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Some non-linear functions fitted to actual volume-age data Sagary-Nokoe, Tertius 1974

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SOME NON-LINEAR FUNCTIONS FITTED TO ACTUAL VOLUME-AGE DATA by T e r t i u s Sagary-Nokoe B.Sc. (For.) Hons. U n i v e r s i t y of Ibadan, N i g e r i a A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY IN THE DEPARTMENT OF FORESTRY UNIVERSITY OF BRITISH COLUMBIA We accept t h i s t h e s i s as conforming t o the req u i r e d standard August 1974 In presenting th is 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 is understood that copying or pub l i ca t ion of th i s thes is for f i nanc ia l gain sha l l not be allowed without my wri t ten permission. Department of PofcfcS T ^ The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date i i A B S T R A C T Seven non-linear functions are f i t t e d to eight groups of data, and the best three functions selected for each group. Selection i s based on the cor-re l a t i o n index, mean bias and the p a r t i a l and ov e r a l l standard error of e s t i -mate. Most of the groups of data are adequately described by the Gompertz "C (X"*G) curve, V = ae~ e where a i s the ultimate l i m i t i n g value, G i s a parameter defining the point of i n f l e c t i o n (maximum current annual increment), c defining the curve shape, V i s volume i n cubic feet (close u t i l i z a t i o n standard) and X i s age i n years. Comparison of the functions with the B r i t i s h Columbia Forest Service volume/age curves (VAC.s) , where appropriate, indicated that these are not very different from each other. I t might, therefore, be possible to describe some of the existing VAC's by suitable non-linear functions. i i i TABLE OF CONTENTS Page TITLE PAGE i ABSTRACT 1% TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i INTRODUCTION 1 LITERATURE REVIEW 5 THE DATA . 13 VOLUME FUNCTIONS AND FITTING PROCEDURE 17 PARAMETER G AND DOUGLAS-FIR 19 LINEAR ESTIMATION . . . 20 SELECTION OF THE BEST EQUATIONS 22 RESULTS 26 DISCUSSION AND SUGGESTIONS 37 CONCLUSIONS 48 REFERENCES 49 APPENDICES: 1. Tests on empirical data: Normality . . . . . 52 2. Tests on empirical data: Homogeneity of variance . . . . . . 53 3. Scattergrams * 54 4. Inflection points: multimolecular (numbers 2 & 3) 58 5. Inflection points: Gompertz curve and Verhulst function . . . 59 i v Page APPENDICES (cont'd.): 6. Pearl and Reed function (Grosenbaugh's suggested parameters) 60 7. Predicted functions - cedar group (medium s i t e - VAC 2 6 5 ) . . . 61 8. Predicted functions - cedar group (poor s i t e - VAC 266) . . . 62 9. Predicted functions - pine group (good s i t e - VAC 578) . . . . 63 10. Predicted functions - pine group (medium s i t e - VAC 79) . . . 64 11. Predicted functions - pine group (poor s i t e - VAC 572) . . . . 65 12. Predicted functions - Douglas-fir group (good s i t e - VAC 757) 66 13. Predicted functions - Douglas-fir group (medium s i t e - VAC 968) 68 14. Predicted functions - Douglas-fir group (poor s i t e - VAC 575) 69 15. 70 LIST OF TABLES Table Page I Data Summary A . . 15 I I Data Summary B 16 I I I Fixing Origin for In t e r i o r Douglas-fir 21 IV S t a t i s t i c s for f i t t e d functions - Cedar, medium s i t e 28 V S t a t i s t i c s for f i t t e d functions - Cedar, poor s i t e 29 VI S t a t i s t i c s for f i t t e d functions - Pine, good s i t e . . . . . . . 30 VII S t a t i s t i c s for f i t t e d functions - Pine, medium s i t e 31 VIII S t a t i s t i c s for f i t t e d functions - Pine, poor s i t e 32 IX S t a t i s t i c s for f i t t e d functions - Douglas-fir, good s i t e . . . 33 X S t a t i s t i c s for f i t t e d functions - Douglas-fir, medium s i t e . . 34 XI S t a t i s t i c s for f i t t e d functions - Douglas-fir, poor s i t e . . . 35 XII Some properties of the selected functions . . . . . 36 XIII Behaviour of parameter estimates - Gompertz and Verhulst functions • 39 v i LIST OF FIGURES Figure Page 1 The Growth of stands - volume/age curve 3 2 The Growth of stands - mean annual increment and current annual increment curves . . . . . 3 3 Pine - Gompertz functions/curves 42 4 Cedar - Gompertz functions/curves . . . 43 5 Douglas-fir - Gompertz functions/curves 44 v i i A C K N O W L E D G E M E N T S This work was undertaken under the supervision of Dr. Antal Kozak to whom I am greatly indebted. I am grateful to the members of my committee — Drs. J . P. Demaerschalk, A. Kozak, D. D. Munro and J . H. G. Smith — for t h e i r guidance, useful sugges-tions and review of the thesis. Permission to use University of B r i t i s h Columbia computing f a c i l i t i e s , and the help given by Mr. Steve Smith i n discussing one of the multiraolecular functions (Function 3) with me, are acknowledged. Financial support i n the form of Faculty of Forestry Teaching A s s i s t -antship and of the B r i t i s h Columbia Forest Service through the Productivity Com-mittee grant for project PC 006 and provision of data for analysis, are apprec-iated. 