-D matrix. The methodology i s outlined i n i t s most general form and i t s f u n c t i o n a l mechanisms examined. Problems encountered with survey estimates of 0-D are discussed and t h i s leads to comments on the nature of the 0-D matrix and on arc volumes. Following t h i s , the methodological perspective of model-building aspects of the work i s established with respect to functional form versus variable considerations. Components of the estimation - 5 -procedure are introduced and s p e c i f i c a t i o n errors a t t r i b u t a b l e to each discussed. The chapter closes with reference to some re l a t e d l i t e r a t u r e . Whether sequential or d i r e c t methods are employed, O-D estimation from arc volumes i s c l e a r l y a procedure which replaces aggregation from O-D to arcs, i n the conventional process, with disaggregation from arcs to O-D. For the l a t t e r to be uniquely defined a. fu n c t i o n a l form which s a t i s f i e s the observed arc volumes has to be postulated and estimated. Thus the r o l e of the next three chapters i s a t h e o r e t i c a l development which r e s t r i c t s and defines functions from much larger classes f e a s i b l e . Only then i s i t possible to estimate parameters to decompose uniquely arc volumes into O-D. Chapter 4 i s the f i r s t of two chapters which describe a class of models c a l l e d d i r e c t unconstrained models. They are " d i r e c t " because the generation and d i s t r i b u t i o n of t r a v e l are re l a t e d i n one estimation step to arc volumes by use of a single equation. They are "unconstrained" because no e f f e c t i v e l i m i t s are imposed on the range of the parameters and comprise no l o g i c a l s t r u c t u r a l constraints of the type employed by sequential models. Di r e c t unconstrained models are p a r t i t i o n e d into two types according to t h e i r form: l i n e a r or non-linear. Chapter 4 i s devoted to non-linear models, whereas chapter 5 develops l i n e a r models, i n i t i a l l y as a s p e c i a l case of the - 6 -non-linear c l a s s , but goes on to demonstrate many extensions po s s i b l e , and necessary, f o r the l i n e a r case. Most of chapter 4 i s taken up by an exposition of m u l t i p l i c a t i v e i n t e r a c t i o n models, of which the f a m i l i a r gravity model i s a s p e c i a l case. Single and multiple parameter models are presented., The l a t t e r allow independent parameters for generation i n addition to d i s t r i b u t i o n (distance-decay) c o e f f i c i e n t s . A general impedance function i s developed which not only u n i f i e s many e x i s t i n g forms but allows estimation of func t i o n a l form at the same time as parametrisation. The chapter ends with a gen e r a l i s a t i o n of the assignment model from the al l - o r - n o t h i n g c r i t e r i o n to multiple path methods. Re s t r i c t e d models are discussed i n the context of a set of nested i t e r a t i v e schemes. Chapter 5 presents the l i n e a r hypothesis. These models are offered as a l t e r n a t i v e functional forms to non-l i n e a r ones on the basis of four s i g n i f i c a n t properties of l i n e a r models. F i r s t , t h e i r s t a t i s t i c a l basis i s much cle a r e r than that of non-linear forms. Second, t h e i r estimation i s e f f i c i e n t and routine. Third, many non-linear forms can be transformed l i n e a r . Fourth, any non-linear function can be approximated by l i n e a r segments. The chapter opens with the de r i v a t i o n of a l i n e a r model. The remainder of the chapter i s devoted to elaborating - 7 -t h i s extremely simple form, while preserving l i n e a r i t y . F i r s t of a l l , power transformations of the Box-Cox v a r i e t y are attempted but i t i s shown that they can be only p a r t l y successful since transformation of the dependent v a r i a b l e (arc volumes) defeats extraction of the O-D estimator. Therefore t h i s a p p l i c a t i o n i s l i m i t e d to transformation of independent variables only. As a r e s u l t of t h i s , the argument turns to switching s t r a t e g i e s to permit n o n - l i n e a r i t i e s over the e n t i r e functional form, but eschewing transformation. Switching strategy involves a p a r t i t i o n of the range of observations and allows d i f f e r e n t l i n e a r parameters to be associated with each p a r t i t i o n . Two a l t e r n a t i v e c r i t e r i a are investigated f o r the p a r t i t i o n j o i n s ; one i s based on the dependent v a r i a b l e , the other on j o i n t consideration of independent v a r i a b l e s . Piecewise methods are very f l e x i b l e from the point of view of estimation. This f l e x i b i l i t y , however, i s a source of weakness where the data e x h i b i t s u b s t a n t i a l error together with few observations per l i n e a r segment. The problem i s that c o e f f i c i e n t s may change slopes d r a s t i c a l l y from one segment to the next as the least-squares c r i t e r i o n attempts to f i t the data. Although t h i s would not necessarily be a problem i n a c u r v e - f i t t i n g exercise, i n the estimation of i n d i v i d u a l O-Ds i t would be disastrous. For example, an - 8 -impedance c o e f f i c i e n t could become l o c a l l y p o s i t i v e f o r a subset of 0-D p a i r s . To eliminate t h i s problem further r e s t r i c t i o n s are necessary. These involve i n e q u a l i t y r e s t r i c t i o n s of two types, one requiring non-negativity (or non-positivity) and the other imposing an ordering on the magnitude of the segment slopes f o r a given v a r i a b l e . These problems lead to a discussion of the a p p l i c a t i o n of l i n e a r and quadratic programming. Although these consider-ations involve estimation, they are c r u c i a l f o r the function forms which may be t h e o r e t i c a l l y a nticipated. Quadratic loss functions for the errors lead to quadratic programming, whereas l i n e a r methods require adoption of l e a s t absolute err o r s . The l a s t of the three t h e o r e t i c a l chapters developing r e s t r i c t e d forms i s chapter 6. Whereas 4 and 5 were devoted to d i r e c t methodologies, 6 presents a c l a s s of multi-staged sequential models i n which generation, d i s t r i b u t i o n and assignment are estimated i n d i f f e r e n t , and usually independent, steps. The question might a r i s e as to why, when a d i r e c t model i s a v a i l a b l e , a t t e ntion should be paid to a sequential one. The problem i s that d i r e c t models have only i m p l i c i t values for parameters such as t r i p generation per c a p i t a and average length of t r i p . In a sequential model, on the other hand, these key parameters can be e x p l i c i t l y c o n t r o l l e d . Consequently, there may be a gain i n a c c e p t a b i l i t y , depending on the - 9 -a p p l i c a t i o n , at the cost of weaker o v e r a l l coherence of methodology. Several types of models are discussed beginning with generation exogenous to d i s t r i b u t i o n with a l l - o r - n o t h i n g assignment to shortest paths. This i s followed by a suggestion for estimating O-D from t r i p ends. A model which combines d i s t r i b u t i o n and assignment thus avoiding some of the s e p a r a b i l i t y problems i s discussed, followed by a d i r e c t model with constrained assignment. Algorithms are systematic procedures which often must be developed i n order to allow models to become computationally f e a s i b l e . In p a r t i c u l a r , i t i s the n o n - l i n e a r i t i e s of the models which requires t h i s . Any device which reduces the computational burden may be regarded as algorithmic i n character, even though i t may be very simple. Chapter 7 i s devoted to these considerations and i s an important aspect since the i t e r a t i v e nature of the majority of the methods e n t a i l s a heavy computational load. The argument so f a r has attempted to e s t a b l i s h the t h e o r e t i c a l basis for estimation, specify a l t e r n a t i v e functional forms and to implement the models by algorithmic development. The next topic which l o g i c a l l y follows i s the s t a t i s t i c a l foundation for estimation. Several aspects of t h i s topic are covered i n chapter 8, including the choice of the loss function, f o r example, l e a s t squares or l e a s t absolute errors; then, given the choice of l e a s t squares, - 10 -discussion of the l i k e l i h o o d function and problems r e l a t e d to the s p a t i a l l y dependent error structures. Methods to deal with s t r u c t u r a l change over time are also developed and discussed. Chapters 9 and 10 present empirical r e s u l t s . These r e l a t e to the Canadian highway network i n i t s e n t i r e t y as well as contiguous USA, and secondly, to B.C. using a f i n e r s p a t i a l grain of anal y s i s . For the Canadian network, co n s i s t i n g of 160 network modes and 300 arcs, an 0-D 2 matrix of (107 -107)/2 c e l l s i s estimated. Only a si n g l e cross-section i s estimated on a nationwide basis. On the other hand, the B.C. data consist of many cross-sections over time. Of these, a set of six consecutive years are selected and the corresponding 0-D matrices estimated for 2 (76 -76)/2 c e l l s . C o e f f i c i e n t s which determine the nature of the 0-D matrix are shown to be stable over time. A l t e r n a t i v e models are tested on one cross-section of B.C. data. Here i t i s shown that a three-parameter model does s u b s t a n t i a l l y better than one with fewer parameters. In addition, an ex post facto analysis i s c a r r i e d out to test the v a l i d i t y of 196 3 c o e f f i c i e n t s when applied to subsequent annual cross-sections. For both the B.C. and the Canadian data, t e s t s are c a r r i e d out to determine whether good parameter estimates can be obtained from using a sample of the O-D matrix. Basic models are extended to include a d d i t i o n a l parameters and p a r t i t i o n s of e x i s t i n g ones. The func t i o n a l form of the O-D matrix i s estimated from the c l a s s defined by Box-Cox transformations of the l i n e a r model. The remainder of chapter 10 extends estimators of the O-D matrix f o r automobile t r a f f i c to estimators f o r bus t r a v e l . Here the r o l e of arc volumes i s replaced by head counts made by bus driv e r s at various staging points. The cost matrix for d i s t r i b u t i o n depends a d d i t i o n a l l y on the existence of tra n s f e r s , on service frequencies, and on the presence and q u a l i t y of competing modes. A s i m p l i f i e d model i s developed and estimated using data f o r Canada, thus producing a f u l l national estimated O-D matrix. Chapters 11 to 13 apply the estimates of Canadian O-D to abstract mode models of i n t e r c i t y t r a v e l . Modal O-D data f o r a i r and r a i l are added to the estimated O-D for car and bus, together with l e v e l of service a t t r i b u t e s , such as fare, t r a v e l times and departure frequency. Abstract mode methodology i s employed and the p r i n c i p a l f u n c t i o n a l forms are exhibited and explained. For a s p e c i f i c form the parameters are estimated together with s e n s i t i v i t y analyses. The conclusion here i s that, using estimated rather than observed O-D gives c o e f f i c i e n t s and e l a s t i c i t i e s that are very s i m i l a r to those from studies which had - 12 -observed O-D a v a i l a b l e . Under parametric updating of the l e v e l of service c h a r a c t e r i s t i c s the demand response from the c a l i b r a t e d model reacts i n a reasonable way. C o e f f i c i e n t s and the nature of the s e n s i t i v i t y experiments are presented. Chapter 13 extends the methods presented i n 11 and 12 to include estimation of the f u n c t i o n a l form within the Box-Cox family of transformations. Here i t i s shown that the choice of form has a decisive influence on the nature of the e l a s t i c i t i e s and on the behaviour of the model i n a number of ways. I t i s concluded that such estimation i s a necessary step i n estimating t r a v e l demand models. 1.2 D e f i n i t i o n s Trip generation involves the p r e d i c t i o n of the t o t a l number of t r i p s per u n i t time leaving an o r i g i n or a r r i v i n g at a d e s t i n a t i o n . Probably the most d i f f i c u l t step to model, t r i p generation i s usually r e l a t e d by a l i n e a r model to a c t i v i t y or socio-economic a t t r i b u t e s of the zones. Trip d i s t r i b u t i o n i s concerned with the estimation of the proportion of generated t r i p s which w i l l be made between an o r i g i n - d e s t i n a t i o n p a i r of zones. This proportion i s a function of a cost matrix and the output of t r i p generation: o r i g i n t o t a l s , destination t o t a l s , and balancing f a c t o r s . The function most commonly employed i s the gravity formulation. - 13 -Modal s p l i t involves the estimation of the proportion of t r i p s between a p a i r of zones which go by a given mode. Its independent variables are various combinations of modal a t t r i b u t e s , such as r a t i o s or differences of cost, t r a v e l time and departure frequency. The most usual dependent variable i s a r a t i o of the observed t r i p s by mode from generation and d i s t r i b u t i o n . Trip assignment i s the process by which the number of o r i g i n -d e stination t r i p s by mode i s loaded on to the highway network or public transport system. Taking the output of generation, d i s t r i b u t i o n and mode-split as exogenous inputs, these demands are al l o c a t e d to an envelope of paths through the network according to behavioural assumptions. A s p e c i a l case, a l l - o r - n o t h i n g assignment', leads a l l 0-D on to a unique path. D e f i n i t i o n s of the network terms are those used in Potts and O l i v e r (1972). In b r i e f , nodes represent points of i n t e r e s t i n the transportation network such as c i t i e s or i n t e r s e c t i o n s , arcs are transport l i n k s such as highway segments, and paths are ordered arc sequences through the network and leading from an o r i g i n node to a destination node. A aentroid or zone i s , i n the i n t e r c i t y case, a c i t y of s u f f i c i e n t s i z e to be included i n the grain of analysis selected for the study. Volume-delay curves are convex functions d e f i n i n g a correspondence between - 14 -t r a f f i c volumes and r e s u l t i n g t r a v e l speeds or times on an arc. A t r i p i s the t r a v e l of one person from one c i t y to another, and i n the demand functions i s treated as an intermediate commodity required for some unspecified f i n a l demand. A mode i s a method of t r a v e l possessing some di s t i n g u i s h i n g c h a r a c t e r i s t i c s i n terms of the var i a b l e s selected to describe i t . If fares and t o t a l t r a v e l time are selected a i r t r a v e l i s f a s t and expensive whereas bus t r a v e l i s cheap and slow. This locates modes or observations i n two-dimensional a t t r i b u t e space. 1.3 Notation th A ^ = observations on the k a c t i v i t y and socio-economic th v a r i a b l e for the i c i t y . Let A^ be a vector of k elements: A. = {P.,Y.,L.,E..} x x x x xk A.., = functions of A. and A. for the ( i , j ) t h c i t y p a i r , x^k x 3 J 2 * Let A.. be a vector of k elements: 13 A. . = {P. . , Y. . , L. . , E. .. } X 3 13 13 X 3 X 3 k where th P^ = population l e v e l of the i c i t y t h t h P^ .. = function of the population of the i and j c i t i e s , e.g., product t h Y^ = income (however defined) of the i c i t y - 15 -Y.. = function of the incomes i n the i t h and j t h 13 c i t i e s , e.g., product, weighted mean = proportion speaking English (mother tongue) • 4.1, -th i n the 1 c i t y L^j = function of l i n g u i s t i c composition i n the i and c i t i e s , e.g., a measure of l i n g u i s t i c s i m i l a r i t y defined as: L . . = 1 - IL.-L.I _.- ' 1 3 An a l t e r n a t i v e d e f i n i t i o n i s L L*. = EL..L. (for L languages). 13 ji 3 * E i k = e m P x ° y m e n t i n t n e a c t i v i t y i n the i t h c i t y E ^ j k = function of employment i n the i and j c i t i e s , e.g., product, mean, etc. = car ownership per capi t a i n c i t y i R. . = function of R. and R. 13 1 3 t tl C.. . = observations on the I " l e v e l of se r v i c e " , 1 jmS, generalised cost, distance or impedance v a r i a b l e by the m mode from i to j . Elements of t h i s set include t r a v e l time, waiting or tra n s f e r time, fare and other money costs, frequency of departure, comfort, convenience, distance and functions of these elements, including perception transformations where applicable D.. = distance i n miles from i to j by mode m 13m J 2 - 16 -C.. = cost or fare from i to j by mode m ljm J J H.. = t r a v e l time i n hours from i to j by mode m xjm J J Z i j k = { A i j k ' C i j J l } (k=l,2,. .. ,K; £=1,2,...,L) t. . = estimate of t r i p volume or t r a v e l demand from ljm i to j by mode m i r r e s p e c t i v e of whether estimate or i g i n a t e s from survey or from synthetic methods t . . = St.. the t o t a l number of t r i p s from i to j , over a l l or a subset of modes, e.g., highway modes t^jp = estimate of t r i p volume using path p from i to j V = estimate of t r i p volume on arc a a r o V a = observed t r i p volume on arc a b = t r i p volume capacity of arc a 3. c^ = cost of using arc a d = distance (in miles) of arc a ci •5 i j a p = arc-path mapping for the ( i , j ) t h c i t y p a i r . Between each i and j there e x i s t p paths each of which consists of non-disjoint subsets of a arcs. Hence 6. . indicates the presence or iDap absence of arc a i n path p from i to j , i . e . , 1 i f arc a contained i n path p 0 otherwise 6 . . iDap > <5 . . = Z6 . . arc-path mapping for a. unique path iDa p iDap from i to j - 17 -sa ( v ) = volume-delay curve f o r arc a CK = estimate of t r i p demand generated by c i t y i Dj = estimate of t r i p demand attracted by c i t y j r ^ , S j = balancing factors f o r c i t i e s i and j , so that endogenous estimates of the 0-D matrix are f e a s i b l e given the constraint set imposed by 0. and D. i 3 f = any function which defines a correspondence between sets of variables g = an objective function to be optimised f o r an estimate of the 0-D matrix. Usually defined e s s e n t i a l l y as a sum of squares or l e a s t absolute e r r o r s . 12*1 h = the h = ( i , j ) t h c i t y p a i r , where d i s t i n c t i and j i d e n t i t i e s are not necessary e = tolerance for convergence of i t e r a t i v e schemes a , B , Y f A . , y , T , etc., parameters to be estimated CHAPTER 2 ON THE STRUCTURE OF TRAVEL DEMAND ESTIMATION 2.1 Background The purpose of t h i s section i s to make a b r i e f review of the f i e l d concerned with modelling the demand for passenger t r a v e l . As the structure of these models w i l l be investigated i n subsequent sections, they can be regarded as candidates for estimators of' the O-D matrix. E a r l i e s t attempts to model the demand f o r t r a v e l are probably to be found i n the f i r s t p r i m i t i v e experiments with the gravity model, some of which date from the nineteenth century. The h i s t o r i c a l development of the gravity model has been reviewed elsewhere i n the more general context of s p a t i a l i n t e r a c t i o n (Olsson, 1965) and there i s no need to duplicate t h i s work here. Despite these early beginnings, the systematic study of t r a v e l demand dates only from the 1950s. The extensive planning and modelling exercises c h a r a c t e r i s t i c of transportation projects are necessitated by the i n d i v i s i b i l i t y of transportation investment and the systematic and wide-ranging impacts on the socio-economic environment of such investment. Such studies were encouraged by i n d i c a t i o n s that the demand f o r t r a v e l was a behavioural - 18 -- 1 9 -phenomenon which could be s u c c e s s f u l l y modelled and eventually understood. The i n d i v i s i b i l i t y of transportation investments has led n a t u r a l l y to an i n t e r e s t i n forecasting the demand for t r a v e l because such out-of-sample extrapolations are enhanced by understanding of the behavioural mechanisms. The systematic impacts and symbiotic i n t e r r e l a t i o n s h i p s involved i n transportation have required, and l e n t themselves to, the construction and a p p l i c a t i o n of mathematical models. The o r i e n t a t i o n of research i n t r a v e l demand has been heavily influenced by the p r e v a i l i n g p o l i t i c a l climate and current perceptions of planning issues and objectives. Thus, i n the 1950s, work was focussed on forecasting t r a f f i c volumes for highway and freeway capacity problems. Since these forecasts were a requirement i n order to j u s t i f y f e d e r a l funds i n the U.S.A. studies of highway t r a f f i c generation and d i s t r i b u t i o n became routine. At the same time, shortest-path algorithms appeared i n the operations research l i t e r a t u r e . This development encouraged a l l - o r -nothing assignment of t r a f f i c to highways, a procedure which unwittingly overrated the importance of freeways, and perhaps contributed to the surge i n freeway construction. By the 196 0s, however, i t was becoming c l e a r that the emphasis on b u i l d i n g roads had decimated public t r a n s i t and had encouraged c i t i e s to explode beyond t h e i r previous boundaries with the r e s u l t that t r a f f i c problems remained - 20 -due to v a s t l y increased t r i p lengths. At t h i s time, the f i r s t mode s p l i t studies appeared (Warner, 1962; Irwin and von Cube, 1962; Kraft, 1963). Since federal money had become a v a i l a b l e to plan t r a n s i t , the objective of the models was f i r s t to explain modal choice, and subsequently to a s s i s t i n the d i v e r s i o n of demand back to t r a n s i t . This required considerable s o p h i s t i c a t i o n and by the end of the 1960s, numerous models were extant which had integrated t r i p generation, d i s t r i b u t i o n , mode-split and assignment. However, t h i s was accomplished e i t h e r by purely d e s c r i p t i v e econometric approaches, or by h e u r i s t i c a p p l i c a t i o n of "subproblems" such as the t r i p d i s t r i b u t i o n problem. As a r e s u l t of t h i s research e f f o r t i n transportation, there remains two d i s t i n c t methodological approaches to modelling the demand fo r t r a v e l : the sequential a p p l i c a t i o n of submodels and the d i r e c t demand approach. The structure of these models w i l l be investigated as candidates f o r estimators of the O-D matrix, along with recent attempts to combine co n s i s t e n t l y the i n d i v i d u a l stages of the sequential process. 2.2 Sequential models The conventional procedure models demand as a ser i e s of sequential, independent choices c o n s i s t i n g of t r i p generation, d i s t r i b u t i o n , mode s p l i t and assignment. - 21 -As suggested i n the previous section, the reason for t h i s presumed s e p a r a b i l i t y i s as much a r e s u l t of the h i s t o r i c a l development of the f i e l d , as of the true nature of the process. However, once established, t h i s framework has allowed the development of a v a r i e t y of quite elaborate, and tested, model s p e c i f i c a t i o n s . Most of the t h e o r e t i c a l l i t e r a t u r e on t r i p d i s t r i b u t i o n and assignment assumes t h i s framework, and thereby assumes that t r i p generation i s exogenous to those steps. The main issue i n sequential models i s the v a l i d i t y of sequential s e p a r a b i l i t y . On an a p r i o r i basis i t seems indefensible. F i r s t of a l l , t r i p generation rates are determined without feedback from d i s t r i b u t i o n or assignment, that i s , without reference to endogenous l e v e l s of a c c e s s i b i l i t y or congestion. Thus a reduction i n t r a v e l time, as computed by the assignment model, would have no e f f e c t on the generation of t r i p s . A s i m i l a r l o g i c a l one-way s t r e e t applies to the r e l a t i o n s h i p between d i s t r i b u t i o n and assignment. Although i t i s well known that the t r a v e l time or cost i s an increasing function of the t r a f f i c volume using the network, t h i s f a c t i s not endogenously incorporated into the s t r i c t sequential methodology. Trip generation, possibly the most d i f f i c u l t step to model s a t i s f a c t o r i l y , involves the p r e d i c t i o n of - 22 -the t o t a l number of t r i p s which are generated from, and attra c t e d to, a zone per unit of time. This quantity i s generally assumed to be a function of the socio-economic and a c t i v i t y c h a r a c t e r i s t i c s of a zone. However, the precise l i s t of variables may depend on the unit of time selected; the shorter the time s l i c e , the more asymmetric the pattern of t r i p s becomes. Thus a one-hour s l i c e i n the morning would involve household type variables f o r o r i g i n s and economic a c t i v i t y variables f o r destinations. As the length of time increases the pattern of t r i p s becomes increasingly symmetrical up to a point where observed asymmetry can be ascribed to migration and errors of observation. Consequently, f o r annual data, the l i s t of variables for o r i g i n s and destinations i s the same. This i s the case throughout t h i s work. Thus t r i p ends may be written as 0 . = D. = f(A..,a.) (k=l , 2 , . . . ,K; i = l , 2 , . . . , I ) X X XK K where A ^ are as defined i n chapter 1, the are parameters, and the 0^ and are obtained by 0-D survey. The f u n c t i o n a l form for t r i p generation i s almost i n v a r i a b l y assumed to be l i n e a r , which implies that the e f f e c t s of the independent variables are additive. Whether th i s i s a reasonable assumption i s not known i n general, but s t r i c t l i n e a r i t y i s very doubtful. There must at l e a s t be some saturation e f f e c t s i n a l l the v a r i a b l e s , even i f - 23 -t h i s i s only due to congestion. . For s i m p l i c i t y , the subsequent exposition assumes l i n e a r i t y i s l o c a l l y approximate without endorsing t h i s form. Tri p d i s t r i b u t i o n , the next step i n the chain, estimates the proportion of generated t r i p s which w i l l be made between an o r i g i n - d e s t i n a t i o n p a i r of zones. This topic has fascinated many workers of various backgrounds with the r e s u l t that there have been more papers written on t h i s topic than on any other i n t r a v e l demand. Part of t h i s i n t e r e s t i s due to the apparent appropriateness of the gravity model, and part stems from the f e e l i n g that t r i p d i s t r i b u t i o n i s amenable to modelling. Repeated contributions from operations research and phy s i c a l sciences have led to the early d e s c r i p t i v e approach being replaced by constrained optimisation. The purpose of most of t h i s work has been the foundation and s p e c i f i c a t i o n of the functional form, but the variables and the e s s e n t i a l problem are the same. I t can be written simply as: T. . = f(0.,D . ,C. .; r. ,s . ,6) 13 1 3 13 1 3 where and D.. are computed by t r i p generation or arc obtained by 0-D survey. i s the average, generalised cost of t r a v e l l i n g from i to j , r ^ and s^ are balancing factors and 3 i s a parameter to be estimated. The form of these functions i s well known. For t h i s , and some var i a n t s , the reader i s r e f e r r e d to chapter 6. - 24 -An e s s e n t i a l point i s that, i n order to estimate t h i s model, conventional p r a c t i c e has been to perform an O-D survey and f i t the model e i t h e r to the observed T.., or to the observed average t r i p length, or to the d i s t r i b u t i o n of t r i p lengths. Since O-D surveys are not undertaken on a routine basis c l e a r l y i t i s d i f f i c u l t to c a l i b r a t e the model conventionally without f i r s t designing and performing such a survey. Mode s p l i t follows t r i p d i s t r i b u t i o n i n the sequence. One of the strangest aspects concerns the manner i n which mode s p l i t i s t r a d i t i o n a l l y achieved. An aggregate O-D matrix from d i s t r i b u t i o n i s exogenous to mode s p l i t . This t o t a l O-D i s then p a r t i t i o n e d among modes on the basis of t h e i r r e l a t i v e l e v e l s of service. To obtain t h i s t o t a l O-D i n the f i r s t place, a t r a v e l cost matrix has to be estimated. Where more than one mode i s involved t h i s cost matrix i s c l e a r l y a weighted average of the mode s p l i t , but what these weights are cannot be determined u n t i l the mode s p l i t step. Thus i t may be concluded that the presumed s e p a r a b i l i t y between d i s t r i b u t i o n and mode s p l i t i s unwarranted, because the cost matrix f o r d i s t r i b u t i o n depends on the r e s u l t of a succeeding step. In the next section, i t w i l l be seen that, not only are the modal weights unknown, but so are the values of components of generalised cost, i n p a r t i c u l a r the t r a v e l time by automobile. - 25 -The c l a s s i c a l form of the mode s p l i t problem i s U P = — ^ m EU m m where U m =. f f f l ( C ^ ,C m 2 C m k) (m=l,2 M) C , are a t t r i b u t e s of the m mode mk P m i s the p r o b a b i l i t y of choosing the th , m mode and the r a t i o of the p r o b a b i l i t i e s of using modes m and n are functions of the at t r i b u t e s of modes m and n alone: P U / U m _ m / n P r o ~ / r u -n m / m m / m U _ _m U n Modes other than m or n are termed " i r r e l e v a n t a l t e r n a t i v e s " . Given an 0-D matrix, whether obtained from generation and d i s t r i b u t i o n models or d i r e c t l y from survey, the purpose of the next step i n the sequence, t r i p assignment, i s to all o c a t e these flows to the network whence they came. To the layman t h i s must seem no doubt a c i r c u i t o u s procedure and with some j u s t i f i c a t i o n . Indeed, i t rout i n e l y proves impossible to reconcile the arc volumes assigned from survey 0-D with the observed arc volumes. Some of t h i s d i f f e r e n c e i s due to the u n r e l i a b i l i t y of the survey 0-D and some to mi s s p e c i f i c a t i o n of the t r a v e l demand problem. - 26 -For s i m p l i c i t y , assume that the O-D and the arc volumes have been measured without e r r o r . Now l e t t h i s O-D be assigned to the network. I f the r e s u l t i n g estimated arc volumes do not correspond with the observed volumes i t implies that the behavioural assumptions behind the assignment mechanism are defective. How should assignment be modelled? I t i s known that t r a v e l time and cost are increasing functions of arc volumes. C l e a r l y , assigning a l l flow to the shortest path would lead i n c e r t a i n cases to gross err o r s . Of course, the d e f i n i t i o n of shortest-path here does not. take account of volume-delay r e l a t i o n s h i p s . In order to spread t r a f f i c away from the shortest-path numerous algorithms have been proposed ranging from di v e r s i o n by hand to D i a l ' s multipath procedure. A more r e a l i s t i c way to do assignment i s presumably to follow Wardrop 1s (1952) p r i n c i p l e s and achieve that: (1) the journey times on a l l routes a c t u a l l y used are equal, and less than those which would be experienced by a s i n g l e v e h i c l e on any unused route; (2) the average journey time i s a minimum. An assignment model based on these p r i n c i p l e s has been developed recently by Murchland (1969) , Dafermos and Sparrow (1969), Nguyen (1974), F l o r i a n , Nguyen and Ferland (1975), and others. - 27 -Since the sequential methodology i s not exploited to any great extent i n the body of t h i s work, i t receives rather scant attention here. Further d e t a i l s are given i n chapter 6, which contains the sole development of sequential methods fo r O-D estimation. The remainder of t h i s review looks at the d i r e c t models which lend themselves rather simply and elegantly to. estimators of the O-D matrix. In the l i t e r a t u r e , the d i r e c t models are fundamentally d i f f e r e n t from the sequential, i n that the former are p r i m a r i l y multimodal, whereas the l a t t e r have concentrated on urban vehicular t r a f f i c . Nevertheless, the two approaches need not be confined to these areas and i n the present search for useful fu n c t i o n a l forms the methodology i s mathematically e c l e c t i c . 2.3 D i r e c t models Whereas the sequential methodology requires the s e p a r a b i l i t y between r e l a t i v e l y well-defined submodels, the d i r e c t approach loosens the d e f i n i t i o n of these submodels to merge them into a more consistent framework. However, where sequential s e p a r a b i l i t y can achieve sub-optimality, d i r e c t estimation i s e s s e n t i a l l y a d e s c r i p t i v e exercise. Nevertheless, the fundamentally unsound assumptions of the sequential hypothesis leaves the d i r e c t approach as a vi a b l e a l t e r n a t i v e . - 28 -Direct models are of two types: purely d i r e c t , using a single estimated equation to r e l a t e the t r a v e l demand by mode d i r e c t l y to modal a t t r i b u t e s , and quasi-d i r e c t , employing a form of s e p a r a b i l i t y between mode s p l i t and t o t a l t r a v e l demand problems. Within t h i s c l a s s i f i c a t i o n a further d i s t i n c t i o n may be made between mode-specific, mode-abstract and quasi-mode-abstract forms. 2.3.1 Direct and s p e c i f i c models One of the e a r l i e s t and best known of t h i s class i s the SARC (196 3) model, the i m p l i c i t form of which may be written as T. . = 6 n A . ^ n n C . ^ (m,n=l,2 , . . . ,M) 13m om£ i 3 k n j l 13 *n This i s c l e a r l y a m u l t i p l i c a t i v e model with p o t e n t i a l l y very many parameters. Generalised impedance i s obtained by an exponentially weighted product of modal a t t r i b u t e s . With n=m own a t t r i b u t e e l a s t i c i t i e s are obtained, for n^ m cross e l a s t i c i t i e s . As i s well-known for m u l t i p l i c a t i v e models of the Cobb-Douglas type, the e l a s t i c i t i e s are constant and independent of the l e v e l of the v a r i a b l e s . A l t e r n a t i v e forms, containing l i n e a r and exponential terms i n addition to m u l t i p l i c a t i v e ones, have been suggested by Domencich, Kraft and Valette (1968) i n an urban context. - 29 -2.3.2 D i r e c t and a b s t r a c t models In 196 6 two papers appeared which argued t h a t i t i s the a t t r i b u t e s o f commodities which are of i n t e r e s t r a t h e r than the commodities themselves ( L a n c a s t e r ; Quandt and Baumol). The i d e a s o f Quandt and Baumol were implemented i n an " a b s t r a c t mode" demand model the c o e f f i c i e n t s o f which were r e l i e v e d o f t h e i r modal s u b s c r i p t s and the o b s e r v a t i o n s on modes p o o l e d . T h i s d e v i c e maps o b s e r v a t i o n s from mode space i n t o a t t r i b u t e space thereby p r o d u c i n g a s i n g l e demand f u n c t i o n f o r a l l modes. Modes d i f f e r o n l y by t h e i r r e l a t i v e weights i n a t t r i b u t e space. Having f r e e d the e q u a t i o n o f i d e n t i f i c a t i o n w i t h s p e c i f i c modes i t was h e l d t h a t i t c o u l d be used f o r new modes w i t h o u t r e s p e c i f i c a t i o n . A l t hough w i d e l y used, the v a l i d i t y o f the pure a b s t r a c t mode model can be q u e s t i o n e d . The a g g r e g a t i o n e r r o r i n t r o d u c e d by imposing what amounts to e q u a l i t y c o n s t r a i n t s on the model c o e f f i c i e n t s may be s i g n i f i c a n t . In f a c t , Young (1969) has shown as such. N o n e t h e l e s s , the a b s t r a c t model, even i n i t s pure form, does seem t o be a b l e t o r e v e a l the s t r u c t u r e o f the t r a v e l market as a whole. Yet i t can be debated whether the r e s u l t i n g e l a s t i c i t i e s , s i n c e they p e r t a i n to an average o f the t r a v e l market, are s u f f i c i e n t l y r e p r e s e n t a t i v e o f any i n d i v i d u a l mode, i n c l u d i n g a new mode. The f u n c t i o n a l form of the o r i g i n a l model o f Quandt and Baumol (1966) may be summarised as: - 30 -3 k yZ ai T . . = 3 nA.K. n c * n(c. .„ / c . . o v ) l j m 0^ l j k ^ 13 &b^ 13 Am' xjHh 1.U where C. .„, i s the 'best' mode, i n the I a t t r i b u t e from i 1 j&b to j . T h i s dependency on the b e s t mode f o r mode s p l i t and g e n e r a l i s e d impedance i s a f u r t h e r weakness which may be p a r t l y removed, however, by use o f the geometric mean over modes i n p l a c e o f the b e s t mode (Crow, Young and Cooley, 1973).. Many d i f f e r e n t v a r i a n t s o f the a b s t r a c t mode model have been attempted on a h e u r i s t i c b a s i s . Some o f these may have l e d t o the q u a s i - d i r e c t models which are sometimes q u a s i - a b s t r a c t as w e l l . 2.3.3 Q u a s i - d i r e c t and s p e c i f i c models Q u a s i - d i r e c t models are so named s i n c e they r e p l a c e the d i r e c t e s t i m a t i o n o f t r a v e l by mode w i t h the e s t i m a t i o n of f i r s t l y , the market share o f mode m, and s e c o n d l y , the t o t a l demand f o r t r a v e l , i r r e s p e c t i v e o f mode. Hence T. . = T. .«S. . 13m i ] l j m One of the f i r s t o f these models i s McLynn's w e l l known composite a n a l y t i c model (McLynn e t a l . , 1968, 1969). In o r d e r t o e s t i m a t e the market share o f each mode, mode s p l i t e q u a t i o n s are e s t i m a t e d , the parameters o f which are then f i x e d and s u b s t i t u t e d i n t o a g e n e r a l i s e d impedance term summed over modes. The form o f t h i s model may be w r i t t e n as a, k r v TT - 1 Y T . . = B nA." f i u l ' |U /SU m xnm o k 1 3 k m' m m _ where U = a IIC. .„ m om^ i]£m - 31 -The advantage of using t h i s model over the SARC model i s that the number of parameters i s reduced while r e t a i n i n g the essence of mode s p e c i f i c i t y . There s t i l l remains, however, a considerable set of parameters. 2.3.4 Quasi-direct and quasi-abstract by Monsod (1966, 1967) which c l o s e l y resembles McLynn's model except that Monsod uses a base mode for the mode s p l i t problem and the form i s predominantly abstract. Again s u b s t i t u t i n g the separable components leads to and a =1 i f m i s the base mode. This device i s due to om the f a c t that mode s p l i t i s estimated as a r a t i o of the base mode's a t t r i b u t e s . The constant i n t h i s equation i s d i f f e r e n t from one only for non-base modes. In the o r i g i n a l formulation Monsod constrained the a equal except for m=b, the base mode, and deleted the a i n the impedance om r term where weighted by Y. A r e l a t i v e l y unknown model has been developed om CHAPTER 3 THEORETICAL SETTING 3.1 Introduction The chapter opens with an o u t l i n e of the d i f f i c u l t i e s encountered i n obtaining an O-D matrix by survey methods. I t i s suggested that t h i s i s an i n e f f i c i e n t and inaccurate way to obtain input data f o r modelling. This leads to the argument that data of s i m i l a r accuracy are obtainable at minimal cost from ubiquitous extant h i s t o r i c a l arc volume records. A method i s advanced which involves several components. Methodological issues a r i s i n g from the model comprise the 1form-versus-variables 1 problem and suggestions about the sources of s p e c i f i c a t i o n e r ror i n the context of various network structures. The discussion ends with a b r i e f survey of previous l i t e r a t u r e which e i t h e r estimated an O-D matrix or appeared to be doing something c l o s e l y r e l a t e d . 3.2 Inadequacy of O-D survey methods Consider the following comments made by D i a l (1973): "The t r a d i t i o n a l o r i g i n - d e s t i n a t i o n survey i s an infamous exercise i n money wasting. I t must be replaced with a more c o s t - e f f e c t i v e t o o l . I t i s t r a g i c that a public agency can spend m i l l i o n s of d o l l a r s surveying t r a v e l behaviour i n an urban area and have none of those data a v a i l a b l e f o r analysis before two years have passed. A t y p i c a l scenario i s the following: a f t e r 3 months of interviewing, a truckload of interviews i s entered into an archaic data processing chain. Months of keypunching and v e r i f y i n g move into months of - 32 -- 33 -e d i t checking. Zone numbers are re l a t e d to addresses. More checking follows more f i x i n g . A year l a t e r a f a c t o r i n g process begins and i s followed by other accuracy checks and general wholesale handwringing on why census numbers and survey numbers do not match, and on and on. F i n a l l y , once the data are a v a i l a b l e , they are r e l a t i v e l y uninformative to demand modellers." Such an indictment raises the question of whether there e x i s t s a better way to obtain an 0-D matrix. At l e a s t there ought to be a cheaper way of obtaining " r e l a t i v e l y uninformative" data. In t h i s work i t w i l l be argued that for i n t e r c i t y highway t r a v e l the whole survey process can be discarded by modelling d i r e c t l y on arc volumes, a source of data which i s c o l l e c t e d continuously, ubiquitously, accurately, cheaply and providing c r o s s - s e c t i o n a l and temporal comparability. Although i t w i l l be " r e l a t i v e l y uninformative" t h i s may be s u f f i c i e n t for c e r t a i n aggregate models and objectives. Before a methodology to replace 0-D surveys i s outlined i t i s pertinent to question the accuracy of t h e i r data. I t i s well known that most surveys have been of doubtful accuracy. In p a r t i c u l a r i t appears that cost considerations have led to too small a sample s i z e being chosen. One problem i s the large number of c e l l s i n the 0-D matrix. I t i s common for 1000 zones to be used i n a large area which r e s u l t s i n about 1000 2 c e l l s i f symmetry i s not imposed. Many small zones are required for accurate assignment of t r a f f i c to physical networks. The usual r e s p o n s e t o p r o b l e m s s t e m m i n g f r o m i n s u f f i c i e n t s a m p l e s i z e i s t o a g g r e g a t e o v e r a g e n t s o r u n i t s u n t i l t h e t h e o r e t i c a l e r r o r b o u n d s a r e t o l e r a b l e . T h i s c o u r s e o f a c t i o n i s i n g e n e r a l i n f e a s i b l e f o r e s t i m a t e s o f t h e O - D m a t r i x o w i n g t o t h e a s s i g n m e n t r e q u i r e m e n t s . A n o t h e r p r o b l e m i s t h e s k e w e d d i s t r i b u t i o n o f demands o v e r t h e p o s s i b l e c e l l s o f t h e m a t r i x . A few c e l l s e x h i b i t t y p i c a l l y v e r y l a r g e t r i p i n t e r c h a n g e s b u t t h e v a s t p r o p o r t i o n c o n t a i n m o d e r a t e a n d s m a l l d e m a n d s . W h e t h e r t h e O - D s u r v e y i s b a s e d o n h o u s e h o l d i n t e r v i e w s o r o n r o a d s i d e q u e s t i o n n a i r e s i t i s i n e v i t a b l e t h a t , i n t e r m s o f t h e r e q u i r e d s a m p l e s i z e , t h e l a r g e i n t e r c h a n g e s a r e o v e r r e p r e s e n t e d i n t h e d a t a a n d t h e s m a l l o n e s u n d e r r e p r e s e n t e d . A s a r e s u l t t h e e r r o r b o u n d s o n s m a l l i n t e r c h a n g e s a r e much l a r g e r t h a n t h o s e o n l a r g e o n e s . T h e e x t e n t t o w h i c h t h i s i s s o i s shown i n T a b l e 3.1 f o r L o n d o n , a s t u d y a r e a c o m p r i s i n g 1000 z o n e s a n d 3 m i l l i o n h o u s e h o l d s ( C r a w f o r d , 19 6 8 i n B e n d t s e n , 19 75). A s e x p e c t e d , f o r s m a l l i n t e r c h a n g e s a n d s m a l l s a m p l e s i z e s t h e e r r o r i s i n t o l e r a b l e . What i s s t r i k i n g , h o w e v e r , i s t h e d e m o n s t r a t i o n t h a t f o r s m a l l i n t e r c h a n g e s t h e e r r o r b o u n d s r e m a i n l a r g e e v e n w i t h a 6 p e r c e n t s a m p l e o f a l l t r i p s . T h e s e f i g u r e s a r e d r a w n f r o m an u r b a n e x a m p l e , w h e r e t h e v a s t m a j o r i t y o f O-D s u r v e y s h a v e b e e n p e r f o r m e d . I t i s o f i n t e r e s t t o know t h e e x t e n t t o w h i c h t h e s e r e s u l t s a r e a p p l i c a b l e t o t h e i n t e r c i t y c a s e . Two f a c t o r s s u g g e s t t h a t t h e s i t u a t i o n w i l l be e v e n w o r s e . F i r s t , t h e z o n e s u s e d i n t h e i n t e r c i t y c a s e , a l t h o u g h w e l l - d e f i n e d g e o g r a p h i c a l l y , - 35 -Trips per day Sample size % 0.2 1.0 6.0 1000 -90+260 60 25 10000 45 20 9 60000 18 10 3 120000 12 6 3 Table 3.1 Percentage error bounds for the 0-D matrix Trips 5% sample 10% sample generated Observed Theoretical Observed Theoretical per day error e r r o r error error 100 76 44 52 30 1000 25 14 17 10 10000 8 4 5 3 100000 2.6 1.4 1-8 1 Table 3.2 Percentage error bounds for t r i p s generated by a zone - 36 -are, due to the r e l a t i v e l y s e l f - s u f f i c i e n t nature of c i t i e s , extremely heterogeneous. This alone precludes small sample s i z e s . Second, the interchange volumes are smaller i n general than i n the urban case. This stems from the greater distances involved and the ubiquitous, e m p i r i c a l l y observable and f a m i l i a r , convex to o r i g i n distance-decay e f f e c t on t r i p volumes. Therefore large and expensive sample sizes are a precondition for estimation of the 0-D matrix by survey methods i n the i n t e r c i t y case. One form of aggregation which might allow 0-D estimation i s to replace an 0-D survey with a two-stage procedure: f i r s t estimate the number of t r i p s generated and attracted by each zone, then use a model to a l l o c a t e o r i g i n s to destinations. Although the expected and observable error i s reduced i n the f i r s t stage i t i s not reduced by much, as Sosslan and Brokke (1960) have suggested. Their r e s u l t s , as presented i n Bendtsen (1975), are shown i n Table 3.2. "Observed" error i s defined as the percentage root mean square error whereas the " t h e o r e t i c a l " error i s the percentage standard deviation e r r o r . In each case these figures are re l a t e d to a base of 100 per cent surveyed. Not only do these r e s u l t s show wide error bounds for small zones and small sample sizes but that the actual error i s much larger than the expected one. This a d d i t i o n a l error appears to be due to the fact that respondents cannot accurately r e c a l l the number of t r i p s made. Further misgivings about the accuracy of these data have been voiced recently by Long (19 74) and Stopher and Meyburg (1975). - 37 -Given that the a c q u i s i t i o n of i n t e r c i t y O-D data possessing acceptable error bounds i s perforce an expensive undertaking i t i s pertinent to ask to what extent such data are avai l a b l e i n Canada, f o r example- In t h i s context several statements by P l a t t s (19 76) are relevant: "The lack of o r i g i n - d e s t i n a t i o n s t a t i s t i c s probably constituted the major data deficiency i n the C.T.C.'s t r a v e l demand model project. Without automobile and passenger bus O-D s t a t i s t i c s , and with incomplete r a i l O-D s t a t i s t i c s i t was impossible to c a l i b r a t e accurately the passenger t r a v e l demand model." In fa c t the s i t u a t i o n r e f l e c t s not so much a "lack", i n the sense of inadequate, as a t o t a l absence of O-D data for highway t r a v e l i n general: "Auto passenger O-D s t a t i s t i c s on a c i t y to c i t y basis do not e x i s t . There are a few c i t i e s where external t r a f f i c surveys have been undertaken as part of an urban transportation study. Such data are l i m i t e d , and w i l l l i k e l y remain so; i n ad d i t i o n , they provide only 'snapshot' i n d i c a t i o n s of passenger movements between p a r t i c u l a r p a i r s of c i t i e s . " and "With a s i n g l e exception... bus c a r r i e r s appeared not to c o l l e c t O-D data. T r a f f i c s t a t i s t i c s were r e s t r i c t e d to (the) number of passengers using a route; i n some instances, the data were disaggregated to show point-to-point flows (arc volumes)...The need for i n t e r c i t y bus O-D s t a t i s t i c s has been recognised for some time." In short, not only i s O-D data inaccurate when obtained with budget-priced sample sizes but there i s so l i t t l e of i t already i n existence for i n t e r c i t y modelling that extensive surveys might seem a precondition f o r any ana l y s i s . Whether the funds would ever be a v a i l a b l e for the - 38 -extensive cr o s s - s e c t i o n a l and time-series analysis e s s e n t i a l for thorough modelling i s extremely doubtful. 