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Urban population density distribution: a contribution from the Vancouver case Tse, Ming-Lan 1976

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URBAN POPULATION DENSITY DISTRIBUTION: A CONTRIBUTION FROM THE VANCOUVER CASE By MING-LAN TSE • S . S c , THE CHINESE UNIVERSITY OF HONG KONG, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS In the School of Community & Regional Planning We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1976 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of Community & Regional Planning The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date January, 1976 i i ABSTRACT C o l i n C l a r k ' s model of the negative exponential p o p u l a t i o n d e n s i t y decay f u n c t i o n i s a p p l i e d to the Vancouver case. Though the model i s claimed to hold true f o r a l l places at a l l times, i t does not o f f e r s u f f i c i e n t explanations why the process i s o c c u r r i n g , nor does i t pay due regards to the topographical e f f e c t s . The a p p l i c a t i o n of the model to the east and south s e c t i o n s of Vancouver may throw some l i g h t to the r a t i o n a l e of the c i t y growth process. We are able to compare the den s i t y gradient of the east and south due to the d i f f e r e n t timing of t r a n s p o r t a t i o n improvement and p h y s i c a l morphology. By examining two s e c t i o n s of the same c i t y we can i s o l a t e the e f f e c t of t r a n s p o r t a t i o n on the density of development, s i n c e both s e c t i o n s are subject to the same growth pressures. The d e n s i t y p r o f i l e s of the whole c i t y , and the eastern and southern s e c t i o n s of i t ( i n the shape of r i n g , s i n g l e a i r l i n e , s e c t o r and band) are prepared; and f o r each p l o t t i n g of po p u l a t i o n d e n s i t y , two p a r a l l e l r e g r e s s i o n runs are made w i t h regard to both r a d i a l d i s t a nce and t r a v e l time. The model i s teste d at four p o i n t s i n time; and i t s goodness of f i t i s measured by the c o e f f i c i e n t s of determination. The conclusions reached are as f o l l o w s : 1. The q u a l i t y of the model i n r e p l i c a t i n g the Vancouver experience i s s i m i l a r to that found f o r a wide range of c i t i e s . 2. The east and the south are marked by the d i f f e r e n t i a l r a t e s of dens i t y d e c l i n e , which are mainly due to the date at which the development takes place. The di s t a n c e parameter measured i n t r a v e l time from the CBD does not give a s i g n i f i c a n t l y b e t t e r f i t to the model than r a d i a l d i s t a n c e from the CBD. The c o e f f i c i e n t s of determination of the model d e c l i n e over time, suggesting v a r i a b l e p a t t e r n of popu l a t i o n d e n s i t y w i t h i n the c i t y over time. The imputed c e n t r a l d e n s i t y does not show a c o n s i s t e n t d e c l i n e over time. The steepness of d e n s i t y d e c l i n e decreases i n the course of time. i v TABLE OF CONTENTS Chapter Page I. INTRODUCTION 1 I I . MODELLING THE URBAN POPULATION DENSITY DISTRIBUTION .. 4 A. Scope of the Study 4 B. D e s c r i p t i o n of the Model 11 C. Commentary on C l a r k ' s Model 15 D e f i c i e n c i e s 15 T h e o r e t i c a l J u s t i f i c a t i o n 16 Footnotes to Chapter I I 17 I I I . ANALYSIS METHODOLOGY 18 A. Hypotheses I n v e s t i g a t e d 18 B. Methodology!..0 24 Measures of A c c e s s i b i l i t y 24 Data Source 24 Data Weaknesses 25 Manipulation of Data 25 . J u s t i f i c a t i o n f o r the use of gross r e s i d e n t i a l d e n s i t y and various geographical set-ups 27 Footnotes to Chapter I I I 33 IV. ANALYSIS OF DATA 34 A. D e s c r i p t i o n of Graphs 34 B. Hypotheses Discussed 41 V. CONCLUSION 50 V Page BIBLIOGRAPHY 53 APPENDIX I : Regression A n a l y s i s 60 Footnotes 63 APPENDIX I I : Graphs 64 APPENDIX I I I : S t a t i s t i c a l Tables 75 APPENDIX IV: Maps 81 vi ACKNOWLEDGMENTS Acknowledgments are due to Dr. M. C. Poulton and Mr. H. Cherniack f o r the advice and supervison of the study. I wish to express to express my gratitude to Mr. T. P. Morris, A r c h i t e c t / Planner of the Central Mortgage and Housing Corporation, f o r reading the manuscript and introducing me to useful sources. I am gre a t l y indebted to those who helped i n typing my thesis and gave me moral support. CHAPTER I INTRODUCTION C i t i e s have s i m i l a r c h a r a c t e r i s t i c s i n i n t e r n a l s t r u c t u r e . Almost a l l of them have a c l e a r l y recognizable CBD, which i s the region of heaviest c o n c e n t r a t i o n of economic a c t i v i t y and a major employment center. R e s i d e n t i a l development tends to f o l l o w a r e g u l a r p a t t e r n . I t i s most dense i n the immediate v i c i n i t y of the CBD, and the d e n s i t y d e c l i n e s towards the periphery. Development tends to take place at the periphery of the c i t y , and so the dwellings of the inner c i t y tend to be older than those of the suburbs. C o l i n C l a r k ^ (1951) has produced an e m p i r i c a l l y - b a s e d model that seems to g e n e r a l i z e the r e s i d e n t i a l d e n s i t y p a t t e r n . The model e f f e c t i v e l y describes the d e n s i t y d i s t r i b u t i o n of r e s i d e n t i a l deve-lopment of North American and European c i t i e s f o r many dates w i t h i n the past 200 years. F u r t h e r , when i t i s a p p l i e d to the same c i t i e s at s e v e r a l p o i n t s i n time, i t i n d i c a t e s that there i s a common pa t t e r n to the change i n the d e n s i t y d i s t r i b u t i o n that has taken place i n the modern era. However, Cla r k ' s model i s purely d e s c r i p t i v e , and the processes that underly the growth p a t t e r n of the c i t y are not c l e a r . We may speculate that t r a n s p o r t a t i o n f a c t o r s and the s e t t i n g of a c i t y are important determinants of the form of the c i t y . 2 I t i s apparent that t r a n s p o r t a t i o n f a c i l i t i e s can i n f l u e n c e urban s p a t i a l s t r u c t u r e by changing the r e l a t i v e a c c e s s i b i l i t y of d i s t a n t and c e n t r a l l o c a t i o n s . However, the demand f o r t r a n s p o r t a t i o n i s i t s e l f i n f l u e n c e d by s p a t i a l s t r u c t u r e and land use. Any change i n t r a f f i c f a c i l i t i e s changes the r e l a t i v e a c c e s s i b i l i t y and hence the r e l a t i v e a t t r a c t i v e n e s s of va r i o u s l o c a t i o n s . On the other hand, the uses at those v a r i o u s l o c a t i o n s generate t r i p s , and these t r i p s i n t u r n define the t r a n s p o r t a t i o n requirements f o r an area. T r a n s p o r t a t i o n can al s o Influence the sequencing of c i t y growth. The h i s t o r y of urban development of almost a l l major c i t i e s i s w e l l marked by v a r i o u s "eras" of transport technology. F i r s t came the horse-drawn buses or s t r e e t - c a r s ; l a t e r , i n the 1880s and 1890s, 2 cable c a r s , e l e c t r i c surface cars and elevated l i n e s . The s t r e e t c a r s and the r a i l r o a d s allowed extensive ribbon development along the l i n e s of the major r a d i a l a r t e r i e s , and the subsequent advent of the motor-cars i n the 1920s allowed the space between the a r t e r i e s to be f i l l e d i n . A f t e r World War I I , extensive c o n s t r u c t i o n of highways and express-ways f a c i l i t i e d the use of p r i v a t e automobiles. The i n c r e a s i n g a v a i l a b i l i t y of the automobile a l s o allowed e a s i e r access to d i s t a n t l o c a t i o n s , causing c i t i e s to "sprawl". As f o r p h y s i c a l morphology, i f a c i t y i s l o c a t e d on an i n f i n i t e f e a t u r e l e s s p l a i n , a c c e s s i b i l i t y to the c i t y center tends to be equal i n a l l d i r e c t i o n s , t h e o r e t i c a l l y at l e a s t . I t i s t h e r e f o r e l i k e l y that p o p u l a t i o n d e n s i t y would be equal i n every d i r e c t i o n from the c i t y center. In a s p a t i a l l y r e s t r i c t e d c i t y where growth i n d i f f e r e n t 3 d i r e c t i o n s from the c e n t r a l core i s assymmetrical, r e g i o n a l centers may develop to c l u s t e r urban f u n c t i o n s c l o s e to r e s i d e n t i a l p o p u l a t i o n and to minimize t r a v e l time. In any event, population d e n s i t y would not be the same f o r a l l p o i n t s at a given distance from the center. The e f f e c t s of both t r a n s p o r t a t i o n and p h y s i c a l morphology on urban form can be observed when the c o n s t r u c t i o n of new t r a n s p o r t -a t i o n f a c i l i t i e s eases t r a v e l across s u b s t a n t i a l p h y s i c a l b a r r i e r s . This has happened i n Vancouver, a c i t y w i t h a h i g h l y r e s t r i c t e d s i t e , where development could take place w i t h r e l a t i v e ease only towards the east. R e s i d e n t i a l development was l a r g e l y r e s t r i c t e d to the Burrard Peninsula by the l a c k of adequate road c r o s s i n g s of the Burrard I n l e t to the no r t h and the Fraser R i v e r (North Arm) to the south u n t i l w e l l i n t o the 1950s. However, continuous development to the east has been r e l a t i v e l y u n r e s t r i c t e d by p h y s i c a l o b s t a c l e s . The d i f f e r e n t timing of t r a n s p o r t a t i o n improvement i n the east and the south has caused c e r t a i n areas i n and around Vancouver to be a c c e s s i b l e f o r r e s i d e n t i a l development at d i f f e r e n t times. As a consequence, the c i t y i s assymmetrical. This assymmetry enables some hypotheses concerning the nature of the c i t y growth process to be examined. Hypotheses on c i t y form, measures of a c c e s s i b i l i t y ( r a d i a l d i s t a nce and t r a v e l t i m e ) , and the v a l i d i t y of Cla r k ' s model, w i l l be considered i n t h i s work. 4 CHAPTER I I MODELLING THE URBAN POPULATION DENSITY DISTRIBUTION A. Scope of the Study In t h i s study we s h a l l concentrate on the r e s i d e n t i a l d e n s i t y d i s t r i b u t i o n . R e s i d e n t i a l a c t i v i t y i s by f a r the predominant land use of c i t i e s . The population d e n s i t y f u n c t i o n of C o l i n C l a r k i s taken as a departure. C l a r k found that f o r many c i t i e s , i n the past and i n the present, p o p u l a t i o n d e n s i t y f a l l s o f f e x p o n e n t i a l l y as d i s t a n c e from the c i t y center i n c r e a s e s . He goes so f a r as to say: "That the f a l l i n g o f f of d e n s i t y i s an exponential f u n c t i o n , appears to be t r u e 3 f o r a l l times and a l l places s t u d i e d , from Los Angeles to Budapest". For the r e l a t i o n s h i p between d e n s i t y and d i s t a n c e i n C l a r k ' s model to hold t r u e , the d e n s i t y p r o f i l e should i d e a l l y be the same i n a l l d i r e c t i o n s i n i n h a b i t e d parts of the c i t y where no "holes" (such as bodies of water) are present, t h i s would r e s u l t i n a c i r c u l a r c i t y , w i t h the C e n t r a l Business D i s t r i c t (CBD) at i t s center. C l a r k found that r e s i d e n t i a l p o p u l a t i o n d e n s i t y d e c l i n e s w i t h increase of distance from the c i t y center, and that the steepness of t h i s d e c l i n e becomes l e s s over time. He argues that t r a n s p o r t a t i o n improvement i s r e s p o n s i b l e f o r t h i s d e c l i n e over time. However, the primary reasons f o r t h i s d e n s i t y p r o f i l e and i t s change through time are not c l e a r . We w i l l use evidence from the Vancouver case to improve our general understanding of t h i s matter. 