1 INTRODUCTION The contribution of forest mensuration to management i s through methods of measurement and estimation appropriate to p a r t i c u l a r problems of management. The p a r t i c u l a r management problem must be appreciated, and a decis-ion made on the method most appropriate to get the required information. In this thesis, an attempt i s made to find a p a r t i a l solution to one of several management problems — that of obtaining the most suitable and simple non-linear volume/age functions with coefficients which can be assigned some important bio-l o g i c a l meaning. Non-linear growth functions have been f i t t e d to several growth forms, but not much use has been made of volume/age non-linear functions, the previous objections being based mainly on the complication i n the computation of the par-ameters. S t a t i s t i c a l objections to the use of these functions had been centered on the lack of exactness i n goodness of f i t . Volume i s one of the most important indices of productivity, being used i n the formulation of forest management p o l i c i e s and i n the assessment of the differences among several s i l v i c u l t u r a l treatments. In p a r t i c u l a r , volume/ age curves or functions have been used to determine the annual allowable cut, and their existence i n the form of y i e l d tables giving average volumes of stands of various ages, dates back to the late eighteenth century (Spurr, 1952). The determination of the annual cut i s based on the culmination of the mean annual increment (MAI) and the rotation age. The maximum MAI coincides with the physi-c a l rotation, that rotation y i e l d i n g the largest quantity of material per unit area over the length of a period, but does not imply maximum return of the i n -vestment. Nevertheless, i t i s one of the several characteristics considered i n 2 deciding on the f i n a n c i a l rotation, which could be defined as the rotation de-termined by f i n a n c i a l considerations. Moreover, because f i n a n c i a l rotation changes with demand, forest managers consider the maximum MAI a more permanent to o l i n the decision on rotation age. Important points on any volume/age curve are the points of i n f l e c t i o n (maximum current annual increment, CAI), the point of tangency with the l i n e passing through the o r i g i n (0,0) (maximum MAI), the upper asymptote (ultimate l i m i t i n g value of the response) and the lower asymptote (lowest value of the response). Figures 1 and 2 i l l u s t r a t e these points. Mathematically, the maxi-mum value i s at the age at which the f i r s t derivative of the function i s zero. For the point of i n f l e c t i o n , the second derivative i s zero, with a further condi-tion that the t h i r d derivative i s not zero. Growth curves are b a s i c a l l y sigmoid i n shape. The curve of the CAI (growth i n a year) continually rises and then f a l l s , intersecting the MAI curve at the l a t t e r ' s maximum point. From this point onwards, the MAI curve i s a l -ways above the CAI curve (Figure 2). The stand begins to retrograde when f u r -ther loss i n volume of trees, and i n increase i n decay of l i v i n g trees, exceed the sound growth of the stand. From this stage, the CAI curve becomes negative. S t r i c t l y speaking, the MAI curve being a curve of averages (growth during a period) does not reach zero unless the stand ceases to e x i s t . Volume/ age curves of the B r i t i s h Columbia Forest Service (BCFS) are hand-fitted, and they are available for each growth type i n a given zone on a given s i t e . • These curves, based on classes of heights of dominant and co-domin-ant trees i n the stands, show average volume per acre i n cubic feet - net with reductions for decay only and a close u t i l i z a t i o n standard. They are used 3 FIGURE 1 T H E G R O W T H O F S T A N D S - VOLUME/AGE CURVE Ul 2 o > I-5 o O AGE ( YEARS ), X FIGURE 2 T H E G R O W T H O F S T A N D S M E A N A N N U A L I N C R E M E N T A N D C U R R E N T A N N U A L I N C R E M E N T C U R V E S CURRENT ANNUAL GROWTH (CAI) AGE (YEARS), X NOTATION : L x — lower asymptote (or value ), (or age of zero growth) U x - upper asymptote (or age of maximum volume growth) Ix — Inflection point (age corresponding to maximum current annual increment) f x — age at which a line through the origin (0,0) touches the curve (tangent) (culmination of mean annual increment - rotation of greatest volume production) NOTE, CAI is growth in volume for a year; MAI is average of volumes at specified intervals, say 5 or 10 years. MAI-not a growth curve but curve of averages. If curve has no inflection, then maximum CAI is in the origin. Figures not drawn to scale. 