3.3 Nature of the 0-D matrix and arc volumes From one point of view the 0-D matrix i s simply an account of the t r i p volumes moving between pai r s of zones. I t s e s s e n t i a l appearance depends, however, on two forms of aggregation, s p a t i a l and temporal. S p a t i a l aggregation involves the c o l l e c t i o n of areas into s p a t i a l units c a l l e d zones. Temporal aggregation i s concerned with the length of time during which t r i p s between these zones are observed. The choice of t h i s time i n t e r v a l has major impact on the configuration of the 0-D matrix. If 0-D i s based on a ti m e - s l i c e of one hour, for example, the r e s u l t i n g d i s t r i b u t i o n of t r i p s i n the matrix i s highly asymmetric. This i s most evident during 'rush-hours' i n urban areas but the same p r i n c i p l e applies i n general. Increasing the length of the time i n t e r v a l increases the symmetry of the 0-D matrix. U n t i l , at some point, d a i l y , weekly, monthly and seasonal cycles are averaged out. Any remaining asymmetry i n the 0-D can be ascribed to trends and cycles of longer amplitude such as migration, to the existence of multiple purpose t r i p s with d i r e c t i o n a l asymmetry and to measurement er r o r . Since the l a t t e r probably dominates i t i s assumed that f o r p r a c t i c a l purposes the 0-D matrix i s aggregated temporally so as to appear symmetric. - 39 -A p r i o r i i t might seem t h a t an asymmetric t r a v e l c o s t m a t r i x would produce an asymmetric O-D m a t r i x . T h i s consequence, however, i s not i n e v i t a b l e . U n l i k e commodity f l o w s , passenger t r a v e l i s fundamentally a two-way t r i p . Hence, even though one d i r e c t i o n o f the t r i p can be cheaper than the o t h e r no e f f e c t on the symmetry o f the O-D m a t r i x i s a n t i c i p a t e d u n l e s s r o u t i n g through a t h i r d zone i s i n v o l v e d . In t h i s case a l t h o u g h s t r i c t symmetry i s m a i n t a i n e d i t may not be measurable owing to the manner i n which the O-D data are c o l l e c t e d . T r i p s r o u t e d through another zone may be d e t e c t e d as an i m p l i c i t decomposition i n t o two t r i p segments. A r c volumes are t r a f f i c counts measured by mechanical d e v i c e s u s i n g p r e s s u r e - s e n s i t i v e s t r i p s p l a c e d a c r o s s the highway. While i t can be arranged t o o b t a i n d i r e c t i o n a l flows i t i s more u s u a l i n an i n t e r c i t y c o n t e x t to o b t a i n n o n d i r e c t i o n a l volumes. No d i s t i n c t i o n i s c u r r e n t l y made between c a t e g o r i e s of v e h i c l e or t h e i r speeds but t h i s would be simple e x t e n s i o n of the t e c h n o l o g y . As y e t t h i s l e v e l of d i s a g g r e g a t i o n i s not a v a i l a b l e f o r h i s t o r i c a l d a t a . I f the O-D m a t r i x f o r a g i v e n c a t e g o r y i s r e q u i r e d , f o r passenger c a r s f o r example, supplementary d a t a on the c o m p o s i t i o n o f the t r a f f i c flow i s n e c e s s a r y . In s p i t e of t h i s requirement the use of a r c volumes b e n e f i t s from s e v e r a l enormous advantages. The most s i g n i f i c a n t o f these i s t h e i r u b i q u i t o u s and c o n t i n u o u s c o l l e c t i o n which f u r n i s h e s abundant c r o s s - s e c t i o n a l and - 40 -time-series data. This advantage i s enhanced by the high degree of comparability due to consistent data d e f i n i t i o n s . Such comparability i s rare indeed outside the physical sciences and consequently permits pooling of data from d i f f e r e n t administrative u n i t s . Furthermore, these data can be c o l l e c t e d with very low l e v e l s of error, a s i t u a t i o n which contrasts markedly with O-D surveys. 3.4 An Estimate of the O-D Matrix 3.4.1 Form of the estimation problem In chapter 2 i t was seen that t r a v e l demand models require an O-D matrix i n order to be c a l i b r a t e d or estimated. In t h i s section i t i s suggested that, by analysis of observed arc volumes, the O-D matrix may be i n f e r r e d thereby allowing the estimation of some t r a v e l demand models. An i l l u s t r a t i o n of the general form of t h i s scheme i s shown i n Figure 3.1. I t i s c l e a r a p r i o r i that arc volumes are the r e s u l t of t r i p generation, d i s t r i b u t i o n and assignment processes as they e x i s t i n t h e i r real-world complexities. Were i t possible to discover the true models underlying such processes i t would be a r e l a t i v e l y straightforward task to deduce the path assignment which led to the arc volumes, the O-D matrix which was assigned and thus the t r i p s generated and attracted by each zone. While the true models are unknown, functional forms which approximate them are quite f a m i l i a r , notably the gravity model. - 41 -generation of t r a v e l demand from socio-economic and a c t i v i t y v a riables d i s t r i b u t i o n among a l t e r n a t i v e destinations (estimated 0-D) assignment of estimated 0-D matrix to arcs I c a l c u l a t i o n of error function on arcs r e v i s i o n of generation and d i s t r i b u t i o n parameters by nonlinear optimising methods to minimise error function convergence. F i n a l parameters give estimate of 0-D matrix Figure 3.1: General scheme to estimate the 0-D matrix from arc volumes. - 42 -The f a c t that s p e c i f i c a t i o n s and estimates of the component models of t r a v e l demand have been made i n the l i t e r a t u r e allows r e s t r i c t i o n s to be placed on the functional forms of such models. These r e s t r i c t i o n s derive not so much from t h e o r e t i c a l considerations as from empirical feedback from t r i a l and error methodologies, yet the large number of such studies endows w e l l - t r i e d forms with a p r a c t i c a l v a l i d i t y . Given that these forms may be assumed known, the estimation problem reduces to one of determining numerical estimates of parameters. In general, the e x p l i c i t f unctional form w i l l not necessarily be assumed. Rather, more general hypotheses can be maintained thereby allowing the estimation of the precise form including, for example, the degree of i n t e r a c t i o n or addivity between terms, i n addition to the estimation of parameters. This unequivocally operational methodology contrasts with that which requires t h e o r e t i c a l d e rivation of the form, then using the r e s t r i c t i o n s thus obtained as an imposed maintained hypothesis. Whereas t h i s more rigorous approach i s t h e o r e t i c a l l y s a t i s f y i n g i t i s not always possible to t e s t the empirical v a l i d i t y of the maintained hypothesis without v i o l a t i n g the t h e o r e t i c a l foundation of the model. On the assumption that useful r e s t r i c t i o n s can be placed on the form of the models, the c r u c i a l question that arises i s whether arc volumes contain s u f f i c i e n t information to estimate the parameters of these forms. This question cannot be answered d i r e c t l y unless the true 0-D matrix i s - 43 -a v a i l a b l e . However, i n the absence of these data, i t i s possible to show that t h e o r e t i c a l l y acceptable models, when estimated, produce unique estimates of the O-D matrix from arc volumes. This i s indeed a major consideration since the procedure i s one which replaces aggregation from O-D to arcs, as i n the conventional modelling process, with a disaggregation from arcs to O-D. If arc volumes were i n s u f f i c i e n t l y informative to permit recovery of O-D the expected r e s u l t would be indeterminacy as manifested by multiple optima. A biased, or otherwise "wrong", estimate would presumably be due to s p e c i f i c a t i o n error i n the forms of the models. One type of bias would be an average t r i p length markedly d i f f e r e n t from the true average t r i p length i n the true O-D matrix. This s t a t i s t i c i s determined p r i m a r i l y by the functional form of the d i s t r i b u t i o n model. Another example would be errors i n the number of t r i p s made for a given c i t y p a i r . A m i s s p e c i f i c a t i o n of t h i s nature could be due to the omission of relevant variables i n addition to functional m i s s p e c i f i c a t i o n . Where s p e c i f i c a t i o n error i s presumed caused by either or both variables and form a methodological problem a r i s e s . This problem revolves around the question of whether as many variables as possible should be added to an equation thus avoiding, for p r a c t i c a l purposes, one type of s p e c i f i c a t i o n error, or whether fewer variables should be used but estimating an approximately correct functional form, thus avoiding s p e c i f i c a t i o n error i n the form. I t seems that the l a t t e r approach i s s l i g h t l y superior on the basis of the following - 44 -argument. I f an equation i s misspecified i n functional form not only are the c o e f f i c i e n t s untrustworthy of variables which should be i n the true model, but which i s worse, variables which do not belong may appear s i g n i f i c a n t . Furthermore, use of a f u n c t i o n a l l y misspecified model for simulation purposes w i l l r e s u l t i n unreasonable behaviour at new l e v e l s of the exogenous v a r i a b l e s . On the other hand, i f key variables are omitted i t w i l l be d i f f i c u l t , i f not impossible, to obtain an unbiased estimate of the form. Also, i f the error term i s not well behaved, being heteroskedastic f o r example, estimates of form w i l l be biased, as Zarembka (1974) has shown. 3.4.2 Some Sources of S p e c i f i c a t i o n Error When an estimation procedure consists of several submodels i t i s of i n t e r e s t to know the extent to which each submodel contributes to the t o t a l error or m i s s p e c i f i c a t i o n of the procedure. I t i s a n o n t r i v i a l task i n general to separate out the e f f e c t s of each model equation. This d i f f i c u l t y i s further complicated by interdependencies between the submodels p a r t i c u l a r l y between the d i s t r i b u t i o n and assignment models; a l t e r n a t i v e assignment mechanisms imply d i f f e r e n t cost matrices which i n turn a f f e c t d i s t r i b u t i o n . These e f f e c t s may, i n addition, be antici p a t e d to extend back to the generation model. I f the number of t r i p s generated by a zone i s re l a t e d to the a c c e s s i b i l i t y of that zone with respect to other zones, that i s , to the average cost of making a t r i p i t follows d i f f e r e n t cost matrices imply d i f f e r e n t numbers of t r i p s . Owing to these complexities, analysis of the sources of s p e c i f i c a t i o n error i s r e s t r i c t e d to c e r t a i n s p e c i f i c network structures. Consider the network shown i n Figure 3.2. Each l i n k flow Vi contains a p a r t i a l sum of O-D t r i p s . For example, vab = tab+tac+tad Given that only the V i can be observed, i t i s cl e a r that the t • are unknown. However, i f estimates of the t- • could be made, i t follows from the t r i v i a l i t y of the network that the assignment of these demands could be made unambiguously to li n k s of the network. Thus, i n the case of t h i s minimally connected network s p e c i f i c a t i o n error originates e n t i r e l y from the generation and d i s t r i b u t i o n models since assignment i s made without e r r o r . Now consider the f u l l y connected network i n Figure 3.3. Although generation w i l l s t i l l be unknown on a zonal basis, the d i s t r i b u t i o n and assignment are known. This assumes of course, that the assignment i s to noncongested l i n k s . D i s t r i b u t i o n i s known because each O-D pai r i s d i r e c t l y connected by a l i n k . Thus, v a c i s i d e n t i c a l to t a £ . Assignment i s known since i t follows t r i v i a l l y from the d i s t r i b u t i o n . Therefore, i n the case of a f u l l y connected network s p e c i f i c a t i o n errors originate s o l e l y i n the generation model. - 46 -a b a d « « # *> Figure 3.2: Minimally connected network. d e Figure 3.3: F u l l y connected network. a Figure 3.4: F u l l y connected subgraph i n a general network. - 47 -I f the l i n k ac i s now deleted from the network, the s i t u a t i o n i s r a d i c a l l y changed. The O-D t which formerl was uniquely c a r r i e d by l i n k ac i s now spread over four a l t e r n a t i v e paths: ado., abc, adbc, abdc. Given that only vl can be observed, i t follows that any vl now contains p a r t i a l sums of t . . . Therefore d i s t r i b u t i o n , i n addition to generation, i s now unknown. Furthermore, the assignment model also comes into question since there i s a set of al t e r n a t i v e paths f o r t ; . to take. I f the l i n k s are of equal lengths and a minimum path assumption can be made c o r r e c t l y , assignment i s known. In t h i s case, t a i s divided equally between adc and abc with nothing assigned to adbc or abdc. Unfortunately, the minimum-path assumption and i t s concomitant all - o r - n o t h i n g assignment cannot u n i v e r s a l l y be maintained. Consequently, the question arises as to the correct form of the assignment model. Thus, i n the case of networks which . are neither minimally connected or f u l l y connected generation, d i s t r i b u t i o n and assignment are a l l unknown and therefore a source of s p e c i f i c a t i o n e r r o r . The network i n Figure 3.4 consists of a combination of simple forms: the minimally connected and the f u l l y connected networks. The subgraph b'-cz'i i s f u l l y connected and as a r e s u l t i t s i n t e r n a l d i s t r i b u t i o n and assignment i s known except for t, which i s merged with external t ; ; . DC -tj However, some information i s obtainable about these external t . . by comparison of the two cut points which i s o l a t e bci^ - 48 -from the remainder of the network. The subgraph abad i s minimally connected with d i s t r i b u t i o n unknown but assignment known. 3.5 Antecedents The previous l i t e r a t u r e i n t h i s area i s sparse indeed. Those who are f a m i l i a r with the transportation planning l i t e r a t u r e i n i t s t h e o r e t i c a l aspects w i l l be aware that the existence of an 0-D matrix, or parameters derivable therefrom, i s assumed rather than demonstrated. There seems to have been only one attempt to estimate e x p l i c i t l y the 0-D matrix elsewhere. This has been done, independently of t h i s work, by R o b i l l a r d (19 75) and by d i f f e r e n t methods. Casting the net somewhat wider i t could be argued that Schneider's (1965, 196 7) d i r e c t estimation of t r a f f i c at a point i s an antecedent, however, Schneider makes no attempt to evaluate the 0-D matrix since h i s exclusive i n t e r e s t i s in highway t r a f f i c volumes. Methods analogous to t h i s were used by K i s s l i n g (1966, 1969) and were extended by W i l l s (1971) to estimate 0-D, though used for d i f f e r e n t purposes. The context there was to provide measures of highway l i n k importance from a crude f i r s t approximation of the 0-D matrix so as to permit inferences about the nature of the regional economic growth process. The methods used i n chapter 3 of W i l l s (19 71) seem to be the e a r l i e s t exposition and a p p l i c a t i o n of an - 49 -0-D estimator from arc volumes. Several aspects of t h i s work are of i n t e r e s t : the s p e c i f i c a t i o n of the model, the cross-c o r r e l a t i o n s between observed and estimated arc volumes f o r d i f f e r e n t time periods, and the a p p l i c a t i o n of the estimated arc volumes as instruments to reveal lead-lag r e l a t i o n s h i p s . The s p e c i f i c a t i o n of the model i s a s p e c i a l case of the d i r e c t unconstrained extended model given i n chapter 9. Using the terminology of that chapter the model may be written as Min!(Vv2 w r t a o , «w v where V = o0+o Z 6 h a t h n t — P D^ h ~ h h' and the model was estimated using, by chance, the algorithm given i n 7 . 1 . This requires an a l t e r n a t i o n of revised values of nonlinear parameters with a multiple regression problem. In W i l l s ( 1 9 71) t h i s a l t e r n a t i o n was c a r r i e d out by hand, a simple l i n e a r search performed for only nonlinear parameter, y, and simple l i n e a r regression performed f o r the a 0 and a j . This scheme fo r a l i n e a r estimate of the l i n e a r parameters i n a nonlinear model was developed independently of that by Lawton and Sylvestre (19 7 1 ) . The study also looks at the c o r r e l a t i o n s between V and V ^, where the t and t ' are time periods for the at at arc volumes. The time periods t and t" are not ne c e s s a r i l y d i f f e r e n t . The V , are used as instruments, based on the at c a l c u l a t e d s t r u c t u r a l properties of the highway network and - 50 -socio-economic environment. In t h i s r o l e the estimated arc volumes look, f o r lead-lag r e l a t i o n s h i p s between expected behaviour, as determined by the V £, and observed behaviour, o as measured by the V a^. As s p e c i f i e d by the model, the V are a function of the estimated O-D matrix and all - o r - n o t h i n g assignment. In turn, the O-D matrix i s a function of the cost matrix which i n t h i s study i s made proportional to the shortest time-path matrix associated with the highway network. I t i s also a function of the summer populations i n i n t e r i o r B.C. communities, as measured by li q u o r scales, a va r i a b l e which aids consistency with the summer observed arc volumes used. Schneider's d i r e c t t r a f f i c estimation method involves the estimation of selected arc volumes as a function of the po t e n t i a l of these arcs f o r a t t r a c t i n g t r a f f i c . Let an arc be selected f o r study. The bundle of paths using t h i s arc i s generated by a set of o r i g i n s i n the domain at one end of the l i n k and the set of destinations at the other. The e f f e c t of a change i n the number of t r i p s attracted at a point on the arc volume i s held to be a function of the a c c e s s i b i l i t y of the arc to the point. Hence, independent of the o r i g i n , increased numbers of t r i p s a t t r a c t e d to a zone by a new development project could be re l a t e d d i r e c t l y to the t r i p volumes on a given arc. Let the a c c e s s i b i l i t y functions of the domains n and s at each end of an arc be I and I , defined as n s I = ZC..T. i ^ ^ where T. i s the number of t r i p ends at i, e i t h e r o r i g i n s or i destinations. Then the t r i p volume on arc cc can be written as an average of I and I weighted by the proportion of TI S the t o t a l represented by each of the domains: 21 21 V = — ( I +1 ) — — I t i s apparent from t h i s form that s p e c i f i c O-D interchanges are not evaluated and remain unknown. R o b i l l a r d (19 75) i s unique to the extent that he was the only worker i n the f i e l d to set out to estimate the O-D matrix as such from arc volumes. Indeed his method i s remarkable f o r i t s ingenuity and o r i g i n a l i t y . Furthermore, in R o b i l l a r d (19 73) he seems to have antici p a t e d , independently from t h i s work, that a sequential and a l i n e a r model might be used for the purpose of estimating O-D. Nevertheless the nature of these contributions i s speculative and i n addition they contain shortcomings which e f f e c t i v e l y preclude p r a c t i c a l empirical estimation. In the 19 75 paper i t i s shown that the elements of an O-D matrix may be treated as c o e f f i c i e n t s i n the l i n e a r regression problem V = St. 6. +e {a-1,2,. . . ,A) a 7 ii na a n where the t T are O-D to be estimated and S, the usual binary h na 2 mapping for the presence or absence of a given t ^ on a s p e c i f i c arc a. This formidable estimation problem i s rendered less a t t r a c t i v e by the discovery that the 6^ are singular, a property which e f f e c t i v e l y f r u s t r a t e s the extraction of s p e c i f i c O-D interchanges but eventually permits l i n e a r combinations of O-D (generation and attraction) to be obtained by a p p l i c a t i o n of generalized inverse techniques. In the 19 73 paper R o b i l l a r d suggests a naive l i n e a r model s i m i l a r to the basic l i n e a r model presented at the beginning of chapter 5. As discussed i n that chapter there are fundamental d i f f i c u l t i e s with using a l i n e a r model for an i n t e r a c t i v e phenomenon when the structure of the problem disallows nonlinear transformations. R o b i l l a r d also b r i e f l y suggests a simple sequential type model but the argument i s not developed nor the procedure s p e c i f i e d . CHAPTER 4 DIRECT UNCONSTRAINED MODELS I. NONLINEAR MODELS The t h e o r e t i c a l basis for t r i p generation, d i s t r i b u t i o n and assignment i s not s u f f i c i e n t l y well developed to supply useful r e s t r i c t i o n s on the functional forms. As a consequence, much of the t h e o r e t i c a l argu-ment i s not deductive but i n t u i t i v e . Lack of t h e o r e t i c a l r e s t r i c t i o n leads also to the d e f i n i t i o n of a very general function, from which can be obtained a v a r i e t y of i n t e r e s t i n g forms, which can be ar r i v e d at by estimation rather than p r i o r considerations. 4.1 General structure of the estimation problem In t h e i r most general form, the class of d i r e c t , unconstrained and nonlinear estimators of the 0-D matrix may be written as: o Minimise g(v , v ) wrt {0 , } d ) a a K where v = E Z S . . t. . (2) a i j 13a. i ] t ± j = f ( 6 k , Z ) (k=l,2,..K) (3) 1 i f path i , j uses arc a 6 . . = I 2-3a 0 otherwise - 54 -That i s , to minimise some function g of the observed and estimated arc volumes by optimising parameters 0 ^ on the d i r e c t generation and d i s t r i b u t i o n function f. S p e c i f i -cation of g i s deferred to a l a t e r chapter on estimation but for the present, i t can be assumed to be the sum of squared e r r o r s . The 6 notation represents the assignment of estimated 0-D to the arcs of the network by the a l l - o r - n o t h i n g method (Potts and O l i v e r , 1972). 4.2 S p e c i f i c a t i o n of the 0-D matrix Given the objective function and the general structure of the problem, the remainder of t h i s section i s devoted to s p e c i f i c a t i o n of the function f and i t s arguments. These arguments are of two types: variables and parameters. The variables are sets of observations on the r e a l world which are thought to have a bearing on the amount of t r a v e l between a p a i r of c i t i e s . These variables are themselves of two d i s t i n c t types. F i r s t , there are variables which represent the p o t e n t i a l for s p a t i a l i n t e r a c t i o n between zones. Foremost i n t h i s l i s t are population l e v e l s , which, i f expressed as a product of the populations i n c i t i e s i and j , r e s u l t i n a matrix of possible i n t e r a c t i o n s . The strengths of these i n t e r -actions can be modified by other socio-economic variables such as income, car ownership, culture, and - 55 -language. Second, another set of variables represents attenuation of t h i s i n t e r a c t i o n p o t e n t i a l by the distance, cost or time separating zones. 4.2.1 Some general forms Each set of variables may be combined i n various ways to produce a composite i n t e r a c t i o n p o t e n t i a l v a r i a b l e or a generalised impedance term. Then these two composite terms can also be var i o u s l y combined to give a func t i o n a l form for the O-D matrix. Obviously there i s considerable v a r i e t y possible. The discussion may be summarised as where f^ i s the form of the a c t i v i t y - i n t e r a c t i o n v a r i a b l e s , f^ the form of cost v a r i a b l e s , and f the function r e l a t i n g these q u a n t i t i e s . In the vast majority of studies, i f not a l l , these functions have been assumed m u l t i p l i c a t i v e . While t h i s i s not unreasonable, i t does not rule out c e r t a i n hypotheses such as the (untransformed) a d d i t i v i t y of e f f e c t s . Consequently, a more general form w i l l be s p e c i f i e d which contains as s p e c i a l cases several i n t e r e s t i n g forms: (4) (5) - 56 where A. - 1 (6) log A i j k and are transformations of the power family o r i g i n a t e d by Tukey (1957), Box and Tidwell (1962) and Box and Cox (1964) . By (A.Q1) i s meant the inverse of the Box-Cox transformation applied to the dependent v a r i a b l e . The function i s s p e c i f i e d t h i s way to leave the t . . untrans-formed since i t must remain i n t h i s form to be assigned to network arcs by aggregation. The equation i n t h i s form i s quite s i m i l a r to the CES production function (Arrow et a l . , 1961). transformation of v a r i a b l e s . One i s that the variables i n the true model are r e l a t e d i n a nonlinear manner. In t h i s case, the objective i s to f i n d a transformation to l i n e a r i s e the function, that i s , to f i n d a scale for each var i a b l e such that i t s e f f e c t s are a d d i t i v e . Secondly, the error term may turn out to be non-normal and hetero-skedastic and, i n t h i s case, i t i s desirable to f i n d a transformation to achieve simultaneously normality and constant variance. Unfortunately, when estimating the functional form from data, these e f f e c t s cannot be unambiguously distinguished. As a r e s u l t , as Zarembka There are two d i s t i n c t elements involved i n the (1974) has observed, the presence of heteroskedasticity i n a nonlinear model may lead to a biased estimate of the true functional form. Postponing estimation problems for the present, the purpose of t h i s discussion i s to c l a r i f y and unify functional s p e c i f i c a t i o n from a s t r u c t u r a l point of view. Thus, equation (5) may turn out to be over-parametrised from an estimation point of view, but t h i s i s immaterial for present purposes. 4.2.2 Special cases of the general model By s e t t i n g the As to predetermined values, i t i s c l e a r that a family of functions i s obtained. One of the more i n t e r e s t i n g members of t h i s family appears when a l l As are equal to zero. This gives the following: t . . - exp(g 0 + | 3 k log A . J K + Z y £ C. j £) 8 Y = e x P ( 6 0 ) n A . . * n c . - J (7) which i s the well-known power family containing the Cobb-Douglas production function and the grav i t y model as sp e c i a l cases. This i s a p a r t i c u l a r l y useful model since i t allows i n t e r a c t i o n s between a l l the variables on the r i g h t hand side of the equation, a property which has in some f i e l d s led to i t being known as the i n t e r a c t i o n - 58 -model. Its e f f e c t s are m u l t i p l i c a t i v e , which means that the impact of any one r i g h t hand side variable on the t ^ j depends on the l e v e l s of the other r i g h t hand side v a r i a b l e s . In the case of t r i p d i s t r i b u t i o n , t h i s allows the e f f e c t of an increase i n population to depend on the distances separating the population. This can be seen by s e t t i n g 8 =0, B,=l, A. . =P.P. , C..=D.., y<0 (8) 0 1 1 3 1 3 13 13 for k=£=l, i n equation (7) Substitution gives t. . = P. P. D T . (9) ID 1 D ID which i s f a m i l i a r , and d i f f e r e n t i a t i o n with respect to P. gives: 3t. . y i i = P . Di . (10) which c l e a r l y shows the dependence on the population at Pj and intervening distance. M u l t i p l i c a t i v e models are often used not so much for t h e i r i n t e r a c t i o n properties as for the existence of convenient log transformation to l i n e a r i t y and the i n t e r p r e t a t i o n of the c o e f f i c i e n t s as constant e l a s t i c i t i e s . However, since the objective - 59 -f u n c t i o n d e f i n e d by (1), (2) and (3) i s i n t r i n s i c a l l y n o n l i n e a r , l o g t r a n s f o r m a t i o n i s p r e c l u d e d , and i n t e r e s t i s c o n f i n e d t o i n t e r a c t i o n e f f e c t s . A s m a l l but s i g n i f i c a n t e x t e n s i o n t o the m u l t i p l i c a t i v e model can be made by s e t t i n g the impedance As t o A^=0, ^ 2 = 1 ^ o r From t h i s i s o b t a i n e d the gamma f u n c t i o n o r i g i n a l l y suggested by Tanner (1961) i n a g r a v i t y model c o n t e x t t . . - exp (3Q) n A . ^ c±]] exp ( Y 2 c . j 2 ) (11) where C.., = C . j 2 - D. . ; Y ]_ <0, Y 2 <0. The n e g a t i v e e x p o n e n t i a l p a r t o f the gamma f u n c t i o n p o ssesses a s t e e p e r n e g a t i v e s l o p e than the power f u n c t i o n . Hence, a t s h o r t e r distances., i t w i l l dominate. S e t t i n g Q^ = °r A.^ .=^ -£=1 an e x p o n e n t i a l model i s o b t a i n e d : t i j - ex P ( 8 0 ) exp(Z ^ A. j k> exp(£ y % C. j £) (12) whereas a l t e r i n g A n = l the l i n e a r model appears: t. . = 8rt + Z 6, A... + Z Y N C. . 0 ID 0 k k iDk i 'I 2.3I (13) - 60 -Equation (13) i s the subject of considerable attention i n section 2. This i s due to the f a c t that the l i n e a r form has a t t r a c t i v e properties of e f f i c i e n t estimation, simple structure and well developed s t a t i s t i c a l foundations but at the same time i s hard put to replace what appears to be an i n t r i n s i c a l l y nonlinear true model. C l e a r l y , a great deal of f u n c t i o n a l v a r i e t y i s obtained by these Box-Cox transformations. In f a c t , r e s t r i c t i n g a l l X^= X^ s t i l l preserves i n t e r e s t i n g v a r i e t y . This scheme gives a 2-A model: A.- on the t. ., A, on the 0 13 1 r i g h t hand side v a r i a b l e s . Such a model i s a t t r a c t i v e because i t could be estimated for optimal Ag and A^, which would allow estimation of the actual amount of i n t e r a c t i o n among the v a r i a b l e s . 4.2.3 S p e c i f i c models Up to t h i s point, the discussion has centred on the f u n c t i o n a l form for t r i p generation and d i s t r i b u -t i o n i n which precise d e f i n i t i o n s of the variables have been avoided. The objective now i s to substitute actual variables into forms to produce s p e c i f i c models. At the same time, s p e c i f i c a t i o n has to make the trade-off between parsimonious parametrisation and m i s s p e c i f i c a t i o n by omission. - 61 -Consider the m u l t i p l i c a t i v e form (7). Substitute 8 Y (P. P-;) for the a c t i v i t y variables and D. . ' for the impedance. This constructs a generalised gravity model: t ± j = exp(a) (P ± P j ) ^ D ± j Y (14) i n which each of the parameters has an acceptable i n t e r -pretation, a i s a constant which adjusts for the d i f f e r e n t units of measurement on each side of the equation. 8 represents the propensity of c i t i e s i and j to generate t r a v e l , whereas y suggests a revealed d i s u t i l i t y associated with distance. The gravity model i s sometimes used with a=0 and 8=1 to produce a single parameter model. This c l e a r l y e n t a i l s a strong assumption that a and 8 are known a p r i o r i . I f t h i s b e l i e f i s unfounded, then the model i s misspecified and any i n t e r p r e t a t i o n of the distance-decay c o e f f i c i e n t y would be subject to question. This shortcoming i s exacerbated by the m u l t i p l i c a t i v e form. Values of these c o e f f i c i e n t s are tested i n a subsequent section. I t i s possible to substitute a s t r i n g of a c t i v i t y v ariables into (7), as well as components of generalised impedance. Above-average t r a v e l demands are l i k e l y to be observed between a p a i r of c i t i e s where they have above-average income, above-average car ownership, s i m i l a r language or c u l t u r a l compositions. Therefore, a f u l l y - 62 -s p e c i f i e d equation might be the following: t . . - exp(B 0)(P. P.) (15) Whether a l l these variables are needed i s another question which depends ultimately on the data and on the convergence experience of the model. d e f i n i t i o n s of the distance v a r i a b l e apart from shortest-path distance i n miles, t r a v e l time or cost. I t has often been argued that the distance-decay e f f e c t should be a function of the p o t e n t i a l destinations located between a c i t y p a i r . The well-known intervening opportunities methodology has attempted to replace the distance variable e n t i r e l y . Nevertheless, i t seems preferable to keep a distance-type variable whilst modifying i t with some measure of intervening opportunity. Thus, the impedance term i n (4) could be written: where P^ i s the population of the k c i t y on the path from i to j . One defect of (16) i s that there i s probably some double-counting of opportunities. Consider a set of small centres as intervening opportunities. These centres possess only a r e s t r i c t e d range of urban functions and F i n a l l y , the discussion turns to a l t e r n a t i v e (16) th given a s u f f i c i e n t number of centres along the path from o r i g i n to destination many of these functions w i l l be duplicated. Now consider the same population clustered into one large urban centre. This centre w i l l have functions not possessed by the smaller centres and thereby represents a more potent intervening opportunity than smaller centres of the same cumulative population. These considerations suggest an a l t e r n a t i v e expression f o r the impedance term i n (16): y y f(D.., P., y,, y 2 ) = D.l (max P R) 2 (17) which uses the largest intervening urban centre rather than the t o t a l intervening population. 4.3 S p e c i f i c a t i o n of assignment 4.3.1 Multipath assignment It i s well known that D i a l (1971) introduced a simple device to escape from the r e s t r i c t i o n s of the a l l -or-nothing assignment p r i n c i p l e . Although i t has been shown recently to possess some u n r e a l i s t i c properties i n ce r t a i n circumstances ( B u r r e l l , 1974; F l o r i a n and Fox, 1976), the method i s s t i l l worth consideration. Dial's algorithm involves a simple recursive diversion mechanism.to d i s t r i b u t e t r a f f i c volumes over a - 64 -s e t o f a l t e r n a t i v e p a t h s i n p r o p o r t i o n t o t h e i r r e l a t i v e e f f i c i e n c i e s compared w i t h t h e s h o r t e s t p a t h . T h r e e e l e m e n t s a r e i n v o l v e d f o r e a c h a r c : a l i k e l i h o o d f u n c t i o n , an a r c w e i g h t and an a s s i g n e d t r i p v o l u m e . The l i k e l i h o o d f u n c t i o n f o r e a c h a r c d e p e n d s on t h e p r o x i m i t y o f t h e a r c t o o r i g i n and d e s t i n a t i o n n o d e s and t h e l e n g t h o f t h e a r c i t s e l f . I t i s s p e c i f i e d as a l o g i t f u n c t i o n , f o r a r c a = ( r , s ) , o r i g i n i and d e s t i n a t i o n j : (18) exp [e ( D i s - D. ^ - D )1 , i f D. <D. , i r r s J i r i s D . <D . S3 r3 o t h e r w i s e where H = t h e l i k e l i h o o d o f a r c a D. i r D. i s D ^3 D . S3 D r s = t h e s h o r t e s t - p a t h d i s t a n c e f r o m o r i g i n i t o node r = t h e s h o r t e s t - p a t h d i s t a n c e f r o m o r i g i n i t o node s = t h e s h o r t e s t - p a t h d i s t a n c e f r o m node r t o d e s t i n a t i o n j = t h e s h o r t e s t - p a t h d i s t a n c e f r o m node s t o d e s t i n a t i o n j = t h e l e n g t h o f a r c a = ( r , s ) The c o n d i t i o n s and D. < D. i r i s D . < D . S3 r3 r e q u i r e t h a t the a r c a be e f f i c i e n t i n terms of l e a d i n g away from o r i g i n i and approaching d e s t i n a t i o n j r e s p e c t i v e l y . Otherwise the a r c r e c e i v e s no t r a f f i c f o r t h a t o r i g i n - d e s t i n a t i o n p a i r . The a r c weight i s c a l c u l a t e d from the a r c l i k e l i h o o d s by w a = i a J 6 a b w b ( 1 9> where 6 . ab 1 i f a r c b immediately precedes a i n the r e c u r s i o n from i to j 0 o t h e r w i s e and E 6 ^ w^ = 1 f o r b=(r,s) and r = i A s s i g n e d t r i p volume i s c a l c u l a t e d from the a r c weights by v = w (E 6* v, / E 6 . w, ) (2.0) a a b a b b b a b b * r where 6 ^ 1 i f a r c b immediately precedes a i n <) the r e c u r s i o n from j to i 0 o t h e r w i s e ye and E 5 , v, = t . . f o r b=(r,s) and s=j b ao b 13 In o r d e r to g e n e r a l i s e the 0-D e s t i m a t i o n model to i n c l u d e m u l t i p a t h assignment, f o r c o n c r e t e n e s s , the b a s i c model i s s p e c i a l i s e d t o an o b j e c t i v e f u n c t i o n which - 66 -minimises the sum of squares and employs the gravity model as estimator of the d i s t r i b u t i o n matrix. This i s written as o 2 Minimise I (v - v ) wrt a, B, y (21) a a a where v =116.. t . . (22) a i j i j a ID t. . = a P 3. D T . (23) ID ID iD and where equation (22) represents a l l or nothing assign-ment. To extend the model to multipath assignment involves replacement of (22) and t h i s i s done by use of the short hand notation of v = £ £ S. . (6) t. . (24) a ± j 13a 13 which i s also meant to indicate the h e u r i s t i c rather than mathematically rigorous nature of the procedure. The d e f i n i t i o n of 6 requires a minor extension. I t can now assume values other than zero or one. Although i n general D^_. should be made a function of 9, since 9 spreads t r a f f i c away from the shortest-path and therefore i m p l i c i t l y increases distances, the present treatment assumes D^_. and 0 are independent up to a constant. I t i s claimed that any updating of D^.. as a function of 0 i s approximately proportional. I f a l l distances are increased p r o p o r t i o n a l l y , then 0 i s - 67 -independent o f to the e x t e n t t h a t a can be decreased t o compensate f o r i n c r e a s e d D „ . N a t u r a l l y , any p a r t i t i o n of 8 would r e q u i r e a c o r r e s p o n d i n g p a r t i t i o n o f a.. 4.3.2 S p e c i f i c models In the most g e n e r a l c o m p u t a t i o n a l scheme, the d i v e r s i o n parameter, 9, i s o p t i m i s e d s i m u l t a n e o u s l y w i t h the parameters which determine the 0 - D m a t r i x , a , 8, y . T h i s model i s s p e c i f i e d by ° 2 Minimise E (v - v ) wrt a , B , y ; a a a where v t . . 13 E E S . . (6) t . . l j 13a 1-a P B . D T . 13 13 > (25) Computational s a v i n g may be o b t a i n e d by a second model which imposes s e p a r a b i l i t y between the o p t i m i s a t i o n of a, 8 and y, and 0. F i r s t the problem (25) i s s o l v e d f o r f i x e d 0, then a, 8/ y are f i x e d and 0 i s o p t i m i s e d by m i n i m i s a t i o n o f the same o b j e c t i v e f u n c t i o n . These s t e p s are a l t e r n a t e d u n t i l convergence. The model may be w r i t t e n as - 68 -o 2 1. Minimise Z (v - v ) wrt a, 8, y a a a where v = Z Z 5 . ._ (9_) t . . i D i j a s x j t . . = a P 3 . D J . i ] x ] I D o 2 2. Minimise Z (v - v ) wrt a a a > (26) where v = Z Z <5 . . (9) t . . i j i D a I D 6 Y t . . = a P . D . s I D I D I D 3. Set s = s + 1 and re turn to step 1 u n t i l convergence of the e n t i r e parameter set i s obtained. Two a d d i t i o n a l models are nested w i t h i n model (26). The f i r s t of these i s obtained i f the i t e r a t i o n i s terminated with s=0. Given an i n i t i a l f ixed e, the a, 8 and Y are opt imised; then a, 6 a n d y are f ixed and the 9 opt imised . There i s no further i t e r a t i o n . Since t h i s model i s nested w i t h i n (26) s t a t i s t i c a l te s t s of the v a l i d i t y of t h i s r e s t r i c t e d ver s ion can be made. This a l so appl ies to the l a s t and s implest model which i s merely step 1 of (26). Given an i n i t i a l estimate of 9, optimise the a , 8 and y . This i s an important model - 69 -for two reasons. Not only does i t permit a s t a t i s t i c a l t e s t of the e l i m i n a t i o n of the i t e r a t i o n step, but more s i g n i f i c a n t l y , s ince i t contains the a l l or nothing assignment as a s p e c i a l case at 0=0, a t e s t of the v a l i d i t y of the assignment procedure i s now p o s s i b l e . CHAPTER 5 DIRECT UNCONSTRAINED MODELS II . LINEAR MODELS I t was shown i n equation (13) of the previous section that a l i n e a r f u n c t i o n a l form could be obtained as a s p e c i a l case of the general nonlinear model (5). The problem which i s addressed now i s whether a f u l l y l i n e a r model can be constructed with the arc volumes as the dependent v a r i a b l e . The answer i s i n the a f f i r m a t i v e , but the s t r i c t l i n e a r i t y which i s e n t a i l e d requires additive e f f e c t s i n the untransformed v a r i a b l e s , and as a r e s u l t , i s a p r i o r i less appealing than the nonlinear version. Linear forms are offered i n general as a l t e r n a t i v e s to nonlinear on the basis of four s i g n i f i c a n t properties of l i n e a r i t y . F i r s t , t h e i r s t a t i s t i c a l basis i s much cl e a r e r than for nonlinear models. Second, t h e i r estimation i s e f f i c i e n t and routine. Third, many non-l i n e a r forms can be transformed l i n e a r . Fourth, any nonlinear function can be approximated by l i n e a r segments. The discussion begins with the d e r i v a t i o n of a l i n e a r model. Subsequent attempts to elaborate and generalise t h i s form by continuous transformations, while maintaining l i n e a r i t y i n the parameters, are shown to f r u s t r a t e the extraction of O-D estimates. As a r e s u l t , t h i s l i n e of argument i s confined to transformation of disaggregate components of the independent v a r i a b l e s . Following t h i s , attention i s focussed on switching strategies f i r s t l y without, then with, constraints on the c o e f f i c i e n t s . 