5 S p e c i f i c a l l y , the aims of the study can be o u t l i n e d as: 1) To t e s t the v a l i d i t y of C l a r k ' s model f o r Vancouver The repeated c o n f i r m a t i o n of the v a l i d i t y of the negative exponential f u n c t i o n of p o p u l a t i o n d e n s i t y d i s t r i b u t i o n has important i m p l i c a t i o n s f o r the planning of m e t r o p o l i t a n r e g i o n s . I t has been found that the i n d i v i d u a l choice of r e s i d e n t i a l l o c a t i o n s conforms c o l l e c t i v e l y to the r e g u a l r p a t t e r n . The negative exponential "law" of mass behaviour i s not a r e s u l t of e x t e r n a l l y imposed p h y s i c a l 4 planning, or l e g a l or a d m i n i s t r a t i v e c o n s t r a i n t s . "The observed behaviour i s c h a r a c t e r i s t i c of urban s o c i e t i e s responding to f o r c e s and subject to c o n s t r a i n t s t h a t , although they are not c o n s c i o u s l y perceived, are nonetheless extremely strong". Should t h i s " n a t u r a l " p a t t e r n of p o p u l a t i o n d e n s i t y decay be the case i n Vancouver, i t must be taken i n t o account i n planning p o l i c i e s . However, i t i s important f i r s t to v e r i f y that i t does i n f a c t hold f o r Vancouver. The c i t y i s unusual when compared w i t h other c i t i e s i n that i t i s e s s e n t i a l l y elongated i n shape and h i g h l y r e s t r i c t e d i n s i t e . I t i s necessary to observe the e f f e c t of p h y s i c a l morphology and the r e s t r i c t i o n on development on the a p p l i c a b i l i t y of the model. C i t i e s that are l o c a t e d on s i t e s that are not unduly r e s t r i c t e d show many s i m i l a r i t i e s of form. Vancouver has developed on a h i g h l y r e s t r i c t e d s i t e and from i t s shape i t would appear to be e x c e p t i o n a l . The slope of the d e n s i t y d e c l i n e should be d i f f e r e n t i n the east and the south. But 6 i f i t . ' i s the same i n every d i r e c t i o n at one point i n time, we can say that Vancouver i s not so d i f f e r e n t from other c i t i e s . 2) To observe the long-run trends on urban form It i s necessary to consider whether the long-run trends i n the development of urban form are i n fac t changing. As indicated i n Clark's model, the urban form would i n time change, showing a r e l a t i v e decline i n imputed ce n t r a l density and a r e l a t i v e increase i n suburban population density. I t i s to be expected that the introduction of planning p o l i c i e s and controls w i l l a f f e c t the pattern of future deve-lopment to some degree. Future urban form would also be affected by changes i n the long-run trend f o r the type of housing constructed, (a) Housing Trends: A housing survey conducted i n 1972 i n Vancouver suggests that 6 there w i l l be more preference f o r smaller units i n the future. The National Housing Act (NHA) finances more mult i p l e dwellings (apartments 7 and row housings) than sin g l e family housing. The cost of land has g been increasing sharply since early 1972. Other economic factors also discourage people from owning s i n g l e detached housing. Further, the " a ssisted home ownership programme" of the Canadian Housing and 9 Mortgage Corporation finances condomium u n i t s . Ownership of single detached dwellings i s becoming r e l a t i v e l y more expensive as compared with multiple housing u n i t s , and as a r e s u l t , apartments are more popular. One r e s u l t i s the emergence of high density l i v i n g i n suburban areas. Townhouses and apartment complexes now coexist with quarter-acre l o t s on the fringes of the c i t y . This suggests suburban deve-lopment i n the f u t u r e at d e n s i t i e s greater than has been the case i n the l a s t two decades. (b) The L i v a b l e Region Growth Strategy: The major planning p o l i c y documented f o r m e t r o p o l i t a n Vancouver i s the s o - c a l l e d L i v a b l e Region Pla n . This plan has s u b s t a n t i a l i m p l i -c a t i o n s f o r the f u t u r e a p p l i c a b i l i t y of the model. The major proposals adopted by the P o l i c y are: (1) The c r e a t i o n of four r e g i o n a l centers a l l to the east and south-east of the CBD: The r e g i o n a l centers are to be l o c a t e d i n Burnaby, New Westminster, Surrey, and the North-east Sector (the exact l o c a t i o n of t h i s f o u r t h area i s as yet to be determined). These would act as f o c i f o r the g e n e r a l l y formless, s c a t t e r e d development to the east of Vancouver. They would help the suburban communities reach a balance of employment and employable r e s i d e n t s . (2) P r o t e c t i o n of a g r i c u l t u r a l land: The housing stock would be expanded p r i m a r i l y w i t h i n the p r e s e n t l y urbanized area by i n f i l l i n g and redevelopment. The net r e s u l t of t h i s p o l i c y i s that d e n s i t i e s would be somewhat higher i n the urbanized p o r t i o n s of the o u t l y i n g m u n i c i p a l i t i e s than would otherwise be the case. (3) A l i g h t r a p i d t r a n s i t system connecting the r e g i o n a l centers and downtown Vancouver: This would put the r e g i o n a l centers on a more equal f o o t i n g w i t h the present CBD i n q u a l i t y of services-and a c c e s s i b l e o p p o r t u n i t i e s . I f these p o l i c i e s are implemented, s e v e r a l consequences can be envisioned. There would be an increase i n suburban pop u l a t i o n 8 d e n s i t i e s , a l e s s uniform p a t t e r n of d e n s i t i e s f o r suburbs at s i m i l a r d istances from the CBD, a comparatively l e s s dominant CBD than b e f o r e , and improvement i n a c c e s s i b i l i t y to s e r v i c e a c t i v i t e s and some of the CBD-type f u n c t i o n s i n the r e g i o n a l centers which may a l t e r the l o c a t i o n preferences of i n d i v i d u a l s . In sum, there already appear to be trends towards higher d e n s i t i e s i n the urbanized areas of the o u t l y i n g suburbs, and these trends w i l l f u r t h e r be enforced by the L i v a b l e Region P o l i c y . I t i s d o u b t f u l whether the proj e c t e d d e n s i t y p a t t e r n to be adopted by the L i v a b l e Region P l a n f i t s C l a r k ' s negative exponential curve, and i f Cla r k ' s model represents a b e h a v i o r a l "law", then the Plan might be more s u c c e s s f u l i f i t were c o n s i s t e n t w i t h t h i s "law". 3) To study the e f f e c t of t r a n s p o r t a t i o n improvement The s i t e of the c i t y poses s i g n i f i c a n t b a r r i e r s to commuting, which have i n the past r e s t r i c t e d t r a v e l to the north and the south. There are s t i l l r e l a t i v e l y few crossings of the major water b a r r i e r s , and these are i n f a c t the places where Vancouver experiences congestion at peak-commuting hours. The CBD i t s e l f i s c l o s e by the Burrard I n l e t Waterfront and i s connected to the northern suburbs by two br i d g e s : one a few miles to the east of the CBD (the Second Narrows), and the other a few miles to the west (the F i r s t Narrows). To the south, the Fraser R i v e r i s a major b a r r i e r . The Fraser River reaches tidewater across a d e l t a to the south of the Burrard P e n i n s u l a . A few major bridges and one tunnel have been constructed i n the l a s t 20 years to improve access 9 to the south. Nevertheless, the Fraser remains a su b s t a n t i a l b a r r i e r . Transport obviously i s a very important factor i n urban deve-lopment, although i t s exact r o l e i s d i f f i c u l t to i s o l a t e because of the m u l t i p l i c i t y of other fa c t o r s . In Vancouver, improvements i n transportation such as the bridges and highway programs of the 50s and 60s generally preceded r e s i d e n t i a l spread. Also, improvements i n transportation through the natural b a r r i e r s i n the east and the south have occurred at d i f f e r e n t times. Because of t h i s , we are able to examine the e f f e c t of transportation on the density of development i n comparison to other explanations of the density p r o f i l e , ;.such as .the-date at wnich^development' takes ' p l a c e t 1 ^ ' ' : n t t a k e s P l a- e-Among the most important projects leading south were the new Gr a n v i l l e Street Bridge, 1954; Oak Street Bridge, 1957; Deas Island Tunnel ( l a t e r named George Massey Tunnel), 1959; Vancouver-Blaine Freeway, 1962, through Richmond, Delta and Surrey to the United States border; and Knight Street Bridge, 1974. To the east and the south-east, the Fraser and i t s t r i b u t a r i e s are s i g n i f i c a n t b a r r i e r s to t r a v e l i n the outer suburbs of Vancouver and beyond. Bridges were b u i l t at Rosdale-Agassiz i n 1956, and at P i t t River i n 1957. The former provided much better inter-connections between the north and south banks of the Fraser River and permitted east-west through t r a f f i c to use the Lougheed Highway as an alternate route to the Trans-Canada Highway, through Burnaby, Coquitlam, Surrey and Langley, both opened i n 1964. ^ Since the t r a v e l r e s t r a i n t s of the various s e c t i o n s of the c i t y were eased at d i f f e r e n t times, we would expect the r a t e of p o p u l a t i o n d e n s i t y d e c l i n e to the south to be d i f f e r e n t from that to the east. This should be the case when the d i s t a n c e parameter i n the model i s measured i n terms of t r a v e l time or r a d i a l d i s t a n c e . However, the l o c a t i o n a l d e c i s i o n s of households may be more responsive to the former than to the l a t t e r . C l a r k ' s model i s t e s t e d w i t h respect to both t r a v e l time and r a d i a l d i s t a n c e i n order to study the e f f e c t of the improvement of transport f a c i l i t i e s on r e s i d e n t i a l spread. I