4 extensively throughout the Province of B r i t i s h Columbia. I n d i v i d u a l curves d i f f e r i n l e v e l , shape, steepness, complexity and the age taken f o r the p a r t i c -u l a r growth type to grow i n t o the 7.1 inches diameter at breast height (dbh) c l a s s . This implies that a p a r t i c u l a r curve may only be used f o r a s p e c i f i c growth type and/or species.group. This i s where the importance of f i t t i n g mathematical functions to the data or to describe e x i s t i n g curves (assuming them to be correct) comes i n . I f s u i t a b l e functions could be f i t t e d to each growth type, t h i s might lead to the formulation of a generalized volume/age function f o r a l l growth types on a l l s i t e s . The study of the problem w i l l involve the f i t t i n g of mathematical functions, mostly non-linear, considered from past studies as being capable of describing sigmoid r e l a t i o n s h i p s e f f e c t i v e l y . S e l e c t i o n of the functions f o r each species group and growth type w i l l be based on the standard e r r o r of e s t i -mate, the mean biases for each age and the c o r r e l a t i o n i n d i c e s , and w i l l be com-pared with e x i s t i n g BCFS volume/age curves (VAC) where appropriate. An attempt w i l l then be made to suggest one non-linear function, which appropriately des-cribes a l l the groups to be investigated. 5 LITERATURE REVIEW The theory of f i t t i n g empirical data has been described by Hartley (1948, 1961), Box and Lucas (1959), Box (I960, 1964), Lipton and McGilchrist (1964), Freese (1964), Gutman and Meeter (1964), Cornell (1965), Draper and Smith (1966) and several other authors. The f i r s t portion of the review i s based on these works, with emphasis on the parts connected with the present study. The l a t t e r part of the review i s on the discussions of past works on models used i n the study. Whenever a mathematical model i s assumed or postulated, our interest i s centered on the possible adequacy or inadequacy of the model, estimates of the various parameters and measures of precision of these estimates. Thus, i n the analysis of the data, one of the main objectives i s the extraction of a l l information contained i n the data relevant to the question of i n t e r e s t . Two main approaches to data f i t t i n g are recognized — l i n e a r least squares estimation for models li n e a r i n t h e i r parameters, and the non-linear least squares estimation for non-linear parameters. In the l i n e a r s i t u a t i o n , i n -clusion of highly correlated variables i n the same model could lead to a non-inversion of the independent variable-matrix, or i f inversion i s obtained, i t might be done at the expense of getting poor intermediate values. For the non-linea r estimation, poor s t a r t i n g estimates of the parameters could lead to a premature stationary point of the sum of squares surface, and i f multiple minima exis t could re s u l t i n several l o c a l minima i n addition to an absolute minima (Draper and Smith, 1966). The use of estimates of parameters which are highly correlated to each other might result i n very large variances. This unsatis-factory state of a f f a i r s could sometimes be the nature of the models themselves, 6 and even the best possible design for the pa r t i c u l a r study could s t i l l lead to highly correlated estimates. There are agreements on the p o s s i b i l i t y that a par t i c u l a r design or model may have i t s parameter estimates highly correlated with large variances, and yet the response i s estimated reasonably accurately. Box and Lucas (1959) stated that " i n s t a b i l i t y i n the parameters which i s comp-ensating, i s by i t s very nature not such as w i l l cause i n s t a b i l i t y i n the e s t i -mate of the response variable." Another common consideration for both approaches i s the assumption of independent normal d i s t r i b u t i o n of the errors, and that of homogeneity of v a r i -ance, 0*2. Based on the assumption of normality, the sum of squares function S(6) defined as S(e) = ^ C v t " V(At,e)~| 2 which measures the differences be-t=l J tween the actual response/]! V t and observed response V (expressed as a function of variable A £ with c o e f f i c i e n t 0 for N observations), i s equivalent to the li k e l i h o o d function and the maximum li k e l i h o o d estimates are least square e s t i -mates. Many of the properties of the li n e a r response