5.1 Derivation of the basic l i n e a r model A l i n e a r estimate for t ^ may be written: fch = 3 p h + y D h i n the context of the objective function min 2(v - v ) 2 (2) a a a where v a = Z 6 h a t h (3) Substituting (1) i n (3) and adding an error term produces °a = £ 5ha ( e P h + Y V + u a < 4 ) which i s recognisable as a standard l i n e a r regression problem i f written as ? a " *<g * h a P h) + Y<£ * h a D h) + u a (5) This l i n e a r model possesses the advantage of s i m p l i c i t y but unfortunately i t i s also s i m p l i s t i c . The e f f e c t s of the r i g h t hand side variables are additive which seems co u n t e r - i n t u i t i v e . Further, i t i s c l e a r l y capable of p r e d i c t i n g negative t ^ i n (1) where |y |>|B P^|, since y<0. Owing to these misgivings, attention i s now turned to transformations of (5) so as to allow i n t e r a c t i v e e f f e c t s while maintaining l i n e a r i t y . For t h i s purpose, the power family of monotonic transformations i s again invoked. 5.2 Transformations of the l i n e a r model The precise s p e c i f i c a t i o n of the power family depends on the model to be transformed, i n p a r t i c u l a r , the presence or absence of a constant (Schlesselman, 1971). As Zarembka has shown, d i v i d i n g by the exponent X gives a simpler Jacobian i n the transformed l i k e l i h o o d function. Tukey considered transformations of the form - (y+d)\ Box and Cox also considered y ^ = y^ and hence suggested two a l t e r n a t i v e d e f i n i t i o n s : (6) _in y X=0 and (the power transformation with s h i f t e d location) YiX) =\ X~[ (7) in (y+X2) Xi=0 Since the unconstrained gravity model has a constant and (6) i s continuous around X=0, the discussion i s s p e c i a l i s e d to t h i s transformation. In the context of the multivariate l i n e a r model: y U ) = xg + e (8) the l i k e l i h o o d function f o r the o r i g i n a l observations may be written as: (2uo - r n / > exp / - ( y U ) - xB) ( y U ) - ( 9 ) V 2 a 2 / where J(X;y) i s the Jacobian of the inverse transformation from y | ^ to the o r i g i n a l observations y^, and n J(X;y) = n i = l d y j ^ N 3L-1 n y (10) i = l 1 Box and Cox (1964) note that, since (9) i s simply the l i k e l i h o o d for a standard least-squares problem, the estimate of a 2 i s obtained a n a l y t i c a l l y by S(X)/n for given X, where S(X) i s the r e s i d u a l sum of squares. This gives the maximised log l i k e l i h o o d as: Lm a v< X> = - i n £ n S ( X ) + m J(X;y) (11) max z n the second term of which vanishes i f , as Box and Cox suggest, the normalised v a r i a t e - 74 -(A) ( A ) J(A;y) i / n y - 1 • n n y . . 1 /n A-1 y - i Ay (12) where y = the geometric mean of the o r i g i n a l observations, i s used. Now consider these transformations applied to (5) . ° i A ) = « + e ( g 6 h a V ( x , + *<£ 6ha V U ) (13) Equation (13) does i n fac t represent a generalisation of (5), i t can be estimated and the optimal A obtained. Unfortunately, when a l l t h i s i s done i t turns out that the estimates of the 0-D cannot be excavated. This i s c l e a r l y due to nonlinear transformation by (nonintegral) A of a sum. As a r e s u l t of t h i s , the best that can be achieved by these means i s merely a s c a l i n g of the h - s p e c i f i c components of the r i g h t hand side sums: v = a + 6 l i s , P, U ) ) + y Z f 6, D, (A) + u (14) which permits the t, to be re t r i e v e d as (A) h = a + B P^'" + Y D (A) h (15) the e f f e c t s of which are s t i l l additive i n the regression equation (14). - 75 -5.3 Switching strategies for a l i n e a r O-D estimator One a l t e r n a t i v e to a nonlinear model for the O-D matrix i s a piecewise l i n e a r approximation or switch-ing strategy (Quandt, 1958; 1960; 1972; Bacon and Watts, 1971; F e r r e i r a , 1975). Given s u f f i c i e n t l i n e a r segments, t h i s method c l e a r l y has the c a p a b i l i t y of approximating any nonlinear function. The main problem, however, concerns how many segments or regimes there should be and the l o c a t i o n of t h e i r j o i n s . I t can be argued that the true model for the data i s not s t r i c t l y nonlinear but consists of l i n e a r segments which are joined at d i s c o n t i n u i t i e s i n the f u n c t i o n a l form. The presence of such d i s c o n t i n u i t i e s may be anticipated i f , f o r example, they depend on gaps i n the hierarchy of c i t i e s . Central place theory and other h i e r a r c h i c a l urban models might support the presence of d i s c o n t i n u i t i e s . There are three fundamental types of switching strategy. F i r s t l y , where there i s a strong t h e o r e t i c a l basis for the estimation, i t i s possible that subsets of observations corresponding to each of the regimes may be i d e n t i f i e d on a p r i o r i grounds. This approach could be employed where d e t a i l e d analysis of a set of c i t i e s has revealed a h i e r a r c h i c a l r e l a t i o n s h i p based on f u n c t i o n a l composition. In t h i s case, not only are the number of regimes known but the l o c a t i o n of the joins can be determined accurately. - 76 -Secondly, the number o f regimes may be known exogenously but the l o c a t i o n s o f t h e i r j o i n s may be unknown. T h i s problem i s one of c l a s s i f i c a t i o n o f o b s e r v a t i o n s i n t o groups where the number o f groups i s known. For example, the q u e s t i o n c o u l d be what t h r e e s t r a i g h t l i n e s b e s t f i t the d a t a . T h i s methodology may be used a l s o where the number of regimes i s unknown but the r e s e a r c h e r can h y p o t h e s i s e the number a t each stage o f an i t e r a t i v e scheme. The f a c t t h a t contiguous o b s e r v a t i o n s b e l o n g e i t h e r t o the same regime o r t o contiguous regimes a i d s the a n a l y s i s c o n s i d e r a b l y . The l o c a t i o n o f new j o i n s may be found h e u r i s t i c a l l y a t the p o i n t o f maximum e r r o r between o b s e r v a t i o n s and r e g r e s s i o n l i n e i n the p r e v i o u s i t e r a t i o n . T h i r d l y , t h e r e i s the c o m p l e t e l y u n c o n s t r a i n e d case where any o b s e r v a t i o n may be r e l a t e d t o any regime. Thus, not o n l y i s the number of regimes unknown but the c o n t i g u i t y c o n s t r a i n t i s a l s o dropped. C o m p u t a t i o n a l l y , t h i s i s an i n f e a s i b l e method i f automatic regime i d e n t i f i c a t i o n methods are to be used. However, h e u r i s t i c ' e y e b a l l ' methods c o u l d be s u c c e s s f u l l y used t o i d e n t i f y such p a t t e r n s i n a s c a t t e r g r a m o f the da t a . T h i s i n d u c t i v e type o f approach i n v o l v e s the s e a r c h f o r c l e a r c r i t e r i a t o r e l a t e an o b s e r v a t i o n w i t h a regime. 5.3.1 Dependent va r i a b l e c r i t e r i o n A simple switching strategy i s to p a r t i t i o n each of the independent variables on the basis of s i z e . Following t h i s method, the basic l i n e a r model: °a ' i l P h ) + D h ) + E a ( 1 6 ) becomes o v a = 5 3k(g Phk) + I \ ( £ D h k ) + £ a <17> and, the 0-D estimate, given k f o r 0-D p a i r h: fchk " 3k Phk + ^k Dhk ( 1 8> This method i s s a t i s f a c t o r y provided the e f f e c t s of the independent variables are a d d i t i v e . Where, however, there are i n t e r a c t i o n e f f e c t s between the va r i a b l e s , i t i s necessary to p a r t i t i o n on the basis of t h e i r j o i n t e f f e c t s . Such i n t e r a c t i o n s are c l e a r l y basic to the gravity model owing to i t s m u l t i p l i c a t i v e form. Consequently, the e f f e c t of any v a r i a b l e depends not just on the variable i t s e l f but on the other variables i t i s combined with. Thus, the e f f e c t s of a population product may be accentuated or damped as a function of the distance involved. 5.3.2 J o i n t l y determined by independent va r i a b l e s Consider the following model: k l h h kl + k El Y k l n °hkl + e (19) and the 0-D estimate: fchkl " 3 k l P h k l + Y k l D h k l (20) where the B. . . , y. . r e l a t e to p a r t i t i o n s of the observa-k l k l tions made on the basis of t h e i r v a r i a b l e s ' j o i n t c h a r a c t e r i s t i c s . Note that t h i s c r i t e r i o n does not maintain cont i g u i t y of observations for any given variable. Let P,D be p a r t i t i o n e d i n t o 2 s i z e ranges: small (S) and large (L) ei t h e r a r b i t r a r i l y , or on a p r i o r i grounds, or s u b j e c t i v e l y . As a r e s u l t of t h i s , a matrix of p o s s i b i l i t i e s i s obtained: SP SD SP LD LP SD LP LD To each of these outcomes i s assigned a c o e f f i c i e n t to obtain a matrix of c o e f f i c i e n t s for each v a r i a b l e : B 11 B 12 '21 '22 Y 11 Y 12 Y21 Y22 Thus, the model becomes where h represents the number of O-D p a i r s associated with the (k,l) regime, i . e . , h ^ . Hence, the O-D p a i r s i n each regime are d i s j o i n t sets. 5.4 Switching strategies with i n e q u a l i t y r e s t r i c t i o n s Although i t i s possible to approximate any non-l i n e a r function by piecewise segments and thereby e s t a b l i s h a switching strategy, serious problems may occur. Consider a two-dimensional scattergram of. a nonlinear process together with a substantial amount of disturbance. Using nonlinear methods, no problems should be incurred i n f i t t i n g a continuous function to these data to obtain a slope of the correct sign. Now consider piecewise approximation of t h i s function by l i n e a r segments. The more segments used, the shorter each segment must be and the case a r i s e s that the l o c a l scatter within a segment w i l l cause the slope to have a sign which i s not the sign for the function as a whole. If the sole objective of the estimation were - 80 -curve f i t t i n g then t h i s aberration would be of l i t t l e consequence. However, i n the case of estimating the O-D matrix i t i s disastrous because i t leads to r a d i c a l l y d i f f e r e n t estimates of O-D f o r c i t y - p a i r s i n contiguous segments. As a r e s u l t of t h i s i n s t a b i l i t y , further elaboration of the l i n e a r model i s necessary, i n p a r t i c u l a r by the imposition of i n e q u a l i t y c onstraints. Unfortunately, t h i s adds s u b s t a n t i a l l y to the computational requirement of the estimation. In order to mitigate t h i s , a l e a s t absolute error (L.A.E.) estimator which can be solved by l i n e a r programming (L.P.) i s introduced f i r s t . A least-squares estimator i s then reintroduced which involves quadratic programming (Q.P.). This i s extended to a r e s u l t i n constrained polynomial regression which also involves Q.P. 5.4.1 Least absolute errors I t i s well-known that the l e a s t absolute errors regression model can be expressed as an L.P. problem (Charnes, Cooper and Ferguson, 1955; Fisher, 19 61; Taylor, 1974). As Taylor points out, L.A.E. i s a p o t e n t i a l l y useful c r i t e r i o n for t h i c k - t a i l e d d i s t r i b u t i o n s but possessing only a rudimentary d i s t r i b u -t i o n theory. Although computationally involved, the - 81 -L.A.E. model, when expressed as an L.P., allows the introduction of inequality constraints i n a most elegant manner, which actually reduces rather than increases the complexity of the problem. The L.P. form can be written as the following: min R = E(u. + w. ) wrt (8, , Y, , k=l, 2, ..K), j_ 1 1 K K (u., w., i = l , 2, ..n) l l (22) subject to y • =6 - y + 3 , x . - y. x . + ...+ 8. x,. - Y, x. . + u. - w. J i o 'o 1 1 1 ' 1 1 1 k k i ' k k i I I ( i = 1, 2, ...n) (23) Clearly t h i s i s an L.P. i n 2(n+k) variables and n constraints. The regression model turns up as the constraint set and L.A.E. i s the objective function. It i s also clear that weights could be applied to the u^ and w^  to obtain a weighted regression without additional e f f o r t . The form of the L.A.E. estimator given i n (2 3) i s unconstrained. This can be seen by noticing that succeeding variables i n (23) , being entered twice, are dependent. Hence, only one of (8 , y ) and (u., w.) for each k and i can be nonzero i n any basic solution. Since th 8^ and Y ^ d i f f e r by a minus sign, the k c o e f f i c i e n t - 82 -can therefore be e i t h e r p o s i t i v e , zero or negative. In order to e s t a b l i s h the constrained model i s dropped i f the k c o e f f i c i e n t i s to.be constrained p o s i t i v e , whereas 8 k i s dropped for n e g a t i v i t y . A f u l l y constrained model removes K variables from the L.P. The basic l i n e a r O-D estimator may be written i n L.A.E. constrained form, with 8 constrained p o s i t i v e and y negative, as min R = Z (u + w ) wrt (8, y, u , w , a=l,2,..,A) (24) a a a a a subject to °a = 6ha V " ^ 6ha V + u a ~ w a <25> (ci = 1 / 2/ • • • f .A.) In order to obtain the dependent va r i a b l e c r i t e r i o n for p a r t i t i o n i n g the constraint set (25) i s replaced by: v„ = Z 6, (Z 6, P^ , ) - Z y, (Z D^ , ) + u - w (26) a k k h ha hk k k h ha hk a a v ' where the segments are to be determined j o i n t l y by the independent variables (25) i s replaced by: ° a = , E 6, 0 (Z 5. P M . ) - I y,.(I S, D. . 0) + u - w ' (27)  kl k£ h ha hki k£ 'kl n ha hki a a - 83 -This ensures that estimates of 0-D cannot have the wrong sign but as w i l l be discussed l a t e r , t h i s does not prevent u n r e a l i s t i c or zero c o e f f i c i e n t s appearing on aberrant segments. 5.4.2 Least squares Although L.A.E. appears to be a p r a c t i c a l c r i t e r i o n for estimation, some authors have voiced doubts about i t s s t a t i s t i c a l e f f i c i e n c y (Ashar and Wallace, 1963; Judge and Takayama, 1966). In these circumstances i t seems prudent to e s t a b l i s h an i n e q u a l i t y constrained l e a s t squares estimator. Let X, = E 6, P., ka n n a x. = E x. k a ka z. = - E 5. D. . ka n n a hk z, = E z, k a ka Then i t follows that the objective function for the Q.P. i s : - 84 -min R Z(v 3 - Z B v x. - Z Y, z. ) 2 a a k k ka k k ka E vf - 2 E v (E 3. x. + Z v, z, ) a a a a vk k ka k rk ka' + E (E 3. x, + Z Yi z, ) a k k ka k k ka' E ° 2 - 2(Z 3. E v x, + E Y, E v z, ) a a k k a a k a k k a a ka + k I 3k h *k X * + J I Yk Y £ Zk Z £ S (28) + 2 ( £ Bk V <j Yk 2 k ) subject to: 0 r u ik « e k * r T k ' r T k * ° ' r i k > 0 r 2 k " 'k ~" "2k (k = 1, 2, ..., K) or, i n matrix notation similar to that of Judge and Takayama: - [B,Y mm R = v v + 2 3,Y x z X X X z Z X z z - ~ 8 Y subject to: r 1 -1 .Y J .u ' 1 I I -I r -r (29) (30) (31) 8, Y >, 0 - 85 -which i s a s t a n d a r d Q.P. problem to minimise a q u a d r a t i c f u n c t i o n w i t h r e s p e c t t o nonnegative v a l u e s o f 8 and y w h i l e s a t i s f y i n g a l i n e a r i n e q u a l i t y c o n s t r a i n t s e t . The c o n s t r a i n t s (31) can be augmented t o a l l o w l i n e a r combinations o f c o n s t r a i n t s (Judge and Takayama). However, something e x t r a i s i d e a l l y r e q u i r e d . For example, i f l a r g e r c i t i e s produce fewer i n t e r c i t y t r i p s per c a p i t a than do s m a l l ones, then a concave f u n c t i o n o f p o p u l a t i o n i s r e q u i r e d i n the continuous f o r m u l a t i o n . L i k e w i s e i n the s w i t c h i n g model, i t would be advantageous to be a b l e t o impose c o n c a v i t y on the f u n c t i o n d e f i n e d by p i e c e w i s e l i n e a r segments. I t does not seem p o s s i b l e t o i n c o r p o r a t e such c o n s t r a i n t s i n t o (31). What i s r e q u i r e d i s 0 < 8K ^ 8 K _ 1 ^ 4 8 k ^...-^ 8 1 (32) where 8 k i s the c o e f f i c i e n t o f the k t h segment. The K t n segment i s t h a t one most d i s t a n t from the o r i g i n . (32) can a l s o be w r i t t e n as the r e c u r s i o n : 0 ^ 6 k 4 1 (k=2, 3, . . . , K ) (33) pk - 1 In a continuous p o l y n o m i a l model the r e q u i r e d c o n s t r a i n t s would perhaps be 3_ . 3x 3x^ 6 o + P k ^ • + Z k P V x k ~ 1 ^ 0 1 k=2 k 8 + Z 6, x o k k k 1 = 28 0 + Z k ( k - l ) R x k ~ 2 0 2 k=3 k - 86 -C l e a r l y , as Hudson (1969) p o i n t s out, these are l i n e a r i n e q u a l i t i e s f o r a g i v e n v a l u e o f x, and g i v e n x not too l a r g e a c o n s t r a i n t c o u l d be e n t e r e d f o r each v a l u e o f x. 5.5 I n t e r a c t i o n p o l y n o m i a l s P u r s u i n g the l i n e a r model v i a s w i t c h i n g s t r a t e g i e s produced a t l e n g t h a r a t h e r i n e l e g a n t and undoubtedly i n e f f i c i e n t model. Most the problems stemmed from attempts t o handle the n o n l i n e a r i t i e s and i n t e r -a c t i o n e f f e c t s i n t r i n s i c t o t r i p d i s t r i b u t i o n . An a l t e r n a t i v e and much s i m p l e r approach i s t o d e f i n e what might be termed an i n t e r a c t i o n p o l y n o m i a l o v = a + a o Z a. Z 6, k k h h a + u (34) where 8 and y a r e b e s t e s t i m a t e s o f 8 and y . These e s t i m a t e s may have been o b t a i n e d from p r e v i o u s s t u d i e s , l o g i c a l n e c e s s i t y o r o t h e r p r i o r i n f o r m a t i o n . The parameter y ^ m o d i f i e s these p r i o r e s t i m a t e s i n the l i g h t o f the da t a thus the model i s a mixed e s t i m a t o r . The r o l e o f y i s p r i m a r i l y t o a d j u s t the mean t r i p l e n g t h . Given 6>0, y<0, as A approaches zero t r i p l e n g t h becomes very l o n g , as A i n c r e a s e s t r i p l e n g t h s h o r t e n s . I f A = l the p r i o r e s t i m a t e s o f 8 and y are unmodified. The a. are weights a t t a c h e d t o each term i n the p o l y n o m i a l . Two d i f f e r e n t schemes are envisaged. F i r s t , the polynomial may be estimated as a l i n e a r multiple regression problem, using perhaps stepwise procedures to weed out inappropriate values of X. To obtain the vari a b l e s f o r t h i s method, a sequence of As centred on A=l i s generated and the r e s u l t i n g values entered simultaneously into the regression. I t i s d i f f i c u l t to know a p r i o r i how many terms of the polynomial to compute and what the i n t e r v a l between successive values of A should be. However, between f i v e and ten terms should be adequate, with an i n t e r v a l between As of 0.1 to 0.3. The second scheme cal c u l a t e s A. for one k at k th a time. Now A^ i s the k i t e r a t i o n on A. A convergent sequence of A^ can be generated by any of the conventional techniques, such as Newton's method or Golden Section. Whereas method one consists of one l i n e a r regression with multiple v a r i a b l e s , method two comprises a one-dimensional minimisation problem with an embedded simple regression problem. For k>l, the f i r s t method leads to the following expression f o r estimates of O-D: fch = J a k ( P h D h ) X k <35> From the second method, with k=l, i t s i m p l i f i e s to t h = a ( P 3 D ^ ) X ( 3 6 ) - 88 -which, f o r A=l, i s a s p e c i a l case of the scheme given i n 7.2 but without i t e r a t i o n on the nonlinear parameters. For Ayi, and the i t e r a t i o n of the second method, i t c l o s e l y resembles 7.2 but with a constant r a t i o of the 8 and y. CHAPTER 6 SEQUENTIAL AND COMBINED MODELS 6.1 Separate generation and d i s t r i b u t i o n models 6.1.1 General form The p r i n c i p a l distinguishing feature of d i r e c t unconstrained models was seen to be the i r a b i l i t y to relate socio-economic and impedance variables to arc volumes i n a single estimated equation. This i s th e i r strength, but i t i s also a weakness. The c a l i b r a t i o n of the t r i p generation and d i s t r i b u t i o n model, since i t depends e x p l i c i t l y only on arc volumes thereby computes i m p l i c i t l y such important quantities as the generation of t r i p s per capita and the average t r i p length. Whether the di r e c t formulation i n fact produces accurate i m p l i c i t estimates of generation and average t r i p length i s not known. This consideration motivates the attempt to control these quantities by estimating them e x p l i c i t l y while separating generation and d i s t r i b u t i o n into two steps but at the expense of some loss of o v e r a l l model coherence. Separation i s achieved by writing separate equations for generation and d i s t r i b u t i o n . Accordingly, a general form of the model may be written minimise g(° a, v &) wrt ( a k , 8 , y ^ k=l,2,..,K; £=1,2,..L) (1) - 89 -- 90 -where v a = z z a.. t , . i j xj (2) t. . = f(0. , D. , A ID i D jk' (3) 0. = D. = (f> (A l x i k ' (4) where <j> i s some function of the a c t i v i t y variables i n the th i c i t y or zone. Equation (4) i s c l e a r l y a generation model, (3) i s t r i p d i s t r i b u t i o n , (2) i s assignment and (1) i s the objective function. These terms are computed i n t h i s order so that (1) depends on (2), which depends on (3), which depends on (4), hence the model i s sequen-t i a l . The remainder of t h i s section i s devoted to s p e c i f i c a t i o n of equations (3) and (4). i t e r a t i v e scheme i s employed which ca l c u l a t e s successively (4) back to (1). Thus, given a s t r i n g of a c t i v i t y v a r i a b l e s , a set of reasonable a, i s chosen. This determines 0. and D., exogenous to (3). For (3), parameters 6, , Y 0 * Assign the computed t. . to arcs and compute g. Revise a^, B^ and i t e r a t i v e l y to minimise g. Inequality constraints can be imposed to ensure that the average t r i p length and per cap i t a t r i p generation are acceptable. In order to estimate the c o e f f i c i e n t s , an i 3 given C^, D., C £ (and A jk' i f necessary) choose i n i t i a l - 91 -6.1.2 Singly constrained t r i p d i s t r i b u t i o n These models are so c a l l e d because the t r i p s d i s t r i b u t e d from i are constrained equal to the observed (or estimated) t r i p s generated at i , whereas the equivalent constraint w i l l not i n general hold for t r i p s a r r i v i n g at a d e s t i n a t i o n . Let the generation at i be estimated as °i = * ( A i k ' a k ) ( 5 ) which can for s i m p l i c i t y and concreteness be s p e c i f i e d as 0. = a P. (6) where a i s interpreted as t r i p s generated per c a p i t a (per unit time). Given t h i s estimate for CK, d i s t r i b u t i o n i s now written as a 'market share' equation due to Wardrop (1961), but also appearing as a shopping model (Huff, 1962): t. . =0. i . . . (7) where IL _. i s the p r o b a b i l i t y that a t r i p o r i g i n a t i n g at i i s destined for j . Several f u n c t i o n a l forms for IT. . have i D been advanced i n the l i t e r a t u r e on consumer choice. The o r i g i n a l model was written as a m u l t i p l i c a t i v e form but more recent work has argued for an exponential form known as the l o g i t model. Whatever the r e l a t i v e t h e o r e t i c a l merits of these two models, i t turns out that the l o g i t model can be generalised to include the m u l t i p l i c a t i v e model, along with many others, as s p e c i a l cases. Hence where (A.) are Box-Cox transformations, with A=l the l o g i t model i s obtained, whereas i f A=0 the 'i n t e r a c t i o n ' form of the market share equation appears: As i s well-known, many e x p l i c i t models are s p e c i a l cases of (9), for example, the si n g l y constrained gr a v i t y model and numerous v a r i a t i o n s on the r e t a i l trade model. u n t i l recently was regarded as i n t r i n s i c a l l y so, being r o u t i n e l y estimated by nonlinear optimising techniques. However, Nakanishi (1972) has demonstrated an ingenious transformation. Following Nakanishi and Cooper (1974), and incorporating m u l t i p l i c a t i v e errors u ; _. , the model II. . may be written generally as: (8) (9) If ever a model looked nonlinear i t i s (9), and - 93 -can be written o TT 13 n x. . . 1 u. k 13k; 13 z ( n x. . m . j Vk ! 3 k / ±3 (10) l e t t i n g X^ ..^  stand for a l l the va r i a b l e s i n (9) F i r s t write log T T . . = £ 8, log X. .. + log u. . y xj k k ^ 13k y 1: - log z f n x. .;- \ u. . (11) Now compute averages by summing over j(j=l,2,...,J) and d i v i d i n g by J: 1 Z log TT . . j i 1 ] Z 8, (1 Z log X. \ + 1 Z log u. . k k j j i D k j j 3 13 - log Z j n x. u, ID (12) noting that the l a s t term i s i n v a r i a n t . Let TT X i k u. 1 n §. . ^ V J \ 3 ' / n x. . , V j !Dk v f ' n u. r - J [3 ^ (13) - 94 -o the geometric means of n \ , and u^ .. . Recognising i n (12) that the f i r s t three terms are the log of the geometric means of these v a r i a b l e s , s u b s t i t u t i o n of (13) into (12) y i e l d s an expression for the market t o t a l term: log /' B k \ j v k 1 3 k ID = £ 6k 1 o ^ hk (14) + log u^ - log T T ^ Hence the l i n e a r form, from s u b s t i t u t i n g (14) i n (11) and rearranging: where . . and T? . are observed proportions or functions of i ] l observed proportions. I t i s i n t e r e s t i n g to note that t h i s n o n t r i v i a l transformation may have been anti c i p a t e d i n the context of Quandt and Baumol's (1966) abstract mode model, as l a t e r elaborated. Crow, Young and Cooley (1973) report that, among the a l t e r n a t i v e s suggested to avoid comparison of each mode with the 'best' mode, was one to use the r a t i o of each mode's at t r i b u t e s with the geometric mean of that a t t r i b u t e over a l l modes. E f f o r t s to use (15) as part of a l i n e a r estimator of the O-D matrix meet the same impasse that - 95 -was encountered i n Chapter 5. A l i n e a r estimator requires not only the r i g h t hand side of (15) to be l i n e a r but also that the l e f t hand side be i n the o r i g i n a l u n i t s . This i s c l e a r l y not the case, and the nonlinear optimisation used i n Chapter 4 must be applied. s p e c i a l case of (8), the usual binary l o g i t form can be used i n place of Nakanishi's involved transformation. This i s written, for X=0 i n (8), as which i s c l e a r l y simpler than (15), as the r a t i o s are based on a pairwise comparison. 6.1.3 Doubly constrained t r i p d i s t r i b u t i o n Although the si n g l y constrained model can be estimated e f f i c i e n t l y , i t has the disadvantage of f a i l i n g to make both the exogeneous t r i p o r i g i n s and the t r i p destinations consistent with the d i s t r i b u t i o n model. This consistency can be achieved by adding a second constraint, and at the same time adding computation cost. I t should be pointed out that, as (9) i s a (16) Let generation be written as before as 0. = a P. l (17) l - 96 -and, s e t t i n g a t t r a c t i o n at i equal to generation at i , write a t t r a c t i o n at j as D . = a P . D D (18) The doubly constrained d i s t r i b u t i o n model i s t . . =0. D. r. s. f (C..) ID x D 1 D ID (19) where r. = I s . = D E s . D . f (C. . )" j D D ID I r. 0. f (C. . ) x i i ID -1 -1 and are balancing f a c t o r s . This model may be c a l i b r a t e d quite e f f i c i e n t l y using an i t e r a t i v e scheme due o r i g i n a l l y to Furness (1965) and modified by Hyman (1969) . 6.2 Estimating O-D from t r i p ends Arc volumes are not nece s s a r i l y the sole source of data with which to estimate the O-D matrix. Conceivably, surveys may have been c a r r i e d out to determine the number of t r i p s o r i g i n a t i n g and terminating i n c i t i e s or zones. A l t e r n a t i v e l y , i t may prove f e a s i b l e to estimate these quantities from socio-economic and other data. R o b i l l a r d (19 75) suggested a method to obtain t r i p ends from arc volumes which, although t h e o r e t i c a l l y ingenious, turns out to be computationally grotesque and impractical for any but small networks. This i s due to the treatment of each 0-D as a v a r i a b l e i n a (singular) regression model. Without pursuing i n any depth at t h i s point how the exogenous estimates of t r i p ends are derived, attention i s now turned to an 0-D estimator which uses an a n a l y t i c a l estimate f o r ..average t r i p length. Wardrop (1961) showed how average t r i p length i s r e l a t e d to t r i p cost i n the case where a large number of zonal p a i r s permitted continuous approximation over a wide range of t r i p costs. In the d i s c r e t e case average t r i p cost i s defined as c = I £ t . . C . . / ? Z t . . (20) 1 ] 13 13 i j 13 K ' The continuous analogue, f o r negative exponential impedance, i s therefore (21) 0 0 oo — ] _ c = / q c exp (-3c) dc / / exp (-3c) dc = 3 This important r e s u l t due to Wardrop has been independently rediscovered at l e a s t three times (Barras et a l , 1971; Baxter, 1972; and by Zaryouni and Liebman, 1976). Equation (21) c l e a r l y r e l a t e s average t r i p length to the r e c i p r o c a l of the distance-decay parameter. As Zaryouni and Liebman point out, i n order to c a l i b r a t e - 98 -a gravity model, for p r a c t i c a l purposes 8 ^ may be set equal to the implied average t r i p length as computed by the model: 8 _ 1 = Z Z t. . C. . / Z E t . . (22) i j ID i : i j ID Using t h i s approximation (22), they suggest the following algorithm to c a l i b r a t e the model. Choose an i n i t i a l 8/ compute t. . (8) and c ( t . . (8)) and, f o r successive values 13 13 of 8 p l o t the functions 8 ^ and c i n the (8,c) parameter space. Where the curves i n t e r s e c t l i e s the s o l u t i o n f o r 8. The s i g n i f i c a n c e of t h i s procedure i s that the t. . which r e s u l t s from the grav i t y model c a l i b r a t i o n i s 13 an estimate of the 0-D matrix. This estimate, i f accurate, can be used independently of i t s gravity model o r i g i n . What t h i s formulation implies i s , since not a l l c i t i e s are equally accessible, a range of t r i p costs i s observable. The d i s t r i b u t i o n of t r i p ends at each c i t y r e f l e c t s i n part the response to these t r i p costs. Hence there may be s u f f i c i e n t information to estimate average t r i p length, thereby providing an estimate of the 0-D matrix. Provided the diffe r e n c e between 8 ^ and c i s unimodal for reasonable values of 8 and c the model can be written as an unconstrained optimisation problem, as follows: - 99 -Minimise F = |c - 1/8 wrt 8 where c = Z Z t. . C. . i j ID ID t - o. D r. s exp(-8 C ) (23) r i = Z s. D exp (-8 C. .) j J D ID -1 -1 s . = D Z r 0 exp(-8 C..) i l l i D -1 This model may be computed by an i t e r a t i v e algorithm which embeds the usual double-constrained gravity model i n an otherwise unconstrained optimisation of F with respect to 3. A computational scheme i s given by: 1. Choose i n i t i a l 8 , k=0 k k 2. Compute t . . =0. D. r. s. exp(-8 C.) * ID 1 D 1 D ID 3. Compute c k = Z Z t k . C . . / Z Z t k . i j !D ID i j ID 4. Compute f = Ic^  - l / 8 k | , k=k+l 5. Revise 8 (by nonlinear methods) to minimise f k k-1 6. I f |8 - 8 | < e convergence, otherwise go to 2 An operational weakness of t h i s method i s that i t requires exogenous estimates of t r i p ends. In general, these data - 100 -are not a v a i l a b l e from surveys. Hence i t would be useful i f estimates could be derived s y n t h e t i c a l l y . Consider the proposition that average t r i p length depends, however crudely, on the d i s t r i b u t i o n of population i n space. Thus t r i p s o r i g i n a t i n g i n areas of sparse population w i l l e x h i b i t greater t r i p lengths than t r i p s o r i g i n a t i n g i n areas of dense population. Ceteris paribus the need which motivates a t r i p w i l l be s a t i s f i e d at a nearer destination than a farther one, and given the denser occurrence of intervening opportunities i n areas of dense population, t r i p lengths w i l l be shorter. This argument could provide the basis for a t r i p generation model which has as i t s arguments a s t r i n g of socio-economic va r i a b l e s such as population, income, language, employment but also a c c e s s i b i l i t y to other c i t i e s . Socio-economic va r i a b l e s determine the number of t r i p s whereas a c c e s s i b i l i t y i s r e l a t e d to the average length of t r i p . 6.3 Combined d i s t r i b u t i o n and assignment A methodological weakness shared by the d i r e c t unconstrained and the sequential models a l i k e i s the absence of l o g i c a l feedback from assignment to d i s t r i b u t i o n . Under the assumption of convex increasing volume-delay functions or capacity constraints, i t follows that the - 101 -assignment of O-D to arcs has a nonnegative e f f e c t on the d i s t r i b u t i o n cost matrix from which the O-D i s computed. This i n t e r r e l a t i o n of d i s t r i b u t i o n and assignment has led to combined models i n a mathematical programming frame-work. I t i s now shown how each of d i s t r i b u t i o n and assign-ment have been expressed as constrained optimisation problems, then how they were combined and generalised. F i n a l l y , i t i s argued that these i n t e r n a l l y coherent models can be simply recast as estimators of the O-D matrix. Murchland (1966) showed that the doubly-constrained gravity model i s equivalent to an entropy maximation problem. This conceptual advance led to Tomlin and Tomlin (1968) who wrote t h i s gravity-entropy model as a g e n e r a l i s a t i o n of the well-known Hitchcock-Koopmans transportation problem: minimise 'l £ t. . C. . + y £ £ t. . In t. . l j I D I D i j I D I D subj ect to £ t. . j ^ £ t. . l I D = 0, D . D t. . > 0 I D wrt {t..} I D (24) On the other hand, Tomlin (1966) had already shown that capacitated assignment was equivalent to a l i n e a r program: - 102 -minimise £ c v = a. a subject to v 4 a a wrt {t.. } > (25) Z t. . = t. . p iDP t. . > 0 13P where v = Z Z Z 5.. t . . a i j p 1D aP iDP and b a are c a p a c i t i e s on arc volumes In order to combine d i s t r i b u t i o n and assignment, the t o t a l system costs i n the objective function of (24) are replaced with the corresponding costs i n the objective function of (25), and the constraints pooled (Tomlin, 1971) minimise Z c v + y Z Z t . . £ n t . a a i D ID ID wrt {t.. } iDP subject to v b z t . . - t. . p iDP ID Z t. . =0. j ^ 1 Z t. . = D . i !3 D t. . 0 iDP = 0 > (26) - 103 -F l o r i a n et a l (1975), Nguyen (1974) have recently shown how to generalise t h i s constant volume-delay function model by use of convex functions of the type studied by Dafermos (1971). Whereas Tomlin suggests the use of cumbersome piecewise l i n e a r approximations to handle t h i s case, F l o r i a n ' s elegant formulation simply requires replacing i n (26) c v a a by v a S, (v) dv a o This, i n e f f e c t , also writes the capacity constraint into the objective function, hence the e x p l i c i t constraint becomes redundant and i s deleted. C l e a r l y , use of (27) includes c v as a s p e c i a l case, a a Given a consistent combined model for d i s t r i b u t i o n and assignment, attention i s refocussed on generation. I t has been assumed that 0. and D. were known exogeneous inputs. This i s frequently a rash assumption. Furthermore, unless the value of y i s known, d i s t r i b u t i o n i s unknown. These quantities usually require an 0-D survey, however, i t i s suggested that observed arc volumes contain enough information to permit inferences about t r i p ends 0^ and D_. , about the t r i p length parameter y, and consequently about the 0-D matrix. - 104 -If the combined model (26 - 27) i s a good one, and there i s no reason to doubt t h i s from a t h e o r e t i c a l point of view, then a convergent sequence of generation parameters a^, and y, so as to reproduce observed arc volumes, w i l l produce a good estimate of the 0-D matrix. For t h i s purpose, (26 - 27) may be extended to minimise g(v , v ) wrt {a. , y} (28) a a K where v are obtained from (26 - 27) as a subproblem i n a which y i s fi x e d and 0. = D. = I a, Z.. (29) where Z ^ are zonal c h a r a c t e r i s t i c s , such as population and automobile ownership. I n t u i t i v e l y , (28) seems a well-defined problem. For i f the a. are set too low there w i l l be i n s u f f i c i e n t k t r i p s generated to explain the observed volumes. Thus, i n order to minimise g, the are revised upwards. S i m i l a r l y , i f y i s too low, the Hitchcock minimum cost problem dominates the objective function which would leave many elements of t^_. e f f e c t i v e l y empty. As a r e s u l t many arcs would be predicted as having near zero volumes. If these were less than the observed, y would be revised upwards. There i s , of course, an i n t e r a c t i o n between y and a^. but t h i s i s common i n m u l t i p l i c a t i v e models and i t - 1 0 5 -seems that the s p a t i a l structure of the network should be s u f f i c i e n t l y informative to permit estimation. 6.4 Direct (unconstrained) demand with constrained assignment The previous section showed that, by imposing s e p a r a b i l i t y between generation and d i s t r i b u t i o n , a combined model for d i s t r i b u t i o n and assignment could be employed to estimate O-D. If t h i s assumption of s e p a r a b i l i t y i s thought unwarranted, i t i s possible to recombine generation and d i s t r i b u t i o n into the d i r e c t unconstrained form. Some semblance of consistency between d i s t r i b u t i o n and assignment can be established by i t e r a t i v e procedures. These may not be necessary e m p i r i c a l l y i f the network i s chosen so as to be w e l l - s u i t e d for a l l - o r -nothing assignment, and the predicted arc volumes are reasonable. However, i f t h i s simple assignment mechanism f a i l s to reproduce accurately the arc volumes, i t c a l l s into question the v a l i d i t y of the estimated O-D matrix. Consequently, some more general assignment procedures containing a l l - o r - n o t h i n g as a s p e c i a l case should be a v a i l a b l e . The remainder of the chapter i s devoted to t h i s problem. - 106 -6 .4 .1 Arc capaci ty cons t ra in t s R e c a l l the l i n e a r programming model for capac i ta ted assignment given i n (25) , the d i r e c t model descr ibed i n Chapter 4 and the usual ob jec t ive funct ion g ( v a , v ) . Combining these elements the fo l lowing model can be obta ined: minimise g(v , v ) wrt {a, 8, y} a a where v i s der ived from: minimise (E c V ) wrt {t. . } a a a xjp subject to v b cl cl where v 13 E t . . = t . . p 13P 13 t . . >. 0 iDP E E E 6 . . t . . i j p i j a p 13P a P 3 . C Y !3 13 c . . = E t . . c . . / E t . . 13 p iDP 13P p 13P > (30) The model obviously contains severa l subproblems. The arc volumes are der ived from the capaci ta ted assignment, the t^j for t h i s assignment are obtained from the d i r e c t g rav i ty model, and aggregate the cost matr ix , which depends - 107 -on the path s e l e c t i o n generated by the capacitated assignment. follows: A suggested algorithm to compute (30) i s as g 1. Compute the C „ as the shortest path matrix, for s = 0. I n i t i a l values of {a, 6, y] s R 2. Compute t . . = a P7. C . 13 13 ID s s 3. Compute fcijp' v a ^ m t h e c a P a c i t a t e d assignment. o s 4. Compute g(v , v ) a a 5. Revise {a s,R s,y S} by unconstrained optimisation. 6. I f l a s - g S " 1 j <e, |B s-g s~ 1| <e, |y s y s " 1 \ <e, stop S - 1 FLS-1 S - 1 a p Y 7. s = s + 1 8. Compute C S. = E t S T 1 C . /E t S T 1 ID p iDP iDP p iDP 9. Go to 2. As before, the capacitated assignment may be replaced by equilibrium assignment, but at the expense of added computation. Where the capacitated problem has an - 108 -L.P. as a subproblem, equilibrium methods require a convex program. Nevertheless, the l a t t e r must be regarded as the superior model. I t i s obtained i n (30) by a p p l i c a -t i o n of the modification noted i n (27). CHAPTER 7 ALGORITHMS FOR O-D ESTIMATION 7.1 Introductory remarks Algorithms are procedures for computing solutions to models. Without e f f i c i e n t algorithms many models would be of l i t t l e empirical a p p l i c a t i o n , and t h i s i s true of most of the nonlinear estimators of the O-D matrix. Any device which reduces the computational burden may be regarded as algorithmic i n character, even though i t could be very simple. The f i r s t such device which i s discussed i s a technique to reduce the dimensionality of the nonlinear optimisation problem. This i s achieved by obtaining a l i n e a r estimate of the constant parameter as a function of the remaining parameters i n the t r i p d i s t r i b u t i o n equation. Computationally, the most expensive part of estimation i s the assignment of estimated t r i p s to arcs. Considerable attention i s focussed on t h i s , and three a l t e r n a t i v e algorithms are proposed which take advantage of d i f f e r e n t computer system configurations. Following t h i s , an attempt i s made to eliminate the assignment stage by expressing the bundles of O-Ds which use each arc as permutations of O-Ds, hence de r i v i n g weights for each O-D p a i r . Next, a novel way of reducing computation cost i s presented. This involves the s e l e c t i o n of those O-D pairs which dominate the objective function and use of only these - 109 -- 110 -data for convergence. The primary aim of t h i s method was to provide good i n i t i a l estimates for optioning parameters but results are s u f f i c i e n t l y accurate to allow such estimates to be regarded as f i n a l . A response surface of one model i s presented. This shows that the l i k e l i h o o d surface i s i n fact unimodal and well-defined. This ensures a unique optimum. 