t i s used i n the study because i t i s "the most appropriate sample d e s c r i p t i v e model f o r r e -presenting the d i s t r i b u t i o n of r e s i d e n t i a l d e n s i t y " . ^ I t i n v o l v e s merely two parameters (A and b ) , which, at f i r s t glance, i s simple but appealing. The d e n s i t y gradient b i s the simplest d e s c r i p t i v e measure of the s p a t i a l d i s t r i b u t i o n of a c t i v i t y . I t can be used to i n d i c a t e the changes i n c i t y form. I t i s e a s i l y computed; and i t represents a c l e a r l y understood d e s c r i p t i v e summary about the s p a t i a l 12 a l l o c a t i o n of r e s i d e n t s i n an urban area. B. D e s c r i p t i o n of the Model Since C l a r k provided a mathematical expression f o r the popu-l a t i o n d e n s i t y d i s t r i b u t i o n i n 1951, other authors have worked w i t h the model and come up w i t h some other formulations. However, i t i s dou b t f u l whether other models have shown an improvement over the 13 o r i g i n a l one. Cla r k ' s model i s : D = D e " b x (1) x o or : Ln D = Ln D - bx (2) x o This s t a t e s that d e n s i t y i s a negative exponential f u n c t i o n of distance from the c i t y center. D q and b are the parameters of the equation. Where: x i s dis t a n c e from the center of c i t y ; D^ i s the d e n s i t y of r e s i d e n t p o p u l a t i o n at a di s t a n c e x; D q i s the ext r a p o l a t e d or imputed c e n t r a l p o p u l a t i o n d e n s i t y ; Ln denotes the n a t u r a l logarithm; e i s the n a t u r a l l o g a r i t h m i c base; b i s the d e n s i t y g r a d i e n t , or the slope of the curve. D q i s the r e s i d e n t i a l d e n s i t y imputed to the c i t y core. Because the CBD i s mostly n o n - r e s i d e n t i a l , t h i s d e n s i t y represents a h y p o t h e t i c a l f i g u r e which i s not r e a l i z e d . The n o n - r e s i d e n t i a l CBD shows up as a ( r e s i d e n t i a l ) " d e n s i t y c r a t e r " not e x p l i c i t l y taken i n t o account i n Clark's model. G r a p h i c a l l y , the model can be represented by: Distance from the c i t y Distance from the c i t y center center C l a r k derived two g e n e r a l i z a t i o n s from h i s study of 36 American and European c i t i e s : 1) In every l a r g e c i t y , excluding the c e n t r a l business zone, which has few r e s i d e n t i n h a b i t a n t s , we have d i s t r i c t s of dense popu-l a t i o n i n the i n t e r i o r , w i t h d e n s i t y f a l l i n g o f f p r o g r e s s i v e l y as we proceed to the outer suburbs; 2) In most (but not a l l ) c i t i e s , as time goes on, d e n s i t y tends to f a l l i n the most populous inner suburbs, and to r i s e i n the outer suburbs, and the whole c i t y tends to "spread i t s e l f out". At one s t a t i c point i n time, the f a l l - o f f i n d e n s i t y from the c i t y center to the periphery f o l l o w s a negative exponential r e l a t i o n -ship w i t h d i s t a n c e . However, d e n s i t y tends to decrease i n the inner suburbs and increase i n the outer suburbs w i t h the passage of time, and t h i s tendency i s r e f l e c t e d i n the parameter b i n the model t a k i n g on smaller values over time. The d e n s i t y gradient may a l s o be c a l l e d the " c o e f f i c i e n t of compactness". The l a r g e r the value of b, the more compact the c i t y , and the smaller the more sprawled the c i t y . ^ C l a r k argues that b i s l a r g e l y dependent upon the costs of int r a u r b a n t r a n s -p o r t , or the cost of t r a v e l l i n g i n r e l a t i o n to the average c i t i z e n ' s 16 income. In other words, i t would get smaller when improvements i n t r a n s p o r t a t i o n a l l o w people e a s i e r access to the suburbs. C l a r k used the f o l l o w i n g methodology i n h i s work: F i r s t , a s e r i e s of c o n c e n t r i c r i n g s were drawn at one m i l e radius from the c i t y center and the average d e n s i t y c a l c u l a t e d f o r each r i n g using the net area e x c l u s i v e of open spaces. Second, the d e n s i t y measure was then p l o t t e d against the mean dis t a n c e of the r i n g from the center. (The population and net area were obtained f o r census t r a c t s or s i m i l a r a d m i n i s t r a t i v e d i v i s i o n s and where the c i r c l e s cut the boundaries of these d i v i s i o n s , apportionments of popu l a t i o n were to be made). T h i r d , the parameters D q and b were obtained by r e g r e s s i o n a n a l y s i s using the l o g a r i t h m i c form of the equation. Having adopted t h i s method, Cl a r k goes on to s t a t e that " i t would be b e t t e r t o / p l o t , f o r each t r a c t , the recorded average d e n s i t y against i t s mean distance from the center of the c i t y , as t h i s would e l i m i n a t e the e r r o r s due to the apportionment process, and give a b e t t e r p i c t u r e of the s c a t t e r about the r e g r e s s i o n l i n e " . This simpler and more e f f e c t i v e procedure was adopted i n t h i s study. C. Commentary on C l a r k ' s Model In very sweeping terms, the model at best describes the popu l a t i o n d e n s i t y d i s t r i b u t i o n i n a snapshot. Though i t i s not p r e d i c t i v e , i t has impMcafcionspfor the growth trend of a c i t y . I t b r i n g s f o r t h the i s s u e of the f r i c t i o n a l e f f e c t of d i s t a n c e , i . e . , a c c e s s i b i l i t y or the ease of i n t e r a c t i o n between the place of r e -sidence and the core area, which i s the primary center of economic a c t i v i t y . However, the model i s not without i t s shortcomings. I t i s t h e r e f o r e the i n t e n t of t h i s s e c t i o n to o u t l i n e i t s d e f i c i e n c i e s i n b r i e f , and to provide a t h e o r e t i c a l j u s t i f i c a t i o n to e x p l a i n the r e l a t i o n s h i p between d e n s i t y and d i s t a n c e . D e f i c i e n c i e s  D e f l u i e n c i e s The model has been claimed to be a p p l i c a b l e 1 to a l l places at a l l times. I t i s derived p u r e l y from e m p i r i c a l observations and i t does not have w i t h i n i t anything to e x p l a i n why the d e n s i t y d i s t r i b u t i o n i s as i t i s or why i t has changed i n such a c o n s i s t e n t manner over the past century. I t i s apparent that as soon as planning i n s t i t u t i o n s such as zoning, taxes, e t c . , are introduced, or a d i f f e r e n t set of market for c e s are i n op e r a t i o n , or the f a c t o r s a f f e c t i n g the choice of r e s i d e n t i a l l o c a t i o n change, C l a r k ' s "law" may no longer apply. On the other hand, i f the "law" does apply, t h i s shows that the " n a t u r a l " f o r c e s may i n f a c t be much stronger than these f a c t o r s . 16 T h e o r e t i c a l J u s t i f i c a t i o n The population d e n s i t y decay f u n c t i o n can be explained i n economic terms. In the a b s t r a c t , i n d i v i d u a l s tend to base t h e i r r e s i d e n t i a l l o c a t i o n d e c i s i o n s upon a t r a d e o f f between land c o s t , the cost of housing and t r a n s p o r t or commuting c o s t s . As d i s t a n c e from the c i t y center i n c r e a s e s , land costs tend to be lower because more land i s a v a i l a b l e , and t r a n s p o r t a t i o n costs to a range of a c t i v i t i e s and job o p p o r t u n i t i e s are higher. As land costs l e s s and l e s s , i t i s l i k e l y that people w i l l consume more of i t ' . I t i s t h e r e f o r e l o g i c a l to expect that the per c a p i t a consumption of land f o r housing increases w i t h d i s t a n c e from the c i t y center. This e x p e c t a t i o n was confirmed by the f i n d i n g s of the study done by the Lower Mainland Regional Planning Board of B r i t i s h Columbia i n 1963 concerning the i n t e n s i t y of land use and i t s a s s o c i a t e d land value of Vancouver. In the main, i t was found that the curves of i n t e n s i t y of land use and land value are downward s l o p i n g . The curve of the r a t i o of occupied land to unoccupied land p l o t t i n g against time-distance zones i s very steep at the center and becomes f l a t t e n e d out at the periphery. This i n d i c a t e d that land i s i n high i n t e n s i t y use at the core and low i n t e n s i t y use at the periphery. ^ Footnotes 1. C. C l a r k , "Urban P o p u l a t i o n D e n s i t i e s " , J o u r n a l of the Royal  S t a t i s t i c a l , S o c i e t y , V o l . CXIV, part IV, 1951 2. Homer Hoyt, "The E f f e c t of the Automobile on Patterns of Urban Growth", T r a f f i c Q u a r t e r l y , 1963, p. 294 3. C. C l a r k , op. c i t . , pp. 490-1. Emphasis i s my own. 4. Rene B u s s i e r e , The S p a t i a l D i s t r i b u t i o n of Urban P o p u l a t i o n s , t r a n s , by the author, P a r i s , CRU, 1970, p. 83 5. I b i d . , p. 83 6. Michele L i o y , S o c i a l Trends i n Greater Vancouver, The United Way of Greater Vancouver, March, 1975, p. 73 7. I b i d . , p. 76 8. I b i d . , p. 77i 9. I b i d . , p. 77 10. A l f r e d H. Siemens (ed.), Lower Fraser V a l l e y : E v o l u t i o n of a  C u l t u r a l Landscape, B.C. Geog. S e r i e s , No. 9, Dept. of Geog., Tantalus Research L t d . , Vancouver, CAnada, p. 86 11. Bernard-Andre Genest, "P o p u l a t i o n D i s t r i b u t i o n Functions f o r Urban Areas", V o l . 2 i n a Series on A i r p o r t L o c a t i o n and Pla n n i n g , Research Rept., R-70-53, Aug. 1970, p. i i 12. Bruce E. Newling, "The S p a t i a l V a r i a t i o n of Urban P o p u l a t i o n D e n s i t i e s " , Geog. Review 59, 1969, p. 248 13. B-A. Genest, op. c i t . 14. C. C l a r k , op. c i t . , p. 490 15. C. C l a r k , "The Lo c a t i o n of I n d u s t r i e s and P o p u l a t i o n " , Town Planning  Review 35, p. 211 16. C. C l a r k , "Urban P o p u l a t i o n D e n s i t i e s " , op. c i t . , p. 491 17. Dynamics of R e s i d e n t i a l Land Settlement, Lower Mainland Regional Planning Board of B r i t i s h Columbia, Supp. Study 2 to Land f o r L i v i n g CHAPTER I I I ANALYSIS METHODOLOGY A. Hypotheses I n v e s t i g a t e d S i x e x p l i c i t hypotheses are considered. The hypotheses are intended to explore c e r t a i n determinants of c i t y growth. They have d i r e c t r e l a t i o n to the model formula: D = D e " b x x o i . e . , Ln D^ = Ln D q - bx ( i n the form f o r l i n e a r regression) where: x dis t a n c e to the c i t y center (whether measured i n r a d i a l d istance or t r a v e l time) D^ = popul a t i o n d e n s i t y at di s t a n c e x Ln D q = the n a t u r a l l o g of the popul a t i o n d e n s i t y imputed at the c i t y center b = d e n s i t y g r a d i e n t , or the slope of the curve i n the l i n e a r r e g r e s s i o n model Hypothesis 1: The d e n s i t y gradient b w i l l be l a r g e r f o r the South than  f o r the East. Reasons: 1. The topographical c o n s t r a i n t s to the south are greater than to the east. The former i s c l e a r l y marked by water bodies, i . e . , the north and south arms of the Fraser R i v e r ; w h i l e the east i s not c h a r a c t e r i z e d by any major b a r r i e r i n the landscape. 19 2. Development to the south has been r e l a t i v e l y slower than to the east. I f r e s i d e n t i a l b u i l d i n g d e n s i t y i s p r i m a r i l y a f u n c t i o n of the time of c o n s t r u c t i o n f o r new p e r i p h e r a l development, and the lower d e n s i t y i s a s s o c i a t e d w i t h l a t e r development, then the slower spread of the urban area to the south w i l l be a s sociated w i t h a l a r g e r b value when dens i t y i s measured against t r a v e l time. I f , on the other hand, t r a n s p o r t i s the dominant f a c t o r i n e x p l a i n i n g d e n s i t y d i s t r i b u t i o n , then the b value to the south should be s i m i l a r to that to the east. Hypothesis 2: There w i l l be a s i g n i f i c a n t d i s c o n t i n u i t y i n the  s c a t t e r i n g of data p o i n t s at about s i x m i l e s from  the c i t y c enter, where the north arm of the Fraser  R i v e r i s l o c a t e d . Reasons: Only i n the 50s-.did h i g h - c a p a c i t y bridges over the Fraser R i v e r improve t r a n s p o r t a t i o n l i n k s to the south. P o p u l a t i o n d e n s i t y should be higher between the c i t y core and the north arm of the Fraser River than f u r t h e r south. R e s i d e n t i a l spread south of the r i v e r s t a r t e d at a l a t e r date, and should be marked by a lower d e n s i t y . There should thus be a sharp break i n the s c a t t e r i n g of data p o i n t s . This break could be due t o : (1) The f a c t that d e n s i t y depends on the date of development. The l a t e development south of 20 the r i v e r would then be re s p o n s i b l e f o r the break i n d e n s i t y at that p o i n t ; or (2) to the f a c t that d e n s i t y responds to t r a v e l time and the detours and delays due to the few r i v e r c r o s s i n g s may cause a sharp leap i n average t r a v e l time over a short increase i n r a d i a l d i s t a n c e from the center. Evidence f o r the former explanation would be continued e x i s t e n c e of a break when d e n s i t y i s considered as a f u n c t i o n of t r a v e l time as w e l l as r a d i a l d i s t a n c e . For the l a t t e r to hold t r u e , the d i s c o n t i n u i t y should be conspicuous only when d e n s i t y i s a f u n c t i o n of r a d i a l d i s t a n c e . When distance i s measured i n terms of t r a v e l time, there should not be a sharp d i s c o n t i n u i t y , and the r e s i d e n t i a l d e n s i t y should be represented by a smoother p r o f i l e . Hypothesis 3: x measured i n t r a v e l time w i l l g i ve a b e t t e r f i t (as 2 evidenced by a higher R ) to the data than x measured  i n r a d i a l d i s t a n c e . Reasons: I t seems reasonable to assume that r e s i d e n t i a l l o c a t i o n d e c i s i o n s are made on the r e l a t e d bases of a c c e s s i b i l i t y and t r a v e l c o s t s . Housing demand i s d i r e c t l y r e l a t e d to a c c e s s i b i l i t y and i n v e r s e l y r e l a t e d to t r a v e l c o s t s . Since t r a v e l time i s probably a b e t t e r measure of t r a v e l cost and of a c c e s s i b i l i t y than r a d i a l d i s t a n c e , i t should be a b e t t e r p r e d i c t o r of de n s i t y . 21 Hypothesis 4: R for the model should decline over time. Reasons: 1. Decline i n the r e l a t i v e dominance of the GBD: The CBD has been the major employment center. Since the journey to work i s the most recurrent t r a v e l movement of a l l the d a i l y t r i p s , residences tend to be located around the CBD. It i s often assumed that r e s i d e n t i a l l o c a t i o n i s the r e s u l t of the attempt to minimize the length of the journey to work, and therefore the costs of work t r a v e l i n -curred. However, the CBD has declined i n r e l a t i v e importance as a center of economic a c t i v i t y and a t t r a c t o r of t r i p s . It follows that r e s i d e n t i a l l o c a t i o n decisions are l e s s influenced by the ease of access to the CBD, and other factors not connected with the CBD w i l l be more i n f l u e n t i a l i n determining the density at which development takes place. 2. Rise of mobility i n a l l d i r e c t i o n s : The increasing a v a i l a b i l i t y of private automobiles has freed personal t r a v e l from dependence on the primary t r a n s i t network. Residences, workplaces and shopping centers can be located away from the major trans-portation routes. -'''3. Decline i n the r e l a t i v e importance of transportation cost through time: Transportation has tended to 22 become a l e s s and l e s s s i g n i f i c a n t factor i n determining personal choice of the type and l o c a t i o n of residence. This may be due to r i s i n g m o b i l i t y caused by the i n -creasing use of automobiles, the d e c e n t r a l i z a t i o n of employment, shopping and r e c r e a t i o n a l opportunities, and the marked increase i n the cost of a v a i l a b l e acco-modation i n recent years. These factors have acted to reduce the transportation cost savings po s s i b l e from selecting one l o c a t i o n over another. A decline i n the importance of a c c e s s i b i l i t y to the CBD and the q u a l i t y of transportation i n general i n l o c a t i o n decisions should lead to a density pattern les s and les s influenced by these f a c t o r s . This would r e s u l t i n a density d i s t r i b u t i o n l e s s conforming to Clark's "law", which would be evidenced by more dispersed pattern of data i n the model. Thus, we would expect a lower R over time. Hypothesis 5: A should decline over time f o r Vancouver. Hypothesis 6: b should decline over time for Vancouver. Reasons: 1. Clark's empirical findings show that these two parameters decline through time f o r many c i t i e s . 2. In Vancouver, as i n a l l major c i t i e s , transportation f a c i l i t i e s have improved over time. As a con-sequence, dista n t i i l o c a t i o n s have become more accessible r e l a t i v e to cen t r a l l o c a t i o n s , and so people have tended to s e t t l e i n places other than the ce n t r a l area. Greater e f f i c i e n c y i n t r a n s p o r t a t i o n has caused a d e c l i n e i n the r e l a t i v e importance of t r a n s p o r t cost as compared to other l o c a t i o n a l c o n s i d e r a t i o n s , as w e l l as the r e l a t i v e advantage of one l o c a t i o n over another i n v a r i o u s s e c t i o n s of the c i t y . 24 B. Methodology In t h i s section we s h a l l focus on the procedure of the study. The source of data and i t s possible weaknesses, the method of aggre-gating the data and i t s j u s t i f i c a t i o n w i l l be discussed i n turn. Measures of A c c e s s i b i l i t y The unit of observation for t h i s study i s the census t r a c t , and distance i n the model i s taken from the census t r a c t ' centroid to the c i t y core. However, distance may be either s p a t i a l or psychorV l o g i c a l , so measurement i s made with respect to both r a d i a l distance and t r a v e l time. (Since the dominant mode of transportation i n Vancouver i s the auto, " t r a v e l time" i s measured by automobile t r a v e l t ime.) Data Source The model involves two variables:" population density and distance ( r a d i a l distance or t r a v e l time). The former i s calculated by taking the t o t a l population count.'of a census t r a c t divided by i t s t o t a l area. I t should be borne i n mind that the value obtained represents the gross r e s i d e n t i a l density of the census t r a c t . The distance "as the crow f l i e s " from the c i t y center i s taken from the census tr a c t map. The census s t a t i s t i c s of 1956, 1961, 1966 and 1971, and the census tr a c t maps of 1961 (for s t a t i s t i c s of 1956 and 1961) and 1971 (for ( f o r s t a t i s t i c s of 1966 and 1971) are used. The d r i v i n g time from the c i t y center f o r 1966 and 1971 i s taken from the T r a v e l Time I s o l i n e s Map of 1968 1 and that f o r 1956 and 1961 i s taken from the T r a v e l Time Map of 1961. 2  Data Weaknesses The data used have some weaknesses: F i r s t , the s i z e s of the census t r a c t s vary considerably. They are small at the c i t y center, but become l a r g e towards the periphery. Second, measurement of r a d i a l d i s t a nce i s made from the c i t y center to the c e n t r o i d of the census t r a c t concerned. However, the l o c a t i o n of census t r a c t c e n t r o i d s are determined by i n t u i t i v e judgment. T h i r d , i n t h i s ' s t u d y research i s not done to l o c a t e the exact c i t y center. I t i s taken to be the G r a n v i l l e - G e o r g i a i n t e r s e c t i o n , which i s g e n e r a l l y agreed to be the "high-value corner" or the center of the CBD. There has been debate on the method of l o c a t i n g the c i t y center e x a c t l y . However, i t has been shown by Genest that the exact center does not s i g n i f i c a n t l y 3 • a f f e c t the q u a l i t y of the model . , and the search f o r i t i s t h e r e f o r e not a worthwhile undertaking. Fourth, the t r a v e l time from the center to the census t r a c t i s an approximation, s i n c e t r a v e l - t i m e study years were not always the same as the census years. A l s o , we assume that there were no major changes i n the road network i n the census years from which the r e s p e c t i v e T r a v e l Time maps are adopted. Manipulation of Data Analyses are made f o r four geographical set-ups ( r i n g s , l i n e s , p i e - s l i c e s , and bands), two measures of distance ( r a d i a l d i s t a n c e and t r a v e l time) and four p o i n t s i n time (1956, 1961, 1966 and 1971). Popu-l a t i o n d e n s i t y of the four types of geographical set-ups are aggregated as f o l l o w s : 1) Rings: The whole m e t r o p o l i t a n area of Greater Vancouver i s d i v i d e d i n t o r i n g s at each m i l e from the c i t y center (Diagram 1). P o p u l a t i o n d e n s i t y f o r each r i n g i s p l o t t e d against the d i s t a n c e of the inner r i n g boundary and against t r a v e l time from the center. (The r i n g boundary i s used i n the study r a t h e r than the mean di s t a n c e of each r i n g as suggested by C l a r k , s i n c e the distance between the successive i n t e r v a l s of the r i n g boundary and the mean distance of each r i n g i s equal.) P o p u l a t i o n d e n s i t y of each r i n g i s found by aggregating the popu l a t i o n counts of a l l the census t r a c t s that f a l l w i t h i n the l i m i t , d i v i d e d by t h e i r t o t a l areas. Density p l o t t i n g of r i n g s i s done f o r 1966 and 1971 only. 2) L i n e s : S t r a i g h t l i n e s are drawn from the center towards east and south to the periphery (Diagram 2). The l i n e s so drawn are along Broadway and G r a n v i l l e S t r e e t s , which are two of the major primary a r t e r i e s i n Vancouver. The r a d i a l d i s t a n c e f o r each t r a c t that f a l l s on the l i n e s i s taken from the c e n t r o i d of the t r a c t to the c i t y c enter, and p o p u l a t i o n d e n s i t y i s p l o t t e d against t h i s r a d i a l d i s t a n c e . I t i s al s o p l o t t e d against t r a v e l time. 3) P i e s l i c e s : Sectors are drawn along Hastings and G r a n v i l l e at some angle at the center to i n c l u d e a considerable p o r t i o n of the metro-p o l i t a n area i n the east and south d i r e c t i o n s (Diagram 3). Density, p l o t t i n g i s done f o r each i n d i v i d u a l census t r a c t against d i s t a n c e and 27 t r a v e l time from the t r a c t centroid to the c i t y center. 