7.2 A l i n e a r estimate of the constant i n the d i r e c t unconstrained model Most models contain a constant term. Often i t i s a t h e o r e t i c a l necessity because the l e f t and right hand sides of an equation are measured i n d i f f e r e n t units. Omission of the constant would bias the remaining c o e f f i -cients. However, the presence of the constant may have deleterious e f fects from two points of view. F i r s t , i t increases the dimension of the solution space. Second, constants are notorious for complicating the response surface, hence retarding or preventing convergence. A common problem encountered i s that a constant has i n t r o -duced a long narrow ridge into the response surface. Since symmetrical and, i f possible, quadratic surfaces, are preferred by most optimising routines, convergence on a ridge i s often poor, r e s u l t i n g i n additional function evaluations or f a i l u r e . Owing to t h i s , at least one method has been developed s p e c i f i c a l l y to optimise on ridges. - I l l -T h i s i s Rosenbrock 1s a l g o r i t h m (Rosenbrock, 1960). In the l i g h t o f these problems, i t would c l e a r l y be advantageous i f the c o n s t a n t c o u l d be e l i m i n a t e d o r , a l t e r n a t i v e l y , approximated i n some way. Lawton and S y l v e s t r e (1971) have shown t h a t a l i n e a r e s t i m a t e o f the c o n s t a n t can be made i f i t e n t e r s the e q u a t i o n i n a l i n e a r manner, t h a t i s , a d d i t i v e l y o r m u l t i p l i c a t i v e l y . T h i s p r o p e r t y was a l s o e x p l o i t e d i n W i l l s (1971). p r e v i o u s l y , i t i s c l e a r t h a t both a d d i t i v e and m u l t i p l i c a t i v e cases are r e p r e s e n t e d . To s i m p l i f y the i n i t i a l e x p o s i t i o n , a simple g r a v i t y model w i t h m u l t i p l i c a -t i v e c o n s t a n t i s shown. R e c a l l t h a t the u n c o n s t r a i n e d d i r e c t model may be w r i t t e n , f o r c i t y p a i r s h and a r c s a, as In the t r i p d i s t r i b u t i o n e q u a t i o n s d e s c r i b e d min g(v , v ) a' a (1) where v a and = 3~ + S 8, A ' h |_ 0 k k hk hk from which can be o b t a i n e d as a s p e c i a l case: t, = a P 3 DJ h h h (2) The f i r s t step i s to i s o l a t e the constant a from the assignment mapping by bringing i t outside the summation. Hence, i t i s c l e a r that v = 2 5 u K a h n a n Z 6, (a DI) h ha h h a £ S h a ( P h D h ) ( 3» Adopting the notation of Lawton and Sylvestre, the s t a t i s t i c a l model may be written as ° a = a g(8, Y ; ' ( P h f Dh>, hea) + u & (4) (a = 1, 2, . . , A) where a i s a l i n e a r parameter, 8 and y nonlinear ones, and u are r e s i d u a l s . Let g be A x 1 with elements CL g a = g a ( 3 ' Y ? { P h ' D h } ' h e a ) (5) Then i t follows that the best estimate of a i n terms of minimising squared error i s - 113 -where g & = £ 6 h a ( P £ D £ ) (7) that i s , by substitution of (7) i n (6), £ v I 6 a = a a lh (8) E E 6, ( P a |_h ha It might be objected at th i s point that this expression requires extensive computation. Such an appearance, however, i s i l l u s o r y since the terms i n square brackets are computed anyway during the assignment. Consequently, the sole addition to each i t e r a t i o n i s A m u l t i p l i c a t i o n s , A squares, 2A additions and one d i v i s i o n , which i s a t r i v i a l amount. parameters from the optimisation procedure has a highly f r u i t f u l consequence: large numbers of dummy variables representing o r i g i n - s p e c i f i c or c i t y - s i z e class constants become computationally f e a s i b l e . Thus, the set of 0-D pairs may be partitioned according to some predetermined c r i t e r i o n and a s p e c i f i c constant assigned to each p a r t i -t i o n . For one p a r t i t i o n , giving two subsets, the model i s therefore The demonstrated a b i l i t y to remove the l i n e a r o v a P ^ D' + a„ E 6 + u (9) a a - 114 -and the l i n e a r estimates of and are obtainable from the e a s i l y recognisable multiple regression subproblem. F i n a l l y , i t i s straightforward to extend t h i s r e s u l t to a constant for every o r i g i n i , provided that the number of arcs with independent information i s greater than the number of o r i g i n s , to obtain v = Z a. H , . pf. D J . + u (10) a i 1 n hia h i h i a v ; Although (10) may involve a large regression problem, t h i s i s s t i l l computationally t r a c t a b l e and many orders of magnitude more e f f i c i e n t than f u l l optimisation, which i n any case would probably f a i l to converge. A question which might a r i s e at t h i s point i s whether there e x i s t s s u f f i c i e n t information i n the arc volumes to provide estimates of o r i g i n - s p e c i f i c constants. In p r a c t i c e , i t may happen that some , i f l e f t uncon-strained, become negative which i s inadmissible since negative O-D estimates would be the r e s u l t . Nonetheless, constants could be estimated for sets of o r i g i n s . An e f f i c i e n t algorithm to compute these estimates i s as follows: 1. Choose i n i t i a l nonlinear parameters 8 and y 2. Compute h i h i h i - 115 -3. Calculate the p a r t i t i o n e d assignment v . = Z 5, • t, . a i h hxa h i 4. Solve the multiple regression problem v = Z a. v . + u a i I a i a Calculate the f u l l assignment v = Z a. v . a X I a i 6. Compute objective function g o g = g ( v a , v a ) , te s t convergence 7. Revise 8 and y, and go to 2. If the constant has not been eliminated from the optimisation by the l i n e a r estimate, i t can nonetheless be s p e c i f i e d so as to reduce the number of i t e r a t i o n s required. This i s done by replacing a with exp(a) so that a l l the parameters become exponents. What t h i s seemingly innocuous change does i s to transform the response surface of the l i k e l i h o o d function so that i t becomes more symmetrical with respect to a . The contours of the l i k e l i -hood e l l i p s e remain eccentric i n general, as concentric contours would imply that the model were l i n e a r . - 116 -Untransformed, the l i k e l i h o o d response s u r f a c e turns out to be markedly skewed, wi t h the t a i l p o i n t i n g away from the 8 and y axes. T h i s asymmetry impedes o p t i m i s a t i o n . R e d e f i n i t i o n of the constant as an exponent has i n p r a c t i c e reduced the number of i t e r a t i o n s by about one t h i r d . 7.3 Assignment algorit h m s f o r p r o p o r t i o n a l methods A p r o p o r t i o n a l assignment method maps the o r i g i n -d e s t i n a t i o n t r i p matrix onto the network arcs i n such a way t h a t any change i n the O-D matrix i s r e f l e c t e d i n a p r o p o r t i o n a l change i n the arc volumes. Three a l g o r i t h m s are proposed f o r t h i s assignment. The d i s c u s s i o n i s s p e c i a l i s e d to a l l - o r - n o t h i n g assignment to s i m p l i f y d e t a i l s . 7.3.1 A l g o r i t h m 1 T h i s method r e q u i r e s the paths between a l l p a i r s of c e n t r o i d s to be s t o r e d . T h i s i n v o l v e s the storage of an MxN matrix (where M are c e n t r o i d s and N nodes) which c o n t a i n s the address of the a r c preceding a node i n the path. I f a s h o r t e s t - p a t h a l g o r i t h m based on D i j k s t r a ' s p r i n c i p l e i s used, t h i s r o u t i n g matrix i s s t o r e d as M v e c t o r s o n l y one of which needs to be i n core a t any one time. T h i s decomposition i s a l s o t r u e f o r the d i s t a n c e matrix used to compute the estimated O-D t r i p s . The a l g o r i t h m i s as f o l l o w s : For o r i g i n i ( i = l , 2 , . . . , I ) Read the r o u t i n g v e c t o r a s s o c i a t e d w i t h o r i g i n For d e s t i n a t i o n j (j=1,2,...,J) do the f o l l o w i n g : 1. Set k=j. 2. Locate the arc a p r e c e d i n g node k. 3 . Compute v •*— v + t . .. a a i ] 4. Set k=h where h i s the node pr e c e d i n g arc a. 5. I f h=i stop, otherwise go to 2. 7.3.2 A l g o r i t h m 2 The purpose of t h i s a l g o r i t h m i s to s t o r e e x p l i c i t l y the f o l l o w i n g mapping as a matrix ^ ^ j a : v = E E S . . t. . a i j 13a 13 where 5. . =< 1 i f u s e s a r c a 13a „ ,, J . J 1 0 otherwise Since 6 . . , i f s t o r e d as a f u l l matrix, would be IxJxA, i 13a i s necessary to e x p l o i t i t s sparseness and s t o r e i t as a set of l i s t s . For each arc a there i s a s s o c i a t e d a l i s t of addresses which l o c a t e o r i g i n - d e s t i n a t i o n p a i r s i n the - 118 -t . . matrix of t r i p s . Where the t . . matrix and/or the ID ID network a r c s are symmetric w i t h r e s p e c t to t r i p volumes and ar c c o s t s r e s p e c t i v e l y , c o n s i d e r a b l e savings are achieved i n computing e f f o r t and s t o r a g e . Formally, the a l g o r i t h m i s as f o l l o w s . Given t h a t 6.. i s a l r e a d y c o n s t r u c t e d and i : a 2 s t o r e d , do the f o l l o w i n g : For each arc a: 1. Read the s u b l i s t of t . . such t h a t 6 . . = 1 . ID ±ja 2 . Compute v v + t . . . a a l j 7.3.3 A l g o r i t h m 3 Whereas a l g o r i t h m 2 s t o r e d the a r c - p a t h mapping e x p l i c i t l y as an a r r a y of l i s t s c o rresponding to addresses i n the O-D matrix, a l g o r i t h m 3 e x p l o i t s the s p e c i a l s t r u c t u r e of these l i s t s . T h i s i s p o s s i b l e owing to the f a c t t h a t the addresses of the O-D matrix are observed to run i n c o n s e c u t i v e sequences. T h i s stems from the s t r u c t u r a l p r o p e r t i e s of the network. An arc l o c a t e d c l o s e to a given c e n t r o i d may be expected to c a r r y paths from t h a t c e n t r o i d to a subset o f d e s t i n a t i o n s . I f c e n t r o i d s w i t h s i m i l a r numbers are i n s i m i l a r l o c a t i o n s i n the network, the subset of d e s t i n a t i o n s reachable by t h a t arc from the o r i g i n c e n t r o i d w i l l r e s u l t i n a c o n s e c u t i v e - 119 -sequence of addresses i n the 0-D matrix. Where the mapping between arcs and many pairs of o r i g i n s and destinations i s to be stored, the l i s t f o r each arc would consist of a set of s u b l i s t s which represent subsequences of contiguous 0-D matrix addresses. Owing to the a d d i t i o n a l l i s t - p r o c e s s i n g software required for t h i s algorithm, the computational e f f o r t i s s l i g h t l y greater than f o r algorithm 2. However, t h i s a d d i t i o n a l cost i s more than recovered i n the savings i n core storage achieved by condensing the arc-path mapping matrix. For large networks with many centroids, the method of algorithm 2 becomes impractical from the point of view of core storage. This problem i s exacerbated by the f a c t that a network with more than 180 o r i g i n s and destinations r e s u l t s i n 0-D matrix addresses larger than the maximum value which can be stored i n an integer h a l f -word on the IBM computer. The precise savings which are r e a l i s e d by using algorithm 3 over algorithm 2 depends on the b i l l i n g algorithm of the system, but they are l i k e l y to be s u b s t a n t i a l . As before, the purpose i s to store the arc-path mapping v = Z Z 6 . . t. . a i j 13a 13 but, i n t h i s case, as a l i s t structure of l i s t s and subsequences. - 120 -Let k= the set of ( i , j ) p a i r s . h£H(a) = the number of subsequences of consecutive addresses associated with arc a. th ieL(h) = the c a r d i n a l i t y of the h set. Then the expression V h a = I 6h£a th& sums the elements within each subsequence h, and v = E V, = E E 5, „ t, „ a h ha h i h£a hi i n addition sums the subsequence t o t a l s for each arc a. Each subsequence i s represented by two items: a) i t s c a r d i n a l i t y , that i s , the number of elements i n the set, and b) the i n i t i a l O-D matrix address i n the set. Thus, a consecutive sequence of 100 addresses i n algorithm 2 would be condensed to a set of two items by algorithm 3. The actual addresses are e a s i l y recovered by incrementing the i n i t i a l address up to the c a r d i n a l i t y of the set. 7.4 Elimination of assignment For simple models of t r i p generation and d i s t r i b u t i o n at least, assignment of th e i r 0-D estimates to the arcs i s the most expensive computational step of each i t e r a t i o n . Three algorithms were described i n the previous section to handle th i s problem, but i t remains a problem. Consequently, the question arises as to whether assignment might be eliminated by d i r e c t l y r e l a t i n g the error function to be minimised with some simpler function of the estimated 0-D. It turns out that t h i s can in fact be done, except that the re s u l t does not seem to be computationally simpler, unless certain assumptions can be made. then the aim i s to obtain a tractable expression for v . F i r s t , expand Let the objective function be sp e c i f i e d as min g(v , v a a £ (v - v ) a a a (13) E (v - v ) a a a' _ o2 _ 2 _ v o = £ v + E v - 2 E v v (14) a a a a a a a where v„ = a h ah h = £ t, (£ v 6 . ) h h a a ah' (15) - 122 -E q u a t i o n (15) s i m p l y r e q u i r e s a m a t r i x o f w e i g h t s w h i c h a r e d e t e r m i n e d from t h e o b s e r v e d a r c volumes and t h e ass i g n m e n t o u t s i d e o f t h e i t e r a t i o n l o o p . However, t h e 2 e x p r e s s i o n f o r Z v i s u n p l e a s a n t : a a Z v 2 = Z (Z 6 . t. ) 2  a a a \ ah h' Z [Z(<5 , t , ) 2 + E Z (6 , t . 5 . t . ) ] a L h ah h h k*=h ah h ak k J Z Z ( 6 , t , ) 2 + Z Z Z (6 , t , 6 v t. ) a h ah h' a h k=i=h ah h ak k' Z Z 5 , t 2 + Z Z t. t. S (6 . 5 .. ) n a a h h H kfh h k a ah ak' Z t 2 5, + Z t. Z t . 5. . (16) h h h h h k^ =h k hk E q u a t i o n (16) t h u s i n v o l v e s a v e r y l a r g e m a t r i x o f w e i g h t s f o r t h e c r o s s - p r o d u c t s o f t ^ and t ^ . Now, as t h e m a t r i x 6 ^ c o n s i s t s o f many z e r o e s , t h e n 6. . = Z (6 , 6 , ) hk a ah ak' w i l l be s p a r s e . Even s o , t h e second term i n (16) c o u l d perhaps be s i g n i f i c a n t . U n l e s s t h i s t e rm c o u l d be shown t o be s m a l l , o r a l t e r n a t i v e l y , t h e l a r g e s t e l e m e n t s e x t r a c t e d and s t o r e d as l i s t s , t h e method appears unpromis-i n g i n p r a c t i c e as y e t . - 123 -S u b s t i t u t i n g (15) and (16) i n (14) to o b t a i n E v 2 + Z tl 6, + E t. Z t. 6. , - 2 E t. Q. (17) a a R h h h h k+h k hk n h ^h i t becomes c l e a r e r t h a t i t may be d i f f i c u l t . t o e l i m i n a t e assignment without f u r t h e r work. 7.5 Maximal interchange s e l e c t i o n procedure f o r c a l i b r a t i o n Although i t i s p o s s i b l e to estimate a l l the c e l l s o f the 0-D matrix and to use the i n f o r m a t i o n from a l l these i n the g o o d n e s s - o f - f i t c r i t e r i o n chosen f o r c a l i b r a t i o n , t h i s f u l l i n f o r m a t i o n approach may t u r n out to be com p u t a t i o n a l l y p r o h i b i t i v e f o r l a r g e t r a n s p o r t a t i o n networks with many c e n t r o i d s . For example, a network of 500 c e n t r o i d s would r e q u i r e 250,000 c e l l s to be recomputed about 100 times d u r i n g c a l i b r a t i o n by i t e r a t i v e o p t i m i s a -t i o n procedures. In order to av o i d these problems, a sampling scheme may be employed i n order t o reduce the number of c e l l s computed. The design o f such a scheme i s f a c i l i t a t e d by the f a c t t h a t the s i z e d i s t r i b u t i o n o f interchange v a l u e s over the 0-D matrix i s markedly skewed. There are many small values but few l a r g e i n t e r c h a n g e s . In the a s s i g n -ment of these values to the network, i t i s c l e a r t h a t the - 124 -l a r g e interchanges w i l l dominate the s m a l l ones. Thus, i t i s probably s u f f i c i e n t f o r approximate c a l i b r a t i o n purposes merely to take the l a r g e s t 10 per cent of the in t e r c h a n g e s , not i n c l u d i n g the d i a g o n a l or i n t r a z o n a l t r i p s . I f an accurate c a l i b r a t i o n i s necessary, u s i n g a l l the i n f o r m a t i o n a v a i l a b l e , t h i s interchange s e l e c t i o n procedure may be used as a p r e - c a l i b r a t i o n phase to determine a good set of i n i t i a l values f o r the c o e f f i c i e n t s f o r the f u l l i n f o r m a t i o n c a l i b r a t i o n phase. T h i s would mean t h a t i t e r a t i o n would begin from a p o i n t a l r e a d y c l o s e to the o p t i o n a l s o l u t i o n and f i n a l convergence would t h e r e f o r e be r a p i d . The c r i t e r i o n f o r the s e l e c t i o n o f maximal interchanges r e q u i r e s the s e l e c t i o n of those interchanges which c o n t r i b u t e most t o the o b j e c t i v e f u n c t i o n . T h i s i n v o l v e s , however, a l i t t l e more than merely the t . . ' s J l j with the h i g h e s t estimated volume. Owing to the a s s i g n -ment of a given interchange to many ar c s along the path s e p a r a t i n g o r i g i n and d e s t i n a t i o n , i t f o l l o w s t h a t the s e l e c t i o n c r i t e r i o n should i n v o l v e the l e n g t h o f the path. These c o n s i d e r a t i o n s motivate the f o l l o w i n g d e f i n i t i o n to s e l e c t the maximal k per cent of i n t e r -changes :determined by the t h r e s h o l d £ i m p l i e d by k, where t . . i s an i n i t i a l estimate of the 0-D and ID a^j i s the path length in arcs from i t to j . Any other c r i t e r i o n for sel e c t i o n , such as t. . alone w i l l T iD be biased at the outset towards contiguous or proximate origins and destinations. A problem remaining concerns the i n i t i a l estimate of the 0-D. Clearly i f the parameters are already known with approximate accuracy for a given model, then these w i l l ensure a good estimate. This s i t u a t i o n might aris e where several related hypotheses are being tested or other minor s p e c i f i c a t i o n changes are made. In contrast, the case w i l l arise where an estimate i s being obtained for the f i r s t time on a new dataset. Here, the best strategy i s probably to employ a two-stage technique which begins with a reasonable estimate, such as that provided by the simple gravity model, obtains new c o e f f i c i e n t s , which are then inserted as new i n i t i a l c o e f f i c i e n t s for a second stage of optimisation. It i s of considerable i n t e r e s t to analyse the re s u l t of using a fixed interchange selection strategy on the f i n a l c o e f f i c i e n t s . As yet the two-stage method i s untested, but the one-stage method has been thoroughly tested, with the simple gravity model as a f i r s t estimate. Computational experience with such an approxima-t i o n i s examined i n some d e t a i l for B.C. and Canadian - 126 -data i n subsequent sections. Using the model given by v • = Z 5. t. + u (18) a h n a n a where t , = a pf h h h a series of estimates for the B.C. data were made with k varying from 5 per cent to 100 per cent of the (76 2-76)/2 interchanges. I t i s apparent that estimates based on 5 per cent of the interchanges are very biased but those on 20 per cent and 30 per cent remarkably good. This alone demonstrates the v a l i d i t y of the technique. More-over, the parameters do not change i r r e g u l a r l y as the percentage i s increased but instead they e x h i b i t a smooth function which i s asymptotic to the true estimates. i The smooth convergence path of the parameters can be exploited to obtain parameter estimates f o r the f u l l set of interchanges. Accordingly, a function could be f i t t e d to several points and the f u l l optimum set obtained by extrapolation. This strategy would prove important where the f u l l problem turns out to exceed computer capacity i n some way. Bias i n the parameters or i g i n a t e s with the omission of many interchanges with not inconsequential volumes. The model given by (18) i s incomplete to the extent that volumes from the omitted interchanges are - 127 -contained i n the observed volumes but do not appear any-where on the r i g h t hand side of the equation. Since these omitted interchanges have by d e f i n i t i o n non-negative flows, t h e i r exclusion i n e v i t a b l y r e s u l t s i n a systematic bias i n the c o e f f i c i e n t s so as to e l i c i t an upward bias i n the estimates of O-D. I t turns out that t h i s problem can be minimised by the i n c l u s i o n of a constant which serves as a receptacle for interchanges not being estimated. Hence the model becomes v = a + Z 6 U I. + u (19) a 0 h ha h a o where a = 1 Z(v - v ) 0 A A A A and v = Z 5, t, a n ha h Thus i s the r e s i d u a l mean assignment due to omitted interchanges and v^ i s the estimated assignment due s o l e l y to included interchanges. Computational experience with t h i s extended formulation r e s u l t s i n marked improvement i n the estimates. For example, whereas for 10 per cent of the interchanges c o e f f i c i e n t s were demonstrably biased i n model (18), estimates very close to the 100 per cent c o e f f i c i e n t s were obtained from model (19) . Furthermore, a was large - 128 -and s t a t i s t i c a l l y s i g n i f i c a n t . F u l l e r e m p i r i c a l d e t a i l s are presented i n subsequent s e c t i o n s . F u r t h e r improvement i n the accuracy of the maximal interchange estimates can be achieved by the use of a two-stage e s t i m a t i o n s t r a t e g y . F i r s t , the i n t e r -change s e l e c t i o n i s based on some f e a s i b l e i n i t i a l c o e f f i c i e n t s , a^, B^, y® i n equation (18). The model then converges u s i n g t h i s i n i t i a l interchange s e l e c t i o n to o b t a i n a new s e t of c o e f f i c i e n t s , a"*", 8"*", y^~. Now the s e l e c t i o n i s r e v i s e d i n the l i g h t of t h i s approxima-t i o n to the f u l l i n f o r m a t i o n c o e f f i c i e n t s and subsequent convergence produces the second stage c o e f f i c i e n t s , 2 2 2 a , 3 , Y • A d d i t i o n a l steps i n t h i s i t e r a t i v e scheme cou l d be attempted e x p e r i m e n t a l l y but i t i s a n t i c i p a t e d t h a t any improvement would be i n c o n s e q u e n t i a l . 7.6 Response s u r f a c e s An i n v a l u a b l e t o o l i n the v a l i d a t i o n and under-standing of a model i s the response s u r f a c e i n parameter space of i t s o b j e c t i v e f u n c t i o n . Examination of the response s u r f a c e allows uniqueness of the optimum to be e s t a b l i s h e d e m p i r i c a l l y . Where the s u r f a c e turns out to - 129 -be multi-modal, the optima may be enumerated and examined provided these are not too many nor too sharp to be caught by the g r i d of function evaluations. The shape of the surface i s of considerable i n t e r e s t . The analyst w i l l f i n d answers to i n t e r e s t i n g questions l i k e how sharp the optimum i s , whether the surface i s asymmetrical, implying n o n l i n e a r i t y , or symmetrical, implying l i n e a r i t y . Existence of elongated ridges i s often substantiated. Interpretation of these topographical features may have considerable relevance for the under-standing of the behaviour of a model. For instance, a ridge i n the surface of a m u l t i p l i c a t i v e model with i n t e r -active e f f e c t s implies the s u b s t i t u t i o n of varying amounts of one variable for another. I f the ridge i s f l a t on top i t implies the variables are, within that range of v a r i a t i o n of the parameters, perfect s u b s t i t u t e s . The more curved the long axis of the ridge, the less perfect the s u b s t i t u t i o n u n t i l , i n the l i m i t the ridge turns into a round h i l l and the variables are orthogonal, the e f f e c t s a d d i t i v e . Response surfaces are most useful for two parameter models because one p l o t i n two dimensions completely describes the behaviour of the function. With models which exceed two parameters surfaces can be - 130 -plotted on a pairwise basis, the remaining parameters constant. Where models have two lev e l s of parameters, such as the As and 8s of the d i r e c t unconstrained model, response surfaces for either set may be obtained. That i s , for a given set of transformations induced by the As, a b i v a r i a t e 8 parameter space i s exhibited. A l t e r n a t i v e l y , for a given set of 8s the biv a r i a t e A space i s examined. This schema would allow the analyst to see how the response changed shape over a family of transformations. A 2-A model would be p a r t i c u l a r l y appropriate i n t h i s context. Figure 7.1 shows the response surface of the l i k e l i h o o d function for the 1966 B.C. data. As a 3-parameter model i s involved, each bivariate parameter space could be shown, but the (8, Y) space, with a fixed, i s selected for analysis. Three features are noteworthy: the extreme asymmetry, the long ridge, and unimodality. The asymmetry confirms expectations that the model i s quite nonlinear. A gently sloping plateau extends from the ridge down to a v e r t i c a l c l i f f produced by a penalty function where the population parameter becomes negative. This contrasts with a steep but l i n e a r slope on the other side of the ridge. The elongated ridge confirms expectations about the i n t e r a c t i v e nature of the gravity type model. It would be very i n t e r e s t i n g to transform i t FIGURE 7.1 BIVARIATE RESPONSE SURFACE OF THE LOG LIKELIHOOD FUNCTION (a FIXED). - 132 -l i n e a r by the Box-Cox family and then examine the surface. F i n a l l y , the surface i s , fortunately, unimodal ensuring that the l o c a l optimum which had previously been found i s a global optimum, at least within reasonable areas of the solution space. In addition, the response surface for the same model and data was constructed but using the maximal interchange strategy to select 10 per cent of the t^_. . It transpired that the surfaces were very s i m i l a r i n configuration, a r e s u l t which supports the proposition that maximal interchange can be used to c a l i b r a t e e f f i c i e n t l y . One scheme might be to locate the position of the ridge by two rough one-dimensional searches by Golden Section p a r a l l e l to the 8 axis. From t h i s could be obtained the orientation of the ridge, which could be followed by another one-dimensional search, based on a l i n e a r combination of 8 and y parameters, for example. 7.7 An improved shortest-path algorithm Three aspects of the computation of shortest paths are s i g n i f i c a n t for the cost matrix used for the estimate of O-D. These are the number of arithmetic and l o g i c a l operations performed, the amount of data required i n core, and the quantity of data required to be - 133 -transferred between d i f f e r e n t storage media. I t i s shown that an algorithm based on that of D i j k s t r a (1958) i s e f f i c i e n t i n these three aspects. Dreyfus (1969) has claimed that the Floyd (1962) algorithm i s more e f f i c i e n t i n the number of operations involved than that of D i j k s t r a fo r s o l v i n g the a l l o r i g i n s problem. Given the improve-ments to the l a t t e r method described i n Yen (1972) and i n t h i s section, t h i s assertion i s now i n v a l i d . Moreover, Di j k s t r a ' s e f f i c i e n t use of storage i s a c r u c i a l advantage over the Floyd algorithm and i t i s to t h i s aspect that the discussion now turns. Tree-building algorithms, including D i j k s t r a ' s , usually solve the a l l o r i g i n s problem by repeated a p p l i c a t i o n of the s i n g l e - o r i g i n problem. This allows a decomposition by o r i g i n which requires that only the vector of the shortest path matrix which r e l a t e s to the current o r i g i n be i n core. Once i n core, t h i s shortest-path problem i s completely solved without the need for further data t r a n s f e r . A further advantage i s obtained i n the construction of the assignment matrix. Whereas i n the Floyd algorithm the routing node i s adjacent to the destination, i n the D i j k s t r a i t i s adjacent to the o r i g i n . The r e s u l t of t h i s i s that to trace the shortest path i n the Floyd requires addressing the assignment matrix i n random order whereas i n the D i j k s t r a , t h i s i s r e s t r i c t e d to addresses i n the same column (or row) of - 134 -the matrix. Not only does t h i s permit decomposition by o r i g i n i n subsequent problems which use the assignment matrix, but can permit the assignment matrix i n t h i s form to be eliminated. Recently, Yen showed how Di j k s t r a ' s algorithm could be improved by avoiding examination, as candidate nodes for tentative l a b e l l i n g , of nodes that had already been permanently l a b e l l e d . Since t h i s discussion was addressed to complete networks, i t i s not s u r p r i s i n g that further gains may be achieved for less strongly connected transportation networks. Yen's algorithm i s now described for single o r i g i n followed by an adaptation. Define the following: (I), I = 1,2,..,N nodes i n the network, (1) i s the o r i g i n H(I) the node number stored i n the I c e l l of H (I,J) directed arc from (I) to (J) D(I,J) the distance matrix F(I) tentative or permanent shortest-distance from (1) to (I). The shortest paths F(J*) are obtained by: 1. Let L=l, K=N, H(I)=I, F(1)=0, F(I)=», 1 = 2,3,.. ,N 2. For 1=2,3,..,K do: A. J=H(I) B. F(J)=min[F(J), F(L) + D(L,J)] wrt. L C. F(J*)=min [F(J)] wrt. J - 135 -3. Permanently l a b e l , L=J* and H(J*)=H(K) 4. Reset K=K-1, i . e . shorten l i s t of candidate t e n t a t i v e l a b e l s . I f K>1, go to 2; otherwise stop. Consider the i n n e r loop of Yen's method. For both steps B and C, the candidate d e s t i n a t i o n s are a l l those not y e t permanently l a b e l l e d . T h i s l i s t can be reduced by making the f o l l o w i n g o b s e r v a t i o n s : 1. a d e s t i n a t i o n J can onl y be permanently l a b e l l e d i n C i f and onl y i f J i s i t s e l f adjacent to permanently l a b e l l e d node L; 2. a s h o r t e r l i s t of candidate d e s t i n a t i o n s J to e s t a b l i s h F(J*) = min F(J) c o n s i s t s of the end-p o i n t s of the c u r r e n t t r e e of s h o r t e s t - p a t h s rooted on the o r i g i n . Define the f o l l o w i n g : DEG(I) the degree of node I. INC ( I , J ) , J=l,2.., DEG(I) the c o n t i g i u i t y matrix (condensed t o a l i s t ) c o n t a i n i n g the nodes adjacent to node I. DNC(I,J), J=l,2,.., DEG(I) the network d i s t a n c e s c o n t a i n i n g the d i s t a n c e s from I of nodes adjacent to I. P(J) the node preceding J i n the s h o r t e s t path from o r i g i n to J . L(J) l o g i c a l s w i tch f o r l a b e l s t a t u s of node J , i . e . whether J i s c u r r e n t l y s t o r e d i n H: i f J=H(I), then L(J)=.TRUE. - 136 -The s h o r t e s t paths F(J*) are o b t a i n e d by: 1. Set L=l, K=0, F(1)=0, F(I)=<=°, 1 = 2,3,. . ,N, L(I)=.FALSE., 1=1,2,..,N, and do steps 2, 3, 4 and 5 e x a c t l y N - l times. 2. For JJ=1,2,.., DEG(L) do: A. J=INC (L,JJ) B. F(J)=min [ F ( J ) , F(L) + DNC (L,JJ)] C. P(J)=L D. For L(J)=.FALSE. add a new end-point of s h o r t e s t path t r e e by: (i) K=K + 1 ( i i ) H(K)=J ( i i i ) s e t L(J)=.TRUE. 3. F(J*)=min [F(J)] wrt. J e H ( I ) , 1=1,2,..,K. 4. Permanently l a b e l , L=J*, H(I)=H(K) 5. Reset K=K-1, L(J*)=.FALSE. The a l t e r e d r o l e o f H i s c l e a r from F i g u r e 7.2. Here H c o n s i s t s only of the end-points of the s h o r t e s t -path t r e e , whereas i n Yen's a l g o r i t h m , i t would c o n s i s t of end-points and u n l a b e l l e d nodes. - 137 --9 f 4 4 — ^ • i 0— -0 f 4 <?> £ 0 _i * * * *-o r i g i n node permanent labels end-points of shortest-path tree unlabelled nodes shortest paths other network arcs FIGURE 7.2 DEMONSTRATION NETWORK FOR A TREE-BUILDING ALGORITHM CHAPTER 8 ASPECTS OF ESTIMATION The purpose of t h i s section i s to discuss some s t a t i s t i c a l problems which a r i s e i n the estimation of O-D from arc volumes. The treatment i s neither comprehensive nor s u f f i c i e n t l y thorough to be completely s a t i s f a c t o r y ; constraints of space determine t h i s . Nevertheless, several i n t e r e s t i n g topics are addressed which may lead to more consistent and e f f i c i e n t estimation. An o u t l i n e of the d e r i v a t i o n of the log concentrated l i k e l i h o o d function for the general case, and for the s p e c i f i c problem i n hand, i s given i n the f i r s t section. Although t h i s development i s s p e c i a l i s e d to the normal model for s i m p l i c i t y of exposition and concreteness, i t i s followed by some ideas on how a l t e r n a t i v e hypotheses could be entertained i n a u n i f i e d context. Among the f e a s i b l e hypotheses i s one that discriminates between unconstrained disturbances due to measurement error and nonnegative residuals due to model s p e c i f i c a t i o n error. The majority of the discussion, however, i s devoted to the estimation and a p p l i c a t i o n of a s p a t i a l l y dependent error covariance matrix. This topic i s thought to be p a r t i c u l a r l y germane owing to the. incongruency between the c l a s s i c a l s t a t i s t i c a l model with independent observations, on one hand, and the i n t e r r e l a t e d arcs of a highway network, on the other. I t i s shown that the - 138 -- 139 -structure of c o r r e l a t i o n between the arcs may be s p e c i f i e d a p r i o r i , or may be estimated where a sequence of cross-s e c t i o n a l data i s a v a i l a b l e . The section closes with the discussion of a novel estimator based on the i n t e r p r e t a -t i o n of observed arc volumes as c a p a c i t i e s and with a note on the estimation of s t r u c t u r a l change for p r e d i c t i o n purposes. 8.1 Structure of the l i k e l i h o o d function 8.1.1 The logarithmic concentrated l i k e l i h o o d function The l i k e l i h o o d function i s defined as the j o i n t density function of the observations c o n d i t i o n a l on a parameter vector: X (u; 9) = n f ( u . , 9) (1) l 1 where u i s a vector of observations f i s the p r o b a b i l i t y density function of u 9 i s a vector of parameters In order to use the l i k e l i h o o d approach to an estimation problem, l e t the u be the errors from a regression equation. Assume also f o r concreteness that these errors are normally d i s t r i b u t e d with zero mean and constant variance a 2 . Then the l i k e l i h o o d function (1) becomes - 140 -d~ (U, e) = (2TT0 ) e x p ( - 2 a - 2 Z u 2 ) (2) where u i = y i - gfx.^, 0) f ( u ± / 9) = a " 1 (2TT) 5 e x p ( - 2 a ~ 2 u?) g = the f u n c t i o n a l form o f the r e g r e s s i o n e q u a t i o n . I t i s c o n v e n i e n t to s i m p l i f y the e x p r e s s i o n i n (2) by working w i t h i t s l o g a r i t h m i c c o u n t e r p a r t , which a t t a i n s i t s maximum a t the same p o i n t i n parameter space, and by o b t a i n i n g an a n a l y t i c a l e s t i m a t e o f the v a r i a n c e . Thus l o g = -n l o g 2TT - n l o g a - 2 a ~ 2 Z u 2 (3) 2 i x a 2 = Z u? /n (4) and the l o g c o n c e n t r a t e d l i k e l i h o o d f u n c t i o n (L) f o r the normal case i s d e f i n e d as L = -n l o g 2ir - n - n l o g ( Z u 2/n) (4) 2 2 2 -1-o b t a i n e d by s u b s t i t u t i o n o f (4) i n (3). E q u a t i o n (4) i s o b t a i n e d by d i f f e r e n t i a t i n g (2) w i t h r e s p e c t t o the v a r i a n c e , s e t t i n g the r e s u l t t o zero and s o l v i n g f o r t h a t v a l u e of the v a r i a n c e which maximises the l i k e l i h o o d f u n c t i o n . _ i - 141 -8.1.2 Maximum l i k e l i h o o d estimation of the 0-D matrix In Chapter 4, the objec t i v e function of the estimation problem was l e f t as an unspecified function g to be minimised. Given p r i o r assumptions about the d i s t r i b u t i o n of the err o r s , s t a t i s t i c a l l y stronger r e s u l t s may be obtained by spe c i f y i n g g as a l i k e l i h o o d function and maximising. Using the log concentrated function given by (5), i t i s apparent that the estimation problem for the normal case becomes Maximise L(u, 8) wrt {0k> (6) where L(u, 9) = -A log 2TT - A - A log (I UVA) 2 2 2 a o u = v - v a a a v = Z 5. t, a h ha h t. = f(Z, . , e, ) h hk k Expression (6) finds i t s maximum when the sum of squared errors i s minimised so that the optimal parameters obtained from (6) are i d e n t i c a l to those obtained from minimising the sum of squared e r r o r s . - 142 -. 8.2 S p e c i f i c a t i o n of the s t a t i s t i c a l model 8.2.1 Estimation of a generalised l i k e l i h o o d function Although the normal model has been found to apply to a wide v a r i e t y of s t a t i s t i c a l phenomena, i t i s desirable to have the a b i l i t y to use a l t e r n a t i v e d i s t r i b u t i o n a l assumptions about the error term. One strategy to adopt could be to compare the l i k e l i h o o d s obtained from d i f f e r e n t models. Thus the normal l i k e l i -hood could be compared with the exponential l i k e l i h o o d and the larger l i k e l i h o o d accepted as the model. A more s a t i s f y i n g s o l u t i o n , however, i s to embed s p e c i f i c l i k e l i h o o d functions i n a more general expression, a composite l i k e l i h o o d function, which allows a blend of d i s t r i b u t i o n a l assumptions. In the complex empirical world, i t i s u n l i k e l y that a s i n g l e model can account f o r the d i s t r i b u t i o n of e r r o r s . Consequently, the simple expression i n (1) i s extended to the following composite function which i s a convex combination of two simple hypotheses (Quandt, 1974; Cox, 1962): <£(u, 9) = -ret (u , 9 ) + (1-T)£ (u , 9 ) (7) 1 1 1 2 2 2 0 -< T -$ 1 Three cases are possible. E i t h e r of the l i k e l i h o o d functions might alone explain the errors, x=0 or 1, or some combination of the two, for 0 < x < l . The s p e c i f i c a t i o n given i n (7) would permit hypothesis t e s t i n g about the most appropriate blend of d i s t r i b u t i o n a l models. A design for such t e s t i n g could proceed i n binary fashion by taking p a i r s of d i s t r i b u t i o n s f o r example, normal versus exponential, normal versus lognormal and exponential versus lognormal. Presumably, these outcomes are t r a n s i t i v e to the extent that, i f normal dominated lognormal and lognormal dominated exponential, then normal would also dominate exponential. Consequently, only c e r t a i n pairs of d i s t r i b u t i o n s need be tested i n order to obtain a comprehensive a n a l y s i s . One shortcoming Of t h i s approach i s that i t s t i l l considers only d i s c r e t e a l t e r n a t i v e s a l b e i t l i n e a r combinations of them. Whereas the normal model minimises sums of squared r e s i d u a l s , and the exponential model minimises sums of the absolute values of re s i d u a l s , there i s no prov i s i o n for f r a c t i o n a l exponents. Abandoning the f a m i l i a r d i s t r i b u t i o n s , an estimator could be developed around the c r i t e r i o n g ( v a , v a) = | |v a - v a | p 0 < p v< 2 (8) which includes sums of squares, and absolute values, as sp e c i a l cases. The p r i n c i p a l property of t h i s estimator i s c l e a r ; the smaller the exponent the less the. dependence of the parameters on the larger r e s i d u a l s . I t provides an a l t e r n a t i v e to the ad hoc "Windsorising" of data by r e j e c t i n g d i s t a n t o u t l i e r s as "obvious" e r r o r s . I f there i s considerable suspicion that large measurement errors could e x i s t i n the data, then l e a s t squares should perhaps be avoided, even though the model ends up i n the s t a t i s t i c a l wilderness. The idea of employing the c r i t e r i o n given by (8) i s not new. Indeed, Newcombe (1886), f i n d i n g that large errors were more frequent than the normal model would i n d i c a t e , suggested that the square exponent of the normal density should be replaced with a less r a p i d l y increasing function. However, t h i s suggestion was not implemented and he adopted a normal model based on a mixture of observations measured with various degrees of p r e c i s i o n (Huber, 1972). 8.2.2 A l t e r n a t i v e d i s t r i b u t i o n a l models Normal: L(u, 6) = - A log 2TT - A - A log (Z u 2/A) 2 2 2 a a = - A log 2TT - A - A log ( E u 2 ) 5 + A log A 2 2 a a 2 - 145 -Exponential: L(u, 6) = - A - A log (||ua|/A) = - A - A log (S|u |) + A log A a a 8.2.3 Truncated models The d i s t i n c t i o n may be made between disturbances, due to measurement errors i n the dependent v a r i a b l e , and resi d u a l s , which o r i g i n a t e with s p e c i f i c a t i o n errors i n the model. Disturbances can be assumed to follow a normal, or at le a s t a symmetrical, d i s t r i b u t i o n whereas the residuals may be bounded and thereby constrained p o s i t i v e or negative, for example. In the case of 0-D estimation from arc volumes, i t i s apparent that p o s i t i v e residuals on the arcs are more acceptable than negative ones. In support of t h i s , i t i s claimed that, 0-D estimation from one point of view i s the a l l o c a t i o n of arc volumes to O-Ds. This being the case, negative residuals are somewhat of an embarrassment since they represent the a l l o c a t i o n to 0-D of flow volumes which were not present on the arcs. Although some negative residuals can j u s t i f i a b l y be ascribed to measurement error on the flow volumes, a considerable proportion stem from s p e c i f i c a t i o n e r r or. Consequently, - 146 -a more acceptable model would r e s u l t i f these r e s i d u a l s were constrained to be nonnegative. One approach to t h i s problem requires that the maximum l i k e l i h o o d problem (6) be solved with the a d d i t i o n a l constraint that the u are nonnegative: CL Maximise L(u, 0) wrt iQ,}, subject to u ^ 0 (9) JC cL (a=l,2,..,A) Even i f t h i s s p e c i f i c a t i o n solves the problem of negative r e s i d u a l s , i t i s d e f i c i e n t to the extent that o i t precludes errors of measurement xn the v . An a a l t e r n a t i v e to t h i s s t r i c t model might be a convex combination of l i k e l i h o o d functions, one constrained, the other unconstrained, with respect to the sign of the r e s i d u a l s . At f i r s t glance, i t would seem that t h i s device would allow the data to determine the optimal mix of constrained r e s i d u a l and unconstrained measurement error. I t i s highly probable, however, that the unconstrained l i k e l i h o o d would dominate the constrained one simply owing to the presence of the constraint which prevents the f u l l optimum being attained. Further d i f f i c u l t i e s are encountered with any truncated models of t h i s type. In p a r t i c u l a r , i t i s unclear as to what are the s t a t i s t i c a l properties of these estimators, since they v i o l a t e one of the r e g u l a r i t y - 147 -conditions for maximum l i k e l i h o o d estimation. The range of the dependent v a r i a b l e v should be independent of the parameters. That t h i s i s v i o l a t e d i s consequent on the fa c t that 0 * u a v a > v a = Z 5 h a t h (10) Whether or not t h i s i s a s i g n i f i c a n t problem i s not known. 8.3 S p a t i a l l y dependent error covariances 8.3.1 Maximum l i k e l i h o o d estimation Let i t be assumed that the in t e r a c t i o n s between the arc volumes on d i f f e r e n t arcs can be written i n matrix notation as: o o V = p W V + £ (11) where v i s an (n. x 1) vector of arc volumes w i s an (n x n) matrix of weights e i s an (n x 1) vector of errors p i s an autoregression parameter Following Ord (1975), Bodson and Peeters (1975), l e t t h i s f i r s t order s p a t i a l process be written as £ = B v (12) where B = I - p w and I i s the (n x n) i d e n t i t y matrix. - 148 -Define the log l i k e l i h o o d function as given by (6) and add °a = p g wab 9 b + £ a ( 1 3 ) The remainder of t h i s section i s devoted to a maximum l i k e l i h o o d estimate of p and to a l t e r n a t i v e i t e r a t i v e estimation schemes. Assume that e i s normally d i s t r i b u t e d with zero mean and a 2 variance. Then s u b s t i t u t i o n of (12) i n t o the log l i k e l i h o o d function for a normally d i s t r i b u t e d v a r i a b l e leads to L(p,a 2) = log |B| - A log a 2 - v / b' B V - A log 2TT (14) 2 2a 2 2 Obtaining an a n a l y t i c a l estimate of the variance gives: (15) O = 1 v B B v A and s u b s t i t u t i o n of (15) i n (14) leads to the log concentrated l i k e l i h o o d L(p) = log |B| - A log (V'B'B V) + constants (16) (16) i s maximised with respect to p i f log |B| - A log (V'B' B' V) (17) 2 i s minimised. Since the f i r s t term i n (17) , the - 149 -determinant of the Jacobian, i s a function of p considerable computation i s involved. Much of t h i s can be avoided by r e c a l l i n g that • , A |B| = n ( i - P A j (18) a a til where A i s the a eigenvalue of the matrix w. In (18) c l A^ i s independent of p so the eigenvalues are calcula t e d once only and leaving j u s t 2A m u l t i p l i c a t i o n s and A additions f o r each p . One estimation procedure for problems s i m i l a r to equations (6) and (13) was devised by Cochrane and Orcutt (1949) and modified for the s p a t i a l case by Ord (1975). For the current problem t h i s scheme can be out-l i n e d as 1. Estimate (6) and obtain OLS residuals u . 3. c o e f f i c i e n t s a ^ . 