4) Bands: Bands are drawn along Hastings and G r a n v i l l e : (1) one band between Hastings and Broadway, and (2) one band of one-mile width with G r a n v i l l e i n the middle (Diagram 4). Any census t r a c t whose cen-t r o i d f a l l s within the boundary of the band i s included; and the same p l o t t i n g i s done as formerly. Since an objective of the study i s to observe the changes i n r e s i d e n t i a l spread before and a f t e r the construction of the bridges and highways, the census years of 1956 and 1961, which are considered as closest to the dates of the construction, are chosen. The s t a t i s t i c s computed for these years can be compared with.those of the l a t e r census years. In t h i s way, we can see the changes i n the r e s i d e n t i a l spread over t ime. It i s apparent from the Census Tract Map of 1961 (See Appendix IV", Map 1) that a great number of the t r a c t s have undergone r e d i s t r i c t i n g s and boundary changes through the decade. The Census Tract Map of 1971 (See Appendix IV, Map 2) shows a more refined breakdown. The census t r a c t s had the same boundaries i n 1956 and 1961; and they were subdivided i n 1966. P i e - s l i c e sectors and bands along Hastings and G r a n v i l l e are drawn on the Census Tract Map of 1961. The census tra c t s that f a l l within the boundary are taken. J u s t i f i c a t i o n . " f o r the use aft.gross r e s i d e n t i a l density and various  geographical set-ups The gross r e s i d e n t i a l density measure i s much simpler than that of the other land use data, such as the net r e s i d e n t i a l density 28 or f l o o r - s p a c e d e n s i t y , and i s used "because the data are more r e a d i l y \ a v a i l a b l e , and the q u a l i t y i s s u f f i c i e n t l y c l o s e to that of net r e s i d e n t i a l 4 d e n s i t y . The use of l i n e s , s e c t o r s , and bands has a f a i r l y s t r a i g h t - f o r w a r d r a t i o n a l e . A r e l a t i v e l y s m a ll sample i s i n v o l v e d i n the use of l i n e s , which may a c c i d e n t a l l y pass through the r e g i o n a l centers of high popu-l a t i o n d e n s i t y and th e r e f o r e biases the r e s u l t s . However, the l i n e V may serve as a y a r d s t i c k against the improvement, or l a c k thereof, of the s t a t i s t i c s computed i n the l a r g e r samples used i n the case of p i e - s l i c e s e c t o r s and bands. The p i e - s l i c e s e c t o r s and bands enlarge the sample, and more accurate estimates of the parameters would presumably be obtained. The p i e - s l i c e sector i n c l u d e s i n i t s sample an approximately equal number of t r a c t s at the center and at the pe r i p h e r y , s i n c e the t r a c t s at the center are s m a l l , and those i n the o u t l y i n g areas l a r g e r . The use of sect o r s may average out the e f f e c t s of the c o n t r i b u t i o n of high p o p u l a t i o n concentrations to the d e n s i t y gradient. The t h e o r e t i c a l c e n t r a l d e n s i t y i s , f o r the p i e - s l i c e s e c t o r s and l i n e s , a value obtained f o r a poi n t at the center. On the other hand, bands would i n c l u d e r e l a t i v e l y more t r a c t s at the center and fewer at the periphery. The c e n t r a l d e n s i t y estimate i s then found not f o r a s i n g l e p o i n t , but f o r a band of some width. I t i s i n t e r e s t i n g to f i n d hhw. the v a r i o u s geographical s e t -ups would give an estimate of the c e n t r a l d e n s i t y and how the d e n s i t y gradient i n the model d i f f e r s according to the use of these geographical set-ups. 29 30 Footnotes 1. T r a v e l Time Map, GVRD, Summer 1968 2. Dynamics of R e s i d e n t i a l Land Settlement, Lower Mainland Regional Planning Board of B.C., Supp. Study 2 to Land f o r L i v i n g , 1963 3. B-A. Genest, op. c i t . , p. 14 4. I b i d . 34 CHAPTER IV ANALYSIS OF DATA A. Description of Graphs The model we employed i n the study i s empirical by nature and s t a t i s t i c a l by design; and the method of analysis i s by simple regression. (For a general d e s c r i p t i o n of regression a n a l y s i s , see Appendix I.) Computation of regression analysis for the study i s by means of the SPSS ( S t a t i s t i c a l Package for the S o c i a l Sciences) computer package of the University of B r i t i s h Columbia. '(;Eor^graphs, see Appendix t l ) Graphs 1 & 2 The graphs of 1966 and 1971 show a f l u c t u a t i n g pattern of scatter points. Population density declines gradually from Ring 1 to Ring 5. These rings incorporate mostly t r a c t s of the C i t y of Vancouver, the density of which are comparatively higher than the rest of the other m u n i c i p a l i t i e s , i n d i c a t i n g a " c r e s t " at the center. The r i n g which i s next highest i n density i s composed of census t r a c t s of Coquitlam, Richmond, and New Westminster three major urban centers. Graphs 3 & 4 Density p l o t t i n g of 1966 and 1971 i s made with respect to the t r a v e l time from the c i t y center. A l l the census t r a c t s of the whole metropolitan region are included (151 observations). Graphs 5 & 6 The graphs represent the density p l o t t i n g of census t r a c t s along Hastings to the east of the region. A sample of 12 census t r a c t s 35 i s considered. Density i s highest at the center; Burnaby has a r e l a t i v e l y higher d e n s i t y i n one of the t r a c t s w i t h i n i t s m u n i c i p a l i t y . The d e n s i t y f l u c t u a t e s i n Port Moody, Port Coquitlam and Coquitlam. In g e n e r a l , d e n s i t y d e c l i n e s w i t h d i s t a n c e from the center, but not uniformly. There i s a sharp d i s c o n t i n u i t y at some census t r a c t s i n the Burnaby m u n i c i p a l i t y , which may be due to the vacant developable land of the Burnaby Park. Graphs 7 & 8 Three c l u s t e r s of data p o i n t s are apparent from the graphs f o r the s o u t h e r l y p o r t i o n of the region. The census t r a c t s at the center undoubtedly occupy the highest d e n s i t y w i t h that of Richmond next i n magnitude and D e l t a s t i l l next. Richmond has been growing at a sub-s t a n t i a l and steady pace i n popu l a t i o n and i n d u s t r y i n the past decade. D e l t a has r e c e n t l y experienced the f a s t e s t growth among the m u n i c i p a l i t i e s i n Canada, yet i t belongs to the lowest d e n s i t y category i n m e t r o p o l i t a n Vancouver. D e l t a i s made up of f i v e g e o g r a p h i c a l l y d i s t i n c t areas: the r e s i d e n t i a l communities of South D e l t a (Tsawwassen), Ladner and North D e l t a , the a g r i c u l t u r a l lands, and the Annacis I s l a n d i n d u s t r i a l . 2 e s t a t e . Graphs 9-12 These graphs show the d e n s i t y p l o t t i n g against t r a v e l time i n the east and south d i r e c t i o n s f o r both census years 1966 and 1971. Approximately the same p a t t e r n of d i s p e r s i o n of data p o i n t s i s depicted when compared to the corresponding d e n s i t y p l o t t i n g against r a d i a l d i s t a n c e . Graphs 13 & 14 The graphs show the d e n s i t y p l o t t i n g of the census t r a c t s of the p i e - s l i c e along Hasings against r a d i a l d i s t a n c e . The data p o i n t s are c l u s t e r e d about the r e g r e s s i o n l i n e . No sharp d i s c o n t i n -u i t y i s apparent i n Burnaby. A census t r a c t i n Port Coquitlam at some 17 miles from the c i t y center records the lowest d e n s i t y . The data p o i n t s become more dispersed from 1966 to 1971 as evidenced 2 by the decreasing R . Graphs 15 & 16 The p i e - s l i c e s e c t o r of the south i s w e l l marked by three c l u s t e r s of po i n t s f o r the year 1966. There i s some d i s c o n t i n u i t y i n the s c a t t e r i n g of data p o i n t s at about s i x m i l e s from the center. The sampled census t r a c t s i n Richmond and D e l t a have r e l a t i v e l y low density. However, d e n s i t y increases considerably i n D e l t a from 1966 to 1971. Graphs 17 & 18 The way i n which the cloud of data p o i n t s i s s c a t t e r e d f o r the p i e - s l i c e s e c t o r along Hastings using t r a v e l time as a parameter i s more or l e s s the same as compared to that when r a d i a l d i s t a n c e i s used. The data p o i n t s are c l u s t e r e d r e l a t i v e l y c l o s e to the regre-s s i o n l i n e f o r 1966. They become more dispersed through the years. The sampled census t r a c t s of Port Coquitlam at about 50 minutes' d r i v i n g time from the center records the lowest d e n s i t y . Graphs 19 & 20 There i s more compact s c a t t e r i n g of data p o i n t s f o r the p i e -s l i c e s e c t o r i n the south. No d i s c o n t i n u i t y i s apparent i n the 37 re g r e s s i o n l i n e at about 20 to 25 minutes' d r i v i n g time from the center, where the north arm of the Fraser R i v e r i s l o c a t e d . Two census t r a c t s i n Richmond record r e l a t i v e l y low d e n s i t y . Graphs 21 $ 22 The p l o t t i n g of r e s i d e n t i a l d e n s i t y i n the band between Hastings and Broadway against r a d i a l d i s t ance i n v o l v e s 19 observations. The number of census t r a c t s i n cluded i n the band i s obviously l e s s than that of the p i e - s l i c e s e c t o r . As seen from the s c a t t e r i n g of data p o i n t s , the d e n s i t y p a t t e r n has changed considerably over the years at 10 miles from the center to the periphery. The d e n s i t y of the census t r a c t s of Coquitlam, F o r t Moody, and Po r t Coquitlam 2 i s higher i n 1971 than i n 1961. The R obtained i s lower than i t s corresponding p i e - s l i c e s e c t o r v a l u e u s i n g r a d i a l d i s t ance as a parameter. The data p o i n t s a l s o show more d i s p e r s i o n over time. Graphs 23 $ 24 The graphs of the band to the south f o r both years show a 2 continuous f a l l - o f f of de n s i t y against r a d i a l d i s t a n c e . The R value f o r both years are i n s i g n i f i c a n t . Graphs 25 $ 26 2 The R values drop o f f considerably from 1966 to 1971 f o r the band to the east using t r a v e l time as a parameter. Density increases at some 20 minutes' d r i v i n g time from the center outwards to the periphery. One census t r a c t i n each of Burnaby and Port Coquitlam has very low d e n s i t y . The Burnaby t r a c t contains vacant 38 'land around C e n t r a l Park. The data p o i n t s a l s o show a co n s i s t e n t 2 p a t t e r n of i n c r e a s i n g d i s p e r s i o n over time. The R f o r the r e g r e s s i o n using t r a v e l time as a parameter i s higher than the corresponding value using r a d i a l d i s t a n c e . Graphs 27 & 28 2 For the band to the south, the R values are very much higher when the model i s regressed upon travel'itime than.>when r a d i a l d i s t a n c e "is used. Two c l u s t e r s of data p o i n t s are apparents i n the graphs. Density i s highest around the c i t y core from 2 to 8 minutes' d r i v i n g time from the Genter. The second bundle of data p o i n t s , r e l a t i v e l y lower i n d e n s i t y , c o n s i s t s of t r a c t s that are 15 to 22 minutes from the center, which corresponds to about 4 m i l e s i n r a d i a l d i s t a n c e . The d e n s i t y p a t t e r n shows great v a r i a b i l i t y . I t f l u c t u a t e s consider-ably; there are data p o i n t s that have d i f f e r e n t l e v e l s of d e n s i t y at the same t r a v e l time from the center. Graphs 29 & 30 The p i e - s l i c e s e c t o r along Hastings f o r the years 1956 and 2 1961 a l s o show a co n s i s t e n t p a t t e r n of decreasing R values through 2 time. I t i s a r e l a t i v e l y b e t t e r f i t f o r the former year (R = 0 . 8 0 ) ; 2 however, the l a t t e r year a l s o gives some support f o r the model (R = 0 . 5 5 ) . The sampled census t r a c t s show higher d e n s i t y around the c i t y core, dropping o f f w i t h distance from the center. A t r a c t i n Burnaby at about 9 m i l e s from the core drops o f f i n density through the years, w h i l e Port Coquitlam increases i n d e n s i t y considerably. The data 39 p o i n t s are c l u s t e r e d about the r e g r e s s i o n l i n e from the core to about seven mil e s out, and become more dispersed towards the p e r i -phery. Graphs 31 & 32 SomeaaiLscoMinui'tyii'h' thetsGateteringiof dataapoints .. i s depicted at the north arm of the Fraser f o r the p i e - s l i c e s e c t o r along G r a n v i l l e . A t r a c t at about 11 miles from the center f a l l s 2 o f f i n d e n s i t y n o t i c e a b l y from 1956 to 1961. The R values s l i g h t l y decrease through time. Graphs 33 & 34 The d e n s i t y p l o t t i n g f o r the band to the east f o r both years 1956 and 1961 show a r e g u l a r r e l a t i o n s h i p betweeen d e n s i t y and d i s t a n c e . Data p o i n t s are r e l a t i v e l y more dis p e r s e d u i n Burnaby, 2 Coquitlam and Port Coquitlam. R decreases over time. Graphs 35 & 36 For the band to the south, the graphs show that there i s a sharp break i n the s c a t t e r i n g of data p o i n t s at the norths of the Fraser f o r the years 1956 and 1961. A census t r a c t at the periphery almost doubles i n de n s i t y through the census years. The R 2 value i s higher f o r 1956 (0.64) than f o r 1961 (0.46). Graphs 37 & 38 When the l o g d e n s i t y i s p l o t t e d against t r a v e l time f o r the p i e - s l i c e sector along Hastings, d e n s i t y drops o f f conside r a b l y i n the census t r a c t s from 19 to 27 minutes' d r i v i n g time from the center. I t i s f a i r to say that there i s more vacant developable land i n Bur-naby, which should account f o r the low average d e n s i t y . The model z. : * t to ;_!>- d a f a s .1° <»v;/ i s a 'good f i t to the data, as evidenced by a consistent r e l a t i o n s h i p between the log of density and t r a v e l time. Graphs 39 & 40 These graphs show the p l o t t i n g of log of. density against t r a v e l time f o r the p i e - s l i c e sector to the south. Density drops o f f for the census t r a c t s at the immediate v i c i n i t y of the water bodies, continues 2 to decrease i n Richmond, but r i s e s again i n Steveston. The R 's for 1956 and 1961 are lower than t h e i r corresponding values when the model i s regressed upon r a d i a l distance, a f i n d i n g contrary to expectation. Graphs 41 & 42 In the band to the east f o r 1956 and 1961, the graph using t r a v e l time shows roughly the same pattern of scatt e r as that using r a d i a l distance. Data points f l u c t u a t e considerably at about 25 to 45 minutes' 2 time from the center. The model also shows a decreasing R over time. Graphs 43 & 44 The graphs of density against t r a v e l time f o r the band to the south show that density drops o f f continuously from the c i t y center 2 outwards. R decreases through time; i t i s 0.57 and 0.43 f o r 1956 and 1961 r e s p e c t i v e l y . B. Hypotheses Discussed Hypothesis 1 Res i d e n t i a l density to the south of the CBD f a l l s more steeply than that to the east. Results: For the hypothesis to he confirmed, the b value i n the model -bx (D e ) f o r the cases considered i n the south should be larger than o fo r the east. Data was analyzed f o r four census years and the b values were computed. These are l i s t e d i n Table (1) along with the relevant 2 R and the A values obtained from the regression equations. Tab"le (1) A & b and the relevant R values of the model f o r various geographical set-ups of the four census years using r a d i a l distance; and t h e i r corresponding values for t r a v e l time i n parenthesis Geographical set-ups 1956 _ _ Sector, S. Sector, E. Band, S. Band, E. 1961 Sector, S. Sector, E. Band, S. Band, E. 9.94927 (10/7,697,6) 9.49519 (10.38463) 9.82152 (10.25926) 9.63216 (10.15263) 10.33737 (11.35954) 9.73567 (10.21197) 9.63519 £ (9.97566) 9.58201 (9.99301) 0.11472 (0.13485) 0.06640 (0.09815) 0.10423 (0.10467) 0.08405 (0.09559) 0.11730 (0.14123) 0.07428 (0.08117) 0.07486 (0.07699) 0.06982 (0.07835) 0.67279 (0.57734) 0.55393 (0.77662) 0.64310 (0.57180) 0.83600 (0.73655) 0.63460 (0.48898) 0.80048 (0.71855) 0.45852 (0.42761) 0.77033 (0.66084) 1966 L i n e , S. L i n e , E. Sector, S. Sector, E. Band, S. Band, E. 1971 L i n e , S. L i n e , E. Sector, S. Sector, E. Band, S. Band, E. 9.66111 (10.24456) 9.84693 (10.35397) 9.85094 (10.65679) 9.84228 (10.40791) 9.54759 (10.57283) 9.38258 (9.84140) 9.44072 (9.92064) 9.45773 (9.82351) 8.95351 (10.37837) 9.59935 (10.24331) 9.62282 (10.63269) 9.28304 (9.64117) 0.28895 (0.09650) 0.24061 (0.08934) 0.25702 (0.09359) 0.22191 (0.08326) 0.16530 (0.08065) 0.14716 (0.05737) 0.23078 (0.07769) 0.18097 (0.06660) 0.08114 (0.07684) 0.17807 (0.06994) 0.17242 (0.08287) 0.10425 (0.04143) 0.54199 (0.45645) 0.71590 (0.70386) 0.43456 (0.48201) 0.66800 (0.65105) 0.17092 (0.59724) 0.62008 (0.65125) 0.40133 (0.34340) 0.58316 (0.56324) 0.35411 (0.39530) 0.51984 (0.55583) 0.18107 (0.52402) 0.43038 (0.43859) The hypothesis i s g e n e r a l l y confirmed. In a l l hut one case (Sector, S., 1971), the b value to the south i s l a r g e r than to the east by 10 to 50 percent. The c o e f f i c i e n t s of determination obtained account f o r over 50 percent of the v a r i a b i l i t y f o r 13 cases out of 20. Since the b values f o r the south are g e n e r a l l y l a r g e r than f o r the east, we suggest that the d i f f e r e n c e i n d e n s i t y i n v a r i o u s d i r e c t i o n s of the c i t y i s due mainly to the age of r e s i d e n t i a l a c t i v i t y r a t h e r than to the t r a n s p o r t f a c t o r . 43 Hypothesis 2 There w i l l be a sharp d i s c o n t i n u i t y i n the s c a t t e r i n g of data p o i n t s at s i x m i l e s south of the c i t y center where the north arm of the Fraser R i v e r i s l o c a t e d . Results: The hypothesis i s not e n t i r e l y confirmed, s i n c e the s c a t t e r i n g of data p o i n t s i s s e n s i t i v e to the way they are aggregated; i . e . , d i f f e r e n t geographical set-ups may show d i f f e r e n t p a t t e r n s of the d i s p e r s i o n of data p o i n t s even f o r the same s e c t i o n of the c i t y . However, when dis t a n c e i s measured i n terms of t r a v e l time, there i s almost no d i s -c o n t i n u i t y . (For r a d i a l d i s t a n c e , see graphs 7, 8, 15, 16, 23, 24, 31, 32, 35, 36; and f o r t r a v e l time, see graphs 11, 12, 19, 20, 27, 28, 39, 40, 43, 44>) 44 Hypothesis 3 Travel time would give a better f i t to the model than r a d i a l distance. Results: The hypothesis i s not confirmed, since only 7 out of 20 2 cases (35.0 percent) have R higher than when r a d i a l distance i s used. The r e s u l t s are l i s t e d i n Table (2) along with the corres-ponding A and b values. TtibV']?e!_(?2) R 2 for the model when regressed upon t r a v e l time and r a d i a l distance distance 2 2 Geographical R (Travel time) R (Radial distance) set-ups 1956 Sector, S. * Sector, E. Band, S. Band, E. 1961 Sector, S. Sector, E. Band, S. Band, E. 1966 Line, S. Line, E. Sector, S. Sector, E. secto r , ^ : Band, S i ' Band, E. 1971 Line, S. Line, E. Sector, S. 0.57734 0.77662 0.57180 0.73655 0.48898 0.71855 0.42761 0.66084 0.45645 0.70386 0.48201 0.65105 0.59724 0.62125 0.34340 0.56324 0,39530 0.67279 0.55393 0.64310 0.83600 0.63460 0.80048 0.45852 0.77033 0.54199 0.71590 0.43456 0.66800 u.aoauu 0;17092 0.62008 0;40133 0.58316 0.35411 Sector, E. 0.55583 0.51984 Band, S. * 0.39530 0.35411 Band, E. * 0.43859 0.43038 2 The geographical set-ups w i t h a s t e r i s k s have R higher f o r the model when i t i s regressed upon, t r a v e l time than against r a d i a l d i s t a n c e . Hypothesis 4 2 R i n the model should d e c l i n e oyer time. R e s u l t s : 2 There i s a co n s i s t e n t p a t t e r n that the R f o r the l o g - l i n e a r equation decreases over the years. As the graphs revealed to us, the data p o i n t s become more dispersed through time. This p a t t e r n would be f u r t h e r dispersed by the c r e a t i o n of the r e g i o n a l centers i n the 2 f u t u r e . The R. values of v a r i o u s geographical set-ups of the four census years are presented i n Table (3) when x i n the model i s measured i n terms of r a d i a l d i s t a n c e , and i n Table (4) when x i s measured i n terms of t r a v e l time. 2 Table (3): R f o r the model when re g r e s s i n g upon r a d i a l d i s t a n c e Geographical ; «set-ups 1956 1961 1966 1971 East L i n e Sector Band Average South Line Sector Band / " a / a r e 0.80048 0.83600 8182 ' / 0.55393 0.77033 0.6621P 0.67279 0.64310 •0.65795-0.63460 0.45852 0 54656 0.71590 0.66800 0.62008 0-5679? 