2. Estimate p s from s s „ s u = p Z w , u, + e a M B ab b a by solving the following problem: Minimise wrt p S : F ( p s ) = log 7 r ( l - p s A J - A log (Z e 2) a a j J a a where e = u S - p s Z w , u f a a K b ab b - 150 -3. Compute new variables os „ _s o o v = E B , v, = v - p a ab b a K o -, E w , v, £ ab b v = £ B i_ v, = v a b ab b ~s v a - p £ w,K v„ k ab a where B a b = - p s w & b OS o s Estimate (6) s u b s t i t u t i n g v f o r v and v a a a for v a , and producing n i s+1 o s. Compute new residuals u = v - v (a . ) a a a K 6. Go to 2, s e t t i n g S=S+1 A simultaneous estimate of (6) and (13) i s made possible by addition of p to the nonlinear parameters to be optimised. Hence (6) becomes Maximise L(u,8,p) wrt {8k,p} where L(u,6,p) = - A log 2TT 2 A - A log(£ u 2/A) T T a a u a - • ( v a - p [ g w a b v b ] ) ( v a " p [ g w a b v b ] ) v = £ 5, t, a h ha h fch = f ( z hk'- V - 151 -Given that a nonlinear optimisation problem i s being solved i n any case, t h i s form of the problem i s probably the most convenient and also i t i s l i k e l y more e f f i c i e n t i n computational terms than the Cochrane-Orcutt procedure. Another a l t e r n a t i v e , when precise estimates of p are not required may involve less computation. This i s to generate a sequence of p , computing L(u,9,p) f o r p known. If t h i s sequence consists of a few t r i a l p set at regular i n t e r v a l s , an approximation of the true p can be had by i n t e r p o l a t i o n . Furthermore, i f the sequence i s a convergent one such as provided by a search strategy l i k e the golden section, then accurate estimates of p may be obtained without increasing the dimension of the parameter space i n the simultaneous part of the problem. Whether or not the imposition of t h i s s e p a r a b i l i t y between p and the other nonlinear parameters r e s u l t s i n computational saving depends p a r t l y on the configuration of the response surface and p a r t l y on the mechanics of the optimisation used. Some algorithms t y p i c a l l y e x h i b i t rapid i n i t i a l convergence followed by slow f i n a l convergence to the optimal s o l u t i o n . These algorithms are suited to simultaneous s o l u t i o n . Strong f i n a l convergence i s a pre-condition for the e f f e c t i v e a p p l i c a t i o n of repeated subproblems with f i x e d p . - 152 -If the matrix W i s r e s t r i c t e d to be summetrical then several s i m p l i f i c a t i o n s r e s u l t . Any symmetric matrix can be diagonalised by an orthogonal transformation so that c'w C = D where D i s a diagonal matrix with the diagonal being the eigenvalues of W. Then, i n matrix notation, (13) becomes £ = (I - p W)v = (I - p C D C')v (19) Premultiplying (19) by C and wri t i n g v as the product CZ leads to C'e = c'(I - p C D C')CZ = c'e (19a) = (I - p D)Z = BZ _ * where B i s diagonal since I and D are, and using c' C = C C/= I for orthogonal matrices. I t follows that E(C'e) = C'E(£) = E(*) = 0, E(C'e £'C) = C /E(£ z' )C = a 2 I Then (19a) leads to the transformed autoregressive process Z a = P X a Z a + e* a which has collapsed the s p a t i a l structure W into i t s eigenvalues. - 153 -Minimising * 9 £ £ = £ a a a (1. - a a 2 (1 a p v 2 z ; £ a a 2p £ Z 2 X, + p 2 £ Z 2 X 2 a a a a a a r e s u l t s i n a major s i m p l i f i c a t i o n as the summation terms are a l l constant and are therefore c a l c u l a t e d once only o for given v and w. Consequently, the maximum l i k e l i -hood estimate of p i s obtained by minimising F(p) = log 11(1-pX ) - A log(£ Z - 2p £ Z 2 X £ Z z X 2) a a a 8.3.2 The structure of s p a t i a l dependence The objective of t h i s section i s to specify a l t e r n a t i v e s for the structure of the weights matrix W using p r i o r information. Only the structure i t s e l f i s being constructed at t h i s point; the actual values are estimated p a r t l y from these p r i o r considerations and p a r t l y from the data. The l a t t e r supplies i t s information v i a the s p a t i a l autocorrelation parameter p which could be a vector with s p a t i a l l y determined p a r t i t i o n s . - 154 -Let W be a (n x n) matrix, not n e c e s s a r i l y symmetric but c e r t a i n l y nonnegative, where n i s the number of observations and the ( i , j ) element represents the degree of s p a t i a l dependence, c o r r e l a t i o n or i n t e r a c t i o n between observations i and j . This matrix i s scaled so that the sum of the weights are equal to one hence each can be interpreted as the proportion of the i th observation's i n t e r a c t i o n which takes place with j observation. In some previous studies W has been s p e c i f i e d using rather s i m p l i s t i c notions such as the presence or absence of c o n t i g u i t y with another observation, the length of zonal boundary which i s common to two adjacent zones, and a l i t t l e more generally, the distance from one observation to a l l others. The f a c t that crude devices have been employed to structure W points up a considerable methodological weakness inherent i n many previous a p p l i c a t i o n s . To be able to specify the structure of W implies that a great deal i s known about the anatomy of s p a t i a l processes. This, however, i s not the case and often p r i o r knowledge i s so d e f i c i e n t as to be unworthy of the name. In these cases W should be estimated from the data. A r e l a t i v e l y strong argument can be made for p r i o r knowledge of the W matrix i n the case of the O-D estimation problem. Here the observations are network - 155 -arcs which contain t r a f f i c volumes. C l e a r l y the t r a f f i c volume on any given arc may have ori g i n a t e d beyond some contiguous arc. Any two adjacent arcs along a highway, such as the Trans-Canada highway, w i l l carry v i r t u a l l y the same pattern of o r i g i n s and destinations. Consequently, i t may be argued that these arc volumes are highly c o r r e l a t e d since they comprise b a s i c a l l y the same t r a f f i c . I t therefore follows that a natural d e f i n i t i o n of the in t e r a c t i o n between any p a i r of arcs i s based on the extent to which they have t r a f f i c i n common. Two s p e c i f i c a t i o n s are advanced: one based on the proportion of p o t e n t i a l shortest-paths which a pairs have i n common, the other on an estimate of the proportion of t r a f f i c common to both arcs. Let the maximum possible number of paths common to arcs a and b be written as which i s patently the upper bound on the set theoret i c i n t e r s e c t i o n of paths. Then the i n t e r s e c t i o n of paths i s 6 a 6 , = £ & , 6 , , ab h ah bh and W may be defined as the proportion W ab ab' a (20) - 156 -The -5 i s the arc path mapping f o r the shortest-path between node p a i r h. The shortest-path i s used here as i t i s consistent with the assignment p r i n c i p l e used i n the O-D estimation model. Although the shortest path i s l i k e l y to be dominant as a general r u l e , i t i s probable that the observed volumes v contain arc in t e r a c t i o n s which are a based on other paths. There i s no conceptual or algebraic problem to a gener a l i s a t i o n of the single path d e f i n i t i o n which can be written as 6 = £ E 6 , (21) a p h ahp 6 . = £ £ 6 . 6. . (22) ab p H ahp bhp where ^ a^p denotes the proportion of any t r a f f i c between node p a i r h which uses path p. D e f i n i t i o n s (21) and (22) are substituted i n (20) to complete the multiple path proportional assignment extension.. This common paths index i s a s p e c i a l case of a more general measure of s p a t i a l association which uses an estimate of the O-D matrix.. In the pure common paths case t h i s estimate i s i m p l i c i t l y set to unity. Define the in t e r s e c t i o n upper bound and the i n t e r s e c t i o n i t s e l f as respe c t i v e l y - 157 -v = Z 5 , t a h ah h vab = £ 5 ah 6bh \ where i s OLS estimate of the O-D for node p a i r h. Then the volume weighted common paths index can be written as W , = v ,/v ab ab 7 a The multiple path counterpart i s simply constructed a l g e b r a i c a l l y by s e t t i n g v = Z Z 6 , t. a p n ahp hp v , = Z Z 6 , 5, , t,. ab p h ahp bhp hp but less e a s i l y calculated unless t. i s known with hp some confidence. 8.3.3 Estimating s p a t i a l error covariances The objective of t h i s section i s to provide a procedure to estimate the structure of s p a t i a l dependence with recourse to a minimum of p r i o r information. It i s shown that i n c e r t a i n circumstances an estimate of some of the elements of the error covariance matrix may be made - 158 -thereby rendering operational the Aitken (1935) generalised l e a s t squares estimation procedure. In order to replace p r i o r information the procedure requires more data, e s p e c i a l l y time series and/or aggregation type r e s t r i c t i o n s on the generality of the covariance matrix. Let the 0-D estimation problem be written as v . = a „ + a „ v . + u , (30 at o 1 at at o where v . i s the arc volume for arc a at time t . The at ordinary l e a s t squares assumptions of E ( u a t ) = a E ( u a t u b t } = 0 and replaced by the assumptions of a c r o s s - s e c t i o n a l l y correlated and s e r i a l l y autocorrelated model (Kmenta, 1971): E(u* ) = a at aa E ( u a t ubt> = aab u a t = P U a t - 1 + £ a t - 159 -where e ^ ~ N(0, <b ) at ' T a a E ( U a t - 1 ' £ b t } = ° E ( £ a t e b t ) = *ab E ( £ a t £ b s ) = 0 ( t ^ s ) Define the covariance matrix as Q = CT11 P 1 1 a i 2 P 1 2 a P a P 21 21 22 22 o P a P A l A l A2 A2 a P 1 A 1 A a P 2A 2A a P AA AA where ab 1 p p ' p i p P P 1 T-1 T - 2 T - 3 P P p T-1 T - 2 T - 3 The estimation procedure i s as follows: 1. Estimate (30), including a l l nonlinear para-meters involved i n v t , as one equation, the observations pooled. - 160 -Calculate the OLS residuals u-,+- = v +^- ~ a - a„ v . (a, ) at at o 1 at k Estimate p by P = I U a t U a t - 1 (t=2,3,..,T) I U i t - 1 Construct new variables as os s s v . = a + a. v . + e . at o 1 at at , os o » o where v & t - v & t - p v ^ v a t = v a t " p V a t - 1 e l t = U a t ~ 5 U a t - 1 (t=2,3,...T; a=l,2,...,A) Estimate equation (31) by OLS, a l l parameters with observations pooled. Calculate the OLS residuals s os -s ~s s ,~ss £ +. = v,. - a - a, v , (a. ) at at o 1 at k - 161 -7. Calculate the s p a t i a l covariances aab = ^ab 1- P : where $ , = 1 Z e s. efL a b T ^ r - t - 2 a t b t 8. Substitution of p and aafa i n ft to produce ft. 9. Estimate (30) by solving the following problem: Maximise L(u, 8 , p, ft) wrt {0 } where L(u ,0,p,H) = - A log 2ir - A - A log 2 2 2 (a U a t I "abt U b t / A ) u a t = v a t " v a t v a t = £ 6 h a t fcht fcht = f ( W 9k ) 8.4 A .Pseudo-capacity estimator In t h i s section an estimator i s o u t l i n e d which i s considerably d i f f e r e n t from the previous models. I t represents an attempt to extract further information from the arc volumes with a minimum of p r i o r assumptions. At - 162 -the same time the problem of r e q u i r i n g residuals to be constrained nonnegative i s addressed. In p a r t i c u l a r , the observed arc volumes are treated as a r c - s p e c i f i c capacity constraints on the volume of estimated t r a f f i c which can be assigned on a given path. Let the O-D estimation problem now be formulated as the following sequence. 1. Maximise E v wrt {0, } a a k subject to v ^ v (a=l,2,...,A) a a where v = Z 6, t. a h ha h fch = f ( z h k ' V 2. Delete a l l arcs where u = v - v ^ e , e >^ 0 ci ci c l and delete a l l O-D pairs h which used these arcs i n any part of t h e i r paths. Set h h v = v a a i f 8, = 1 and u ^ e ha a 3. Construct a new arc-path mapping 6 ^ a 0 i f u < £ (h=l,2,...H) (a) a. 6 h a = \ ° i f 6ha 6ha = °' 5ha = 1 6, otherwise ha (b) (c) - 163 -Note that condition (b) depends on (a), This operation i s equivalent to d e l e t i n g any row-column pa i r where t h e i r i n t e r s e c t i o n has been set to zero under condition (a). In the case where one arc i s used by a l l remaining paths h i f t h i s arc encounters condition (a) the computations terminate at t h i s point. 4. Replacing with 6^ repeat steps 1, 2, 3 and 4 u n t i l ''ha 6vL = 0 (h-1,2,...,H; a=l,2,...,A), at which point u a = ^a - v a « e u a > 0 CL and the estimated 0-D and arc volumes are s V = V a a where s i s the i t e r a t i o n step at which conditions 3(a) and 3(b) were encountered. - 164 -8.5 Estimation of s t r u c t u r a l change In section 8 .3 ,3 , where observations on the arc volumes were pooled, a s i n g l e set of parameters was estimated for a l l time periods. This i m p l i c i t constraint i s now relaxed to the extent that each time-period i s allowed to have i t s own parameters. The objective of t h i s i s to obtain an estimate of s t r u c t u r a l change. I t i s most u n l i k e l y that successive cross-sections of data would produce i d e n t i c a l parameters. So the a b i l i t y to free the parameters can y i e l d information about how the model i t s e l f i s changing over time. Thus, attention can be focussed on (a) the s t a b i l i t y of the parameters over time, and (b) the s t a b i l i t y of the f u n c t i o n a l form over time. Let the parameters be estimated i n the following model at ot i t at at £ 6 h a t fcht f t ( Z h k t ' 9 k t } Then the { Q i ^ } and the {a f c } w i l l form a sequence which reveals s t r u c t u r a l trends. I f these trends appear reasonably well-behaved, even monotonic, then the parameters may be estimated by a function such as where v at 'ht - 165 -T) or transformations of t h i s . Where t h i s function was able to describe the sequence of parameters well i t could be used for p r e d i c t i o n of future parameters, using As i s shown i n Chapter 4, a monotonic parameter change i s present as a function of time so that p r e d i c t i o n of parameters, i n addition to p r e d i c t i o n (or projection) of exogenous v a r i a b l e s , seems mandatory i n t h i s series of cross-sections. 8 t+1 = 8 q + 6 1 ( t + 1 ) CHAPTER 9 EMPIRICAL RESULTS I: B.C. DATA 9.1 Two models The e m p i r i c a l r e s u l t s are based on two models, one r e l a t i v e l y simple with few parameters, the o t h e r , owing to i t s a d d i t i o n a l parameters i s thereby more complex. The simple model, s i n c e i t i s a s p e c i a l case of the extended model, may be compared d i r e c t l y w i t h the l a t t e r . 9.1.1 A b a s i c model The most d e s i r a b l e model i s one which possesses the q u a l i t y of m i n i m i s i n g both the number of f r e e parameters and the amount of exogenous i n f o r m a t i o n r e q u i r e d w h i l s t at the same time maximising e x p l a n a t i o n . Although i t i s d i f f i c u l t t o d e f i n e such an o b j e c t i v e f u n c t i o n i n r i g o r o u s terms i t remains nonetheless a d e s i r a b l e o b j e c t i v e . A simple model which appears to s a t i s f y t h i s o b j e c t i v e can be based on a g e n e r a l form of the g r a v i t y model. Using the n o t a t i o n g i v e n p r e v i o u s l y one s p e c i f i c model i s g i v e n by Minimise 2, < v a - V ) w.r.t. a , 8, Y a (1) where V = J 6 U t, a £ ha h and t, = apf D Y n h h A number of i n t e r e s t i n g models are o b t a i n e d as nested hypotheses by c o n s t r a i n t s on the parameters. These are presented i n s e c t i o n 2. - 167 -9.1.2 An extended model I t i s p o s s i b l e to extend the b a s i c model i n a convenient manner f o r e s t i m a t i o n . T h i s i s achieved by m a i n t a i n i n g the number of n o n l i n e a r parameters at two but i n c r e a s i n g the parameters which enter l i n e a r l y . Thus the l i n e a r estimate f o r a i s augmented to become a m u l t i p l e r e g r e s s i o n subproblem. The extended model may be w r i t t e n as 2 Minimise Y Y Y ( v -V ) w.r.t. a , a . , 3, y L L L apq apq °Pq ± a p q ^ W h e r e V a p q = a o p q + K p h i a p q * h i (2) lhi - P h i °hi and the m u l t i p l e r e g r e s s i o n subproblem i s V = a + Ta. V. -HU (3) apq opq V l lapq apq w h e r e V i a p q = J; 6hiapq fchi . The t h i are p a r t i a l e stimates of the O-D s i n c e the a i which enters l i n e a r l y i s separated i n t h i s computational scheme. The f u l l estimate i s t. • = a . p j . DI. (4) h i I h i h i y ' T h i s extended model employs two a d d i t i o n a l concepts: an estimate of f a c t o r s s p e c i f i c to each of s e v e r a l a r c c l a s s e s and p a r t i t i o n s of the O^D m a t rix, each - 168 -p a r t i t i o n having i t s own ou . Arc class s p e c i f i c factors are composed of three elements as shown i n equation (5). a = e + 1 + m (5) opq P q q a c o r r e c t i o n factor associated with province - or r e g i o n - s p e c i f i c measurement procedures, for the p f ck cl a s s of arcs th l o c a l t r a f f i c associated with the q class of arcs a c o r r e c t i o n f a c t o r r e l a t e d to the omission of interchanges involved i n the maximal interchange s e l e c t i o n procedure. The intent of a i s , to some extent at l e a s t , to p u r i f y opq the arc volumes of errors which o r i g i n a t e from l o c a l or regional sources. Furthermore, m i s designed to absorb q the i n t e n t i o n a l omission of numerous interchanges, which, although very small i n d i v i d u a l l y , may accumulate l o c a l l y on an arc thereby introducing bias. As the maximal i n t e r -change s e l e c t i o n becomes small the c o r r e c t i o n f a c t o r becomes correspondingly large. P a r t i t i o n s of the O-D matrix are obtained by the a.. I f these constants are associated with o r i g i n s or destinations alone the r e s u l t i s an asymmetric estimated O-D matrix. This can be avoided by associating a given a.. where £ P 1 q m q - 169 -with a class of interchanges, for example, big c i t y to big c i t y , big c i t y to small c i t y , small c i t y to small c i t y . A l t e r n a t i v e l y , the p a r t i t i o n could be regional or c u l t u r a l . The purpose of the a i s to adjust upward or downward definable classes of interchanges i n order to improve the explanation of the observed arc volumes, thereby obtaining improved estimates of the 0-D volumes. 9.2 B r i t i s h Columbia data 9.2.1 Introduction The most extensive t e s t i n g of methods for estimating the 0-D matrix have been performed on the highway flow volumes i n B.C. This area i s thought to be well suited to the methods used, p a r t i c u l a r l y the all-or-nothing method of assignment of 0-D volumes to arcs. Owing to the separation of urban settlements by considerable distances i n which there i s l i t t l e or no population, the representa-t i o n of zones by spaceless centroids r e s u l t s i n an i n s i g n i f i c a n t information loss due to s p a t i a l aggregation. Furthermore, the s i m p l i c i t y of the highway network allows a r e l a t i v e l y complete representation as an abstract graph. 9.2.2 The basic model 9.2.2.1 Hypothesis testing F i r s t of a l l a complete cross-section of data i s selected for 1966. Using the basic model a variety of - 170 -alternative, but nested, models were tested. These re s u l t s are given i n Table 9.1. Whereas models 1 and 4, and perhaps 7, give good and reasonable values for the free parameters and the goodness of f i t to the arc volumes, the remaining models are unambiguously rejected by the l i k e l i h o o d r a t i o t e s t at the required number of degrees of freedom. Let A be the parameter space under a maintained hypothesis, such as that produced by the functional form of the simple model with a, B , and y a l l free parameters. Let ui be the r e s t r i c t e d parameter space generated by a n u l l hypothesis, involving constraints on the parameters. Then the l i k e l i h o o d r a t i o t e s t s t a t i s t i c i s known to be 2 [L(fi) - L(w) ] where L(fi) i s the maximum logarithmic l i k e l i h o o d under the maintained hypothesis and L(to) i s the r e s t r i c t e d hypothesis. This test s t a t i s t i c has a x 2 d i s t r i b u t i o n with a degree of freedom for each r e s t r i c t i o n i n to but not in Q. Restrictions may be of two types: equality constraints between parameters, and fi x e d values for parameters such as zero or one. The rejected models are as i n t e r e s t i n g as the accepted ones. In p a r t i c u l a r , i t should be noted that model 6 i s the most commonly seen simple form of the gravity model containing only the product of the populations and the inverse square of the distance. Remarkably enough - 171 -PARAMETERS MODELS X X X X VX XX • • ' -L, J_l X \ k^J 1 2 3 4 a 36.34 14588. 57.28 35.95 3 0.7199 0.0 0.0030 0.7242 Y -1.988 -1.325 0.0 -2.0 L -1510.14 -1698.39 -2161.17 -1510.17 R 2 0.8609 0.1191 0.0 0.8608 A 86 86 86 86 H 5700 5700 5700 5700 D.F. 0 1 1 1 TABLE 9.1 THE BASIC MODEL: UNCONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF ALTERNATIVE HYPOTHESES. - 172 -PARAMETERS MODELS 5 6 7 8 a 0.23 1.0 5.82 1.0 8 1.0 1.0 1.0 1.0 Y -2.0 -2.0 -2.561 -2. 246 L -1659.93 -2015.06 -1581.95 -1627.29 R 2 0.3958 0.0 0.7187 0.5612 A 86 86 86 86 H 5700 5700 5700 5700 D.F. 2 3 1 2 TABLE 9.1 (CONTINUED) - 173 -t h i s model was not markedly superior to model 3 which eliminated distance altogether as an explanatory v a r i a b l e . This r e s u l t cautions against using model 6, as i t i s some-times used, as a rough estimate of i n t e r a c t i o n s , without c a l i b r a t i o n . Improvement on model 6 i s obtained by freeing the distance parameter thereby obtaining another frequently encountered form of the gravity model. Adding the constant to t h i s set of free parameters r e s u l t s i n considerable improvement as shown by model 7, as does the population parameter i n model 1. Comparison of models 2 and 3 demonstrates that the explanatory power of the population product i s superior to that of distance: elimination of population, by s e t t i n g 3 = 0 , degrades the l i k e l i h o o d function by more than elimination of distance, by s e t t i n g y = 0. One of the most i n t e r e s t i n g findings i s that i n the f u l l y parametrised model, with a , 8, and y free, the estimate of the distance parameter y i s not s i g n i f i c a n t l y d i f f e r e n t from -2. By analogy with physical phenomena i t has often been conjectured that there e x i s t s i n socio-economic i n t e r a c t i o n an inverse-square law. Model 4 shows that such, a conjecture cannot be rejected at even the lowest l e v e l s of s t a t i s t i c a l s i g n i f i c a n c e . Moreover, i t i s of p a r t i c u l a r s i g n i f i c a n c e that unless a and 8 are free to vary the inverse-square law i s soundly rejected. This i s shown by models 2, 7 and 8. - 174 -9.2.2.2 Maximal interchange The re s u l t s given i n 9.2.2 were, without exception, computed using the complete 0-D matrix which was estimated and assigned to the arcs. Since many of these interchanges contain l i t t l e or no t r a v e l volume the maximal interchange selection c i r t e r i o n i s pertinent to test how much of these data are necessary to obtain reasonable estimates of the parameters. As previously described, the selection c r i t e r i o n i s based on p r i o r estimates of the magnitudes of the interchanges and on the number of arcs i n the network which carry these flows, that i s , the path length i n arcs from o r i g i n to destination. Maximal interchange results are given i n Table 9.2. For t h i s simple model they show that reasonable parameters are obtainable using twenty per cent or more of the i n t e r -changes. Below this c r i t e r i o n the parameters become quite seriously biased. Nonetheless, i n view of the computational and core storage saving of t h i s procedure, i t s p r a c t i c a l i t y and v a l i d i t y seem to be established. Further values of the parameters are obtainable from the graph i n Figure 9.1. 9.2.2.3 Structural change over time In order to evaluate the usefulness of the model as a predictive t o o l , a series of six cross-sections of data was analysed. Unless there exists complete s t a b i l i t y i n the s p a t i a l socio-economic environment the parameters obtained for each year of data w i l l be d i f f e r e n t . The best that can - 175 -PARAMETERS MODELS 1 2 3 4 5 a 2981.55 202.25 59.10 43.75 36. 39 6 0.2059 0.4964 0.6415 0.6855 0.7203 Y -1.0611 -1.5536 -1.8145 -1.9067 -1.9904 L N/C N/C N/C N/C -674.028 <R2 0.6390 0.8219 0.8480 0.8469 0.8523 A 62 81 85 85 86 H 106 250 539 855 i 1 2850 k . 05 .1 . 2 1 • 3 j .—. L 1.0 TABLE 9.2 THE BASIC MODEL: UNCONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES WITH MAXIMAL INTERCHANGE EXPERIMENTS. - 176 -3000 2 5 0 0 H 2000 -A a 1500 1000 -I 500 -J PROPORTION (k) OF INTERCHANGES SELECTED FIGURE 9.1 CONVERGENCE PATHS OF PARAMETERS WITH INCREASING MAXIMAL INTERCHANGE SELECTION SIZE. - 177 -be hoped for i s a well-defined trend, preferably monotonic. This allows functions to be f i t t e d to the parameters them-selves, as a function of time, and new parameters to be obtained by extrapolation. Moreover, given that s t a b i l i t y or reasonable trends are found i n the parameters, then such a r e s u l t enhances the v a l i d i t y of the model and the procedures used i n i t s estimation. Results from the O-D estimation model for 1963-1968 are presented i n Table 9.3 and graphed i n Figure 9.2. The parameters c l e a r l y show evidence of gradual s t r u c t u r a l change which i s b a s i c a l l y monotonic i n form. Whereas the population parameter becomes larger positive over time, the distance parameter becomes larger negative, thus compensat-ing each other. As the marginal e f f e c t of the population parameter i s greater than that of the distance parameter, further compensation i s supplied by the constant which decreases four-fold over the period. Owing to the m u l t i p l i c a t i v e form of the model and to the existence of opposite signs on the parameters, considerable i n s t a b i l i t y could e a s i l y have been the outcome. The nonspherical shape of the l i k e l i h o o d function augments the p o s s i b i l i t y of just such an eventuality. However, the fact that t h i s did not occur empirically i s an encouraging outcome for the v a l i d i t y and usefulness of the model i n a predictive mode. The ef f e c t s of s t r u c t u r a l change on the parameters as revealed by d i f f e r e n t models can be seen by comparing - 178 -PARAMETERS DATA SETS 1963 1964 1965 1966 1967 1968 a 56. 20 63.12 40.53 36.31 19.14 10.11 8 0. 718 0.706 0. 710 0. 720 0.739 0.762 Y -2. 03 -2. 02 -1.99 -1. 99 -1.93 -1.89 R 2 0.844 0.831 0.791 0.861 0.822 0.833 A 86 86 86 86 86 86 H 2850 2850 2850 2850 2850 2850 TABLE 9.3 THE BASIC MODEL: UNCONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF A SERIES OF ANNUAL CROSS-SECTIONS. - 179 -100 80 -I a -1.90J Y -1.95. •2.00 J -2.05 1963 1964 — r ~ 1965 1966 I 1967 1968 FIGURE 9.2 THE BASIC MODEL: UNCONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF A SERIES OF ANNUAL CROSS-SECTIONS. - 180 -results i n Tables 9.3 and 9.4. In section 9.2.2 i t was shown that a common form of the gravity model with a single parameter, on distance, performed r e l a t i v e l y poorly i n terms of goodness of f i t to the observations. Table 9.4 and Figure 9.3 show that the trend of the distance parameter i s opposite to that i n the more f u l l y s p e c i f i e d model. This suggests that the single distance parameter i s biased by effects properly ascribable to the population variable on which the parameter i s constrained to unity. Ex post facto prediction i s related to the question of s t r u c t u r a l change. It involves the v a l i d a t i o n of a model by using i t to predict conditions for a h i s t o r i c a l period using parameters obtained from estimation in a p r i o r h i s t o r i c a l period. The extent to which the model i s able to reproduce the data indicates the v a l i d i t y of the model for predictive purposes. Naturally t h i s i s a very demanding task for a model and poor r e s u l t s are commonly obtained. The prediction may be achieved i n two ways, either with parameters fixed at some p r i o r l e v e l or a l t e r n a t i v e l y with parameters predicted themselves. Demonstration of the f i r s t case i s given by Table 9.5 which shows remarkably that model was e a s i l y able to predict the arc volumes for 1964 and s t i l l obtained good estimates for 1968. Moreover, i f parameters are themselves predicted, by using the smooth functions given i n Figure 9.2, estimates - 181 -PARAMETERS DATA SETS 1963 1964 1965 1966 1967 1968 a 1.0 1.0 1.0 1.0 1.0 1.0 8 1.0 1.0 1.0 1.0 1.0 1.0 Y -2.191 -2.199 -2.247 -2.247 -2.247 -2.252 R 2 0.455 0.406 0.464 0.561 0. 652 0.726 A 86 86 86 86 86 86 H 2850 2850 2850 2850 2850 2850 TABLE 9.4 THE BASIC MODEL: CONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF THE SIMPLE GRAVITY MODEL FOR A SERIES OF ANNUAL CROSS-SECTIONS. - 182 -1963 1964 1965 1966 1967 1968 FIGURE 9.3 THE BASIC MODEL: CONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF A SERIES OF ANNUAL CROSS-SECTIONS. - 183 -PARAMETERS DATA SETS 1963 1964 1965 1966 1967 1968 a 56.20 56.20 56.20 56. 20 56.20 56.20 3 0. 718 0.718 0.718 0. 718 0.718 0.718 Y -2.03 -2. 03 -2.03 -2. 03 -2.03 -2.03 R 2 0.844 0. 830 0. 605 0.718 0. 660 0.504 A 86 86 86 86 86 86 H 2850 2850 2850 2850 2850 2850 TABLE 9.5 THE BASIC MODEL: EX POST FACTO PREDICTION USING 1963 PARAMETER VALUES. - 184 -of the parameters which are very close to those from 1968 data are obtained. Needless to say that these parameters would give very accurate predictions for future periods, and i n the case of 1968 would almost exactly reproduce the data. Further work could reveal the extent and form of these trends i n the future, which i f maintained i n terms of t h e i r r e g u l a r i t y , would permit accurate predictions. 9.2.3 Aspects of the extended model There are numerous extensions which could be made to the basic model. A few of these are now explored. Beginning with the maximal interchange set at 10 per cent for the basic model a single arc parameter i s added. Following t h i s the o r i g i n constant i s partitioned by sets of o r i g i n s . After t h i s the arc constant i s partitioned on the basis of volume. The section i s concluded with an examination of f i n e r p a r t i t i o n s of the o r i g i n constant while maintaining a single arc constant. 9.2.3.1 Par t i t i o n s of the o r i g i n constant Model 1(a) i n Table 9.6 shows the re s u l t s of the addition of a single arc parameter to the model given i n Table 9.2. Several changes have taken place. The arc parameter turns out large and s i g n i f i c a n t . The remaining parameters, which alone constitute an estimate of the O-D, resemble more closely the parameters of the f u l l model. - 185 -PARAMETERS MODELS PARTITION ORIGINS IN K a ) K b ) K c ) K d ) RANGES PARTITION a o 759.30 777.43 806.73 788.39 FULL M.I. a l 58.03 125.83 97.71 27.30 35000 3 3 a2 58.03 73.70 57.05 18.67 35000 2000 6 5 a 3 58.03 73.70 57.05 21.35 2000 950 10 10 a4 58.03 66.75 45.15 23.22 950 700 4 4 a 5 58.03 66.75 45.15 8.27 700 500 4 4 a6 58.03 66.75 45.15 -36.31 500 400 5 5 a 7 58.03 66.75 59.32 28.08 400 300 8 6 a8 58.03 66.75 59.32 5.26 300 200 9 6 a 9 58.03 66.75 59.32 41.06 200 100 14 5 a10 58.03 66.75 59.32 -110.02 100 0 13 2 6 0.6589 0.6594 0.6913 0.7721 Y -1.8783 -1.9405 -1.9995 -2.0358 L -623.885 -622.205 -621.909 -613.814 R 2 0.8869 0.8915 0.8923 0.9118 A 81 81 81 81 k .10 .10 .10 .10 TABLE 9.6 THE EXTENDED MODEL: PARTITIONS OF THE ORIGIN CONSTANT - 186 -This indicates that the arc constant has absorbed much of the e f f e c t of the interchanges omitted by the maximal i n t e r -change method. Over and above t h i s , i t now provides an estimate of the l o c a l t r a f f i c or 'noise' from interchanges below the grain of analysis selected for the network. The o r i g i n constant i s p a r t i t i o n e d into three sets for model 1(b). The impact i s somewhat curious i n that a l l three parameters increase i n magnitude, as does the arc parameter. However, these increases are compensated by an increase i n the magnitude of the negative distance parameter whereas the population parameter remains e s s e n t i a l l y unchanged. Addition of a fourth constant obtained by p a r t i t i o n i n g the class with the smallest populations has the e f f e c t of reducing the magnitude of a l l the o r i g i n constants but increasing the arc constant. The decrease i n the o r i g i n constants i s compensated by the increase i n the population parameter. F i n a l l y , 1(d) demonstrates the r e s u l t i f the p a r t i t i o n i n g i s taken too f a r . Not only i s there large and unreasonable v a r i a t i o n i n the constants but some turn negative, an i n f e a s i b l e s o l u t i o n . This outcome was due to the i n s u f f i c i e n t number of interchanges which were l e f t i n some p a r t i t i o n s . In the s i x t h p a r t i t i o n , for example, although f i v e o r i g i n s were covered t h i s generated only 17 interchanges a f t e r the maximal interchange procedure had dropped a l l but those with s u b s t a n t i a l flow p o t e n t i a l . Again only two o r i g i n s remained i n the tenth p a r t i t i o n . - 187 -9.2.3.2 Estimation of l o c a l t r a f f i c The extended model allows an estimate to be made of the proportion of the arc volumes which might properly be ascribed to l o c a l t r a f f i c , omitted interchanges or simply noise. This estimate i s achievable by the addition of either a single additive constant for a l l arcs or a set of constants s p e c i f i c to a set of arc classes partitioned on the basis of the magnitude of t h e i r volumes or some other c l a s s i f i c a t i o n . A single arc constant produces an estimate of the local/O-D r a t i o which i s averaged over a l l arcs. Comparison of models 2(c) and 2(d) i n Table 9.7 shows the e f f e c t of introducing a single arc constant. The p r i n c i p a l r e s u l t i s perhaps the improvement i n the goodness of f i t suggesting that there i s indeed a certain proportion of the t r a f f i c volume more cor r e c t l y interpreted as noise rather than as O-D. Of course, since these re s u l t s are computed using the maximal interchange methodology the constant also contains the e f f e c t of interchanges i n t e n t i o n a l l y omitted. Model 2(b) shows the r e s u l t of p a r t i t i o n i n g the o r i g i n constant with a maintained arc constant. Very l i t t l e improvement i s obtained i n the f i t to the extent that the l i k e l i h o o d r a t i o test f a i l s to rej e c t the n u l l hypothesis and 2(d) the n u l l , the l a t t e r i s unequivocally rejected. A p a r t i t i o n of the arc constant by volume leads to further enhancement i n the a b i l i t y of the model to r e p l i c a t e the observed volumes. This i s demonstrated by model 2(a). I f - 1 8 8 -PARAMETERS MODELS 2(a) 2(b) 2(c) 2(d) a o i 2 9 4 5 . 3 7 7 7 . 4 3 7 5 9 . 3 0 a 0 2 1 5 7 6 . 5 7 7 7 . 4 3 7 5 9 . 3 0 a 0 3 5 1 5 . 4 2 7 7 7 . 4 3 7 5 9 . 3 0 a l 3 1 . 7 9 1 2 5 . 3 0 5 8 . 0 3 2 0 2 . 3 5 a 2 1 4 . 9 5 7 3 . 7 0 5 8 . 0 3 2 0 2 . 3 5 a 3 1 9 . 6 9 6 6 . 7 5 5 8 . 0 3 2 0 2 . 3 5 3 0 . 8 6 4 3 0 . 6 5 9 4 0 . 6 5 8 9 0 . 4 9 6 3 Y - 2 . 4 6 4 5 - 1 . 9 4 0 5 - 1 . 8 7 8 3 - 1 . 5 5 3 4 L - 5 9 2 . 2 0 6 - 6 2 2 . 2 0 5 - 6 2 3 . 8 8 5 - 6 4 2 . 2 9 5 R 2 0 . 9 4 8 3 0 . 8 9 1 5 0 . 8 8 6 9 0 . 8 2 1 9 A 8 1 8 1 8 1 8 1 k . 1 L-. 1 .1 . 1 TABLE 9.7 THE EXTENDED MODEL: PARTITIONS OF THE ARC CONSTANT - 189 -2(a) i s d e f i n e d the maintained, and 2(b) the n u l l h y p o t h e s i s , the l a t t e r i s soundly r e j e c t e d by a l i k e l i h o o d r a t i o s t a t i s t i c of 30 with two degrees of freedom. From models 2(a) and 2(d) a c o n s i s t e n t t r e n d may be observed i n the n o n l i n e a r parameters. Both 8 and y a l t e r t h e i r magnitudes by about f i f t y per cent, an outcome which demonstrates t h a t the l i n e a r parameters have an important r o l e to p l a y i n the model. Trends i n the l i n e a r parameters are l e s s t r a n s p a r e n t . N e v e r t h e l e s s some p a t t e r n s emerge; the i m p o s i t i o n of the arc constant e q u a l i t y c o n s t r a i n t leads t o a decrease i n the local/O-D r a t i o . The r e d u c t i o n i n the a r c constants shows t h i s . A s i m i l a r r e s u l t i s ob t a i n e d by the e q u a l i t y c o n s t r a i n t on the o r i g i n c onstant i n 2 ( c ) . Reduction of the arc constants to zero by d e f i n i t i o n reduces the l o c a l t r a f f i c estimate to zero and thereby a s s i g n s a l l a r c volumes t o O-Ds t h a t can be so assign e d w i t h i n the c o n f i n e s of the remaining three parameter models. 9.2.3.3 Maximal interchange i n the extended model In s e c t i o n 9.2.2.2 the e f f e c t of s e l e c t i n g the p r i n c i p a l i nterchanges o n l y f o r the e s t i m a t i o n of parameters f o r the b a s i c model was d i s c u s s e d . I t was seen t h a t below ten per cent the estimates were b i a s e d . One of the reasons f o r t h i s b i a s i s presumably the omission of an a r c constant w i t h the c a p a b i l i t y of absorbing the e f f e c t s of the - 190 -i n t e n t i o n a l l y omitted interchanges. As the extended model contains t h i s constant two results are to be anticipated: the remaining parameters exhibit less bias, and the greater the number of omitted interchanges the greater the magnitude of the arc constant. Both these expected re s u l t s are observed i n Table 9.8. In the case of the arc constant the parameter shows a consistent increase as the number of omitted i n t e r -changes grows. Thus when the f u l l set of interchanges are used, as in model 3(e), the constant r e f l e c t s s o l e l y the estimate of l o c a l t r a f f i c or noise. As the f i r s t forty per cent of interchanges are eliminated the constant increases but s l i g h t l y , a finding which i s consistent with the fact that very many of the interchanges have e f f e c t i v e l y zero t r a f f i c . F i n a l l y , at the l e v e l of ten per cent included and ninety per cent excluded, the arc constant has increased by 150, t h i s being the contribution to the average arc of the omitted interchanges. Any further reduction i n the percentage of interchanges used would no doubt r e s u l t i n rapid increase i n the arc constant since only the larger interchanges now remain. In t h i s way, i t i s seen that the arc constant absorbs the e f f e c t of the omitted interchanges and thereby p u r i f i e s the remaining parameters of elements of t h i s bias. Comparison of the parameters i n Table 9.8 with those of Table 9.2 demonstrates that the bias has indeed been reduced. - 191 -PARAMETERS MODEL 3(a) 3(b) 3(c) 3(d) 3(e) a Q 759.75 649.36 620.17 614.00 605.46 a i 59. 97 26. 84 20.29 19.11 18.90 e 0.659 0. 749 0.792 0. 803 0.804 Y -1.879 -2.04 -2.13 -2.16 -2.16 L -662.18 -659.30 -660.80 -661.08 -661.01 R 2 0.8879 0.8952 0.8914 0.8907 0.8909 k 0.1 0.2 0.4 0.6 1.0 TABLE 9.8 MAXIMAL INTERCHANGE IN THE EXTENDED MODEL: ESTIMATED PARAMETERS. - 192 -Although b i a s i n the parameters may have been reduced by the a r c c o n s t a n t the a c i d t e s t must u l t i m a t e l y be the v a r i a t i o n i n the O-D e s t i m a t e s . These are shown, f o r the corresponding models, i n Table 9.9. Here i t i s shown t h a t , apart from the f i r s t two i n t e r c h a n g e s , i n v o l v i n g C algary, the estimated O-D, even as low as the ten per cent l e v e l , are q u i t e s i m i l a r to the complete e s t i m a t i o n . The estimate of the C a l g a r y - L e t h b r i d g e O-D c o n t a i n s a t h i r t y per cent b i a s u s i n g ten per cent of the i n t e r c h a n g e s , and a f i v e per cent b i a s u s i n g f o r t y per cent. Other i n t e r -changes, i n c o n t r a s t , seem to be l o c a t e d near a f i x e d p o i n t i n t h a t they change almost not at a l l . Vancouver-P r i n c e George shows t h i s . Larger percentage changes are not a p p a r e n t l y c o n f i n e d to the l a r g e r O-Ds. The O-D from Kelowna to P r i n c e George i s b i a s e d upward by about seventy per cent. The average t r i p l e n g t h remains remarkably constant f o r a l l of the maximal interchange l e v e l s . T h i s constancy i s p a r t i c u l a r l y encouraging between models 3(a) and 3(b) where the estimated O-Ds are b e g i n n i n g to d i v e r g e s u b s t a n t i a l l y . The magnitude of the t r i p l e n g t h i s reasonable although there are no known observed data to check t h i s by. I f t h i s estimate appears somewhat hi g h , i t should be r e c a l l e d t h a t i t a p p l i e s to i n t e r c i t y t r a v e l o n l y and t h a t the n e a r e s t c i t y t o Vancouver i n the d e f i n i t i o n of the study network was Hope at some 100 m i l e s . Hence, the - 193 -CITY-PAIR MODEL 3(a) 3(b) 3(c) 3(d) 3(e) Calgary t o : Vancouver 597 697 739 751 753 Lethbridge 1972 2326 2547 2606 2620 Kamloops 175 165 159 158 158 Vancouver t o : Edmonton 417 472 492 498 499 Hope 1884 2124 2292 2337 2349 Kamloops 641 693 719 726 728 P e n t i c t o n 458 470 477 479 479 Pr. George 262 267 266 266 266 Vernon 277 272 269 269 269 Kelowna t o : Vernon 1653 1679 1750 1769 1776 Pr. George 28 21 18 17 17 Lethbridge 21 15 13 12 12 Avg. T r i p Length (Miles) 133.2 132.9 131. 6 131.7 131.9 TABLE 9.9 MAXIMAL INTERCHANGE IN THE EXTENDED SAMPLE ESTIMATED O-D'S. MODEL: - 194 -minimum t r i p length for Vancouver i s bounded below by t h i s , the nearest destination. The average t r i p length given i n Table 9.9 i s averaged over a l l included interchanges. 9.2.3.4 Alternative extended models and O-D estimates The question which now arises i s the extent to which d i f f e r e n t model s p e c i f i c a t i o n s give r i s e to d i f f e r e n t estimates of the O-D matrix. Since there exists no unique model for such estimation the question i s consequently an interes t i n g one. Table 9.10 compares a model with the arc constants constrained equal, 4(a), and a model without t h i s constraint, 4(b). The parameters are comparable without being p a r t i c u l a r l y close. Also, the log l i k e l i h o o d functions are s u f f i c i e n t l y d i s t i n c t to require r e j e c t i o n of the n u l l hypothesis i n 4(a) at any conventional l e v e l of s i g n i f i c a n c e . Given the d i s s i m i l a r i t y of these two models, i t is unexpected to discover the s i m i l a r i t y of t h e i r O-D estimates. In view of the error encountered i n survey estimates of the O-D matrix, estimates which d i f f e r by less than twenty per cent, for example, could be regarded as e s s e n t i a l l y the same. The estimates given i n Table 9.11 are a l l of the same order of magnitude. Ignoring the small flows, under one hundred persons, the p r i n c i p a l deviations constitute a difference of no more than f i f t y per cent. Moreover, the largest interchanges d i f f e r by less than f i v e to f i f t e e n per cent. - 195 -PARAMETERS MODEL 4(a) 4(b) a o i 605.47 2743.53 a02 605.47 1458.26 a03 605.47 430.95 a 18.86 6.570 8 0.8046 0.9322 Y -2.1633 -2.4844 L -661.01 -630.35 R 2 0.8909 0.9465 k 1.0 1.0 TABLE 9.10 ALTERNATIVE EXTENDED MODELS - 196 -CITY-PAIR MODEL 4(a) 4 (b) Calgary t o : Vancouver 753 544 Lethbridge 2620 2224 Kamloops 158 88 Vancouver t o : Edmonton 499 339 Hope 2349 1941 Kamloops 728 512 P e n t i c t o n 479 315 P r i n c e George 266 162 Vernon 269 162 Kelowna t o : Vernon 1776 1375 P r i n c e George 17 7 Lethbridge 12 5 Avg. T r i p Length (Miles) 131.9 119 .7 TABLE 9.11 ESTIMATED 0-D FROM ALTERNATIVE EXTENDED MODELS - 197 -The only s i g n i f i c a n t d i f f e r e n c e perhaps i s to be found i n the average t r i p length which decreases by twelve miles when the arc constant i s p a r t i t i o n e d , 4(b), This model, i n addition to s h i f t i n g t r a f f i c volumes from 0-D to l o c a l t r a f f i c has also t r a n s f e r r e d 0-D from longer interchanges to shorter ones. Although t h i s t r a n s f e r has not been applied p r o p o r t i o n a l l y over a l l c i t y p a i r s , i t i s c l e a r that most, i f not a l l , of the estimates have f a l l e n . - 198 -9.2.4 E s t i m a t i n g t h e f u n c t i o n a l f o r m o f t h e 0-D m a t r i x T h e b a s i c m o d e l p r e s e n t e d i n s e c t i o n 9.1.1 c a n be g e n e r a l i s e d i n t e r m s o f t h e f u n c t i o n a l f o r m o f t h e t r i p d i s t r i b u t i o n m a t r i x . T h i s i s o b t a i n e d by t h e a p p l i c a t i o n o f B o x - C o x t r a n s f o r m a t i o n s t o t h e v a r i a b l e s . H e n c e t h e e q u a t i o n g i v e n by (1) now becomes v 0 2 M i n i m i s e >(V - V ) w . r . t . ( a,8 ,Y? X) a a a w h e r e V = j&, t , a £ h a h n and t h = [a -t- ( 3 P H yD h (6) S e v e r a l s p e c i a l c a s e s may be o b t a i n e d f o r g i v e n v a l u e s o f t h e A ' s . F o r X0 = Xl = X2 = 0 ( 7 ) t h e t h r e e - p a r a m e t e r g r a v i t y m o d e l i s o b t a i n e d . S e t t i n g XQ = 0; X1 = A 2 = 1 (8) t h e l o g i t f o r m i s d e r i v e d , a n d f o r A0 ~ Xl ~ A2 = 1 ( 9 ) t h e l i n e a r h y p o t h e s i s a p p e a r s a l s o as a s p e c i a l c a s e o f t h i s g e n e r a l f o r m . - 199 -In t h i s s e c t i o n an e q u a l i t y c o n s t r a i n t i s main-t a i n e d on the X's. Although t h i s e l i m i n a t e d the l o g i t model (8) from c o n s i d e r a t i o n i t n e v e r t h e l e s s allows the important g r a v i t y ( m u l t i p l i c a t i v e ) (7) and l i n e a r ( a d d i t i v e ) (9) models to be t e s t e d a g a i n s t each other and a g a i n s t more gen e r a l f u n c t i o n a l forms. The us u a l l i k e l i h o o d r a t i o t e s t s can be a p p l i e d to r e s t r i c t i o n s on the X's. Resul t s o f these e s t i m a t i o n s are given i n Tables 9.12 and 9.13. I t i s shown i n Table 9.12 t h a t the o p t i m a l model i s l o c a t e d with the X's at 0.01 (to 2 s i g n i f i c a n t f i g u r e s ) . That t h i s model i s not s i g n i f i c a n t l y d i f f e r e n t from the m u l t i p l i c a t i v e g r a v i t y model i s shown by comparison of models (b) and ( c ) . In terms of the l i k e l i h o o d r a t i o t e s t XQ = X^ = X2 = 0 cannot be r e j e c t e d a g a i n s t XQ = X^ = X2 t 0 even a t the 10 per cent l e v e l of s i g n i f i -cance. Such a c o n c l u s i o n i s very important because i t suggests t h a t the g r a v i t y model, i n i t s f u l l y p a r a metrised form, i s e f f e c t i v e l y the bes t model from t h i s f a m i l y of models. A v a r i e t y of a l t e r n a t i v e models i s shown i n Table 9.14. The l i n e a r model, the other main a l t e r n a t i v e to the m u l t i p l i c a t i v e model, i s demonstrably i n f e r i o r being unambiguously r e j e c t e d by the l i k e l i h o o d r a t i o t e s t . The same v e r d i c t a p p l i e s to the square r o o t model and a l s o t o models i n the negative range of X. From the t a b l e s i t i s - 200 -PARAMETERS MODELS (a) (b) (c) (d) (e) Ao -0.05 0.0 0.01 0.02 0.05 h -0. 