0.54199 0.43456 0.17092 0.38249 0.58316 0.51984 0.43038 0.40133 0.35411 0.18107 47 Table (4): :R 2 f o r the model when r e g r e s s i n g upon t r a v e l time Geographical set-up 1956 1961 1966 1971 East Line 0.70386 0.56324 Sector 0.77662 0.71825 0.65105 0.55583 Band 0.73655 0.66084 0.65125 0.43859 Average_ . f,. 75659 0.68969 6587:. "w -:~ South Line 0.45645 0.34340 Sector 0.57734 0.48898 0.48201 0.39530 Band 0.57180 0.42761 0.59724 0.52402 .; veragec. 0„574r./- 0.4582S. C51190 (j /• •' " The R 's f o r the GVRD when x i s measured i n terms of r a d i a l d i s t a n c e are 0.56914 and 0.51460 f o r 1966 and 1971 r e s p e c t i v e l y ; and when x i s measured i n terms of t r a v e l time, they are 0.43245 and 0.38894. r e s p e c t i v e l y . Hypotheses 5 & 6 A & b values i n the f i t t e d model should decline over time. Results: There i s not a consistent pattern that A value i n the model declines over time. However, i n almost a l l cases, the b value shows a decrease as time progresses. This indicates that population den-s i t y gradient has become f l a t t e r . The A and b values f o r the various geographical set-ups at four points i n time when the model i s regressed upon r a d i a l distance are to be found i n Table (1), page 41. As f o r the A and b values when the model i s regressed upon t r a v e l time, they are l i s t e d i n Table (5). Table (5) A & b values i n the model when regressed upon t r a v e l time Geographical set-ups A 1956 Sector, S. Sector, E. Band, S. Band, E. 1961 Sector, S. Sector, E. Band, S. Band, E. 1966 Line, S. Line, E. Sector, S. Sector, E. Band, S. ' Band, E. 10.76976 10.38463 10.25926-10.15263 11.35954 10.21197 9.97566 9.99301 10.24456 10.35397 10.65679 10.40791 10.57283 9.84140 0.13485 0.09815 0.10467 0.09559 0.14123-'" 0.08117 0.07699 0.07835 0.09650 0.08934 0.09359 0.08326 0.08065 0.05737 1971 L i n e , S' L i n e , E. Sector, S. Sector, E. Band, S. Band, E. 9.92064 9.82351 10.37837 10.24331 10.63269 9.64117 0.07769 0.06660 0.07684 0.06994 0.08287 0.04143 The r e l a t i v e increase i n suburban d e n s i t i e s would lower the b value. The r e s u l t s obtained are comparable to previous ex-perience w i t h C l a r k ' s model. Thus, i t seems f a i r to conclude that Vancouver i s s i m i l a r to other major c i t i e s , at l e a s t as f a r as the b value i s concerned, and Cla r k ' s model a p p l i e s to Vancouver f a i r l y w e l l . CHAPTER V CONCLUSION The a p p l i c a t i o n of the model to the Vancouver case enables us to h i g h l i g h t the nature of the c i t y growth process, and to o f f e r exp-l a n a t i o n s f o r the po p u l a t i o n d e n s i t y d i s t r i b u t i o n . In a capsule, the f i n d i n g s can be o u t l i n e d as f o l l o w s : J. The d i f f e r e n t i a l r a t e s of d e n s i t y d e c l i n e : The model i s found to apply to the south and east s e c t i o n s of the c i t y at v a r i o u s p o i n t s i n time. However, the b value to the south i s l a r g e r than to the east. A higher b value means that d e n s i t y d e c l i n e s more shar p l y w i t h i n c r e a s i n g d i s t a n c e from the c i t y center; and a lower b value means that d e n s i t y d e c l i n e s more slo w l y . The d i f f e r e n t b values of both s e c t i o n s of the c i t y suggest that r e s i d e n t i a l d e n s i t y i s more a f u n c t i o n of the age of development than t r a n s p o r t f a c t o r . I n the past r e s i d e n t i a l development i n the south has been l a r g e l y r e s t r i c t e d to the Burrard Peninsula as a r e s u l t of the water b a r r i e r s . Though improvements i n t r a n s p o r t a t i o n occurred i n the l a t e 50s, r e s i d e n t i a l b u i l d i n g a c t i v i t y has not been able to grow f a s t enough to compensate f o r i t s l a t e s t a r t of development. For tr a n s p o r t to be a . r e l a t i v e l y more important f a c t o r than the time at which develop-ment took p l a c e , the b value to the south should be s i m i l a r to that towards the east when sep a r a t i o n from the CBD i s measured i n terms of r a d i a l d i s t a n c e or t r a v e l time. In f a c t t h i s i s not the case. The higher b value to the south i s a l s o a s s o c i a t e d w i t h a higher A value i n the model. For $5 out of 20 cases considered, the south has 51 higher imputed c e n t r a l d e n s i t y than the east. The r e l a t i v e l y higher A and b values f o r the south i n d i c a t e that d e n s i t y i s more c l u s t e r e d and concentrated between the c i t y center and the north arm of the Fraser River as a r e s u l t of the topographical r e s t r i c t i o n . 2. T r a v e l time vs r a d i a l d i s t a n c e : Density i n the south seems to respond more to t r a v e l time than to r a d i a l d i s t a n c e . There i s almost no d i s c o n t i n u i t y i n the s c a t t e r i n g of data p o i n t s i n the region where the north arm of the Fraser R i v e r i s l o c a t e d when the model i s regressed upon t r a v e l time. On the other hand, there i s no co n c l u s i v e evidence that t r a v e l time gives a b e t t e r f i t to the model than r a d i a l d i s t a n c e . 3. The model as a good f i t : The f a i r l y good f i t of the model f o r 1966 and 1971 i s i n d i -c a t i v e of the f a c t that the negative e x p o n e n t i a l decay f u n c t i o n can describe the den s i t y d i s t r i b u t i o n even f o r a c i t y which i s h i g h l y r e s t r i c t e d i n s i t e and elongated i n shape. However, there i s evidence that the model has d e c l i n e d i n i t s a p p l i c a b i l i t y over time, which 2 i s r e f l e c t e d by the d e c l i n i n g R values. Data have become more d i s -persed and random. The more dispersed p a t t e r n of data may r e f l e c t the' appearangfePof s nodes1"o£^po;p^latiqn cone exit r a t i o n i n some of the o u t l y i n g m u n i c i p a l i t i e s , such as Burnaby and New Westminster. 4. The A and b values i n the model: The A value has not shown a c o n s i s t e n t p a t t e r n of d e c l i n e over the years even w i t h i n the same geographical set-up. The b value has shown a d e c l i n e as time progresses. There i s a r e l a t i v e increase i n suburban d e n s i t i e s , which i s a r e s u l t of the improvement i n t r a n s p o r t a t i o n and the r e l a t i v e decrease, i n im-52 portance i n transport cost as compared w i t h other l o c a t i o n a l c o n s i d e r a t i o n s . As C l a r k says: "{.There are] two p o s s i b i l i t i e s f o r development, i f the p o p u l a t i o n i s i n c r e a s i n g . E i t h e r t r a n s p o r t costs are reduced, enabling the c i t y to spread out, or they cannot be reduced, i n which case d e n s i t y has to increase at a l l p o i n t s . " 5. Methodology: There i s no d e c i s i v e evidence as to the s u p e r i o r i t y i n the use of one geographical set-up over the other. Conceptually, the use of s e c t o r s i s best because i t i s able to i n c o r p o r a t e s i m i l a r -numbers of census t r a c t s at d i f f e r e n t d i s t a n c e s from the center. An approximately equal number of census t r a c t s both at the center and at the periphery i s included i n i t s l i m i t . The s i z e of census t r a c t s u s u a l l y r e f l e c t s i t s d e n s i t y , i . e . , a smaller p a r c e l of t r a c t i s u s u a l l y a s s o c i a t e d w i t h a higher d e n s i t y ; and v i c e v e r s a . 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Review, Vol. LVI, 1966, pp. 213-225. • , "The S p a t i a l V a r i a t i o n of Urban Population Densities", Geog. Review 59, 1969, pp. 242-252. Pendleton, William C., "Relation of Highway A c c e s s i b i l i t y to Urban Real Estate Values", HRR 16, 1963, pp. 14-23. P u r n e l l , J. Stanley, "Planning Transportation F a c i l i t i e s to Guide Urban Development", T r a f f i c Quarterly 20, 1966, pp. 277-287. Row, Arthur & Ernest J.urkat, "The Economic Forces Shaping Land Use Patterns", JAIP XXV, May 1959, pp. 77-81. Schneider, Jerry B. & Joseph R. Beck, "Reducing the Travel Requirements of the American C i t y : An Investigation of A l t e r n a t i v e Urban S p a t i a l Structures", Transportation Research Record 449, Wash., D.C, 1974, pp. 12-33. Smith, Wilbur S. & Theodore Matson, " W i l l Large C i t i e s F i n a l l y Succumb to Transportation Crises?", T r a f f i c Quarterly 6, 1952, pp. 402-415. Stegman, M.A., " A c c e s s i b i l i t y Models and Res i d e n t i a l Location", JAIP 35, pp. 22-39. Stein, Martin 0., "A Planning Model for Transportation i n Urban A c i t i v i t y Centers", HRR 1971. Stewart, J.Q., "Empirical Mathematical Rules Concerning the D i s t r i b u t i o n and Equilibrium of Population", Geog. Review, Vol. 37, 1947, pp. 461-485. & W i l l i a m Wartz, "Physics of Po p u l a t i o n D i s t r i b u t i o n " , J o u r n a l of Reg. Sc., V o l . 1, 1958, pp. 99-123. Sussna, Stephen, "Zoning as a T r a f f i c Remedy", T r a f f i c Q u a r terly 16, 1962, pp. 433-441. Swerdloff, C a r l N., " R e s i d e n t i a l Density S t r u c t u r e : An A n a l y s i s and Forecast w i t h E v a l u a t i o n " , HRR 207, Urban Land Use: Concepts and Models, 1967. Voorhees, Alan M., "Urban T r a v e l and C i t y S t r u c t u r e " , HRR 1970, pp. 121-135. , Charles F. Barnes, J r . , F r a n c i s E. Coleman, " T r a f f i c P a t t e r n and Land Use A l t e r n a t i v e s " , HRB B u l l 347, 1962, pp. 1-9. Weiss, Herbert K., "The D i s t r i b u t i o n of Urban P o p u l a t i o n and an A p p l i c a t i o n to a S e r v i c i n g Problem", Operations Research 9, 1961, pp. 860-873. Wolfe, R.I., " E f f e c t of Ribbon Development on T r a f f i c Flow", T r a f f i c Quarterly 18, 1964, pp. 105-117. W o l f o r t h , John R., " R e s i d e n t i a l L o c a t i o n and the Pl a c e of Work", Geog. S e r i e s , No. 4, 1965. REPORTS Bartholomew, Harland, & Associates, A Plan f or the C i t y of Vancouver, B r i t i s h Columbia, Including a General Plan of the Region 1928, 1928. Genest, Bernard-Andre, "Population D i s t r i b u t i o n Functions f o r Urban Areas", Vol. 2 i n a Series on Ai r p o r t Location and Planning, Research Rept. R-70-53, Aug. 1970. Jackson, J.N., et a l . , The Impact of Highway Development on Land Use,  A Study of Selected L o c a l i t i e s i n the Greater Vancouver Area, Research Rept., No. 1, Comm. & Reg. Planning, UBC, June 1963. Our F i f t y Years 1890-1940, A brochure issued by the P a c i f i c Coast F i r e Insurance Co., Vancouver, B.C., 1940. VATS, 1968, Ci t y of Vancouver, B r i t i s h Columbia. Technical Rept., Part 2: The Urban F r o n t i e r , Lower Mainland Regional Planning Board of B r i t i s h Columbia, Supp. Study 4 to Land f o r L i v i n g . Dynamics of Re s i d e n t i a l Land Settlement, Lower Mainland Regional Planning Board of B r i t i s h Columbia, Supp. Study 2 to Land for L i v i n g . Downtown Vancouver, Rept. f o r Discussion, C i t y of Vancouver, Sept. 1974. Rept. on the Greater Vancouver Area, Rapid T r a n s i t Study, prepared for the j o i n t Transportation Committ ee, Greater Vancouver Regional D i s t r i c t & B.C. Hydro & Power Authority, Sept., 1970, by De Lewn Catha & Co. of Canada Ltd. i n assoc. with P h i l l i p s Barratt, H i l l e r , Jones & Partners, Hans Blumenfeld. The Livable Region 1976/1986, The Greater Vancouver Regional D i s t r i c t . UNPUBLISHED MATERIALS Jones, Gerald Ieuen Howell, A Century of Settlement Change: A Study  of the Evolution of Settlement Patterns i n the Lower Mainland of B r i t i s h Columbia, unpublished M.A. t h e s i s , Dept. of Geography, The Univ. of B r i t i s h Columbia, A p r i l , 1966. Roy, P a t r i c i a E., Railways, P o l i t i c i a n s and the Development of the  C i t y of Vancouver as a Metropolitan Centre 1886-1929, unpublished M.A. t h e s i s , Univ. of Toronto, Oct. 1963, NEWSPAPERS Province: 3 A p r i l 29, 1937, " I n d u s t r i a l Expansion Supplement: Greater Vancouver of the Future w i l l be World-Famous Metropolis". 60a Appendix I : Regression A n a l y s i s REGRESSION ANALYSIS Regression a n a l y s i s i s concerned w i t h the problem of d e s c r i b -ing or e s t i m a t i n g the value of one v a r i a b l e , c a l l e d the "dependent v a r i a b l e " , on the b a s i s of one or more other v a r i a b l e s , c a l l e d "independent v a r i a b l e s " . I t i s hypothesized that the behaviour observed f o r a dependent v a r i a b l e can be accounted f o r by the independent v^rjuablres.., The model used i n t h i s work i n v o l v e s two parameters: extrapolatedd c e n t r a l d e n s i t y and d e n s i t y g r a d i e n t , which are used to i n d i c a t e the negative exponential d e c l i n e of p o p u l a t i o n d e n s i t y w i t h the i n c r ease i n d i s t a n c e . I t i s important to note that the inferences from the data are derived from a conceptual and t h e o r e t i c a l i n t e r -p r e t a t i o n of the s t a t i s t i c a l r e s u l t s ; and from a s t a t i s t i c a l s tandpoint, there should not be any i m p l i c a t i o n of causation, or d i r e c t i o n of "cause and e f f e c t " i n v o l v e d . 1 The Method of Least Squares i s most o f t e n used i n determining the probable values of the dependent v a r i a b l e from the independent v a r i a b l e ( s ) . I t r e q u i r e s f i t t i n g a c e n t r a l l i n e , or the b e s t - f i t t i n g l i n e , to the data, where the sum of squares of the v e r t i c a l d e v i a t i o n s of the observations f o r the dependent v a r i a b l e from the l i n e i s a minimum. In other words, the Method of Least Squares gives the minimum e r r o r v a r i a n c e f o r the purpose of p r e d i c t i n g one property 2 from a knowledge of the other. The r e g r e s s i o n equation can be i d e n t i f i e d by: y = a + bx + u where: y i s the dependent v a r i a b l e x i s the independent v a r i a b l e a, b are constants, which are to be determined from the equation u i s the " d i s t u r b a n c e " or " e r r o r " term, which may take on p o s i t i v e or negative values. Among the many f a c t o r s that c o n t r i b u t e to the i n s e r t i o n of the u term, the more important ones are the measurement e r r o r and s p e c i f i c a t i o n e r r o r ; the l a t t e r means that the a c t u a l phenomena may not be f u l l y represented by the expressmond b T h e h s p e e i f i e a t i o n e r r o r i s i n v e r s e l y r e l a t e d to the number of v a r i a b l e s involved i n the expression, w h i l e the measurement e r r o r i s d i r e c t l y r e l a t e d to the number of v a r i a b l e s . The assumptions i n v o l v e d i n the s p e c i f i c a t i o n of the disturbance term are homoscedasticity and independence, t h a t , i s , the v a r i a n c e i s a constant and independent of x, and the values 3 are independent of one another. I t i s u s u a l l y a common p r a c t i c e to give a f u l l a p p r a i s a l of the r e l a t i o n s h i p by means of a " s c a t t e r diagram" or "cor-r e l a t i o n r e l a t i o n diagram". The c e n t r a l l i n e f i t t e d to the data p o i n t s i s c a l l e d a " r e g r e s s i o n " l i n e . The r e l a t i o n s h i p between the dependent v a r i a b l e and the independent v a r i a b l e s (s) i s expressed by a tendency f o r one to increase or decrease w i t h an increase i n the other. For the v a r i a b l e s to be h i g h l y c o r r e l a t e d , a l l the data p o i n t s are c l u s t e r e d about the r e g r e s s i o n l i n e , or n e a r l y so. Often i n the case of n o n - l i n e a r i t y , a transformation by way of lo g a r i t h m , r e c i -p r o c a l , or square root of one or both v a r i a b l e s can be made to 4 e l i m i n a t e the curvature. R e f e r r i n g to the model we a p p l i e d , there i s a l o g a r i t h m i c transformation of the popu l a t i o n d e n s i t y at a distance x. ( D x ) , and imputed c e n t r a l d e n s i t y ( D Q)• The extent of the p r e c i s i o n of the r e g r e s s i o n depends on the f o l l o w i n g f a c o r s : 1) the number of observations; and 2) the extent of the s c a t t e r about the r e g r e s s i o n . For any value of y, there i s a corresponding value of x, where y^ = a + bx^ + u^ y = a + b x + u . The discrepancy between the t h e o r e t i c a l n n n /A value of y, denoted by y, and the observed value of y may be regarded as e r r o r s , the measure of which i s known as the "variance". tnotes Donald J . Bogue & Dorothy L. H a r r i s , "Comparative P o p u l a t i o n and Urban Research v i a M u l t i p l e Regression and CcVariate Analy-s i s " , Scripps Foundation f o r Research i n P o p u l a t i o n Problems, 1954, p.7 Owen L. Davies (ed.), S t a t i s t i c a l Methods i n Research and Pro- d u c t i o n , O l i v e r and Boyd, Edingburg, 1947, p. 123 J . Johnston, Econometric Methods, McGraw-Hill Book Co., New York, 2nd Ed., 1972, pp. 8-12 Owen L. Davies, op. c i t . , p. 120 Appendix I I : Graphs 64b (3)') GVRD, T r a v e l Time, 1966 (4) GVRD, Tr a v e l Time, 1971 ( I ; ot s' ,65 66 (9) Line, East?lTi?avel Time, (10) Line, East", Travel Time, 1966 1971 (11) Line, South, Travel Time, (12) Line, South, Travel Time, 1966 1971 67 (15) Sector, South, R a d i a l Distance, 1966 (16) Sector, South, R a d i a l Distance, 1971 68 (17) Sector, East, T r a v e l Time, 1966 (18) Sector, East, T r a v e l Time 1971 (19) Sector, South, T r a v e l Time, 1966 (20) Sector, South, T r a v e l Time, 1971 69 (23) Band, South, R a d i a l Distance, (24) Band, South, R a d i a l Distance 1966 1971 (25) Band, East, T r a v e l Time, (26) Band, East, T r a v e l Time, 1966 1971 (27) Band, South, T r a v e l Time, 1966 (28) Band, South, T r a v e l Time, 1971 (31) Sector, South, R a d i a l Distance, 1956 (32) Sector, South, R a d i a l Distance, 1961 72 (33) Band, East, R a d i a l Distance, 1956 ] 1 1 — i 1 6 -4- * ^ (35) Band, South, R a d i a l Distance, 1956 (34) Band, East, R a d i a l Distance, 1961 (36) Band, South, RAdial Distance, 1961 73 (39) Sector, South, T r a v e l Time,> 1956': (40) Sector, South, T r a v e l Time, 1961 74 (4,3) Band, South, T r a v e l Time, 1956 (44) Band, South, T r a v e l Time, 1961 Appendix I I I : S t a t i s t i c a l Tables Relationship Between Density and RadialoDistance From the Cit y Centre  Geog. Set- „ ups/1966 A b r No. R Graph GVRD 9.12414 0.14715 -0.75441 23 0.56914 1 LINE,::"E. 9.84693 0.24061 -0.84611 12 0.71590 5 LINE,"S. 9.66111 0.28895 -0.73620 14 0.54199 7 SECTOR, E. 9.84228 0.22191 -0.81731 33 0.66800 13 SECTOR SECTOR, S. 9.85094 0.25702 -0.65921 32 0.43456 15 BAND, E. 9.38258 0.14716 -0.78745 19 0.62008 21 BAND, S. 9.54759 0.16530 -0.41343 21 0.17092 23 Relationship Between Density and Radial Distance From the Cit y Centre  Geog.Set- 2 ups/1971 A b r r No.r - R Graph GVRD 9.15360 0.13229 -0.71736 23 0.51460 : 2 LINE, E. 9.45773 0.18097 5017,6365 •12-- 763550.58316 6 LINEj, S. 9.44072 0.23078 -0.63350 14 0.40133 8 SECTOR, E. 9.59935 0.17808 -0.72100 33 0.51984 14 SECTOR, S. 8.95351 0.08114 -0.59507 32 0.35411 16 BAND, E. 9.28304 0.10425 -0.65603 19 0.43038 22 BAND, S. 9.62282 0.17242 -0.42552 21 0.18107 24 R e l a t i o n s h i p Between Density and Travel Time From the C i t y Center Table 3 Geog. Set-ups/1966 A b r No. R2 Graph GVRD 10.41882 0.08161 -0.65761 151 0.43245 3 LINE, E. 10.35397 0.08934 -0.83896 12 0.70386 9 LINE, S. 10.24456 0.09650 -0.67561 14 0.45645 11 SECTOR, E. 10.40791 0.08326 -0.80688 26 0.65105 17 SECTOR, S. 10.65679 0.09359 -0.69427 29 0.48201 19 BAND, E. 9.84140 0.05737 40.78820 21 0.65125 25 BAND, S. 10.57283 0.08065 -0.77282 28 0.59724 27 Relationship Between Density and Travel Time From the Cit y Center Table 4 Geog. Set-ups/1971 A b r No. R 2 Graph GVRD 10.26468 0.07217 -0.62365 151 0.38894 4 LINE, E. 9.82351 0.06660 -0.75049 12 0.56324 10 LINE, S. 9.92064 0.07769 -0.58600 14 0.34340 12 SECTOR, E. 10.24311 0.06994 -0.74554 26 0.55583 18 SECTOR, S. 10.37837 0.07684 -0.62873 29 0.39530 20 BAND, E. 9.64117 0.04143 -0.66226 21 0.43859 26 BAND, S. 10.63296 0.08287- -0.72389 28 0.52402 28 00 Relationship Between Density and Radial Distance From the City Center Table 5 Geog. Set-ups/1956 A b r No. R2 Graph SECTOR, E. 9.73567 0.07428 -0.89469 17 0.80048 29 SECTOR, S. 9.94927 0.11472 -0.82024 16 0.67279 32 BAND, E. 9.63216 0.08405 -0.91433 15 0.83600 33 BAND, S. 9.82152 0.10423 -0.80193 15 0.64310 35 Relationship Between Density and Radial Distance From the Cit y C e n t e r — -Table 6 Geog. Set-ups/1961 A b r No. R2 Graph SECTOR, E. 9.49519 0.06640 -0.74426 17 0.55393 30 SECTOR, S. 10.33737 0.11730 -0.79662 16 0.63460 31 BAND, E. 9.58201 0.06982 -0.87769 15 0.77033 34 BAND, S. 9.63519 0.07486 -0.67714 15 0.45852 36 Relationship Between '. Density and ' Travel Time From the Cit y Center Table 7 Geog. Set-ups/1956 A b r No. R 2 Graph SECTOR, E. 10.38463 0.09815 -0.88126 17 0.77662 37 SECTOR, S. 10.76976 0.13485 -0.75983 16 0.57734 39 BAND, E. 10.15263 0.09559 -0.85823 15 0.73655 41 BAND, S. 10.25926 0.10467 -0.75618 15 0.57180 43 Relationship Between Density and Travel Time From the Cit y Center Table 8 Geog. Set-ups/1961 A b r No. R2 Graph SECTOR, E. 10.21197 0.08117 -0.84767 17 0.71855 8 SECTOR, S. 11.35954 0.14123 -0.69927 16 0.48898 40 BAND, E. 9.99301 0.07835 -0.81292 15 0.66084 42 BAND, S. 9.97566 0.07699 -0.65392 15 0.42761 44 OO o Appendix IV: Maps Map of Census Tract s of M e t r o p o l i t a n Vancouver, 1971 Map 2 I M E T R O P O L I T A N V A N C O U V E R T r a v e l Time In GVRD Source: Dynamics of R e s i d e n t i a l  Land Settlement, LMRPB of B.C., 1963 M a p 3 T r a v e l Time i n GVRD Summer 1968. Source: T r a f f i c Volumes T r a v e l Times, 1967/1968 Planning Dept., 'GVRD. Map 4  

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