05 0.0 0 . 01 0.02 0. 05 a 0.430 36.31 82. 74 182.33 1786.12 8 1.2574 0.7217 0.6419 0.5734 0.4079 Y -1.9158 -1.9943 -1.9988 -2.0107 -2.0441 L -678.89 -674.03 -673.10 -673.13 -675.06 R 2 0.8347 0.8519 0.8555 0.8554 0.8487 A 86 86 86 86 86 H 2850 2850 2850 2850 2850 TABLE 9.12 THE BASIC MODEL. ESTIMATING THE FUNCTIONAL FORM. LOCATION OF THE OPTIMUM. - 201 -PARAMETERS MODELS (a) (b) (c) (d) (e) X 0 0.0 0.1 0.2 0.5 1.0 X 1 0.0 0.1 0.2 0.5 1.0 a 36.306 62830 2.04xl0 7 565340 2.29xl0 6 3 0.7217 0.2235 0.06058 -0.000658 2.5 x 10~ 7 Y -1.9943 -2.0426 -1.8355 -0.2839 -0.01913 L -674.03 -680.56 -697.16 -737.76 -739.46 R 2 0.8519 0.8281 0.7471 0. 3500 0.2867 A 86 86 86 86 86 H 2850 2850 2850 2850 2850 TABLE 9.13 THE BASIC MODEL. ESTIMATING THE FUNCTIONAL FORM. A COMPARISON OF ALTERNATIVE FORMS WITH XI 0. - 202 -PARAMETERS MODELS (f) (g) (h) (i) X 0 -0.3 -0.15 -0.1 0.0 X 1 -0.3 -0.15 -0.1 0.0 a 1.80 x l O " 3 0 6.30 x 10~ 7 0.001608 13.306 e 22.7910 3.9747 2.2139 0.7217 Y -1.6670 -1.7891 -1.8429 -1.9943 L -717.70 -696.27 -686.88 -674.03 R 2 0.5634 0.7523 0.8009 0.8519 A 86 86 86 86 H 2850 2850 2850 2850 TABLE 9.14 THE BASIC MODEL. ESTIMATING THE FUNCTIONAL FORM. A COMPARISON OF ALTERNATIVE FORMS WITH X < 0. - 203 -c l e a r that the optimum i s quite sharp and that the l i k e l i h o o d function i s unimodal and reasonably symmetrical about the optimum. CHAPTER 10 EMPIRICAL RESULTS I I : CANADIAN DATA 10.1 I n t r o d u c t i o n The o b j e c t i v e o f t h i s s e c t i o n i s to p r e s e n t the models and t h e i r r e s u l t s from an a n a l y s i s of a 1972 highway network c o v e r i n g Canada and contiguous USA. One o f " t h e s e models i s s e l e c t e d to estimate the c a r 0-D m a t r i x f o r a multimodal model which i s subsequently e s t i m a t e d , another computes an e s t i m a t e of the bus 0-D m a t r i x f o r a s i m i l a r purpose. For each of the c a r and bus modes an 0-D matrix-of dimension 107 by 107 i s computed. A l i s t of the 107 c i t i e s i s g i v e n i n the Appendix. The r e s u l t s are o r g a n i s e d i n t o two s e c t i o n s : c a r and bus, and emphasis i s p l a c e d on h y p o t h e s i s t e s t i n g of a l t e r n a t i v e models, and i n the case of c a r , on the performance of the maximal i n t e r c h a n g e s e l e c t i o n procedure. As b e f o r e , the primary highway network of each p r o v i n c e i s r e p r e s e n t e d by a s e t of nodes and c o n n e c t i n g l i n k s or a r c s . P a r a l l e l l i n k s are e l i m i n a t e d by a g g r e g a t i o n to a s i n g l e a r c or s e r i e s of a r c s . Each arc i s a s s i g n e d a d i s t a n c e and an observed t r a f f i c volume. D i s t a n c e s are e x t r a c t e d from p r o v i n c i a l highway maps and ar c t r a f f i c volumes are d e r i v e d from average annual d a i l y t r a f f i c data (A.A.D.T.) p u b l i s h e d by p r o v i n c i a l departments. In a d d i t i o n to these data an estimate of the p r o p o r t i o n of nonpassenger v e h i c l e s i n the t r a f f i c stream and of average car occupancy l e v e l s were o b t a i n e d . These were c a l c u l a t e d to be 15 percent and 2.0 r e s p e c t i v e l y . Bus route flows were o b t a i n e d d i r e c t l y from the b u s . c a r r i e r s . - 204 -- 205 -10.2 The basic model 10.2.1 Hypothesis testing Using the basic model a set of nested hypotheses are tested with the usual l i k e l i h o o d r a t i o t e s t . The l i s t of tests performed i s detailed i n Table 10.1, and the re s u l t s are shown i n Tables 10.2 and 10.3. The primary objective of the tests i s to show that a l l three parameters are needed, that the model i s not overparametrised. This fact i s shown by the three unconditional tests with single r e s t r i c -tions: U(a), U(8) and U(y). Each n u l l hypothesis i s vigorously rejected which implies that, at least when tested separately, none of the parameters i s superfluous. Not only i s the zero r e s t r i c t i o n re-jected but so i s the 8 = 1.0 r e s t r i c t i o n . One hypothesis which i s not rejected, even at the 10 per cent l e v e l of s i g n i f i c a n c e , i s that y i s equal to -2.0. Once again the inverse-square hypothesis cannot be rejected, as was found i n the B.C. data. This i s a remarkable finding as the Canadian and B.C. datasets d i f f e r considerably i n scale. It also adds considerable credence to the v a l i d i t y of the O-D model which i s estimated since - 206 -Test Hypothesis(H Q) A l t e r n a t i v e ( H ^ ) T e s t S t a t i s t i c C r i t i c a l Region U(cx) a = a a * 3 L ( a * a , 6 , Y ) U ( a ) > § X 2 ( l ) - L(a=a , 0,y) U(6) 6 = 8 6 t 6 L ( a , 6 ^ 6 , Y ) U ( 6 ) > i x 2 ( D - L(a , 6 = 3 , Y ) U(y) Y = Y Y * Y L ( a , 6 , Y * Y ) U ( Y ) > l x 2 ( D - L(a , 6 , Y = Y ) U(a, a = a a t a L(a^a,6^6,y) U(a ,6)> 6) 8 = £ 8 t 8 - L(a=a ,6=3 ,Y) i x 2 ( 2 ) U(6, 8 = 3 8 t 8 L(a, 6 ^ 6 , Y * Y ) U (6 ,Y )> Y) Y - y Y * Y - L(S,6=6,Y=Y) l x 2 ( D U ( a , a = a a * 5 L ( a * a , 6 * 6 , Y * Y ) U ( a , 6 , Y ) > 6 ,Y ) 8 = 6 8 2 8 - L(a=a ,6=6 ,Y=Y) I X 2(3) Y - Y Y t Y C(a) a = a a ;r a L(a*a,6=6,Y) C ( a ) > | X 2 ( l ) given 6 = 6 given 8=6 - L(a=a ,6=6 ,Y) C(6) 8 = 8 8 * 6 L(a , 6 5 * 6 , Y - Y ) C ( 6 ) > | X 2 ( D given y=y given Y=Y - L(a,6=6 ,Y=Y) C ( Y ) Y - Y Y * ! L(a=a,6=6,Y^Y) C ( Y ) > | X 2 ( D given 8-8 given 6=6 - L(a=a,6=6 ,Y=Y) a=a TABLE 10.1 LIKELIHOOD RATIO TESTS FOR THE BASIC MODEL - 207 -PARAMETERS MODELS (a) (b) (c) (d) (e) a 1051.5 1.0 4.18 x 10 6 51.367 627.28 8 0.3751 0.6127 0.0 9.91 x 10" 8 0.3847 Y -2.055 -1.9216 -1.8352 0.0 -2.0 L -1786.88 -1800.57 -1828.31 -1891.25 -1787.35 R 2 0.7459 0.7069 0.6087 0.2463 0.7446 k 0.2 0.2 0.2 0.2 0.2 A 192 192 192 192 192 H 1091 1091 1091 1091 1091 D.F. 0 1 1 1 1 TABLE 10.2 THE BASIC MODEL: UNCONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF ALTERNATIVE HYPOTHESES. - 208 -PARAMETERS MODELS (f) (g) (h) (i) (j) a 3.14 x 10" 5 1.0 8.172 1.0 1.0 8 • 1.0 1.0 1.0 1.0 0.0 Y -2.0 -2.0 -2.951 -2.346 0.0 L -1835.93 -3918.93 -1815.48 -1822.73 -1957.10 R 2 0.5734 0.0 0.6440 0.6283. 0.0 k 0.2 0.2 0.2 0.2 0.2 A 192 192 192 192 192 H 1091 1091 1091 1091 1091 D.F. 2 3 1 2 3 TABLE 10.2 THE BASIC MODEL: UNCONSTRAINED MAXIMUM LIKELIHOOD ESTIMATES OF ALTERNATIVE HYPOTHESES (CONTINUED). - 209 -Te s t-R e s t r i c t i o n s L i k e l i h o o d R a t i o Test A d d i t i o n a l Maintained A l t e r n a t i v e Hypothesis S t a t i s t i c D.F. U(a) a = 1.0 none -1786.88 -1800.57 13.69 1 U(8) 8 = 0.0 none -1786.88 -1828.31 41.43 1 U(8) 8 = 1.0 none -1786.88 -1815.48 28 .60 1 U(y) Y = 0. 0 none -1786.88 -1891.25 104.37 1 U ( Y ) Y =-2.0 none -1786.88 -1787.35 0.47 1 U (a , 6) a = 1.0 6 = 1.0 none -1786.88 -1822.73 35 . 85 2 U(B, 8 = 1.0 none -1786.88 -1835.93 49.05 2 Y ) Y =-2.0 U (a , a = 1.0 none -1786.88 -3918.93 2132.05 3 8 / Y ) 8 - 1.0 Y =-2.0 C (a) a = 1.0 6 = 1.0 -1815.48 -1822.73 7. 25 1 C(B) 8 - 1.0 Y =-2.0 -1787.35 -1835.93 48.58 1 C ( Y ) Y =-2.0 6 = 1.0 -1815.48 -1835.93 20.45 1 C ( Y ) Y =-2.0 a = 1. 0 8 = 1.0 -1822.73 -3918.93 1096.20 1 Percentage p o i n t s of the ^x d i s t r i b u t i o n w i t h n degrees of freedom. L e v e l 0.1 0.05 0. 025 0. 01 0.005 0.001 0.0001 n=l 1. 35 1.92 2.51 3.32 3.94 5.41 7.57 Value n=2 2. 30 3.00 3. 69 4.61 5. 30 6.91 9.21 n=3 3. 89 4.74 5.57 6.64 7.43 9.23 11.76 TABLE 10.3 THE BASIC MODEL. LIKELIHOOD RATIO TEST RESULTS. - 210 -i t i s consistent with the distance parameter found i n studies based on observed 0-D data. As i n the B.C. data when the a and 8 are r e s t r i c t e d the inverse square hypothesis i s l o s t and Y = - 2 . 0 i s soundly rejected. This confirms the need for a l l parameters and displays the e f f e c t of s p e c i f i c a t i o n error i n one aspect of the model on hypotheses based on other aspects. Among the rejected hypotheses i t i s c l e a r that the uncalibrated models U(a ,8,Y) are nonstarters as i s any model without both population and distance v a r i a b l e s . Improvement on model (g) i s made by freeing the distance parameter from i t s - 2 . 0 constraint to the extent that the conditi o n a l t e s t C(Y) on y = - 2 . 0 i s rejected when the r e s t r i c t i o n s on a and 8 are maintained. Although t h i s model (i) i s an improvement on (g), i t s apparently biased estimate of Y makes t h i s frequently encountered model rather suspect. The t e s t C(a), which computes the gain from (h) over (i) by allowing an un r e s t r i c t e d constraint to be estimated, c l e a r l y r e j e c t s the n u l l hypothesis a= 1 . 0 . Models (b), (c) and (d), and (f) to (j) are thus a l l rejected and t h e i r ranking i n terms of proximity to the unr e s t r i c t e d model i s given by the ranking of t h e i r log l i k e l i h o o d functions. - 211 -10.2.2 Maximal interchange The hypothesis te s t i n g i n section 10.2.1 was carried out using the maximal interchange c r i t e r i o n to screen out inconsequential interchanges thereby achieving major computational saving. Twenty per cent of the complete 0-D matrix was used i n the t e s t s . In order to assess the accuracy of the maximal interchange procedure at t h i s l e v e l of selection models based on 10, 20, 30 and 100 per cent of the 0-D matrix are shown i n Table 10.4. As i n the B.C. data twenty per cent or more of the 0-D matrix gives accurate estimates of the parameters for the complete matrix. Although i t i s tempting to do so i t i s not possible to use l i k e l i h o o d r a t i o tests on these models since they are not nested. Maximal interchange procedures can be conceptualised as a p a r t i t i o n of the constant a. Interchanges for which estimates are computed received the unrestricted a which are however constrained equal for each included interchange. Those interchanges deleted by the maximal interchange c r i t e r i o n are as i f t h e i r a = 0. Now the procedure i s only nested i f the maximal interchange re s u l t s are compared with a model with two a's which are partitioned by the maximal - 212 -PARAMETERS MODELS (a) (b) (c) (d) a o 0.0 0.0 0.0 0.0 a l 749.19 1051.5 1215.6 1578.1 3 0.3768 0.3751 0.3716 0.3651 Y -1.974 -2.055 -2.073 -2.0994 L -1797.03 -1786.88 -1781.69 -1772.98 R2 0.7175 0.7459 0.7592 0.7801 k 0.1 0.2 0.3 1.0 A 192 192 192 192 C 0 0 0 0 Included Interchanges 518 1091 1663 5671 TABLE 10.4 MAXIMAL INTERCHANGE IN THE BASIC MODEL: ESTIMATED PARAMETERS. - 213 -interchange c r i t e r i o n . Then the t e s t would involve the l i k e l i h o o d function for a (included), a (excluded), 8, y and a (included), a = 0 (excluded), 8 , . Y• Thus the maximal interchange comparisons as presently c a r r i e d out i n Table 10.4 does not involve nested hypotheses.' A comparison of d i f f e r e n t l e v e l s of s e l e c t i o n , other than with the 100 per cent l e v e l , n ecessarily involves several l e v e l s of p a r t i t i o n . Since these p a r t i t i o n s have no empirical s i g n i f i c a n c e they are not imposed and the associated l i k e l i h o o d r a t i o tests not computed. Since i n t e r e s t i n the maximal interchange procedure i s focussed on the values of the parameters rather than the goodness of f i t an a l t e r n a t i v e assess-ment presents i t s e l f . The parameters from models (a), (b) and (e) can be substituted into the f u l l 0-D matrix model and the l i k e l i h o o d function c a l c u l a t e d on the f u l l dataset without c a l i b r a t i o n . I d e a l l y , the constant should not be substituted i n t h i s way since i t absorbs much of the e f f e c t of omitted i n t e r -changes. Hence 8 and y should f i r s t be substituted, then e i t h e r a simple one-dimensional search or l i n e a r regression problem c a r r i e d out to estimate a new a, given 8 and Y- This new model would almost c e r t a i n l y not be rejected for parameters from not less than twenty per cent of the 0-D matrix, against the - 214 -f u l l u n r e s t r i c t e d model. I t should be r e i t e r a t e d that there i s major computational saving from t h i s device. It i s of econometric i n t e r e s t since some parameters (the nonlinear ones) may be estimable from a subset of the data, whereas the l i n e a r para-meter i n t h i s case requires the f u l l set. - 215 -10.3 Aspects of the extended model Exploration of the extensions which are possible to the basic model are confined to the addition and par-t i t i o n i n g of an arc constant thus providing an estimate of the contribution of l o c a l t r a f f i c , and to the performance of maximal interchange i n t h i s more general context. Comparisons are made, i n terms of estimated O-D, between the models, on one hand, and between the models and some composite estimates o r i g i n a t i n g p r i m a r i l y from survey sources, on the other. 10.3.1 Estimation of l o c a l t r a f f i c The extended model allows an estimate to be made of the proportion of the arc volumes which might properly be ascribed to l o c a l t r a f f i c , omitted interchanges or noise. This estimate i s obtained by in c l u s i o n of an additive constant which i s applied to a l l arcs, or a set of constants s p e c i f i c to a set of arc classes p a r t i t i o n e d on the basis of the magnitude of t h e i r volumes, t h e i r regional l o c a t i o n , the method used to measure the volumes i f varying accuracies are involved or some other c r i t e r i o n . Models given i n Table 10.5 show the e f f e c t of an arc constant which i s p a r t i t i o n e d on the basis - 2 1 6 -PARAMETERS MODELS (a) (b) (c) a o i 2 3 2 3 . 8 2 6 7 3 . 1 0 . 0 a 0 2 a 0 3 2 4 4 5 . 4 2 6 7 3 . 1 0 . 0 : 2 4 4 3 . 0 2 6 7 3 . 1 0 . 0 a 0 4 j 3 1 9 3 . 9 2 6 7 3 . 1 0 . 0 a 0 5 1 2 3 3 0 . 8 2 6 7 3 . 1 0 . 0 a l : 1 4 5 . 7 6 1 6 2 . 2 2 5 7 1 . . 3 4 8 0 . 4 6 1 5 0 . 4 6 0 1 0 . 3 9 0 8 Y - 2 . 1 4 1 - 2 . 1 5 9 - 1 . 9 9 7 L - 1 6 5 2 . 6 6 - 1 6 5 4 . 2 5 - 1 7 0 6 . 4 6 R 2 0 . 8 5 8 3 0 . 8 5 5 8 0 . 7 4 5 6 k 0 . 1 0 . 1 0 . 1 H 5 1 8 5 1 8 5 1 8 A 1 8 4 1 8 4 1 8 4 C 2 0 2 0 2 0 TABLE 10.5 THE EXTENDED MODEL: REGIONAL PARTITIONS OF THE ARC CONSTANT Maritimes Quebec Ontario West Others KEY: a 01 a 02 a 03 a 04 a 05 - 217 -of the province i n which an arc i s p r i n c i p a l l y l o -cated. The most general model i s given by (a). Models (b) and (c) are obtainable by r e s t r i c t i o n s on the values of the arc constants and are therefore embedded. This permits two l i k e l i h o o d r a t i o tests to be performed: f i r s t l y , for the s i g n i f i c a n c e of the equality constraint on the constants and secondly, for the s i g n i f i c a n c e of the single equality-con-strained arc constant. Test 1 involves the difference of the l i k e l i h o o d functions i n models (a) and (b), with one degree of freedom. Using the values given i n Table 10.3 i t i s apparent that the n u l l hypothesis of the equality constraint can only be rejected at the 10 per cent l e v e l of s i g n i f i c a n c e . This implies that the p r o v i n c e - s p e c i f i c arc constant set i s super-fluous to some extent even though the constants are quite d i f f e r e n t numerically. Test 2 compares models (b) and (c) and shows that the single arc constant i s very s i g n i f i c a n t . The n u l l hypothesis that the constant can be con-strained to zero i s rejected at the highest l e v e l s of s i g n i f i c a n c e against the a l t e r n a t i v e of a non-zero constant. From goodness of f i t considerations the arc constant belongs i n the model. - 218 -10.3.2 Maximal i n t e r c h a n g e i n the extended model One o b j e c t i v e i n e x t e n d i n g the b a s i c model was to p r o v i d e an a r c c o n s t a n t t o absorb the e f f e c t s of o m i t t e d i n t e r c h a n g e s i n the maximal i n t e r c h a n g e procedure. As a r e s u l t i t was expected t h a t the parameters f o r s m a l l s e l e c t i o n s from the 0-D m a t r i x would be l e s s b i a s e d f o r the extended than f o r the b a s i c model. For the B.C. data t h i s was found t o be the case. I t w i l l now be seen t h a t the Canadian data support t h i s p r e l i m i n a r y v e r d i c t . The r e s u l t s f o r s e l e c t i o n l e v e l s o f 10, 20 and 30 per cent are g i v e n i n Table 10.6. Any r e d u c t i o n i n b i a s can be measured by a comparison of these models w i t h the c o r r e s p o n d i n g models i n Table 10.4. Both the a and y, the b i a s i s indeed g r e a t l y reduced, but s l i g h t l y g r e a t e r f o r 8 . Whereas the b i a s i n i s reduced from 466.41 to 104.98, t h a t i n 8 i s i n c r e a s e d from 0.0052 to 0.0114, and t h a t i n y reduced from 0.099 to 0.054. On the whole the b i a s i s l e s s i n the extended model. I f the a r c c o n s t a n t i s i n f a c t a b s o r b i n g the e f f e c t o f the o m i t t e d i n t e r c h a n g e s i t w i l l i n -cr e a s e i n s i z e as the number of o m i t t e d i n t e r c h a n g e s i n c r e a s e s . A l s o , the amount of the i n c r e a s e w i l l - 219 -MODELS (a) (b) (c) (d) (e) a o 2673.1 2201.7 2085.9 2807.3 0.0 a l 162.22 264.54 267.20 237.46 571.34 8 0.4601 0.4478 0.4487 0.4414 0.3908 Y -2.159 -2.203 -2.213 -2.130 -1.997 L -1654.25 -1649.51 -1655.84 -1746.39 -1706.46 R 2 0.8558 0.8752 0.8781 0.8333 0.7456 k 0.1 0.2 0.3 0.1 0.1 c 20 20 20 0 20 A 184 185 186 192 184 H 518 1091 1663 518 518 TABLE 10.6 MAXIMAL INTERCHANGE IN THE EXTENDED MODEL: ESTIMATED PARAMETERS. - 220 -increase monotonically as the maximal interchange s e l e c t i o n c r i t e r i o n deletes larger and larger i n t e r -changes as the selected portion of the O-D matrix diminishes i n s i z e . Both these expected r e s u l t s are observed i n Table 10.6 j u s t as they were for the B.C. data. The models (a) to (c) were computed on a subset of the data defined by deletion of arcs sub-tended by a node within 20 miles of a c i t y of 100,000 population or greater. The objective of t h i s was to attempt to remove the bias due to high volume commuter arcs. Comparison of models (a) and (d) shows the e f f e c t of t h i s r e s t r i c t i o n . The e f f e c t of bias i n the parameters due to maximal interchange i s manifested i n the O-D estimates which r e s u l t . A comparison of the selected O-D e s t i -mates i n Table 10.7 for models (a) to (c) shows the extent of t h i s bias. It i s immediately apparent that there i s very l i t t l e d i fference i n the estimates, e s p e c i a l l y between models (b) and (c). Obviously the estimates, as a function of k, the interchange selec-t i o n parameter, have converged approximately even at the 20 per cent l e v e l . Such a r e s u l t i s s u b s t a n t i a l l y improved over the r e s u l t s obtained for the B.C. data and are very encouraging. Hence the maximal i n t e r -- 221 -CITY-PAIRS MODEL (a) (b) (c) (d) Toronto to: Oshawa 19086 19355 19348 18806 Montreal 480 423 414 479 St.Catherines 4564 4416 4380 4538 London 1881 1787 1766 1893 Ottawa 434 394 386 441 Winnipeg 12 10 10 13 Quebec 95 84 82 99 Montreal to: Ottawa 2549 2396 2368 2529 Quebec 1155 1073 1057 1162 Winnipeg 10 9 8 11 Edmonton to: Calgary 32 6 308 302 342 Vancouver 2 3 20 20 25 Average T r i p 105 112 116 108 Length TABLE 10.7 MAXIMAL INTERCHANGE IN THE EXTENDED MODEL: SAMPLE ESTIMATED 0-D. - 222 -change procedure, i n the extended model at l e a s t , i s established. Average t r i p length increases as the i n t e r -change s e l e c t i o n increases but i n general i s quite constant between these models. This constancy i s less evident than i t was i n the B.C. data where average t r i p length changed by 0.3 miles between 10 and 20 per cent l e v e l s . In the Canadian data the corresponding change i s 7 miles. The B.C. estimates are reasonably close to the Canadian estimates and the longer t r i p length for B.C. i s a function of the geographically more widely spaced c i t i e s found there. Also the B.C. data i s based on summer d a i l y arc volumes whereas the Canadian data contains most annual d a i l y averages. Summer t r a v e l can be ex-pected to have greater t r i p lengths than i n winter. T r i p lengths given i n Table 10.7 are averaged over the included interchanges. This accounts for the dependency of t r i p lengths on interchange s e l e c t i o n l e v e l s . - 223 -10.4. Estimation of a bus O-D matrix 10.4.1 Available sources of data It i s well-known that O-D data on bus t r a v e l i n Canada are very sparse indeed ( P l a t t s , 1976) . What data do e x i s t s u f f e r from problems of being confined to a few bus c a r r i e r s or to a small region. They also s u f f e r from sampling d e f i c i e n c i e s due to i n s u f f i c i e n t sample size and r e s t r i c t e d tem-poral a p p l i c a b i l i t y . Furthermore, even where some O-D data have been c o l l e c t e d these data are not necess a r i l y r e a d i l y a v a i l a b l e to the public f or competitive or c o n f i d e n t i a l i t y reasons. The province of Ontario has compiled a small sample of bus O-D for parts of Ontario. These data are d e f i c i e n t from several points of view. F i r s t l y , the samples taken were very small, c o n s i s t i n g of t i c k e t l i f t s which ranged i n duration from two weeks down to one day. Secondly, an analysis of t i c k e t s produces modal O-D, that i s , from bus s t a t i o n to bus s t a t i o n , and therefore does not neces s a r i l y r e f l e c t the t r a v e l l e r ' s r e s i d e n t i a l l o c a t i o n . T h i r d l y , the small samples are r e s t r i c t i v e temporally to the extent that seasonal bias i s i n e v i t a b l e . Seasonal bias appears not only i n the volume of t r a f f i c but - 224 -also i n the d i s t r i b u t i o n of t r a v e l l e r s i n the 0-D matrix; d i f f e r e n t seasons favour d i f f e r e n t des-t i n a t i o n s . I t would be a formidable problem to factor such a small sample con s i s t e n t l y up to annual t o t a l s . F i n a l l y , the samples are r e s t r i c t i v e s p a t i a l l y so that many longer c i t y - p a i r s are omitted (zero observations i n the sample) and many others are of suspect r e l i a b i l i t y . Indeed, u n t i l bus 0-D surveys are r e g u l a r l y and co n s i s t e n t l y c a r r i e d out the expectation of using observed 0-D except i n r e s t r i c t e d cases i s quite remote. An a l t e r n a t i v e source of data are the route volumes which are compiled by the bus c a r r i e r s them-selves i n order to a s s i s t t h e i r own system planning and to supply summary data required by law for S t a t i s t i c s Canada. These consist of the t o t a l number of passengers which used a given bus route for any portion of the length of the route. I t turns out that these route volumes can be used as con t r o l t o t a l s for the estimated 0-D matrix i n an analogous way to highway arcs i n the car 0-D matrix estimation case. 10.4.2 A comparison of the bus and car estimation problems Any comparison of the two problems, while recognising t h e i r formal s t r u c t u r a l homogeneity i n - 225 -mathematical terms, leads to a recognition of funda-mental differences which are almost s u f f i c i e n t l y great as to question the v a l i d i t y of the analogy. Formally, the volume of t r a v e l l e r s on a given bus route consists of a set of O-D demands assigned by some p r i n c i p l e . Yet t h i s s i m p l i c i t y i s complicated by the existence of transfers between routes and the heterogeneity of service l e v e l s between d i f f e r e n t bus operators. The a d d i t i o n a l complexity e n t a i l e d i n the bus problem has not been addressed as f u l l y as i t might have been. Some h e u r i s t i c methods have been used to obtain some of the e f f e c t of tr a n s f e r s . This was p a r t i a l l y achieved by merging that e f f e c t with the intervening opportunity e f f e c t of larger centres. Much remains to be done i n t h i s area to extend the methods su c c e s s f u l l y applied to the car mode to estimate an O-D matrix for bus t r a v e l . Nevertheless the procedures implemented show i n i t i a l promise and lead to not unreasonable estimates. 10.4.3 D e f i n i t i o n s of terms A route segment i s the portion of a bus service or set of bus services for which a single passenger volume i s given. - 226 -A path i s a sequence o f a r c s which lead s from o r i g i n t o d e s t i n a t i o n . A s t a g i n g l i n k i s d e f i n e d by a s e t of a r c s which are aggregated t o the l e v e l a t which the r o u t e segment data are a v a i l a b l e . The path bundle o f an a r c o r s t a g i n g l i n k i s the number of paths u s i n g t h a t a r c or s t a g i n g l i n k : 10.4.4 The b a s i c model The o b j e c t i v e f u n c t i o n of the model comprises the observed and e s t i m a t e d volumes f o r s t a g i n g l i n k s . The aim i s to minimise some f u n c t i o n o f these so t h a t the e s t i m a t e d volumes s h o u l d r e p l i c a t e as c l o s e l y as p o s s i b l e the observed v a l u e s . The e s t i m a t e d volumes are a f u n c t i o n of an assignment model r e f l e c t i n g the r o u t e s t r u c t u r e of the bus network and of a d i s t r i b u -t i o n model. Whereas the assignment i s an unparamet-r i s e d v e r s i o n o f the a l l - o r - n o t h i n g p r i n c i p l e , the d i s t r i b u t i o n model c o n t a i n s s e v e r a l parameters which the o b j e c t i v e f u n c t i o n i s o p t i m i s e d w i t h r e s p e c t t o . The model may be w r i t t e n as o min g(V , V ) w.r.t. {a,H,y} S o - 227 -where V s = H z a h s & a h s t h a h fch = « p h D h ^ahs ^ a r c a ^ s c o n t a i n e d i - n path h and i n staging l i n k s ,0 otherwise Zahs =( 1 i f 6 i h s 6 j h s = 0 ; i = i + 1 ' 1 + 2 ' ' 0 otherwise Subscript i i s defined as the f i r s t arc for which ^ a n s - 1, and the j are a l l subsequent arcs. The expression 6 . . 6 ... =0 lhs jhs i f and only i f the path bundles of the i t h and j t h arcs are d i s j o i n t . I f 6 .. 6 ., = 1 lhs ]hs then 6 = 6 = 1 lhs jhs and arcs i and j possess the common path h. Since arc i i s assumed to have mapped h into s, the path h i n j i s annihilated by s e t t i n g z , = 0, ahs - 228 -10.4.5 Transfers and intervening opportunities The d e f i n i t i o n of D, i n the basic model was h l e f t unspecified although simple distance i n miles would be a reasonable f i r s t approximation. This d e f i n i t i o n may be extended to account for some of the e f f e c t of transfers and of intervening opportunities both of which occur i n larger c i t i e s . Hence may be written as D, = 6 -h M, -t- T 7 P. . h h ,L kh k where 0 = a headway constant r e f l e c t i n g waiting time, access and egress penalties for terminals P kh = the population of the kth intervening centre along path h x = a parameter for the sum of intervening op-por t u n i t i e s M^ = distance i n miles for c i t y - p a i r h. The objective function i s now minimised with respect to. 9 and x i n addition to a, 8 and y . 10.4.6 Empirical Results Based on the same c i t i e s and nodes as the highway network, a matching network of bus staging l i n k s was constructed. This led to a set of 54 ag-gregate staging l i n k s on which a f i v e parameter model - 229 -was estimated. Results of t h i s estimation under various assumptions and constraints are shown i n Tables 10.8 and 10.9. Equation (a) i s a s p e c i a l case, the basic model, where 8 and x are constrained to be zero. The i n t e r p r e t a t i o n of the a, 6 and y are s i m i l a r to the model i n the case of car. Moreover, i t i s appa-rent that the parameters are quite s i m i l a r i n mag-nitude and sign. Whereas the car arc volumes are d a i l y t o t a l s , those for bus are annual so that some differences may be expected from t h i s source. This i s p a r t l y reduced by the much larger t r a v e l volumes which go by car. In general the basic model seems to have performed well as a f i r s t approximation. It has produced reasonable parameters, a good f i t to the data and a set of o r i g i n - d e s t i n a t i o n estimates, which, i f not necessarily accurate, are of the same orders of magnitude as the estimates from the other equations. Models (b) and (c) show the e f f e c t of increasing x, the intervening population parameter. The goodness of f i t i s s i g n i f i c a n t l y improved and the e f f e c t of the distance parameter i s weakened, as might be expected since x i s also a kind of gener-a l i s e d distance parameter. The population parameter - 230 -MODEL (a) (b) (c) (d) 8 0.0 0.0 0.0 10.0 T. 0.0 1.0 x 10~ 4 5.0 x 10~ 4 1.0 x 10~ 4 a 4068.38 572.49 133.75 1207.13 8 0.3821 0.4575 0.5027 0.4632 Y -1.797 -1.709 -1.573 -1.842 L -664.84 -662.03 -658.42 -662.42 R 2 0.9099 0.9188 0.9290 0.9177 A 54 54 54 54 Estimated O.D. Toronto To: Ottawa 9030 15212 18623 16752 Oshawa 210816 295102 364828 274238 Montreal 9819 18166 22121 19722 Winnipeg 463 958 1774 899 Vancouver 155- 344 631 295 Montreal To: Ottawa 39425 69987 111113 79154 Calgary To: Edmonton 7136 11007 15578 12218 TABLE 10.8 ESTIMATION AND TESTS OF A BUS O.D. MODEL (i) - 231 -PARAMETERS MODEL (e) (f) (g) (h) 6 10.0 20.0 30.0 40.0 T 5.0 x 10" 5 1.0 x 10" 5 1.0 x 10" 5 1.0 x 10~ 5 a 2773.87 21525.6 54176.4 135944. 6 0.4343 0.3924 0.3904 0.3883 Y -1.888 -2.082 -2.217 -2.350 L -663.76 -666.12 -666.93 -667.63 R 2 0.9135 0.9055 0.9027 0. 9001 A 54 54 54 54 Estimated O.D. Toronto To: Ottawa 14085 11114 11478 11755 Oshawa 247598 195387 184849 177112 Montreal 16036 11711 11853 11910 Winnipeg 682 409 364 324 Vancouver 217 113 91 74 Montreal To: Ottawa 64826 51638 54184 56265 Calgary To: Edmonton 10544 8990 9442 9827 TABLE 10.9 ESTIMATION AND TESTS OF A BUS O.D. MODEL ( i i ) - 232 -i s larger to compensate for the ad d i t i o n a l distance s p e c i f i c a t i o n . As i s c l e a r l y shown i n the tables the O-D i s quite s e n s i t i v e to T. In fac t a l l of the selected O-D values increase quite sharply i n (b) and (c) compared with (a). These increases are compensated by some decreases i n O-D interchanges which are short i n length and pass through large c i t i e s , Hamilton to Oshawa for example, which passes through Toronto. The l i k e l i h o o d r a t i o t e s t , c o n d i t i o n a l on 9 being zero, r e j e c t s the constraint T = 0 e a s i l y . This i s shown by the difference i n the log l i k e l i -hoods between models (a) and (c)... The optimal T has not been computed, however, for the present objec-t i v e i s l i m i t e d to demonstrating the appropriateness of including the intervening population term i n the functional form i n addition to a standard distance term. Addition of a non zero distance constant 9 uniformly worsens the r e s u l t s i n terms of goodness of f i t . Although small non zero values of 9 cannot be rejected i t does nevertheless appear to be a superfluous parameter. Its e f f e c t on the O-D values, while noticeable, i s of a s u b s t a n t i a l l y lower mag-- 233 -nitude than T. In summary, these bus r e s u l t s suggest that an O-D matrix can be constructed for bus passengers from staging l i n k volumes. These are only preliminary experiments and further work i s needed to e s t a b l i s h the v a l i d i t y of the technique i n t h i s context. CHAPTER 11 APPLICATION TO INTERCITY TRAVEL DEMAND I METHODS 11.1. Objectives The objective of t h i s paper i s to take the car and bus O-D matrices estimated i n previous papers and to use them to complete the estimation of a multimodal t r a v e l demand function. I t seems that t h i s a p p l i c a t i o n i s the f i r s t complete multimodal t r a v e l demand model to be estimated on Canadian data. A previous study (Canadian Transport Commission, 1970) omitted the car mode thus r e s t r i c t i n g t h e i r analysis to common c a r r i e r s within the Windsor-Quebec C i t y c o r r i d o r . Estimation of t h i s model permits an i n d i r e c t assessment of the v a l i d i t y of the estimated O-D matrices. For i f the use of estimated, instead of observed, O-D gives c o e f f i c i e n t s and e l a s t i c i t i e s which are very s i m i l a r to those c a l c u l a t e d i n other studies, which had observed O-D av a i l a b l e , the estimated O-D i s approximately correct. The demand model i s cr o s s - s e c t i o n a l i n nature, and t h i s f a c t together with a c e r t a i n amount of ar b i t r a r i n e s s i n the s p e c i f i c a t i o n of the model, means that the estimates of e l a s t i c i t i e s are not conclusive. Nonetheless they may be compared with studies employing s i m i l a r methodologies. - 234 -- 235 -11. 2. Methodology 11. 2.1 Cross-sectional models The nature of the extant data and the macro aspect of the objectives determine that an aggregate model be used. Similar considerations lead to the adoption of cross-s e c t i o n a l rather than time-series or pooled methods. As a consequence the p r i n c i p l e methodological question centres around the appropriateness of a model based on cross-s e c t i o n a l inference for the various components of the a n a l y s i s . In general, where a long-run stable r e l a t i o n s h i p e x i s t s amongst the v a r i a b l e s i n a model i t i s v a l i d to r e l y on c r o s s - s e c t i o n a l methods. I t i s , however, very u n r e a l i s t i c to expect a l l r e l a t i o n s h i p s to be free of s t r u c t u r a l change. Nevertheless i t i s argued that some components of t r a v e l demand models are much les s subject to s t r u c t u r a l change than others and that, for these aspects at l e a s t , c r o s s - s e c t i o n a l analysis i s appropriate. The analysis of t r a v e l demand by mode involves three main subproblems: (i) the estimation of t o t a l t r a v e l demand; ( i i ) d i s t r i b u t i o n of t h i s demand between c i t i e s ; and ( i i i ) the assignment of the t o t a l t r a v e l for each c i t y p a i r to a mode (mode s p l i t ) . . Cross-sectional analysis seems to be appropriate for the d i s t r i b u t i o n of t r a v e l problem since t h i s i s l a r g e l y dependent on d i s t r i b u t i o n of population. The d i s t r i b u t i o n of population usually undergoes only slow transformation. A s i m i l a r but less convincing case can be made for a cross-s e c t i o n a l approach to mode s p l i t . Whereas the four p r i n c i p a l modes have been i n competition for several decades i t i s c l e a r that there have been at the same time major technological advances, e s p e c i a l l y i n a i r c r a f t . In addition, c i t i e s have become increasingly orientated to the automobile as urban sprawl has become the dominant growth aspect of c i t i e s . As a r e s u l t , t r a d i t i o n a l l y r a i l bound passengers on long t r i p s have been captured by a i r t r a v e l whereas the automobile has taken short-haul passengers which formerly would have t r a v e l l e d by bus or r a i l . Faced with these trends, inference from c r o s s - s e c t i o n a l analysis seems to be on weak methodological ground. Some of these problems are a l l e v i a t e d by using abstract mode models which submerge i n d i v i d u a l modal i d e n t i t y into a set of modal a t t r i b u t e s : fare, time and departure frequency. The burden of explanation i s thereby s h i f t e d to the t r a v e l l e r s ' evaluation of these three modal a t t r i b u t e s . Cross-sectional methodology i s probably l e a s t suited to estimation of the l e v e l s of t o t a l t r a v e l demand. This i s p a r t l y due to the inherent time-series nature of the problem but also to the f a c t that the a l t e r n a t i v e s to i n t e r c i t y t r a v e l - telecommunications and other forms of correspondence, and l o c a l (within c i t y ) s a t i s f a c t i o n of needs, have been e n t i r e l y omitted from the model. Owing to the rapid evolution of telecommunications and transportation i n general, i t would seem that inference based on a single cross-section of data i s based on weak foundations. The essence of the problem i s that i n the c r o s s - s e c t i o n a l approach, c a l i b r a t i o n i s achieved by comparison of the l e v e l s of service and t r a v e l demand for d i f f e r e n t c i t y -p a i r s . Thus, i f the model i s used fo r simulation and service i s improved between a c i t y - p a i r , the simulated t r a v e l p r o f i l e would resemble that of another c i t y - p a i r which already has that l e v e l of service. C l e a r l y , nowhere i n the model i s there any account taken of s t r u c t u r a l change a f f e c t i n g both the transportation system and the socio-economic environment as a whole. Accordingly, pure cr o s s - s e c t i o n a l models are incomplete for forecasting purposes but are suitable for short-run impact studies. .2.2 Abstract mode models The inadequacy of the t r a d i t i o n a l approach has resulted i n a major modification of concepts. Owing to work by Lancaster (1966) and Quandt and Baumol (1966) among others the t r a v e l l e r i s now viewed as demanding c h a r a c t e r i s t i c s or a t t r i b u t e s such as fare, t r a v e l time and departure frequency. This i s what i s known i n transporta-t i o n demand analysis as the abstract mode model. As the modal a t t r i b u t e s are common to a l l modes, i t follows that - 238 -any number of modes can be represented by performance i n these three categories. Hence, i n d i v i d u a l modal i d e n t i t y can be submerged by mapping observations r e l a t i n g to mode m int o parameters r e l a t i n g to modal c h a r a c t e r i s t i c k. An a d d i t i o n a l d i s t i n g u i s h i n g feature of the abstract mode approach i s the hypothesis that the demand for t r a v e l by several d i f f e r e n t modes i s determined by a single demand function. This allows observations on t r i p s between c i t y pairs r e l a t i n g to d i f f e r e n t modes to be pooled and a singl e regression equation estimated. The advantage of t h i s form i s that the demand for new modes can be estimated using c o e f f i c i e n t s estimated from e x i s t i n g modes, that i s , without redefining the demand function. 11.3. S p e c i f i c a t i o n of a demand function 11.3.1 A two-stage model In a previous paper, two qu a s i - d i r e c t models were b r i e f l y introduced. A generalised model including these and many other models i s now developed. Consider the following t r a v e l demand expression: T = T(A, C , ) • S (C . ) (1 m mk m mk where T m = t r i p s by mode T = t o t a l t r i p s - 239 -= share of t r i p s using mode m A = socio-economic and a c t i v i t y variables C , = k*"*1 a t t r i b u t e of mt*1 mode mk Thus, the demand f o r t r a v e l by mode i s derived from the product of a t o t a l demand term and a mode s p l i t term which c l e a r l y implies that the e l a s t i c i t i e s of the two parts are add i t i v e : 3T m 3C mk C , > mk m 3T 'mk 3C mk <• 3S + m 3C mk 'mk m (2) Hence, the fare e l a s t i c i t y of demand by mode i s composed of the sum of two e l a s t i c i t i e s : the e l a s t i c i t y of t o t a l i n t e r c i t y demand and the e l a s t i c i t y of the market share of mode m with respect to the fare of mode m. 11.3.2 U t i l i t y function Let the set of modal a t t r i b u t e s be designated as C m k (m = 1, 2, ..., M; k = 1, 2, K) and define the (indirect) u t i l i t y function U = U(C , ). The question m mk which immediately a r i s e s i s how should these a t t r i b u t e s be combined to obtain a u t i l i t y function. Let the u t i l i t y derived from a mode be a function of the a t t r i b u t e s of that mode alone and thus ad d i t i v e with respect to mode, but not necessarily with respect to modal a t t r i b u t e s . Hence, U m may be written as - 240 -U(C m k) = exp (a om (X) "mk m^k ^ i ^ r r t l / - ^ r r i \r ^ (3) where - - J . / / « . I t i s c l e a r that with X = 1, the l o g i t form i s obtained whereas the product form a r i s e s i n the l i m i t as X approaches zero. .3.3 Mode s p l i t The mode s p l i t term i s required to express the aggregate p r o b a b i l i t y of t r a v e l l e r s demanding a given mode i n terms of the r e l a t i v e u t i l i t i e s of a l t e r n a t i v e modes. Define a market share expression as -1 Sm = ( V I u. Lm m which becomes, a f t e r s u b s t i t u t i o n of (3) i n (4) (4) S -m exp om la C(X ) mk mk [ exp (a + ya , C ^ Lm * \ om Lkamk mk , (5) E l a s t i c i t y of modal share with respect to one of the att r i b u t e s i s therefore 3S_ m 3C mk 'mk m = a . c \ mk mk ( 1 - V (6) This completes the s p e c i f i c a t i o n of S i n (1) m - 241 -11.3,4 Impedance and t o t a l demand Up to t h i s point, the discussion has covered the following problem: given a demand for t r a v e l between a pa i r of c i t i e s , determine the proportion which would use each a l t e r n a t i v e mode. Attention i s now focussed on the structure of demand independent of s p e c i f i c modes which i s designated i n (1) as T = T (A, C^) (7) (7) i s defined to comprise two separate problems: the aggregate impedance of the transportation system and the le v e l s of c e r t a i n socio-economic and a c t i v i t y variables which act as a measure of the latent demand for t r a v e l . Impedance i s derived from an averaging procedure applied to the modal-specific u t i l i t y function contained i n the mode s p l i t term. Since the average i s to be taken over a l l a v a i l a b l e modes, i t i s defined as the denominator of (5), that i s : I U n = I exp( a o m + l« ) m m k (o) The p a r t i c u l a r s p e c i f i c a t i o n of the u t i l i t i e s or of the impedance implies no r e s t r i c t i o n on the form of the expression for t o t a l i n t e r c i t y t r a v e l demand. Nevertheless, i t i s probable that the untransformed e f f e c t s of the socio-economic and transportation impedance variables are not independent and additive. Very l i t t l e t h e o r e t i c a l - 242 -r e s t r i c t i o n can be put on the form of t h i s expression. Hence, i t i s defined generally as L k m J which i s g e n e r i c a l l y r e l a t e d to the addilog production function (Houthakker, 1960). The e l a s t i c i t y of (9) with respect to one of the modal a t t r i b u t e s i s (9) 3T 3C mk 'mk = a , C , mk mk -X T ° Y I u. •-m m m (10) 11.3.5 Special cases Two s p e c i a l cases of the general model are of p a r t i c u l a r i n t e r e s t : the l o g i t model and the product model The l o g i t model i s obtained by s e t t i n g X = 1 i n (3) and (5) so that the market share equation now becomes exp (a + Ja . C , ) ^ om f mk mk S„, — ^ m I exp ( a Q m + l a ^ C^) m k In the case of the t o t a l demand component, the corresponding case i s obtained from (9) by l e t t i n g X Q approach zero and se t t i n g X^ = X^ = 1: T = exp (6 + I B kA k + Y [ I U j ) ( 1 2 ) k m The market share e l a s t i c i t y corresponding to (6) becomes: - 243 -as m 9C mk 'mk = a , C , (1 - S ) mk mk m (13) m and for t o t a l demand e l a s t i c i t y , corresponding to (10) ac mk 'mk mk mk ' L L mJ m m (14) The product model i s obtained by l e t t i n g X approach zero i n (3) and (5) which r e s u l t s i n the f a m i l i a r m u l t i p l i c a t i v e market share form: a o exp (a ) IT c . Sm = ^ our K mk m . mk amk )' exp (a ) H e , m ^ our K mk (15) The t o t a l demand component i s obtained by l e t t i n g X Q, X , and X2 approach zero thus: 6 k rv „ iY T = exp ( 6 Q ) n A k" [I U J k m (16) Accordingly, the market share e l a s t i c i t y s i m p l i f i e s to as m ac mk 'mk m = amk <1 " S m) (17) and the t o t a l demand e l a s t i c i t y to 3T ac mk 'mk mk ' m ' (18) The general model, as given i n . (5) and (9) may be further s p e c i a l i s e d by the imposition of equality constraints - 244 -FIGURE 11.1: GEOMETRIC INTERPRETATION OF THE ELASTICITIES a) Own e l a s t i c i t y <3 -»• 1 - 245 -on the a c o e f f i c i e n t s . This allows varying degrees of mode abstractness to be introduced. F u l l y abstract models are obtained i f , f o r a given a t t r i b u t e k, c o e f f i c i e n t s are constrained equal across modes. A class of quasi-abstract forms r e s u l t i f the a remain unconstrained. om 11.4. Estimation procedures A l l the models described i n 3 can be estimated by ordinary l e a s t squares and using the l i n e a r s t a t i s t i c a l model. Two rela t e d techniques are needed: seemingly unrelated methods for the case of i d e n t i c a l explanatory var i a b l e s , and simple pooling of the observations for r a t i o s of the modal a t t r i b u t e s . Two separate estimation steps are required, one for the mode s p l i t term and one for the t o t a l demand component. Estimation of the mode s p l i t problem i n (5) proceeds as follows. Use i s made of the s e p a r a b i l i t y of the independence axiom. By t h i s r u l e , the r a t i o of the p r o b a b i l i t i e s of choosing two modes i s dependent only on the s t r i c t u t i l i t i e s of these modes and independent of "i r r e v e l a n t a l t e r n a t i v e s " . Setting one m = b to designate a base mode the following i s obtained: S T / T, T _m = m / b - _m TT / YT T, b L m/ L m b m ' m = exp ( a o m + ) exp ( a Q b + l % k ) ( 1 9 ) T b k k - 246 -Af t e r s e t t i n g a Q b = 0 and taking logs (19) s i m p l i f i e s to log f j a ] = a o m + I a m k + £a b k ( 2 0 ) |T bJ k k For a given X (20) i s a l i n e a r equation system c o n s i s t i n g of M-l equations with m ^ b. Whereas the a . mk are s p e c i f i c to each equation, the a b k are constrained equal. Such a structure may be handled simply by Zellner's (1962) seemingly unrelated regression technique. S p e c i a l i s -ing the discussion to the four mode and two a t t r i b u t e case, the following scheme may be constructed: o o c 1 2 0 0 c 2 1 o o c 2 2 0 o c 3 1 0 0 c. a o i a02 a03 a l l V T 4 1 0 0 c l l T 2 / T 4 0 1 0 0 T./T, 0 0 1 0 41 a42 C41 C42 C41 C42 C41 C42 Imposition of equality constraints on the c o e f f i c i e n t s further s i m p l i f i e s the estimation. In t h i s case, the following scheme r e s u l t s : a o i a02 a03 a l l a 2 2 V T 4 1 0 0 C l l / C 4 1 C12 / C42 T 2 / T 4 0 1 0 C 2 1 / C 4 1 C22 / C42 T 3 / T 4 0 0 1 C 3 1 / C 4 1 C32 / C42 In both cases, the number of observations i s N*(M-1) where N i s the number of c i t y pairs and M the number of modes. - 24 7 -Est imat ion of the t o t a l demand proceeds by s u b s t i t u t i n g the computed a and a , i n (8) which i s subs t i tu ted i n om mk (9). This component i s then estimated by standard l i n e a r methods for given A^, X , X^. Up to t h i s p o i n t , the X have been regarded as constant . However, i f the A are allowed to vary , and i n fact estimated i t i s c l e a r that the func t iona l form of the equations i s a l so being est imated. Hence, the least-squares problems are now treated as subproblems w i t h i n a g loba l scheme to estimate the A from the data. This ob jec t ive i s achieved by maximum l i k e l i h o o d methods. Define the l i k e l i h o o d funct ion for the m u l t i -var i a te normal model as _n £ z (2IIa2) exp ( - l i y ^ - £ a k x ^ } ) 2 2a* 2 ) J (A; y, x) where y . = T . /T, . : m i mi' b i x mki = C m k i / C b k i J (A ;y ;x ) = the determinant of the Jacobian of the inverse transformations induced by the A . The A may be optimised by nonl inear techniques so as to maximise the concentrated log l i k e l i h o o d . CHAPTER 12 APPLICATION TO INTERCITY TRAVEL DEMAND II EMPIRICAL 12. 1. Nature of the data 12.1.1 Problems of aggregation and data a v a i l a b i l i t y The amount of disaggregation employed for any s t a t i s t i c a l analysis i s fundamental to the i n t e r p r e t a t i o n of the numerical r e s u l t s . In studies of a s p a t i a l nature such as transportation, there are severe constraints on the amount of aggregation which can be employed. This stems e s s e n t i a l l y from the point-to-point c h a r a c t e r i s t i c of transportation. If a set of c i t y - p a i r s are aggregated s p a t i a l l y to obtain an area-to-area set of observations, much of the useful information i s l o s t i n the ravages of aggregation bias. As a r e s u l t , s p a t i a l s i m p l i f i c a t i o n of the data i s achieved by omission rather than aggregation of d e t a i l . Selection c i t y - p a i r s for analysis i s made prim a r i l y on the s i z e of the populations of the c i t i e s and on t h e i r proximity. Whilst such a f i l t e r i s necessary to reduce the data requirements, i t i n e v i t a b l y r e s u l t s i n a r e s t r i c t i o n i n the v a l i d i t y of the conclusions to lar g e r c i t i e s and shorter lengths of t r i p . It i s not always a simple matter to decide on the geographical extent of a c i t y . In t h i s context, a c i t y may appear as a c l u s t e r of smaller centres which are s u f f i c i e n t l y close to defy any attempt to separate them for the purposes - 248 -- 249 -of i n t e r c i t y t r a v e l yet s u f f i c i e n t l y d i s t i n c t that the analyst f e e l s uncomfortable t r e a t i n g them as one c i t y . The agglomeration Guelph-Kitchener-Waterloo i s an example. This problem i s severe for a i r t r a v e l . One a i r p o r t may serve a number of centres which, i n t h i s context, act as one c i t y . Furthermore, the problem i s exacerbated by the-fact that the catchment area of an a i r p o r t varies with the d e s t i n a t i o n . For example, Toronto a i r p o r t i s , i n general, the p r i n c i p a l point of departure by a i r for Hamilton residents. This i s not the case, however, for t r i p s between Hamilton and Montreal since there i s a d i r e c t f l i g h t from Hamilton a i r p o r t . Recognition of the fa c t that c i t i e s are often not c l e a r l y separable, d i s t i n c t e n t i t i e s i s fundamental to the d i s t i n c t i o n between i n t e r c i t y and i n t r a - c i t y t r a v e l . To tre a t a c l o s e l y - k n i t c l u s t e r of urban centres as separate c i t i e s would r e s u l t i n a great deal of d a i l y commuting being classed as i n t e r c i t y t r a v e l . In t h i s way a serious bias might unwittingly be introduced i n the data. A natural solution to t h i s problem i s to aggregate these centres s p a t i a l l y u n t i l d a i l y commuting i s no longer a s i g n i f i c a n t element i n the t r a v e l . This i s e s s e n t i a l l y the approach which has been taken i n t h i s study. The problem has not been solved, of course, but a reasonable compromise seems to have been made. - 250 -Paucity of data necessitates a considerable degree of aggregation. Unfortunately, i f aggregation i s c a r r i e d too f a r i t may completely obscure the underlying causal factors and patterns of v a r i a t i o n . Although some types of transportation data are a v a i l a b l e i n abundance, the key variables for analysing t r a v e l demand are more e l u s i v e . This r e f e r s to the number of t r i p s between each o r i g i n -d e stination p a i r by mode and purpose of t r a v e l . No usable national data e x i s t on the purpose of t r a v e l , not even a simple business - non-business breakdown. A l l that i s a v a i l a b l e are t o t a l aggregated t r i p s by o r i g i n and destination for a i r and r a i l only. As a r e s u l t , there i s no way to disaggregate by purpose of t r a v e l and t h i s must be regarded as a major data and conceptual d e f i c i e n c y . A much more serious data de f i c i e n c y e x i s t s . Although approximately 90 per cent of i n t e r c i t y t r a v e l i s accomplished by car or bus, there e x i s t s no o r i g i n -destination data for these modes. This was the objective of a previous paper where estimates were made of an 107 x 107 matrix of 0-D t r i p s for car and bus modes. The c i t y pairs required for the demand model were subsequently selected from these matrices. Abundant data e x i s t for describing modes i n terms of t h e i r fares, t r a v e l times and frequencies of departure. In t h i s case, however, there i s such a d i v e r s i t y of service c h a r a c t e r i s t i c s that the r e s u l t i n g plethora of data i s quite - 2 5 1 -unmanageable. Consequently, service c h a r a c t e r i s t i c s or modal a t t r i b u t e s must be averaged or aggregated. Although t h i s standard device makes the problem t r a c t a b l e , there i s always the danger of averaging away d e t a i l e d modal a t t r i b u t e s which may be s i g n i f i c a n t to a s p e c i f i c sector of the t r a v e l market, or important to government p o l i c y . 12.1.2 Transportation modal a t t r i b u t e s The purpose of t h i s section i s to specify the modal a t t r i b u t e s defined i n the general formulation of the model, that i s , the C . . F i r s t of a l l , the d i s t i n c t i o n can mk be made between modal c h a r a c t e r i s t i c s which are a function of the mode s p e c i f i c a l l y such as the l e v e l of comfort or convenience or the type of service c h a r a c t e r i s t i c of a p a r t i c u l a r mode, and those which, i n addition vary by c i t y -p a i r , such as fare, t r a v e l time and frequency. Whereas the at t r i b u t e s c h a r a c t e r i s t i c of modes rather than c i t y - p a i r s can be handled e f f e c t i v e l y by mode-specific constants, the remaining a t t r i b u t e s are more appropriately treated as ordinary v a r i a b l e s . Accordingly, for each of car, a i r , r a i l and bus modes the average fare, t r a v e l l i n g time and departure frequency are computed for each of 230 c i t y - p a i r s . Car costs are based on the average d i r e c t operating costs, the average l e v e l of car occupancy and the overnight accommodation charges for long t r i p s . Ten hours - 252 -d r i v i n g per day was assumed. The following formula was used: C. . = 1 1 D P a M. . + b -M. . I • d J (1) where p = number of passengers per car = 2.0 a = $/mile = $0,095 b = $/night/car = $2 0 d = average distance t r a v e l l e d per day = 500 I^j = distance i n miles from c i t y i to c i t y j M. C i i = cost per t r a v e l l e r by car i n $ from -1 c i t y i to c i t y j T r a v e l l i n g times are calcula t e d from an average highway speed of 5 miles per hour below the posted speed l i m i t plus overnight stays based on 10 hours d r i v i n g per day. A i r fares are defined as the economy fare and were obtained from the c a r r i e r t a r i f f schedules. Time and departure frequencies were also obtained from these schedules. Departure frequency i s defined as a weighted average of summer and winter scheduled frequencies. The weighting i s the number of weeks for which the summer and winter services were operative. The t y p i c a l winter week was taken as the t h i r d week of January 1972, whereas the t y p i c a l summer week was the t h i r d week of July 1972. Travel - 253 -time i s defined as the average of t r a v e l times between the o r i g i n and the destination for a l l service departures from the o r i g i n . Travel time i s averaged between summer and winter. Total t r a v e l time includes three types of elapsed time: (a) access time incurred i n t r a v e l l i n g to the terminal of o r i g i n ; (b) t r a v e l l i n g time between the terminal of o r i g i n and destination terminal; (c) egress time incurred i n t r a v e l l i n g to the f i n a l destination from the destination terminal. R a i l fares are defined as the 'white' coach fare, modified to account for the increased use of sleeping accommodation with increasing length of t r i p . Their source was the c a r r i e r t a r i f f schedules. Travel time and departure frequencies were also obtained from c a r r i e r schedules. The d e f i n i t i o n s of t r a v e l time and frequencies are the same as for a i r . Bus fares, t r a v e l times and departure frequencies were obtained from the c a r r i e r schedules. Fare i s defined as the standard bus fare. Travel time i s a frequency weighted average of express and l o c a l bus services and includes stopping and tra n s f e r times. - 254 -12.1.3 Socio-economic v a r i a b l e s Whereas the t r a n s p o r t a t i o n v a r i a b l e s l a r g e l y measure the c o s t s of one type o r another i n v o l v e d i n t r a v e l l i n g , the socio-economic v a r i a b l e s are r e l a t e d t o the b e n e f i t s which accrue from t r a v e l . Which v a r i a b l e s from the socio-economic environment have the p r i n c i p a l e f f e c t on t r a v e l p r o p e n s i t i e s i s not a l t o g e t h e r c l e a r . The o n l y obvious f a c t o r i s the d i s t r i b u t i o n o f p o p u l a t i o n which g i v e s the number o f people a v a i l a b l e f o r t r a v e l i n each c i t y . Yet i t i s c l e a r t h a t the d i s t r i b u t i o n o f t r a v e l demand amongst the v a r i o u s segments o f the p o p u l a t i o n of a g i v e n c i t y i s f a r from uniform. For t h i s r eason, many i n t e r c i t y t r a v e l demand models have added an income v a r i a b l e i n the b e l i e f t h a t h i g h e r incomes w i l l l e a d t o more t r a v e l . Some s o c i o -economic v a r i a b l e s which have been employed a r e : (a) per c a p i t a income pr o d u c t s (Quandt & Baumol 1966; Young, 1969) (b) p o p u l a t i o n weighted a r i t h m e t i c mean of per c a p i t a income (Quandt & Young, 1969) (c) number o f households exceeding a g i v e n income l e v e l (McLynn & Watkins, 1967; McLynn & Woronka, 1969; Cheslow & Ku, 1969) (d) p r o p o r t i o n o f households exceeding a g i v e n income l e v e l (CTC, 1970) (e) index of c u l t u r a l a t t r a c t i o n (Monsod, 1969) (f) index of l i n g u i s t i c p a i r i n g (CTC, 1970) These are proxy v a r i a b l e s , rather than the actual causal variables which may be not only unknown as yet, but unobservable i n p r i n c i p l e . For the purpose of c a l i b r a t i n g the demand model, the relevant variables were i d e n t i f i e d as population, language and income. Population was taken from the 1971 census and was expressed as a product, t h i s being an i n d i c a t i o n of the t o t a l number of possible i n t e r a c t i o n s . Language c h a r a c t e r i s t i c s , also taken from the 1971 census are defined as the percentage i n each c i t y with English as mother tongue. Using these percentages, an index was constructed by taking 100 minus the absolute value of the difference i n the percentage English between a p a i r of c i t i e s . That i s : L.. = 100 - L i " L j (2) where L^ = percentage English i n c i t y i This l i n g u i s t i c p a i r i n g index i s at i t s maximum when two c i t i e s are s i m i l a r i n t h e i r percentages of English and at i t s minimum when they are completely d i f f e r e n t . Mother tongue, rather than "language spoken at home", was used as t h i s accounts f o r ethnic and c u l t u r a l o r i g i n s which may - 256 -i n f l u e n c e t r a v e l p a t t e r n s over and above p u r e l y l i n g u i s t i c c o n s i d e r a t i o n s . Income i s d e f i n e d f o r the purposes of the demand model experiments as weighted average per c a p i t a income. T h i s i s w r i t t e n as: Y i j = ( P i * i + P j Y j ) / ( P i + Pj) (3) where = average per c a p i t a income i n c i t y i P^ = p o p u l a t i o n i n c i t y i - 257 -12-1.4 Form of the model estimated Substitution of the variables described i n 1.2 and 1.3 into the model defined by (15) and (16) i n the previous section and imposing quasi-abstractness gave the following model for market share: a.. a„ exp (a ) C . H. D.J  c _ our 13m 13m 13m b i j m " l 4 ; a, a„ a., I exp (a ) C.. H. D. t r om 13m 13 m 13 m m and for t o t a l demand: (5) a, ou a. 3-, &2 3o T. .- exp (6 ) (P.P.) L. Y. y exp (a J C 1 H.2 D.3 , ^ c om ljm ljm ijm| IB 4 Recall that the complete model i s thus the product of (4) and (5): T. . = T. . S. . (6) ijm 13 ljm 12. 2. Results, experiments and comparisons 12.2.1 Introduction The demand model was estimated f o r : (a) a l l modes (car, a i r , r a i l , bus) (b) surface modes (car, r a i l , bus) (c) common c a r r i e r modes ( a i r , r a i l , bus) For each of these s i t u a t i o n s various experiments were performed such as t e s t i n g the e f f e c t - 258 -of length of t r i p and the si z e of t r i p volumes on the c o e f f i c i e n t s . The e f f e c t of dropping variables of marginal s i g n i f i c a n c e was also tested. Af t e r des-c r i b i n g the nature of the experiment an attempt i s made to state a p r i o r i what should happen to the c o e f f i c i e n t s . These a p r i o r i expectations are then compared with what a c t u a l l y happened to the coef-f i c i e n t s i n the empirical t e s t . Some basic con-s t r a i n t s were maintained throughout the ana l y s i s . C i t y - p a i r s with t r i p volumes of le s s than 30 or greater than 150,000 per annum were not considered. The f i r s t constraint removes c i t y - p a i r s of minor importance which lack e f f e c t i v e service. The l a t t e r constraint eliminates c i t i e s which are s u f f i c i e n t l y close that d a i l y commuting to work i s a major factor such as Toronto-Oshawa together with c e r t a i n major a t y p i c a l l i n k s such as Montreal-Toronto, Ottawa-Toronto, Ottawa-Montreal. From p r i o r reasoning i t seems that more sp e c i a l i z e d model i s needed to analyze these c i t y -p a i r s . I t should be noted that the sheer magnitude on t r a v e l on these few l i n k s might unduly influence the model's c o e f f i c i e n t s . However, the model can be applied to these l i n k s when i t has been c a l i b r a t e d on the rest of the system. Under these circumstances one would expect that t r a v e l volume on the most major i n t e r - c i t y l i n k s would be consistently under-predicted. This i s - 259 -because the model assumes that t r a v e l demand occurs i n response to the time and fare associated with a p a r t i -cular t r i p , whereas on the l i n k s between Montreal-Toronto-Ottawa, government and business t r a v e l (at l i t t l e d i r e c t cost to the user) dominate. 12.2.2 Experiments with A l l Modes'^ " Equation 1(a), using car as the base mode, included fare and t r a v e l time but not departure f r e -quency as vari a b l e s . A p r i o r i i t i s expected that the fare c o e f f i c i e n t would be greater i n absolute value than the t r a v e l time c o e f f i c i e n t as three of the four modes are surface modes which dominate the r e s u l t s . The argument i s that surface mode t r a v e l l e r s are cost-savers whereas the a i r t r a v e l l e r s are time-savers. Modal constants that measure those s p e c i f i c modal att r i b u t e s which are not covered by fare and time are included i n the model. The modal constant of the base mode, car, i s defined as unity. Hence i t might be expected that the common c a r r i e r modal constants would be smaller, and possibly ordered, i n magnitude with decreasing s i m i l a r i t y or r e l a t i v e attractiveness to car mode. The population c o e f f i c i e n t i s generally 1. See Tables 12.1 and 12.2. - 260 -close to the square root and that of the impedance term approximates the cube root. The r e s u l t s of estimating the c o e f f i c i e n t s of equation 1(a) support the a p r i o r i expectations. Fare and time, whilst having the manditory correct sign (negative), also e x h i b i t the ordering anticipated together with acceptable magnitudes. Modal constants for the common c a r r i e r s are as expected smaller than for car, bearing i n mind these constants are exponents of the base of natural logarithms. The population and impedance c o e f f i c i e n t s are as expected. Consequently i t may be concluded that the model i s a v a l i d one. This conclusion i s reinforced by the fact that a l l the c o e f f i c i e n t s are c l e a r l y s i g n i f i c a n t s t a t i s t i c a l l y . Equations 1(b) and 1(d) p a r t i t i o n the ob-servations by length of t r i p . Taking a i r fare as the c r i t e r i o n 1(b) includes those t r i p s under $50 whereas 1(d) includes those over $50. Equation 1(c) contains the 32 c i t y - p a i r s over $50 but not over $100. Since i t i s known that a i r t r a v e l i s the dominant mode f o r the long t r i p s and that t r a v e l time i s the key a t t r i b u t e of a i r t r a v e l the following hypothesis i s advanced. The r a t i o of the fare c o e f f i c i e n t i n terms of the time c o e f f i c i e n t w i l l be lower for equations 1(c) and 1(d) than for 1(b). That i s , the longer the t r i p , the more s i g n i f i c a n t w i l l t r a v e l time become r e l a t i v e to t r a v e l - 261 -cost. Inspection of the estimated c o e f f i c i e n t s for these equations shows that a p r i o r i expectations are more than s a t i s f i e d . Not only are the r a t i o s of fare/time c o e f f i c i e n t s lower for 1(c) and 1(d) than for 1(b) but i n addition they are higher for 1(b) than for 1(a). A l l t h i s confirms the hypothesis that short-distance t r a v e l i s cost-orientated and long-distance time-orientated. Furthermore, these c o e f f i c i e n t s are a l l s t a t i s t i c a l l y s i g n i f i c a n t . An unexpected discovery i s that the population and impedance c o e f f i c i e n t s both vary systematically with length of t r i p . The population c o e f f i c i e n t increases whereas that on impedance decreases. The e f f e c t s of increasing the lower bound on t r i p volumes and the addition of departure frequency are demonstrated by a comparison of equations 1(e) and 1(f) estimated for t r i p s costing under $100 by a i r . It was anticipated that increasing the lower bound on t r i p volumes would have the e f f e c t of de l e t i n g more long-distance than short-distance t r i p s with the con-sequence that the fare/time c o e f f i c i e n t r a t i o would increase. This i n fact turned out to be the case. As for departure frequency i t was expected that i t s c o e f f i c i e n t would be p o s i t i v e but marginally s i g n i f i c a n t - 262 -i f at a l l . Previous studies have not found departure frequency such a straightforward variable as fare and time and have resorted to various transformations to achieve s i g n i f i c a n c e . Equation 1(f) shows that depar-ture frequency i s not s i g n i f i c a n t . The same equation also gave an ordering to the mode-specific constants which permits a simple i n t e r p r e t a t i o n . With the constant for the car mode defined as 1.0, those for the common c a r r i e r s were -roughly 0.7 for a i r , 0.4 for r a i l and 0.2 for bus.* As expected t h i s ordering seems to represent a comfort and convenience of t r a v e l r e l a t i o n s h i p . The e f f e c t of dropping the mode-specific constant i s shown by a comparison of 1(f) and 1(g). The l a t t e r has one constant which d i f f e r e n t i a t e s car mode from the common c a r r i e r modes. Although the remaining c o e f f i c i e n t s i n th i s equation are r e l a t i v e l y unaffected, the departure frequency c o e f f i c i e n t i s unacceptable i n terms of si g n i f i c a n c e and sign. Certainly t h i s v a r i a b l e can be dropped without loss of explanatory power. A comparison of equations 1 ( f ) , 1(h), l ( i ) and l ( j ) shows the e f f e c t of ad d i t i o n a l socio-economic varia b l e s . Whereas 1(f) uses only the population product, 1(h) adds a l i n g u i s t i c p a i r i n g v a r i a b l e , l ( j ) adds average annual per capi t a income and l ( i ) includes them a l l . No strong a p r i o r i expectations - 263 -were held about these c o e f f i c i e n t s except that they should probably be p o s i t i v e and not p a r t i c u l a r l y s i g n i f i c a n t . Equation 1(h) demonstrated that the l i n g u i s t i c p a i r i n g index i s quite s i g n i f i c a n t whereas l ( j ) showed that income as measured i s marginally s i g n i f i c a n t . In combination, however, while the income c o e f f i c i e n t becomes small negative and i n s i g n i f i c a n t , the l i n g u i s t i c variable increases i n magnitude as i f to compensate. I t suggests that income i s a less valu-able variable i n c r o s s - s e c t i o n a l analysis than l i n g u i s -t i c or c u l t u r a l d i f f e r e n c e s . Although language i s l i k e l y to be correlated with income i t should be noted that the l i n g u i s t i c p a i r i n g index measures l i n g u i s t i c d i f -ferences. This variable therefore seems to be something conceptually d i f f e r e n t from income l e v e l s . Equation l(k) shows the s e n s i t i v i t y of the parameters to lowering the upper bound on t r i p volumes from 150,000 to 100,000. Whereas the e f f e c t on the fare and time parameters i s extremely small, that on the other parameters i s not i n s u b s t a n t i a l . 12.2.3 Experiments with Surface Modes"'" It i s possible to analyse surface transpor-t a t i o n as a d i s t i n c t market for t r a v e l . If surface 1. See Tables 12.3 and 12.4 - 264 -t r a v e l l e r s are cost minimisers whereas a i r t r a v e l l e r s are t r a v e l time minimisers i t follows that dropping a i r mode w i l l decrease s i g n i f i c a n t l y the time c o e f f i -ient r e l a t i o n to the fare c o e f f i c i e n t . This hypothesis i s supported by a comparison of the equations 1(a) through to l(k) with equations 2(a) through to 2 ( i ) . In every case i s the expected trend to be found. Depar-ture frequency, included because i t i s a p o l i c y v a r i -able, i s consistently i n s i g n i f i c a n t ; t r a v e l time i s much less important than fare. The mode-specific constants i n equations 2(a) through to 2(i) p e r s i s t e n t l y maintain the ordering found i n equations 1(f) through to l ( k ) . This adds weight to an i n t e r p r e t a t i o n of these constants as a convenience of t r a v e l factor. They are mostly s i g -n i f i c a n t , and, when the departure frequency variable i s dropped, as i n equation 2 ( i ) , highly s i g n i f i c a n t . The surface mode experimental equations are pa r t i t i o n e d into those with t r i p lengths under 10 0 hours by the slowest mode, equations 2(a) - 2(d), and those with,in addition, t r i p costs under $100 by the most expensive surface mode, equations 2(e) - 2 ( i ) . Within the set 2(a) - 2(d) the lower bound on t r i p volumes i s successively raised (30, 60, 100, 200) i n order to perform a s e n s i t i v i t y analysis. The r e s u l t s of t h i s show small or systematic changes i n the c o e f f i c i e n t s . - 265 -The fare c o e f f i c i e n t increases, whereas the time coef-f i c i e n t decreases, i n absolute value. The population c o e f f i c i e n t decreases where the impedance c o e f f i c i e n t increases. There i s l i t t l e o v e r a l l change i n the c o e f f i c i e n t s and so i t may be concluded that the model i s not unstable with respect to the lower bound on t r i p volumes. A s i m i l a r s e n s i t i v i t y analysis i s performed for the set of equations with t r i p s bounded by 100 hours and $100 that i s , equations 2(e) - 2 ( i ) . Equa-tions 2(e) to 2(g) show the e f f e c t s of a l t e r n a t e l y r a i s i n g the lower bound and decreasing the upper bound on t r i p volumes. 2(e) and 2(f) are bounded below by 30 and 2(g) - 2(i) by 200 whereas 2(f) - 2(i) are bounded above by 150,000. As before, the fare coef-f i c i e n t systematically increased whereas the time coef-f i c i e n t decreased. In addition, both modal constants showed a systematic increase i n magnitude. A concomi-tant decline i s noted i n the population and impedance c o e f f i c i e n t s . Equation 2(h) i s a version of 2(g) without mode-specific constants. 12.2.4 Experiments with Common Carriers"'' In the experiments on surface modes i t was found that the fare/time c o e f f i c i e n t r a t i o increased when the a i r mode was omitted. Maintaining the hypo-- 266 -thesis that surface t r a v e l l e r s are cost-orientated and a i r t r a v e l l e r s time orientated implies that dropping the p r i n c i p a l surface mode, car t r a v e l , would r e s u l t i n a decrease i n the fare/time c o e f f i c i e n t r a t i o . With t h i s expectation, and using a i r as the base mode, equations 3(a) - 3(e) were estimated. Expectations were f u l f i l l e d i n every case to the extent that the time c o e f f i c i e n t became dominant i n equation 3(e) and approximately equal i n 3(a) - 3(d) the l a t t e r set based on the same constraints as 2(g). At the same time, the c o e f f i c i e n t on t r a v e l time i s of higher s t a t i s t i c a l s i g n i f i c a n c e than that on fare which reverses the pre-vious pattern. The most s t r i k i n g expression of t h i s r e s u l t i s found i n 3(e) which i s based on a l l c i t y -p airs with t r i p volumes i n excess of 30 and t r i p lengths under 100 hours duration. 1. See Table 12.5 - 267 -Equation # 1(a) 1(b) 1(c) 1(d) 1(e) Parameter constant 6.58 9.98 4.44 1.92 8.05 (8.93) (12.4) (1.41) (1.15) (10.6) population 0.541 0.402 0.682 0.771 0.480 (17.4) (14.9) (10.3) (14.0) (16.7) language xncome impedance 0.386 0.410 0. 344 0. 298 0.417 (21 .6) (14 .3) (3 .65) (7 . 62) (18 .8) a i r -2. 85 -2. 42 -5 .34 -5 .78 -2. 84 constant ( -6. 42) ( -4. 25) ( -3 .84) ( -4 .87) ( -5. 95) r a i l -1. 88 -2. 05 -1 . 64 -1 .64 -1. 96 constant ( -13 .6) ( -13 .6) ( -3 .24) ( -3 .81) ( -13 .7) bus -2. 15 -2. 11 -2 .01 -2 .02 -2. 14 constant ( -16 .1) ( -14 • 3) ( -4 .51) ( -5 .22) ( -15 .6) fare -2. 33 -2. 27 -2 .54 -2 .86 -2. 13 ( -13 .0) ( -7. 48) ( -3 .71) ( -5 .70) ( -9 . 86) time -1. 91 -1. 66 -2 .99 -3 .35 -1. 80 ( -8. 60) ( -6. 14) ( -4 .48) ( -5 .92) ( -7. 54) departure frequency R 2 t r i p s by mode .5454 .7738 .6595 .6597 .6633 observations 508 292 128 216 420 TABLE 12.1 LOG LINEAR ESTIMATES OF COEFFICIENTS AND T- STATISTICS 1. Experiments on a l l modes (i) - 268 -Equation # 1(f) 1(g) 1(h) l ( i ) l ( j ) 1(k) Parameter constant 7.66 9.18 4.12 7.28 -1.56 4.38 (8.76) (11.16) '(4.07) (1.72) (-0.36) (4.13) population 0.448 0.445 0.492 0.50 0.439 0.484 (12.3) (13.63) (15.2) (14.7) (12.3) (13.6) language 0.520 0.578 0.499 (5.20) (4.62) (4.83) income -0.446 1.16 (-0.768) (2.19) impedance 0.348 0.404 0.339 0.340 0.343 0.328 (13.3) (15.26) (15.1) (15.1) (13.4) (13.2) a i r -0.377 -2.25 -0.377 -0.377 -0.377 -1.035 constant (-0.426) (-3.74)(-0.426)(-0.426)(-0.426) (-1.16) r a i l -0.979 -2.25 -0.979 -0.979 -0.979 -1.47 constant (-1.38) (-3.74)(-1.38) (-1.38) (-1.38) (-2.00) bus -1. 52 -2. 25 -1. 52 constant (-2. 40) ( -3. 74) (-2. 40) fare -2. 72 -2. 53 -2. 72 (-13 .0) ( -15 .2) (-13 .0) time -1. 31 -1. 65 -1. 31 (-5. 50) ( -20 .3) (-5. 50) departure 0.128 — 0.044 0.128 frequency (1. 27) ( -0. 49) (1. 27) t r i p s by mode .7578 .7621 .7553 -1.52 -1.52 -1.94 (-2.40) (-2.40) (-2.97) -2.72 -2.72 -2.70 (-13.0) (-13.0) (-12.96) -1.31 -1.31 -1.38 (-5.50) (-5.50) (-5.50) 0.128 0.128 0.056 (1.27) (1.27) (0.545) .7572 .7525 .8138 296 296 276 observations 296 296 296 TABLE 12.2. LOG LINEAR ESTIMATES OF COEFFICIENTS AND T-STATISTICS 1. Experiments on a l l modes ( i i ) - 269 -Equation # 2(a) 2(b) 2(c) 2(d) Parameter constant 10.59 10.98 11.22 10.99 (23.7) (23.8) (23.1) (21. population 0.428 0.428 0.422 0.415 (33.4) (31.5) (29.8) (26. language 0.104 0.095 0.10 0.154 (1.85) (1.64) (1.57) (2.25) income impedance 0.651 0.688 0.6 39 0.585 (66.1) (61.5) (56.0) (45. a i r constant r a i l -2.06 -2.18 -2.38 -2.2 constant (-2.83) (-2.99) (-3.35) (-2.9 bus -2.24 -2.40 -2.68 -2.6 constant (-3.45) (-3.66) (-4.21) (-3.8 fare -2.12 -2.15 -2.23 -2.5 (-13.4) (-13.3) (-13.8) (-13. time -0.63 -0.445 -0.533 -0.426 (-1.91) (-1.30) (-1.60) (-1.24) departure 0.039 0.023 -0.03 -0.016 frequency (0.37) (0.215) (-0.29) (-0.15) R2 t r i p s by mode .9289 .9267 .9288 .9211 observations 489 462 417 351 TABLE 12.3 LOG LINEAR ESTIMATES OF COEFFICIENTS AND T-STATISTICS 2. Experiments with surface modes (i) - 270 -Equation # 2(e) 2 (f) 2(g) 2(h) 2(i) Parameter constant 11.28 (29.3) 11.59 (28.9) 12.19 (26.7) 14.63 (31.89) 11.84 (28.06) population 0.401 (38.1) 0.395 (34.3) 0. 382 (27.0) 0.383 (28.34) 0.382 (27.03) language 0.065 (1.32) 0.060 (1.22) 0.112 (2.06) 0. 048 (0.926) 0.113 (2.087) income impedance 0.750 (75.1) 0.763 (59.1) 0. 613 (42.1) 0.712 (44.2) 0. 621 (42.15) a i r constant r a i l constant -1.95 (-2.73) -2.23 (-2.89) ( -2.60 -3.45) -3.95 (-6.34) -2.14 (-17.8) bus constant -2. 05 (-3.20) -2.39 (-3.42) ( -3. 01 -4.40) -3.95 (-6.34) -2.59 (-21.0) fare ( -1.81 -10.24) -1.83 (-9.37) ( -2.45 -11.8) -2.50 (-11.8) -2.45 (-11.8) time -0.636 (-1.96) -0.499 (-1.51) ( -0.29 -0.93) -0.442 (-1.4) -0.253 (-0.819 departure frequency 0.065 (0.53) 0.021 (0.187) ( 0. 068 -0.63) -0.244 (-2.61) 2 I t r i p s by mode .9490 .9417 .9494 .9368 .9501 observations 441 402 294 294 294 TABLE 12.4 LOG LINEAR ESTIMATES OF COEFFICIENTS AND T-STATISTICS 2. Experiments with surface modes ( i i ) - 271 -Equation # 3(a) 3(b) 3(c) 3(d) 3(e) Parameter constant 0.409 (0.321) 1. 22 (0.970) 1.17 (0.936) 0.409 (0.322) -1. 37 (-1.33) population 0.578 (11.2) 0.585 (11.7) 0.582 (11.7) 0.578 (11.2) 0.676 (16.3) language income impedance 0.281 (6.90) 0. 319 (7.55) 0. 314 (7.56) 0.281 (6.93) 0.388 (12.6) a i r constant r a i l constant 1.32 (2.02) 1.22 , (1.88) 1.11 (1.76) 1.31 (2.05) 2.26 (3.87) bus constant 1.01 (1.51) 1. 22 (1.88) 1.11 (1.76) 0.991 (1.56) 1.99 (3.51) fare -1.85 (-4.80) -1.86 (-4.81) -1.89 (-4.93) -1.85 (-4.85) -1.36 (-3.87) time -1.91 (-8.08) -1.94 (-8.23) -1.90 (-8.33) -1.91 (-8.40) -2.15 (-11.2) departure frequency 0.0012 (0.010) -0.076 (-0.712) 2 R t r i p s by mode .6323 .6147 .6150 .6344 .5997 observations 252 252 255 255 417 TABLE 12.5 LOG LINEAR ESTIMATES OF COEFFICIENTS AND T- STATISTICS 3. Experiments with common c a r r i e r s - 272 -12.2.5 A comparison of e l a s t i c i t i e s with other models and data The purpose of t h i s section i s to compare the c o e f f i c i e n t s and e l a s t i c i t i e s of the model c a l i -brated using estimated car and bus 0-D with those of si m i l a r models which obtained t h e i r 0-D i n other ways, pr i m a r i l y through surveys. I t i s not e s s e n t i a l that the e l a s t i c i t i e s be j u s t i f i e d or otherwise shown to be accurate. In order for the estimated 0-D to be accep-ted as reasonable i t i s s u f f i c i e n t to demonstrate the s i m i l a r i t y b e tween.elasticities. The comparison i s confined to fare and time e l a s t i c i t i e s i n the modal choice mechanism. These are shown i n Table 6. Results from two-stage separable models are given i n sections a) and c ) . The e l a s t i c i t i e s shown are maxima since the actual e l a s t i c i t y depends on the market share of each mode. Thus for equation 1(a) the fare e l a s t i c i t y for a zero share i s -2.33, for a 50 per cent share i s -1.16 and for a 100% share i s zero. Consequently these quantities are pure mode s p l i t e l a s t i c i t i e s . Results i n section b), since the models used are single stage equations, i n e v i t -ably contain a mixture of mode s p l i t and t o t a l t r a v e l e f f e c t s . The t o t a l t r a v e l e f f e c t s are never s u f f i c i e n t to compensate for the decrease due to the market share and therefore the e l a s t i c i t i e s i n a) and c) are prac-- 273 -t i c a l upper bounds for t h e i r combined mode s p l i t and t o t a l t r a v e l e l a s t i c i t i e s . This i s shown i n Table 7 for equation 1(f) and i s due to a > a ( l - S) + ayS, a < 1, 0 < S ^ 1 from equations (17) and (18). The s i m i l a r i t y between the e l a s t i c i t i e s obtained from estimated 0-D and those from other studies i s evident. - 274 -a) Two-stage separable models' Monsod (1969) Crow and S a v i t t (1974) CTC (1970) 6 •Fare Time •2.99 -2.15 •2.34 -1.94 •2.59 -1.35 Departure Frequency 0.65 0. 36 b) Other models 2 Quandt and Baumol Kraft and Kraft (1974) Jung and F u j i i (1976) 3 Quandt and Young (1969)^ Quandt and Young (1969) b c) Two-stage separable model with O-D estimation! •2.17 •2.89 •2.73 •2.26 •2.79 •0.53 •1.84 •0.55 •1.95 0.80 0.22 (i) A l l modes (car, r a i l , bus) ai r , Equation ( i i ) ( i i i ) K a ) -2. 33 -1.91 -K b ) -2.27 -1.66 -K c ) -2.54 -2.99 -K d ) -2.86 -3. 35 -K e ) -2.13 -1.80 K f ) -2.72 -1.31 0.13 Surface modes (car, r a i l , bus) 2(a) -2.12 -0.63 0.04 2(b) -2.15 -0.45 0.02 2 (c) -2. 23 -0.53 -0. 03 2(d) -2.50 -0.43 -0. 02 2(e) -1.81 -0. 64 0.07 2(f) -1.83 -0.50 0.02 Common c a r r i e r s ( a i r , r a i l , bus) 3(a) -1.85 -1.91 0.001 3(b) -1.86 -1.94 -0.08 3(c) -1.89 -1.90 -3(d) -1.85 -1.91 -3(e) -1.36 -2.15 -TABLE 12.6 COMPARISON OF ELASTICITIES WITH MODELS USING DATA OBTAINED BY OTHER METHODS - 275 -Notes: 1. For two-stage separable models these are upper bounds on the maximum e l a s t i c i t i e s obtainable from the model with each arc having i t s own e l a s t i c i t y . 2. As reported i n Gupta, Monsod and Young (19 67), a one-stage model, Northeast Corridor data. 3. A i r f a r e e l a s t i c i t i e s for O-D pairs around Memphis, Atlanta and New Orleans. 4. Northeast Corridor data. 5. C a l i f o r n i a data. 6. R a i l own e l a s t i c i t i e s for Montreal - Toronto. - 276 -a) Fare e l a s t i c i t i e s T r i p Length (miles) 77-766 766-1455 1455-2144 2144-2833 2833-3526 b) Time e l a s t i c i t i e s T r i p Length (miles) 77-766 766-1455 1455-2144 2144-2833 2833-3526 Car A i r 0.94 -2.19 0.96 -2.25 1.0 6 -2.34 1.72 -1.56 2.32 -1.33 Car A i r 0.45 -0.37 0.46 -0.42 0.50 -0.54 0.48 -0.36 0.55 -0.37 R a i l Bus -2.16 -2.08 -2.40 -2.40 -2.48 -2.57 -2.48 -2.60 -2.93 -2.66 R a i l Bus -0.92 -0.97 -1.03 -1.08 -1.02 -1.14 -1.13 -1.14 -1.06 -1.20 TABLE 12.7 COMBINED MODE SPLIT AND TOTAL TRAVEL ELASTICITIES CHAPTER 13 APPLICATION TO INTERCITY TRAVEL DEMAND I I I : EXTENSIONS AND TESTS 13.1. I n t r o d u c t i o n I n v e s t i g a t i o n s have shown t h a t the e l a s t i c i t i e s which are o b t a i n e d from t r a v e l demand models depend c r i t i c a l l y on the f u n c t i o n a l form used i n the models. T h i s suggests t h a t any model, the form of which has been s e l e c t e d on a r b i t r a r y grounds, i s indeed suspect and w i l l probably g i v e i n a c c u r a t e estimates o f e l a s t i c i t i e s . I t i s w e l l known t h a t , i n the absence of c l e a r t h e o r e t i -c a l underpinning, the m a j o r i t y of t r a v e l demand models n e c e s s a r i l y f a l l i n t o t h i s category. The o b j e c t i v e s of t h i s paper are t h r e e f o l d . F i r s t l y , i t i s shown t h a t s e v e r a l e x i s t i n g models are s p e c i a l cases o f a more g e n e r a l form which can be d e r i v e d u s i n g a f a m i l y of monotonic t r a n s f o r m a t i o n s . T h i s more ge n e r a l context allows the e s t i m a t i o n of the exact f u n c t i o n a l form together w i t h i t s a s s o c i a t e d e l a s t i c i -t i e s . Secondly, c e r t a i n e x p l i c i t forms are o b t a i n e d by e s t i m a t i o n i n two i n t e r r e l a t e d c o n t e x t s : a market share equation f o r modal c h o i c e and an e q u a t i o n f o r the . t o t a l demand f o r t r a v e l . F i n a l l y , i t i s demonstrated em-p i r i c a l l y from an e x t e n s i v e and d e t a i l e d d a t a s e t t h a t the e l a s t i c i t i e s o b t a i n e d from these models can.be h e a v i l y de-pendent on the f u n c t i o n a l form which has been s p e c i f i e d . I n c o r r e c t forms l e a d i n v a r i a b l y t o i n c o r r e c t e l a s t i c i t i e s - 277 -- 278 -and thus to erroneous simulated responses to p o l i c y v a r i a b l e s . Numerous models for the estimation of passenger t r a v e l demand have been proposed during the l a s t twenty years. Their p r i n c i p a l objective has been to estimate the demand which could be expected on p o t e n t i a l new or improved transportation configurations. For t h i s purpose a model or set of models with fi x e d functional forms i s f i t t e d to h i s t o r i c a l data to obtain estimates of parameters. These parameters are then fixed and the chosen model used to simulate demand responses by systematically a l t e r i n g modal a t t r i b u t e s derived from proposed system configurations. Simulations of t h i s nature involve e i t h e r i n t e r p o l a t i o n s or extrapolations from the h i s t o r i c a l data. C l e a r l y , use of an i n c o r r e c t functional form i n these contexts constitutes a s p e c i f i -cation error of unknown proportions. The class of admissible functions for t r a v e l demand estimation i s i n t r a c t a b l y wide. One way to . r e s t r i c t t h i s class i s to derive forms t h e o r e t i c a l l y from more fundamental propositions. Although such methods have proved f r u i t f u l i n the physical sciences t h e i r a p p l i c a t i o n has been noticeably less productive i n the socio-economic f i e l d s . An a t t r a c t i v e a l t e r n a t i v e i s to express the estimation problem i n as general a - 279 -form as possible and to l e t the data themselves place r e s t r i c t i o n s on t h i s form. Although e x i s t i n g models are varied the majority are i n fac t v a r i a t i o n s on e i t h e r a l i n e a r form, i n which the variables have independent additive e f f e c t s , or a product form, i n which complete i n t e r -action i s presumed. Linear or product, these forms are i n fact just d i s c r e t e points on a continuum allowing varying amounts of independence and i n t e r a c t i o n . There i s no theory to provide the optimal combination, i t can only be estimated. One way to estimate t h i s i s to use a class of transformations known as the power family. These monotonic transformations have been studied successively by Anscombe, Tukey, Box and Tidwell, Box and Cox, and by Zarembka. .1.1 C r i t e r i a f or the comparison of models Within the generalised functional con-text adopted here the fundamental c r i t e r i o n for the comparison of models i s how well they are able to explain the data. Hence the natural t e s t i s to compare the values of the maximised l i k e l i h o o d functions by means of the l i k e l i h o o d r a t i o t e s t . Since the maximisation of the l i k e l i h o o d function i s e s s e n t i a l l y the minimisation of residuals a - 280 -l o g i c a l extension i s to examine the d i s t r i b u t i o n of these residuals i n terms of i t s skewness and kurtosis. This c r i t e r i o n i s appropriate as a preliminary t e s t for s p e c i f i c a t i o n error. Should the error d i s t r i b u t i o n at the point of maximum l i k e l i h o o d be s i g n i f i c a n t l y d i f f e r e n t from normal m i s s p e c i f i c a t i o n with respect to form, variables or d i s t r i b u t i o n a l model may be indicated. A second general area for model com-parison l i e s i n the s i g n i f i c a n c e and magnitude of the estimated parameters. This leads to an inspection of t - s t a t i s t i c s f or s i g n i f i c a n c e and of e l a s t i c i t i e s . The v a r i a t i o n i n the computed e l a s t i c i t i e s from models of d i f f e r e n t functional forms i s , from an applied point of view, probably the most c r u c i a l c r i t e r i o n for d i s t i n g u i s h i n g the behaviour of one model from another. Except i n the purely m u l t i p l i c a t i v e model the e l a s t i c i t i e s are not immediately apparent but can be derived a n a l y t i c a l l y i n a s t r a i g h t forward manner. - 281 -13.2. Theoretical framework 13.2.1 General form of the mode s p l i t equation Let the modal generalised cost function be defined as the following: U m ( C W \ e x e ( a o m + I amk ( Cmk^^) U k ) ) ™ , ( X k } , xk where ( C ^ + y *) = ( C ^ y k) - 1 Then the market share demand equation -1 S = U T Yu ] (2) m mL L mJ v ' m becomes, by su b s t i t u t i o n of (1) into (2): U k ) (3) S = exp (a + ya , (C , + y,1 ) \ m * om f mk mk Hk' ) k '__ Texp(a + Ya . (C , •+• y,1) ) u * ova. f mk mk Kk m k The market share model given by (3) can be converted to a quasi-abstract mode model by imposing the equality constraint amk = ank ( m = 1'2"-'^> n = 1/2,..,M) (4) for each k(k = 1,2,..,K) Application of (4) to (3) re s u l t s in - 282 -Sm = e x P ( a o m + | ak ( Cmk + > I e x p(a o m + K ( C m k + y k ) , m k which may be further r e s t r i c t e d to a pure abstract mode by the ad d i t i o n a l equality constraint on the mode s p e c i f i c constants i n (5): °om = aon { m = i^ z-'/M; n = 1,2,..,M) (6) The own e l a s t i c i t y derived from (3) i s found to be an expression containing not only the parameters but also the market share, S^, and the values of the cost v a r i a b l e s : A. * - l % ' ffm_ (Snkl = a m k ( C m k + y>) k C^-U-SJ (7) 3C , Is mk m ' 13.2.2 Special cases and t h e i r e l a s t i c i t i e s At t h i s point i t i s of i n t e r e s t to hig h l i g h t some of the s p e c i a l cases of t h i s gen-e r a l i s e d model by s e t t i n g the transformation para-meters A and y i n (5) to s p e c i f i e d values. Setting y k = 0 for a l l k,. the more r e s t r i c t e d Box-Cox form i s obtained: ( X k } Sm = e x ? ( a o m + K Cmk ) ( 8 ) k v I r i k T ~ £ e x P ( a o m + K Cmk } m k - 283 -I f , i n a d d i t i o n , the A^+O f o r a l l k the m u l t i -p l i c a t i v e o r p r o d u c t form i s d e r i v e d : a k S = exp(a ) n C £ m * v om' k mk ( 9 ) aT" Texp(a ) n C , L * om' , mk m k F i n a l l y , i f , instead of l e t t i n g X^+0, they are set to unity the l o g i t model i s obtained: Sm = e xP ( aom + K ^ n k 1 k (10) Jexp (a _ + Ta, C , ) L r om f k mk m k The own market share e l a s t i c i t y cor-responding to (8) i s seen to be % = a k C I <1-sm) <n> and that corresponding to the m u l t i p l i c a t i v e model (9) i s simply n s = a k ( l - S m ) (12) and the same e l a s t i c i t y for the l o g i t model given by (10) i s ^s = a k Cmk <1-Sm> • d3) - 284 -13.2.3 General form of the t o t a l t r a v e l demand equation The estimation of the t o t a l demand for t r a v e l between a given c i t y - p a i r involves a set of socio-economic a c t i v i t y variables A and the general impedance or cost estimated i n the market share equation. This i s given by (14) -1 T = [B 0 + ZB k(A k+- U k ) U k ) + Y ( U + u k + 1 ) k r l U ^ y 2 ) (14) where U = £u. m m U = exp (a +• la . (C , + LU1 ) ) k -1 and (XQ,\XQ) indicates the inverse of the Box-Tukey transformation applied to the l e f t hand side of (14). 13.2.4 Special cases and e l a s t i c i t i e s As i n the market share equation several s p e c i a l cases may be obtained by applying r e s t r i c -t i o n s . Irrespective of the functional form adopted in the market share equation any of the following are simply derived. Setting u k = 0 leads to the Box-Cox case: T = r i X k ] 3 0 + | 8 kA k + YU k -1 (15) - 285 -If , i n addition, the A k tend to zero for a l l : k a product form i s obtained, which includes the gravity model as a further s p e c i a l case: 3k Y T " exp(80) £ A k u ( 1 6 ) Setting a l l A k = 1 leads to a l i n e a r form for the t o t a l t r a v e l equation: T = 8Q +• l3 kA k +• Y U (17) 3c whereas maintaining AQ-»-0 leads to a l o g i t trans-formation: T = exp(8Q + l3 kA k + YU) (18) k where the exponential function i s c l e a r l y the inverse of the Box-Cox transformation for Ag->-0. As i n the market share equation, the own e l a s t i c i t y with respect to t r a v e l costs or att r i b u t e s contains not only parameters but varies with market share and values of the cost v a r i a b l e s : n T = 3T 9C , mk C > mk (19) A 1 -1 A 2 - l 1-A2 = "mk^ mk^ k^^  c,nkYV^yJ + 1) k t l ( T + y 2 ) V 1 -1 A2 A2 A2 where T = [l+A Q (8Q + ~S k (\ + y£) k + Y ( U + y 2 r l ) °-y 2 - 286 -and U and U m are as defined for equation (14). Allowing a general transformation for the market share equation, the t o t a l t r a v e l own el a s -t i c i t i e s f or s p e c i a l cases (15) to (18) can be ex-pressed as follows. For the Box-Cox case (19) si m p l i f i e s to The corresponding e l a s t i c i t y for the product form (16) i s obtained by, i n addition, l e t t i n g A£-*•() for a l l k thus reducing (20) to where T n T = amk ( Cmk + wk> - l 2 (20) n T = amk ( Cmk+V 'mk Y S. m (21) and for the l i n e a r model, the same e l a s t i c i t y i s n T = amk ( Cmk + uk> mk -1 (22) where T = (8Q + 1) -H l8 kA k + yU k whereas for the l o g i t model given by (18) n T = amk(Cmk+ V mk Y U m (23) - 287 -13.3. Approach to estimation 13.3.1 Rearrangement for estimation The market share equation can be e s t i -mated as a l i n e a r regression problem for given X and u i f i t i s estimated not as i t stands but i n r a t i o form. Thus r a t i o s of the market shares by mode and of the modal a t t r i b u t e s lead to the f o l -lowing form l o g ( T m / T b ) = ct f-Ia k X.1 X^-< Cmk +^> K " < C b k + ^ (23) + £ Using the c o e f f i c i e n t s and variables from (23) a new variable i s constructed and sub-s t i t u t e d i n the t o t a l t r a v e l demand equation (14) U = £ e x P a o " i + K ( C m k + 4 ) m ^ k (24) Equation (14) can now be estimated as (25) ( X0> ( Xk> (XLi> < T +^» = eo +pk< Ak-rUk> k + y ( u ^ + 1 ) k + 1 + £ which again i s a l i n e a r regression problem for given X and y . The X and u are, i n each case, estimated by embedding the l i n e a r problem i n a more general l i k e l i h o o d function which i s maximised by nonlinear optimisation methods or more simply by scanning. - 288 -In the case of the market share equation the log concentrated l i k e l i h o o d function for nor-mally d i s t r i b u t e d errors i s L 1 (A) y1) = - i n • log("e?/n) + log J(Xj) (26) where £ i = l o 9 < V V - aom + K k ( C m k + ^ " k - ( Cbk + ^ ) X k and J(Aj) = n y i J-For the t o t a l t r a v e l equation the appropriate l i k e l i h o o d function i s L 2 ( A j u j A 2 y l = -§ n log(]>e?/n) .+ log J ( A 2 , y 2 ) (27) i where (X 2) (A 2) ( A 2 , ) c ± = (T +yJ) - 3 0 t l B k ( A k ^ y 2 ) k + T(U + y k + 1 ) n A 2 - l and J ( A 2 , y 2 ) = n (y ± + y 2) u i 13.3.2 Models selected R e s t r i c t i n g the set of possible models to those of the quasi-abstract c l a s s , the f o l -lowing subset of models are selected for estimation and scrutiny for the market share equation: (a) S = U / Tu ^ ' m m/ L m m - 289 -(Ai) ( A 2 ) ( A 3 ) where IT = exp(a +a C + a H + a-D ) m c om 1 m 2 m 3 m and AQ = 0, A ^ = A ^ = A ^ are constraints. ( b ) Sm = V ^Um m . u i ) (Al) where U m = e x p f a ^ + a ^ C ^ u J ) + ^ ( H ^ l ^ ) (M) + a 3 ( D m + u ^ ) ) and Aj = 0, = \\- \\, =.0, 1 ^ = = y3 are constraints, (c) As (b) but removing the constraints A^ =A2 = X 3 The following models are estimated for the t o t a l t r a v e l equation: -1 (A 2) r (XV ( x 2 ) (A23) 1 (a) T = |_3n + 3,P + 60L + 6 0 U J 0 K l H2 M3 e x P ( a o m + a l ( C m + *V m where U= £ p ( Q m + ]_ (C m  y ^ (Ah (A*) (A 1) + a 2 ( H m + . y 2 ) + a 3 ( D m + y 3 ) ) with { a o m' a]_' a2 ' a 3 ; A^,A 2,A 3, y^,y 2,y 3> set at t h e i r optimal values from the market share equation, 2 2 2 and subject to A^ = A 2 = A 3 > 2 2 2 (b) As (a) but removing the constraint A^ =A2 - 290 -13.4. Results for the mode s p l i t equation The mode s p l i t or market share equation i s es-timated by maximising the l i k e l i h o o d function given by (26) for models (a), (b) and (c). The impact of the tra n s f o r -mations defined by X and y i s demonstrated i n the contexts of the s t a t i s t i c a l s i g n i f i c a n c e of the va r i a b l e s , the d i s -t r i b u t i o n of errors i n terms of t h e i r skewness and kur t o s i s , and on the modal a t t r i b u t e own e l a s t i c i t i e s of market share. Likelihood r a t i o tests are used to tes t the si g n i f i c a n c e .of X and y. 13.4.1 Estimation and tests of the functional form Model (b) i s estimated for a g r i d of X's and y's; the corresponding l i k e l i h o o d function (L) i s p l o t t e d i n Figure 13.1. C l e a r l y there i s considerable v a r i a t i o n due to d i f f e r e n t values of X, reaching a well-defined maximum for each value of y. At y= 0 X = -0.2 i n d i c a t i n g strong i n t e r a c t i o n e f f e c t s max 3 between the variabl e s . Whereas the l i k e l i h o o d function i s symmetrical at y = 0, increasing values of y lead to increasing asymmetry. That the e f f e c -tiveness of y i s a function of X i s c l e a r l y shown. At A = 1, nonzero values of y r e s u l t i n a simple t r a n s l a t i o n of a l l variables by the same value, a property which leaves the re s u l t s unaltered. The - 291 -FIGURE 13.1 LIKELIHOOD SURFACE FOR THE MODE SHARE EQUATION, CONDITIONAL ON U-- 292 -same r e s u l t i s shown on a portion of the l i k e l i h o o d surface near the optimum, i n Figure 13.2. In order to t e s t the hypothesis about X and y the l i k e l i h o o d r a t i o t e s t i s used. I f ft i s the parameter space under a maintained hypothesis and ui the r e s t r i c t e d space generated by a n u l l hypothesis the t e s t s t a t i s t i c i s well known to be 2[L(ft) - L(co)] (8) where L(ft) i s maximum log l i k e l i h o o d under the main-tained hypothesis and L(w) i s maximum log l i k e l i h o o d under the r e s t r i c t e d hypothesis. The t e s t s t a t i s t i c (8) has a x 2 d i s t r i b u t i o n with q degrees of freedom one for each r e s t r i c t i o n i n OJ but not i n ft. Within this framework two types of tests are ca r r i e d out. Conditional tests assume a value for one of the parameters and then tests the other. Uncondi-t i o n a l tests make no assumption about the untested parameter which i s unres t r i c t e d . Five s p e c i f i c hypo-theses are tested. F i r s t l y , given u, X i s tested for being s i g n i f i c a n t l y d i f f e r e n t from a given value, X. This t e s t i s simply C(X) = L U , u =U) " L U " X,y = y) > i x * ( U O) with 1 degree of freedom and s i g n i f i c a n c e l e v e l a. A second te s t reverses the roles of X and y i n (9). - 293 -- 294 -The v a l i d i t y of these two tests may be strengthened by allowing the parameter not involved i n the t e s t to take on i t s maximum l i k e l i h o o d value. Hence C(u) becomes U(y) where U(y) = L (A,y) - L (X ,y = y) > I x j d ) (10) Likewise C(X) becomes U(X) . F i n a l l y , both X and y may be tested simultaneously by combining U(X) and U(ii) to obtain .,U(X,y) where U(X,y) = L(X,y) - L(X = X,y = y) > ^X*(2) (11) with two degrees of freedom since two new r e s t r i c -tions are imposed by the n u l l hypothesis. Further d e t a i l s on tests and r e s u l t s are given i n Tables 1 3 . 1 - 1 3 . 4 . In Table 13.2 i t i s shown that, conditional upon y= 0, X i s equal to -0.2 at the optimum but also that X l i e s between -0.3 and -0.1 with a p r o b a b i l i t y of 95%. That X i s zero i s re-jected at the 0.5% l e v e l and X equal to one i s unam-biguously rejected. Similar conclusions are obtained for y = 10 and 50. Consequently the model with func-t i o n a l form given by X = - 0 . 2 , y = 0 or by X = - 0 . 3 , y = 10 i s c l e a r l y superior to ei t h e r the m u l t i p l i c a -t i v e form (X = y ^ 0) or the l o g i t form (X= 1, y= 0). - 295 -Test Hypothesis(H Q) Alternative(H 1) Test S t a t i s t i c C r i t i c a l Region C(X) X = X given y = y X * X given y = y L(X,y=y)-L(X=X,y=y) C ( X ) > | x 2 (1) C(y) y = y given X = X y * y given X = X L(y,X=X)-L(y=y,X=X) c ( y ) > i x a ( D U(X) X = X X * X L(X , C)-L(X=X,y) U(X)>JxJ(D U(y) y = y y * y L (£,y)-L (X\y=y) u(y ) > i x „ ( D U(X,y) X = X,y = y X £ X,y ^  y L(X,y)-L(X=X,y=y) U(X,y)> §X*(2) 1 TABLE 13.1 LIKELIHOOD RATIO TESTS FOR FUNCTIONAL FORM - 296 -C U ) y = 0 y = 10 y = 50 X L(X,y) Stat. L (X,y) Stat. L (X,y) Stat. -1. 0 484. 10 48. 86 495. 03 44. 78 515. 95 18 .16 -0. 9 491. 18 41. 78 503. 36 36. 45 518. 62 15 .49 -0. 8 498. 86 34. 10 511. 98 27. 83 521. 58' 12 .53 -0. 7 506. 86 26. 10 520. 39 19. 42 524. 83 9 .28 -0. 6 514. 77 18. 19 527. 92 11. 89 528. 21 5 .90 -0. 5 521. 97 10. 99 533. 95 5. 86 531. 30 2 .81 -0. 4 527. 78 5. 18 537. 99 1. 82 533. 48 0 .63 -0. 3 531. 57 1. 39 539. 81 0 534. 11 0 -0. 2 532. 96 0 539. 27 0. 54 532. 73 1 .38 -0. 1 531. 96 1. 00 536. 39 3. 42 529. 25 4 .86 0 528. 71 . 4. 25 531. 38 8. 43 523. 94 10 .17 0. 1 523. 51 9. 45 524. 71 15. 10 517. 32 16 .79 0. 2 516. 82 16. 14 516. 96 22. 85 509. 95 24 .16 0. 3 509. 20 23. 76 508. 67 31. 14 502. 28 31 .83 0. 4 501. 14 31. 82 500. 24 39. 57 494. 66 39 .45 0. 5 492. 97 39. 99 491. 95 47. 86 487. 29 46 .82 0. 6 484. 95 48 . 01 483. 98 55. 83 480. 28 53 .83 0. 7 477. 21 55. 75 476. 41 63. 40 473. 69 60 .42 0. 8 469. 84 63. 12 469. 27 70. 54 467. 51 66 .60 0. 9 462. 87 70. 09 462. 58 77. 23 461. 72 72 .39 1. 0 456. 32 76. 64 456. 32 83. 49 456. 32 77 .79 TABLE 13.2 CONDITIONAL TEST FOR X. Percentage Points of | x 2 d i s t r i b u t i o n with 1 degree of freedom Level . 0. 1 0. 05 0.025 0. 01 0.005 0.001 0.0001 Value 1. 35 1. 92 2.51 3. 32 3.94 5.41 7.57 - 2 9 7 -C(y) X : : - l . 0 X -- 0 . 2 X - o .X = 1 . 0 y L(X,u) Stat. L(X,y) Stat. L(X,y) Stat. L(X,y) Stat. - 1 . 0 4 8 2 . 4 1 3 3 . 5 4 5 3 0 . 1 6 9 . 2 9 5 2 6 . 4 8 5 . 4 3 4 5 6 . 3 2 0 0 4 8 4 . 1 0 3 1 . 8 5 5 3 2 . 9 6 6 . 4 9 5 2 8 . 7 1 3 . 2 0 4 5 6 . 3 2 0 1 . 0 4 8 5 . 3 8 3 0 . 5 7 5 3 5 . 1 1 4 . 3 4 5 3 0 . 1 2 1 . 7 9 4 5 6 . 3 2 0 5 . 0 4 8 9 . 6 2 2 6 . 3 3 5 3 9 . 0 3 0 . 4 2 5 3 1 . 9 1 0 4 5 6 . 3 2 0 8 . 0 4 9 2 . 8 1 2 3 . 1 4 5 3 9 . 4 5 0 5 3 1 . 7 2 0 . 1 9 4 5 6 . 3 2 0 1 0 . 0 4 9 5 . 0 3 2 0 . 9 2 5 3 9 . 2 7 0 . 1 8 5 3 1 . 3 8 0 . 5 3 4 5 6 . 3 2 0 1 2 .0 4 9 7 . 3 2 1 8 . 6 3 5 3 8 . 9 4 0 . 5 1 5 3 0 . 9 5 0 . 9 6 4 5 6 . 3 2 0 1 5 . 0 5 0 0 . 7 8 1 5 . 1 7 5 3 8 . 3 3 1 . 1 2 5 3 0 . 2 7 1 . 6 4 4 5 6 . 3 2 0 2 0 . 0 5 0 6 . 1 7 9 . 7 8 5 3 7 . 2 7 2 . 1 8 5 2 9 . 1 4 2 . 7 7 4 5 6 . 3 2 0 5 0 . 0 5 1 5 . 9 5 0 5 3 2 . 7 3 6 . 7 2 5 2 3 . 9 4 7 . 9 7 4 5 6 . 3 2 0 TABLE 1 3 . 3 CONDITIONAL TEST FOR u. - 298 -X y L (X,y) Stat. -1.0 0 484.10 55.71 -0.5 0 521.97 17.84 -0.4 0 527.78 12.03 -0.3 0 531.57 8.24 -0.2 0 532.96 6.85 -0.1 0 531.96 7.85 0 0 52 8.71 11.10 0.5 0 492.97 46.84 1.0 0 456.32 83.49 -1.0 10 495.03 44.78 -0.5 10 533.95 5.86 -0.4 10 537.99 1.82 -0.3 10 539.81 0 -0.2 10 539.27 0.54 -0.1 10 536.39 3.42 0 10 531.38 8.43 0.5 10 491.95 47.86 1.0 10 456.32 83.49 TABLE 13.4 . UNCONDITIONAL TEST FOR X AND y . Percentage Points of 3j X 2 D i s t r i b u t i o n with 2 degrees of freedom Level 0. 1 0. 05 0.025 0. 01 0. 005 0.001 0.0001 Value 2. 30 3. 00 3. 69 4. 61 5.30 6.91 9.21 - 299 -Conditional tests for y, given X are given i n Table 13.3. For X - 0, a value of 5.0 for u i s s i g -n i f i c a n t l y d i f f e r e n t from y = 0. For X - -0.2 the n u l l hypothesis of u - 0 i s rejected at the 0.1% l e v e l . The larger negative that X becomes the more strongly i s the hypothesis of zero y rejected. How-ever for X - 1 no d i s t i n c t i o n can be made between al t e r n a t i v e values of y. The simultaneous unconditional t e s t given i n Table 13.4 provides a more v a l i d t e s t of the func-t i o n a l form. Whereas the condi t i o n a l t e s t for X accepted X = -0.2 given y = 0 t h i s i s rejected by the unconditional t e s t at the 0.5% l e v e l . The only forms which are accepted by t h i s t e s t i n Table 13.4 l i e between X - -0.4 and -0.2 for y = 10. The s p e c i a l case of the m u l t i p l i c a t i v e (log-linear) form with X = y = 0 i s unequivocally rejected, as i s the log model with y = 10. I t may be concluded from these tests that the optimal functional form l i e s with a high degree of confidence i n a narrow i n t e r v a l centred on X = -0.3 and y = 10. Up to t h i s point the X's have been con-strained equal when applied to the independent v a r i -ables. This constraint i s now relaxed. In addition, the p r e c i s i o n of the X's and y which were estimated to the nearest 0.1 are now estimated with 5-figure - 300 -accuracy and a f u l l y optimal estimate with d i s t i n c t X's and u i s calculated. The r e s u l t s obtained from t h i s estimation problem are given i n Table 13.5. The most s i g n i f i c a n t feature of these re-s u l t s i s that the transformation parameters are r e l a -t i v e l y d i f f e r e n t once the equality constraint i s removed. Whereas the fare X, which i s the dominant variable i n t h i s case, i s s i m i l a r , the time and frequency X's are quite d i s t i n c t . That these differences are s t a t i s t i c a l l y s i g n i f i c a n t i s substantiated by the r e j e c t i o n at the 5 percent l e v e l of a l l models with r e s t r i c t i o n s imposed on the parameters. The product model, given by (e), which had been c l e a r l y rejected previously by (c), i n Table 13.4, i s more c e r t a i n l y rejected by t h i s t e s t . 13.4.2 Analysis of residuals Transformations of the o r i g i n a l v a r i a b l e s into new variables are also transformations of the error term to the extent that i t s d i s t r i b u t i o n i s al t e r e d . Although t h i s d i s t r i b u t i o n should i d e a l l y be approximately normal these conditions are not com-monly encountered i n p r a c t i c e . However i t turns out that maximum l i k e l i h o o d estimation of the functional form also has the property of r e g u l a r i s i n g the d i s -t r i b u t i o n of the residuals i n terms of skewness and - 301 -PARAMETERS MODELS (a) (b) (c) (d) (e) (f) X X 1,1 1,2 1,3 y a. 10 a 20 a 30 a 40 a, a . a . L (A,y) L1(X,u)-L1(X,y) R Skewness Kurtosis D.F. -0.2399 -1.0982 0.0298 35.757 4.7986 (3.629) 5.0241 (4.230) 4.2821 (3.761) 0.0 (-) -1.8164 (-14.13) -0.0153 (-5.941) 1.3779 (5.735) 543.87 0 0.7301 3.155 0.566 0 -0.2660 -0.2660 -0.2660 8.6862 0.7288 (1.044) 0.5087 (1.124) -0.3085 (-0.793) 0.0 (-) -1.7231 (-14.96) -0.3932 (-5.105) 0.4414 (5.067) 539.95 3.94 0.7223 3.331 0.711 2 -0. -0. 0. 0. 21. (5. 21. (4. 20. (4. 0. (--1. (-17. -0. (-4. 13. (5. 2626 0513 5712 0 762 001) 115 957) 354 837) 0 ) 8274 09) 8358 596) 503 368) -0.1930 ! -0.1930 i | -0.1930 I 0.0 0.9343 (1.462) 0.0380 (0.097) -0.8106 (-2.504) | 0.0 | (-) ! -2.2254 |(-18.70) | -0.3605 ! (-4.144) t i 0.1331 538.66 5.21 0.7197 3.268 0.624 1 (5.022) 532.97 10.90 0.7079 3.466 0.765 3 0. 0. 0. 0. 1. (1. 1. (1. 0. (0. 0. (--2. (-15. -1. (-5. 0. (4. .0 .0 .0 .0 .3910 .695) ,0136 ,618) ,2516 ,452) ,0 -) ,9653 .57) ,3148 i 576) ! ,4221 j 607) 528.71 15.16 0.6987 1. 1. 1. 0. 113. (3. 112. (3. 111. (3. 0. (--0. (-11. -0. (-.9. 0. (3. 456. 87. .0 .0 .0 .0 .0 ,486) ,28 ,459) ,76 ,448) ,0 •) ,841x10 ,31) ,0141 ,092) ,0114 ,521) ,32 ,55 -3! 0.4909 3. 0. 4 731 | 6. 990 \ 2. 4 888 667 * Numbers in parentheses are t-statistics 1 2 Percentage points of j x distribution with 4 degrees of freedom Level 0.1 0.05 0.025 0.01 0.005 0.001 | 0.0001 ! Value 3.89 4.74 5.57 6.64 7.43 9.23 11.76 TABLE 13.5: EXTENDED BOX-TUKEY TRANSFORMATION OF THE MODE SPLIT EQUATION* - 302 -kur t o s i s . Consider the graph i n Figure 13.3 where i t i s apparent that the maximum l i k e l i h o o d estimate of X coincides with minimising skewness and kur t o s i s . For the normal d i s t r i b u t i o n these quantities are equal to zero. 13.4.3 Functional form and t - s t a t i s t i c s The choice ,of func t i o n a l form may be c r i t i -c a l for the s i g n i f i c a n c e l e v e l of variables i n the equation. Indeed s p e c i f i c a t i o n error with respect to form may lead to the erroneous i n c l u s i o n or ex-clusio n of variables and thereby to spurious r e s u l t s . That such an outcome i s a r e a l i t y i s shown by Figure 13.4. The s i g n i f i c a n c e of fares i s unequivocal even though there i s considerable v a r i a t i o n i n the l e v e l of s i g n i f i c a n c e . In the case of t r a v e l time, however, given y - 0, for X - 1 the t - s t a t i s t i c implies s i g n i -ficance whereas for X - -1 i t i s i n s i g n i f i c a n t . Thus the improper p r i o r choice of form could lead to t r a v e l time being eliminated as an explanatory v a r i a b l e . I t i s of considerable i n t e r e s t to note that the s i g n i f i c a n c e of the modal constants i s at a minimum around the point of maximum l i k e l i h o o d for y = 0, that i s , X - -0.2. This implies that s p e c i -f i c a t i o n error i n form i s taken up i n these constants which were intended to capture q u a l i t i e s i n t r i n s i c to modes themselves. I t i s d i f f i c u l t to make much of in m m o o © O Q O o o • i in cn r-i c r v O r — in cn in m in T O Z as o M H U « 2 J O H 3 — FIGURE 13.3 RESIDUAL DISTRIBUTION AS A FUNCTION OF A . - 304 -- 305 -such an i n t e r p r e t a t i o n when t h e i r s i g n i f i c a n c e d i -verges towards X = -1, f a l l s almost to zero around X = -0.2 and converges to equality at X = 1. The s i t u a t i o n i s greatly c l a r i f i e d when y i s permitted to increase from zero. This r e l a t i v e l y i n e f f e c t u a l parameter with respect to the l i k e l i h o o d function turns out to have a c r u c i a l r o l e i n damping down the wide v a r i a t i o n s i n the t - s t a t i s t i c s . A comparison of Figures 13.4 and 13.5 shows t h i s . In Figure 13.5, where the t - s t a t i s t i c s are plotted for y = 20, the r e l a t i v e s i g n i f i c a n c e of the variables for a l l X i s cl e a r . The reduction i n the s i g n i f i c a n c e of the con-stants suggests that further s p e c i f i c a t i o n error i n form has been eliminated. I t i s these properties i n addition to the l i k e l i h o o d r a t i o tests that j u s t i f i e s the i n c l u s i o n of the s h i f t parameter y. 13.4.4 E l a s t i c i t i e s Probably the most s i g n i f i c a n t comparison between models i s that with respect to t h e i r e l a s t i -c i t i e s . Models with d i f f e r e n t e l a s t i c i t i e s are d i f -ferent models. Where models are to be used for out-of-sample extrapolation i n order to simulate or fore-cast the correct estimation of e l a s t i c i t i e s i s of primary importance. - 306 -o E-i 13.5 T-STATISTICS AS A FUNCTION OF A, GIVEN y- 2 - 307 -Figures 13.6 and 13.7 show market share e l a s -t i c i t i e s for fare and t r a v e l time for four modes as a function of X, given u = 0. There i s considerable v a r i a t i o n i n these e l a s t i c i t i e s which to some extent matches the pattern found with the t - s t a t i s t i c s for fare and time. The generally lower e l a s t i c i t i e s for car t r a v e l are l a r g e l y due to the e x i s t i n g large market share taken by the car. Bus and r a i l t r a v e l have remarkably s i m i l a r e l a s t i c i t i e s and t h i s i s a l l the more remarkable since t h e i r O-D data were derived from widely d i f f e r e n t sources. In terms of the over-a l l pattern a i r t r a v e l appears the odd man out. In r e l a t i o n to fares a i r i s r i s i n g when the other modes are f a l l i n g whereas for time a i r i s f a l l i n g when the others are r i s i n g . Further d e t a i l s on the e l a s t i c i -t i e s computed from various X's and y's are shown i n Tables 13.6, 13.7, and 13.8. - 310 -U - 0 CAR AIR X FARE TIME FREQUENCY FARE TIME FREQUENCY -1.0 -0.48 -0. 02 -0.50 -0.16 0.14 -0.5 -1.03 -0.16 -1.31 -0. 61 0.24 -0.2 -1.25 -0.37 N/A -1.86 -0.86 0.28 0 -1.26 -0.56 -2.07 -0.92 0.29 0.2 -1.21 -0.65 -2. 22 -0.77 0. 31 0.5 -1.09 -0.66 j i -2. 37 -0 .48 0. 32 1.0 -0.82 -0.56 -2.36 -0.17 0.26 N/A = NOT APPLICABLE y = 0 RAIL BUS X FARE TIME FREQUENCY FARE TIME FREQUENCY -1.0 -1. 34 -0.03 0 .52 -1.32 -0.03 0.24 -0.5 -2.48 -0.29 0.53 -2.50 -0.29 0. 36 -0.2 -2.84 -0.75 0.46 -2.91 -0.77 0.40 0 -2.74 -1.21 0.39 -2.83 -1.26 0.40 0.2 -2.53 -1.52 0. 33 -2.64 -1.59 0.40 0.5 -2.14 -1. 74 0. 25 -2.27 -1.82 0. 38 1.0 -1. 46 -1.75 0.12 -1.59 -1.84 0.28 TABLE 13.6 MARKET SHARE ELASTICITIES FOR y = 0. - 311 -u =10 CAR AIR X FARE TIME FREQUENCY FARE TIME FREQUENCY -1.0 -0.48 -0.18 -0.49 -0.8 3 0. 32 -0.5 -0.88 -0.51 -1.09 -1.08 0.32 -0.2 -1.14 -0.53 N/A -1.65 -0.76 0.34 0 -1.22 -0.56 -1.99 -0.63 0.35 0.2 -1.21 -0.60 -2.23 -0.53 0.35 0.5 -1.09 -0. 63 -2.39 -0. 38 0.33 1.0 -0. 82 -0.56 -2.36 -0 .17 0.26 y =10 RAIL BUS X FARE TIME FREQUENCY FARE TIME FREQUENCY -1.0 -1. 31 -0.30 0.58 -1.29 -0.30 0.49 -0.5 -2.11 -0.94 0.43 -2.12 -0.96 0.46 -0.2 -2.55 -1.09 0. 37 -2.61 -1.12 0.46 0 -2.65 -1.24 0. 32 -2.73 -1.28 0.46 0.2 -2.53 -1.45 0.28 -2.64 -1.50 0.43 0.5 -2.15 -1.69 0.21 -2.28 -1. 77 0. 38 1.0 -1.46 -1.75 0.18 -1.59 -1.84 0.28 TABLE 13.7. MARKET SHARE ELASTICITIES FOR y = 10. - 312 -X r-0. 2 CAR AIR V FARE TIME FREQUENCY FARE TIME FREQUENCY 0 -1.25 -0. 37 -1.86 -0.86 0.28 2 -1.20 -0.47 -1.76 -0.96 0.30 5 -1.16 -0.53 N/A -1. 68 -0.93 0.31 10 -1.14 -0.53 -1.65 -0.76 0.34 20 -1.16 -0.47 -1. 68 -0.51 0.38 50 -1.23 -0.36 -1. 81 -0.25 0.40 90 -1.28 -0. 30 -1.93 -0 .17 0.39 X =-0.2 RAIL BUS y FARE TIME FREQUENCY FARE TIME FREQUENCY 0 -2.84 -0.75 0.46 -2.91 -0.77 0.40 2 -2.71 -0.95 0.43 -2.77 -0.98 0.42 5 -2. 60 -1.08 0.40 -2.66 -1.11 0.44 10 -2.55 -1. 09 0.37 -2.61 -1.12 0.47 20 -2.60 -0.99 0.33 -2.65 -1.02 0.49 50 -2.11 -0.81 0.27 -2.82 -0.83 0.49 90 -2.90 -0.72 0.23 -2.96 -0.75 0.46 TABLE 13.8 MARKET SHARE ELASTICITIES FOR X = -0.2. - 313 -13. 5. Results for the t o t a l t r a v e l demand equation The remainder of t h i s section i s devoted to estimation of the equation given by (31) . Again use i s made of the family of monotone power transformations de-fined by X and y. A feature of i n t e r e s t stems from the sub s t i t u t i o n of the denominator of the estimated form of equation (30) as an argument of equation (31). Thus the l i k e l i h o o d function for the t o t a l t r a v e l demand equation i s a function not only of the parameters of that equation but also of the p r i o r mode s p l i t equation. For the present analysis these f i r s t stage parameters are not allowed to vary i n the t o t a l t r a v e l l i k e l i h o o d function. Instead they are fi x e d at t h e i r optimal values from the mode s p l i t equation. This imposed s e p a r a b i l i t y r e s u l t s i n a considerable s i m p l i f i c a t i o n i n the estimation prob-lem. The shape of the logarithmic concentrated l i k e -lihood function i s shown i n Figure 13.8 for a two- A2 model with \i- 0, given the X's from the f i r s t stage are equal and y 1 = 10. C l e a r l y the surface i s quite regular with a s l i g h t asymmetry r e f l e c t e d i n steeper slopes due to high values of X 2 combined with low values o of X 2^, k 0. Contours form nearly concentric e l l i p s e s rather than spheres, the major axes of the e l l i p s e s being orientated along the plane where X2 = X2 . In fa c t - 314 -FIGURE 13.8 LIKELIHOOD SURFACE FOR THE TOTAL TRAVEL DEMAND EQUATION. - 315 -the optimal s o l u t i o n for X 2 = X 2appears c lose to the 0 1 optimal s o l u t i o n for X 2 f X 2 , leading to the subject ive o 1 expectat ion that the a d d i t i o n a l free parameter i s s l i g h t l y superfluous. That t h i s conclus ion i s premature i s shown by equation (a) i n Table 13.9. Here the optimal set of X's are revealed to be qui te d i s t i n c t from each other . The l i k e l i h o o d r a t i o t e s t with the appropriate degrees of freedom e a s i l y r e j ec t s models (b), (c) and (d). - 316 -PARAMETERS MODELS (a) (b) (c) (d) 0.1685 0.1310 0. 06 0.0 h 0.1973 0.0550 0.06 0.0 1.9396 0.0550 0.06 0.0 *3 0.0355 0.0550 0.06 0.0 60 24.462 10.8875 9.230 4.0628 (29.57) (7.244) (12.803) (4 .438) 61 0.8060 1.4124 0.6172 0.4937 (27.69) (23.00) (21.53) (17.95) 62 0.0014 1.6025 0.7358 0.5222 (8.360) (6.266) (6.061) (5.492) B3 2.5130 1.8088 0.8289 0.3509 (36.23) (31.10) (29.06) (23.74) L -863.53 -877.23 -880.81 -896.13 R 2 0.9544 0.9360 0.9277 0.8937 skewness 4.954 4.442 2.192 N/C kurtosis 0.065 0. 380 0.448 N/C D.F. 0 2 1 1 TABLE 13.9 ALTERNATIVE MODELS FOR TOTAL TRAVEL DEMAND CHAPTER 14 CONCLUSIONS In summary, t h i s d i s s e r t a t i o n has attempted to e s t a b l i s h f i v e main points: (a) e x i s t i n g 0-D survey methods are expensive, cumbersome and s t i l l u n r e l i a b l e , (b) f a m i l i e s of models can be constructed to estimate the 0-D matrix from arc volumes, (c) these estimates can be shown to give good approximations of the 0-D matrix together with reasonable parameters, (d) the construction of the f i r s t i n t e r c i t y car and bus 0-D matrices f o r Canada permits the estimation of the f i r s t i n t e r c i t y multimodal demand model f o r Canada passenger t r a v e l , (e) e x i s t i n g multimodal demand models are r e s t r i c t e d s p e c i a l cases of a more general model which can be developed and estimated. Point (a) was addressed i n Chapter 3 where the process of 0-D survey i s outlined and i t s u n r e l i a b i l i t y documented. Objective (b) was taken up i n Chapters 3, 4, 5 and 6 where numerous types of models were advanced from a t h e o r e t i c a l point of view. I t was shown there how models to estimate the 0-D matrix from arc volumes could indeed be constructed. E f f i c i e n t algorithms to r e a l i s e computationally some of the most promising of - 317 -- 318 -these models were described i n Chapter 7. Some econometric aspects of the models were addressed i n Chapter 8. Point (c), that the models a c t u a l l y work, was demonstrated i n Chapters 9 and 10 where data from B.C. and for the whole of Canada were thoroughly analysed. That the r e s u l t s of t h i s empirical work could lead to a multimodal demand model of i n t e r c i t y t r a v e l , point (d), was shown i n Chapters 11 and 12 i n the context of a simple model. F i n a l l y , that point (e) could be established was demonstrated i n Chapter 13 where a s u b s t a n t i a l l y generalised model i s presented and estimated. There i t was shown that the e x i s t i n g models, which are s p e c i a l cases of t h i s general model, make strong i m p l i c i t assumptions about the form of the e l a s t i c i t i e s - I t was shown that these assumptions are unfounded. Given that the p r i n c i p a l objectives have been attained, i t i s appropriate to conclude with some perspectives on the work as a whole and on the d i r e c t i o n s for further i n v e s t i g a t i o n . The f i r s t aspect can be addressed with a methodological perspective. The methodology used i n the 0-D estimation procedures may be described generally as a probable disaggregation of some observed aggregate, given p r i o r knowledge about the general construction of the aggregate. This approach i s not claimed to be unique to t h i s study. In f a c t , just the opposite i s asserted, that t h i s i s an i n t r i n s i c c h a r a c t e r i s t i c of modelling: an - 319 -observed dependent v a r i a b l e i s r e l a t e d to a combination of independent variables which are assigned weights which best "explain" the dependent v a r i a b l e . Where there i s some di f f e r e n c e , however, l i e s i n the i n t e r p r e t a t i o n of "to explain" as "to disaggregate" i n the sense that an explanation of observed arc volumes may also be a disaggregation into component O-Ds, although both terms can imply the decomposition of e f f e c t s or q u a n t i t i e s . As f a r as further work i s concerned, i t seems natural to extend these methods to other problem areas. One t e n t a t i v e extension was made to estimate a bus O-D matrix and further work i s needed there. Analogous problems e x i s t with other passenger transport modes. One example w i l l demonstrate t h i s . The O-D data published by the A v i a t i o n S t a t i s t i c s Centre of S t a t i s t i c s Canada and the O-D data produced by the CN passenger reservation system (PARRS) have a common methodology: they are derived from t i c k e t analyses. Hence the true o r i g i n s and destinations of t r a v e l l e r s are unknown. Passengers can enter and leave the system by connecting to a non-participating c a r r i e r or by taking another mode. Consequently these O-D data are as much l i n k flows as O-D, and i t would be an i n t e r e s t i n g extension to the present work to address the disaggregation of these "O-D" data. - 320 -In terms of refinement of techniques the most promising avenues seem to be implementation of the pseudo-capacity estimator. Recent experiments have suggested that t h i s device may be superior to e x i s t i n g methods but considerable e f f o r t i s s t i l l required to substantiate t h i s conclusion. 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This leads to a passenger t r i p s model with objective function: min "(V - V a r a or to a passenger-miles model with objective function: min d" 1 Id (V - V a ) 2 , d = £d a a where d a - the length of arc a, i n miles. o V, = the observed arc volumes a V„ - ToV t, = the estimated arc volumes a ^ ha h 8 Y fch ~ a P n h = t h e e s t i m a t e d 0-D 5^ a = the arc-path binary assignment mapping. The passenger-miles objective function i s equi-valent to a weighted regression model where the weights are the arc lengths. Choice between passenger t r i p s and passen-ger-miles versions depends on the units required for the 0-D matrix, whether i n passengers or passenger-miles. If the 0-D matrix i s estimated on passenger t r i p s i t i s a maximum - 329 -- 330 -l i k e l i h o o d estimate only for t r i p s and not.for passenger-miles. The passenger-miles model should be used to obtain the l a t t e r estimate. If the passenger-miles objective function i s rear-ranged and modified i t leads to a simpler, although more aggregate, function, which may or may not need a d d i t i o n a l r e s t r i c t i o n s to be s a t i s f a c t o r y . I t minimises the difference between observed t o t a l and estimated t o t a l passenger-miles leading to t h i s i m p l i c i t e limination of assignment. Total passenger-miles (q) i s observable, computed as ° vo * = £ V a d a a Estimated t o t a l passenger-miles (q) can be written i n an analogous manner, as q = • As written, minimising the diff e r e n c e of these two expressions i s not an advantage computationally since to obtain V"a requires a complete assignment. However, q can be reexpressed as * = ^ a d a a = V J"6, t.d L t ha h a a h - It. Y5. d t hL ha a h a - 331 -= I t D h h h where 6 ^ a i s the usual binary assignment mapping and i s the associated shortest-path distance i n miles. This ex-pression for estimated passenger miles does not require an assignment since t h i s information i s i m p l i c i t l y contained D ^ . Hence the objective function may be written as Minimise F = I I v a d a - 1 ^ ° ^ ! ' w.r.t. a, B , y i n t^= a P h D h a h If the squared difference i s taken instead, the gradient i s calculated e a s i l y . Let e = § - q F = £ 2 9 " ( a , B , Y ) • Then 3F _ - 2 e~3t. D. , • w - h h 3F _ - 2 3 e l t D P " 1 , 36 " h n n n 1^  - - 2 y e l t , D, D - 1 3y ' £ h. h h - 2 Y e I t h h As i t stands t h i s convergence c r i t e r i o n may not be usable i n practice since some of the observed volumes, 0 V , may be missing data. The expression for estimated pas-senger miles can be modified to allow for t h i s , by reducing the length of the shortest-path matrix used to turn passen-gers into passenger-miles. Define s { _ i 1 i f path h uses arc a ha " ] o 0 otherwise, or, V a missing data. D* = 76* d h t ha a a Then F can be redefined as F* where F* = I Jv d - Tt, D* I 1 L a a t h h 1a h which i s again a simple and e a s i l y computed expression. It requires the storage of an a d d i t i o n a l distance matrix and i t s computation once. The assignment procedure subsumed i n the expression for estimated passenger miles can be generalised from a l l - o r -nothing to multiple paths. This i s achieved by w r i t i n g q as q(p) where t ^ = estimated t r i p s by path p for c i t y - p a i r h D^p = distance by path p for c i t y - p a i r h. D ^ i s constructed by D. =-T6. d hp f; hap a ci which represents the distance matrix associated with a - 333 -multiple-path assignment. The d e f i n i t i o n of &na-p "*"s <5v, = t. /It. hap hp' £ hp that i s , the proportion of t r i p s for c i t y - p a i r h which takes path p. APPENDIX B A n a l y t i c a l gradient of the d i r e c t unconstrained O-D model Define the model as 0 2 minimise F = £.(V - V ) w.r.t. a , 3, y a where V = cl 8 Y fch = « P h D h 0 Let 9 = (ct/B/y) / £ = v - V , then the general form of the ' 3 3 . 3 . gradient i s = I ^ a 39 a-^ Q = 2 E e a 3 e a a 39 a 39 = -2 j>a p h a ^ h 38 Setting 9 = a, t 8 Y 3 rh _ P^ D,r 3a -1 and 3F _ -2a £E V — — a a 3a a Setting 8 = 8, 38 and 3F _ - 2 3 y £ v; • • — d a 38 a - 334 -- 335 -V' = ~6-. t,P~ a t ha h h h -1 S e t t i n g 8 - y , 3y and 3F _ -2y £e V " 37 " a a a V " = "6. t, D, 1 a M a h h h APPENDIX C LIST OF CITIES AND CITY-PAIRS Appendix C . l ; Canadian C i t i e s 1. St. John's 48 . Windsor 2. Grand Falls/Windsor 49. Barrie 3. Cornerbrook 50. Pembroke 4. Sydney 51. North Bay 5. Truro 52. Sudbury 6. Halifax/Dartmouth 53. Sault Ste. Marie 7. Charlottetown 54. Fort E r i e 8. Moncton 55. Kirkland Lake 9. Saint John 56. Timmins 10. Fredericton 57. Kapuskasing 11. Chatham/Newcastle 58. Thunder Bay 12. Bathurst 59. Winnipeg 13. Yarmouth 60. Portage La P r a i r i e 14. N. Glasgow/Stellarton 61. Brandon 15. Gander 62. Thompson 16. Campbellton 63. Kenora 17. Edmunston 64. Saskatoon 18. Montreal 65. Regina 19. St. Hyacinthe 66. Melville/Yorkton 20. Drummondville 67. Moose Jaw 21. Sherbrooke 68 . Swift Current 22. Thetford Mines 69. Prince Albert 23. Trois Rivieres 70. North B a t t l e f o r d 24. Quebec 71. F l i n Flon 25. La Tuque 72. Lloydminster 26. Chicoutimi 73. V i c t o r i a 27. Val D'Or 74 . Vancouver 28. Riviere Du Loup 75. Nanaimo 29. Rimouski 76. Prince Rupert 30. Baie Comeau/Hauterive 77. Terrace/Kitimat 31. Sept l i e s 78 . Prince George 32. Gaspe 79. Dawson Creek 33. Rouyn/Noranda 80. Kamloops 34. Cornwall 81. Kelowna 35. Ottawa 82. Penticton 36. B r o c k v i l l e 83. Cranbrook 37. Kingston 84. Jasper 38. B e l l e v i l l e 85. Banff 39. Peterborough 86. Grande P r a i r i e 40. Oshawa 87. Edmonton 41. Toronto 88. Red Deer 42. Hamilton 89. Calgary 43. St. Catherines 90. Lethbridge 44. Kitchener 91. Medicine Hat 45. London 92. Nelson/Trail/Castleqar 46. Sarnia 93. Fort St. John 47. Chatham 94. White Rock - 336 -- 337 -Appendix C.2: U.S. C i t i e s 95. Boston 96. New York/New Jersey/Philadelphia 97. Syracuse 98. Buffalo/Niagara F a l l s 99. Cleveland 100. Detroit 101. Chicago 102. Minneapolis/St. Paul 103. Spokane 104. Seattle 105. Portland (Oregon) 106. Albany 107. Rochester - 338 -Appendix C.3: C i t y - P a i r s f o r T r a v e l Demand 1 Sydney H a l i f a x 2 Sydney Moncton 3 Sydney S a i n t John 4 Sydney F r e d e r i c t o n 5 Sydney B a t h u r s t 6 Sydney Campbellton 7 H a l i f a x Moncton 8 H a l i f a x S a i n t John 9 H a l i f a x F r e d e r i c t o n 10 H a l i f a x B a t h u r s t 11 H a l i f a x Yarmouth 12 H a l i f a x Campbellton 13 H a l i f a x M o ntreal 14 H a l i f a x Quebec 15 H a l i f a x Ottawa 16 H a l i f a x Toronto 17 H a l i f a x Hamilton 18 H a l i f a x London 19 H a l i f a x Windsor 20 H a l i f a x Sudbury 21 H a l i f a x Thunder Bay 22 H a l i f a x Winnipeg 23 H a l i f a x Saskatoon 24 H a l i f a x Regina 25 H a l i f a x Vancouver 26 H a l i f a x Edmonton 27 H a l i f a x C a l g a r y 28 Moncton S a i n t John 29 Moncton F r e d e r i c t o n 30 Moncton B a t h u r s t 31 Moncton Campbellton 32 S a i n t John F r e d e r i c t o n 33 S a i n t John B a t h u r s t 34 S a i n t John Campbellton 35 S a i n t John Montreal 36 S a i n t John Quebec 37 S a i n t John Ottawa 38 S a i n t John Toronto 39 S a i n t John Hamilton 40 S a i n t John London 41 S a i n t John Windsor 42 S a i n t John Sudbury 43 S a i n t John Thunder Bay 44 S a i n t John Winnipeg 45 S a i n t John Saskatoon 46 S a i n t John Regina 47 S a i n t John Vancouver - 339 -48 S a i n t John Edmonton 49 S a i n t John C a l g a r y 50 F r e d e r i c t o n Campbellton 51 Chatham Campbellton 52 Campbellton Montreal 53 Campbellton Quebec 54 Montreal Quebec 55 Montreal C h i c o u t i m i 56 Mon t r e a l V a l D'or 57 Mont r e a l Rimouski 58 Mon t r e a l Gaspe 59 Mo n t r e a l Rouyn/Noranda 60 Montreal Ottawa 61 Mont r e a l Toronto 62 Montreal Hamilton 63 Mon t r e a l London 64 Montreal Windsor 65 Mont r e a l Sudbury 66 Mo n t r e a l Thunder Bay 67 Montreal Winnipeg 68 Mon t r e a l Saskatoon 69 Mon t r e a l Regina 70 Mon t r e a l Vancouver 71 Mon t r e a l Edmonton 72 Mont r e a l C a l g a r y 73 Quebec C h i c o u t i m i 74 Quebec V a l D'or 75 Quebec Rimouski 76 Quebec Gaspe 77 Quebec Rouyn/Noranda 78 Quebec Ottawa 79 Quebec Toronto 80 Quebec Hamilton 81 Quebec London 82 Quebec Windsor 83 Quebec Sudbury 84 Quebec Thunder Bay 85 Quebec Winnipeg 86 Quebec Saskatoon 87 Quebec Regina 88 Quebec Vancouver 89 Quebec Edmonton 90 Quebec C a l g a r y 91 Rimouski Gaspe 92 Rimouski Rouyn/Noranda 93 Ottawa Toronto 94 Ottawa Hamilton 95 Ottawa London 96 Ottawa S a r n i a 97 Ottawa Chatham .- 34 0 -98 Ottawa Windsor 99 Ottawa North Bay-100 Ottawa Sudbury 101 Ottawa Kapuskasing 102 Ottawa Thunder Bay 103 Ottawa Winnipeg 104 Ottawa Saskatoon 105 Ottawa Regina 106 Ottawa Vancouver 107 Ottawa Edmonton 108 Ottawa C a l g a r y 109 Toronto London 110 Toronto S a r n i a 111 Toronto Chatham 112 Toronto Windsor 113 Toronto Pembroke 114 Toronto North Bay 115 Toronto Sudbury 116 Toronto K i r k l a n d 117 Toronto Timmins 118 Toronto Kapuskasing 119 Toronto Thunder Bay 120 Toronto Winnipeg 121 Toronto Saskatoon 122 Toronto Regina 123 Toronto Vancouver 124 Toronto Edmonton 125 Toronto C a l g a r y 126 Hamilton Chatham 127 Hamilton Windsor 128 Hamilton Winnipeg 129 Hamilton Regina 130 London Windsor 131 London Pembroke 132 London North Bay 133 London Sudbury 134 London K i r k l a n d 135 London Kapuskasing 136 London Thunder Bay 137 London Winnipeg 138 London Saskatoon 139 London Regina 140 London Vancouver 141 London Edmonton 142 London C a l g a r y 143 S a r n i a Windsor 144 S a r n i a North Bay 145 S a r n i a Sudbury 146 S a r n i a K i r k l a n d 147 S a r n i a Timmins 341 -148 Chatham Windsor 149 Windsor Pembroke 150 Windsor North Bay 151 Windsor Sudbury 152 Windsor Timmins 153 Windsor Kapuskasing 154 Windsor Winnipeg 155 Windsor Saskatoon 156 Windsor Regina 157 Windsor Vancouver 158 Windsor Edmonton 159 Pembroke Thunder Bay 160 North Bay- Sudbury 161 North Bay Sault Ste Marie 162 North Bay Thunder Bay 163 Sudbury Thunder Bay 164 Sudbury Winnipeg 165 Sudbury Saskatoon 166 Sudbury Regina 167 Sudbury Vancouver 168 Sudbury Edmonton 169 Sudbury Calgary 170 Sault Ste Marie Thunder Bay 171 Sault Ste Marie Winnipeg 172 Sault Ste Marie Regina 173 Sault Ste Marie Vancouver 174 Sault Ste Marie Edmonton 175 Sault Ste Marie Calgary 176 Thunder Bay Winnipeg 177 Thunder Bay Brandon 178 Thunder Bay Kenora 179 Thunder Bay Saskatoon 180 Thunder Bay Regina 181 Thunder Bay Vancouver 182 Thunder Bay Edmonton 183 Thunder Bay Calgary 184 Winnipeg Brandon 185 Winnipeg Thompson 186 Winnipeg Kenora 187 Winnipeg Saskatoon 188 Winnipeg Regina 189 Winnipeg Prince A l b e r t 190 Winnipeg Vancouver 191 Winnipeg Edmonton 192 Winnipeg Calgary 193 Brandon Saskatoon 194 Thompson Saskatoon 195 Thompon Regina 196 Kenora Saskatoon 197 Kenora Regina - 342 -198 Saskatoon 199 Saskatoon 200 Saskatoon 201 Saskatoon 202 Saskatoon 203 Regina 204 Regina 205 Regina 206 Regina 207 Vancouver 208 Vancouver 209 Vancouver 210 Vancouver 211 Vancouver 212 Vancouver 213 Vancouver 214 Vancouver 215 Prince Rupert 216 Prince Rupert 217 Prince Rupert 218 Prince Rupert 219 Prince Rupert 220 Terrace 221 Terrace 222 Prince George 223 Prince George 224 Prince George 225 Kamloops 226 Edmonton 227 Edmonton 228 Edmonton 229 Red Deer 230 Calgary Regina Prince A l b e r t Vancouver Edmonton Calgary Prince A l b e r t Vancouver Edmonton Calgary Prince Rupert Terrace Prince George Kamloops Edmonton Red Deer Calgary Medicine Hat Terrace Prince George Kamloops Edmonton Calgary Prince George Edmonton Kamloops Edmonton Calgary Edmonton Red Deer Calgary Medicine Hat Calgary Medicine Hat P u b l i c a t i o n s 1976 A Comparative A n a l y s i s of S t r a t e g i e s f o r I n t e r c i t y Passenger T r a n s p o r t i n Canada. Canadian T r a n s p o r t Commission, Research Report No. 226, January, (with J'.C. Rea, M.D. Dahl, T . J . Hooper, K.D. Ku, J.B. P l a t t s , & L . J . Ranger) 1976 I n t e r c i t y T r a n s p o r t i n Canada. A n a l y s i s of the consequences of a l t e r n a t i v e p r i c i n g and network s t r a t e g i e s . Canadian T r a n s p o r t Commission, Research Report No. 254, March. (with J.C. Rea, M.D. Dahl, T . J . Hooper, K.D. Ku, J.B. P l a t t s , & L.R. Ranger) 19 7 7 The P o t e n t i a l f o r R a i l Passenger S e r v i c e s i n the Windsor-Quebec C o r r i d o r . RTAC Forum, Summer, 9-1.6. (with J.C. Rea and J.B. P l a t t s ) 1977 E s t i m a t i n g the f u n c t i o n a l form of t r a v e l demand models. C.R.T. p u b l i c a t i o n No. 63, U n i v e r s i t y de Montreal. Forthcoming i n T r a n s p o r t a t i o n Research (with M.J.I. Gaudry) 1977 E v a l u a t i o n of p o t e n t i a l p o l i c i e s f o r i n t e r c i t y t r a n s -p o r t a t i o n i n Canada. T r a n s p o r t a t i o n Research Board, 637, 77-81. (with J.C. Rea and J.B. P l a t t s ) 1977 CANPASS: A s t r a t e g i c p l a n n i n g c a p a b i l i t y f o r i n t e r c i t y passenger t r a n s p o r t a t i o n . In E . J . V i s s e r (ed.) Tran s p o r t D e c i s i o n s i n an Age of U n c e r t a i n t y , Proceedings of the 3rd World Conference on Transport Research, Rotterdam. (with J.C. Rea and J.B. P l a t t s ) 1978 A Foundation f o r I n t e r c i t y Passenger T r a n s p o r t Systems A n a l y s i s i n Canada. Forthcoming i n RTAC Forum. (with J.C. Rea and J.B. P l a t t s ) 1978 T e s t i n g the DOGIT model wit h aggregate t i m e - s e r i e s and c r o s s - s e c t i o n a l t r a v e l data. C.R.T. p u b l i c a t i o n No. 94, U n i v e r s i t e de Montreal. Forthcoming i n T r a n s p o r t a t i o n Research (with M.J.I. Gaudry). 

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