function — unbiasedness, smallest variance — are independent of the normality assumption. I f the assump-tion of homogeneity of variance i s not met but an examination of the residuals reveals a part i c u l a r trend - increasing or decreasing band - a weighting proce-dure could be employed. Test of accuracy of models could be done by examining the residuals or calculating the sum of squares due to lack of f i t and pure error. Precision of the estimates i s based on the theory of confidence i n t e r v a l s , the assumption being that the model i s adequate and the variance i s an unbiased estimate of the population variance, *f^-, Sometimes an examination of the data indicates a li n e a r relationship up to a certain point, and then i t curves smoothly. In such cases, polynominals could be usefully employed, and f i t t e d using the l i n e a r least square approach. A major problem with the polynomials i s that they might be applicable to the data being f i t t e d but might not necessarily give reasonable estimates i n between a set of data. I t i s therefore strongly recommended (Kozak, A. - personal com-munication) that the intermediate values be calculated to give an indication of the extent to which polynomials can be used, and not only the p l o t t i n g of bias or residuals since the l a t t e r does not give any indication of the intermediate values. Most growth functions are known to follow the S- or sigmoid shape, but for quite a few the parabolic form prevails. Once the true functional form could be postulated, i t might be preferable to use a non-linear function having parameters which could describe the changing forms. Previous objections to the use of the non-linear functions were based on the d i f f i c u l t y i n the computation of the parameters, the lack of adequate tests of goodness of f i t and the d i f f i c u l t y of establishing the sampling d i s -t r i b u t i o n of the f i t t e d s t a t i s t i c s . The l a s t two objections could also be the case for l i n e a r functions since the assumptions are rarely met. No d i f f i c u l -t i e s should be envisaged with the computation of the parameters for carefully planned and examined data. Apart from being able to attach meanings to para-meters of non-linear functions, the p o s s i b i l i t y of obtaining a generalized func-tion always e x i s t s . Non-linear least squares estimation of the parameters could be accom-plished by several methods. These include the l i n e a r i z a t i o n technique (Taylor's series) which involves a series of l i n e a r approximations, the method of steepest descent and Marquardt's compromise. As the name suggests, the Marquardt's com-promise represents a compromise between Taylor's series and the method of 8 steepest descent, combining the best and rejecting the most serious l i m i t a t i o n s . A major problem with non-linear least squares estimation, i s the selec-tion of the i n i t i a l estimates of the parameters. These are supposed to be meaningful guesses but with up to f i v e or more parameters to be estimated, i t takes more than ordinary guessing. Good s t a r t i n g values result i n f a s t e r con-vergence to a solution than poor ones, implying smaller computing expenses than for poor ones. In deciding on the i n i t i a l parameters, several suggestions have been made. Box (1960) suggested the p l o t t i n g of the l i k e l i h o o d function over the whole range of parameter values ( l i k e l i h o o d contour diagrams) to give a v i s u a l representation of what the data make us believe. Such v i s u a l representation i s , however, not possible for more than three parameters i n which case the logarithm of the l i k e l i h o o d function i s suggested. The Spearman's estimation (Cornel, 1965)^method of P a r t i a l Totals (Speckman and Cornell, 1965) and the Muench's method (Muench, 1959) have been used frequently for the estimation of the para-meters of the simple exponential model. Whereas the Spearman and P a r t i a l Total methods are used f o r equally spaced values of the independent v a r i a b l e , Muench's method does not rely on t h i s , but has been found to lead to a biased estimator and wide confidence l i m i t s . Demaerschalk (1973) f i t t e d non-linear functions for volume on diameter and height using as s t a r t i n g values the coefficients obtained from a l i n e a r least squares estimation. Several models have been described for growth characteristics but not so much has been written on stand volume (V) and age (X) relationships, except with some additional variables. Individual tree volume estimations are not of interest i n this study. Basic mensuration texts and a r t i c l e s written by Haack 9 (1963), Gregory and Haack (1964), Munro (1964), Evert (1969), Smalley and Beck (1971) and others are convenient references. In the discussion that follows, b i o l o g i c a l and mechanical considerations i n the derivations of the functions w i l l not be discussed. Stand volume versus age curves have been constructed and used as y i e l d tables since the l a t t e r part of the eighteenth century (Spurr, 1952). The v o l -ume functions were i n most cases estimated from the relationships with any or a l l of average age, dbh, t o t a l height of average trees, basal area, s i t e index and transformations of these variables. Despite the fact that volume plotted against age gives valuable information, Spurr (1952) was of the opinion that future stand dimensions can more often than not be more accurately predicted i f independent variables other than age are used. The logarithmic equation, Log Y = a + b (1/X), (Schumacher, 1939), where a, b are coefficients to be estimated, Y i s the response and X the age, and functions involving a weighting procedure have been found to describe many cumulative growth patterns. In recent years, non-linear functions have been f i t t e d s a t i s f a c t o r i l y to height-age/site index data, and as an attempt to stimulate further interest i n t h e i r uses, Nelder (1961) and Grosenbaugh (1965) suggested the use of general-ized growth functions. Three growth functions are well-known (Richards, 1959). These are the monomolecular V = a (1 - be °^), the autocatalytic V = a / ( l + be ~ and the - cX Gompertz V = ae ~ D e where a, b, c represent parameters to be estimated. The monomolecular d i f f e r s from the other two i n that i t has no point of i n f l e c -tion. In a l l cases, the parameter a i s the ultimate l i m i t i n g value, b i s a 10 parameter f i x i n g zero V, and c the spread of the curve. Von Bertallanfy (1938, 1957) formulated a growth model V = a ( l - e ~ c ^ ) 3  1 or V = a(l-e~ c X)*~2/3 , which s t i r r e d up interest and controversy. Richards (1959) applied the growth functions i n studying plant growth while Chapman (1961, 1967) applied this to the growth of f i s h . Both f e l t that the constant 2/3 should be undefined, and replaced t h i s by a parameter, m. Recent works i n height and s i t e index studies with non-linear least square estimation have been published using the modifications of Richards and Chapman to the Von Bertallanfy model, and the o r i g i n a l suggested model. The results indicate that the model and i t s variation could be applied to a wide variety of curve forms (Cooper (1961), B r i c k e l l (1966, 1968), Beck (1971), Pienaar and Turnbull (1973)). In some of the works (Lundgren and Dolid, 1970), the r a t i o -JL. i s replaced by the parameter d and the model cal l e d the m u l t i -1-m molecular function. Moser and H a l l (1969) i n deriving growth and y i e l d functions for uneven-age forest stands suggested a volume equation of the form V = bQ|>(Bo,t)3 1 where B, the basal area and F(Bo,t) i s a function correspondending to B = [~n/c - b e ~ ( l ~ m ) c X J l A " " m where bo, b are parameters to be estimated, and t the time. This function could be considered as another v a r i a t i o n of the modi-fie d Von Bertallanfy function. Other growth functions suggested include the Verhulst (Pearl-Reed, 1923), Gauss and i t s modification,the Johnson-Schumacher (Johnson, 1935; Schumacher, 1939) and the Pearl and Reed (1923) functions. Once an unwieldy but most f l e x i b l e function, the Pearl and Reed Function was given meaningful parameters which could* be e a s i l y estimated by Qrosenbaugh (1965). Whereas e x p l i c i t 11 solution of the point of i n f l e c t i o n was not feasible i n the o r i g i n a l suggested model, the p r i n c i p a l point of i n f l e c t i o n could be located using the suggested parameters (Appendix 6). Comparison of linea r and non-linear equations i s a major problem i n s t a t i s t i c s . The standard error of estimate (overall and for sections of curve), the mean biases (by age classes or curve sections), the multiple c o e f f i c i e n t of determination (R 2) and the correlation indices (CI) have been widely used (Ezekiel and Fox (1967), Demaerschalk (1973)). The R 2 and CI are i d e n t i c a l i n that when both are expressed as percentages, they represent the percentage v a r i -ation of the dependent variable accounted f o r by the independent variables. Hejjas (1967) concluded from a study of the comparison of absolute and r e l a t i v e standard errors and estimates of tree volumes by li n e a r functions, that no one part i c u l a r s t a t i s t i c could be considered as being enough i n deciding which of the many possible functions i s suggested best by the data. The Furnival index (Furnival, 1961), which i s based on the lik e l i h o o d function, has also been widely used i n the comparison of volume equations. For, the same dependent variable, however, th i s index i s equivalent to the standard error of estimate as used i n forest mensuration. While comparing curves obtained by different techniques, C u r t i s , et a l . (1974) were able to show that different dependent variables lead to different systems of curves. This i s an important discovery which has to be recognized i n comparing equations with different dependent variables. For instance, i f volumes computed from hand-fitted curves or estimated by back-transforming a log volume function are compared with those f o r , say, volume/age function, some deviations , from each other should be expected. 12 Free-hand fitting has been found to be less laborious while giving satisfactory results for most purposes (Husch, 1963). The British Columbia Forest Service curves are hand-fitted. Some of these curves have been in use since 1964. 13 THE DATA Volume/age data for t h i s study were obtained from the B r i t i s h Columbia Forest Service for three mature i n t e r i o r species and species groups — Douglas-f i r , F (Pseudotsuga menziesii (Mirb.) Franco), Lodgepole pine, PI (Pinus contorta Dougl.) and Western redcedar, C (Thuja p l i c a t a Donn.). A t o t a l of eight groups representing good, medium and poor s i t e s for the three species groups (except for C where no data were immediately available for good site ) were selected for the study. The data are presented i n Appendix -3 as scattergrams with average volume per acre i n cubic feet - net of decay only as the dependent variable and age class mid-point i n years as the independent v a r i -able. The dots (.) represent the actual observations and the asterisk (*) i n d i -cates the mean for the set of observations where more than one observation was available. A second group of data comprised eight BCFS Volume/age curves . ~: corresponding to the selected groups, four of which had been f i t t e d with the means of the same set of data used i n this study (Table I ) . Data for the re-maining four were taken from a mixture of species and species groups representing the same BCFS-VAC: example, data available for analysis of Cedar and Cedar mix-tures had been taken from a mixture of Cedar, Cedar mixtures, Hemlock and Hemlock mixtures (Table I I ) . In a l l cases, the BCFS-VAC*. were f i t t e d to pass through most of the--means of the sets of observation. . The data were tested s t a t i s t i c a l l y to find out whether the:'assumptions of normality and equal variances for each set of observations were met. I t was, however, not possible for these tests to be carried out on a l l the sets of obser-vation due to lack of s u f f i c i e n t r e p l i c a t i o n s . Where there were enough 14 replications, the assumptions were usually not met, implying that some statis-tical conclusions (confidence intervals, for instance) may be invalid. The Bartlett's test and the Chi-square dispersion test were used for the test on equal variances and normality, respectively (Appendices 1 and 2). An important requirement in fitting functions to empirical data is the presence of equally spaced data, but this was not possible for most of the sets of data available especially between ages 130 and 195. TABLE I - DATA SUMMARY A Group PURE F - 757/GOOD F + Fmix - 575/POOR PI conif - 578/GOOD PI conif - 572/POOR Location YALAKOM P. S.Y.U. WILLIAMS LAKE P.W.C. WILLIAMS LAKE P.W.C. OKANAGAN S.Y.U. rolume Cu.Ft. MIN. MAX. MEAN N MIN. MAX. MEAN N MIN. MAX. MEAN N MIN. MAX. MEAN N Age 30 - - - - 0 703 165 5 0 0 0 1 0 0 0 1 50 1470 1751 1610 2 0 581 265 9 880 2095 1502 4 204 296 248 3 70 3041 3041 3041 1 181 1712 695 10 1893 3376 2633 6 264 1147 787 4 90 3267 8398 6183 9 738 1698 1266 5 1564 4904 3290 4 1082 1475 1238 3 110 4734 8688 6482 5 875 2394 1513 3 2540 3693 3116 2 167 1272 720 2 130 6395 8777 7586 2 14455 3145 2289 4 3485 3485 3485 1 - - - -195 3286 12472 8783 6 380 4011 :2429 10 4164 1164 4164 1 2269 2919 2594 2 290 - - - - 2050 2050 2050 1 - - - - - - - -Total 25 47 19 15 Observations Note: ( i ) 757/G00D refers to BCFS - Unit 757 on Good s i t e , etc. ( i i ) mix - mixtures, conif - coniferous, ( i i i ) min., max. and mean refer to minimum, maximum and mean average volumes per acre i n cubic feet - close u t i l i z a t i o n , respectively; N number of observations for each age class, (iv) - indicates no observation, (v) BCFS-VAC* 572 and 575 were replaced A p r i l 1974 by 1440 and 1443-A respectively. Species/Site Group C + Cmix - 265/MED. Location SALMO S.Y.U. -sc ; Volume Cu.Ft. MIN. MAX. MEAN Age 30 454 1482 968 50 778 778 778 70 3793 3793 3793 90 984 6428 3429 110 - - -130 5245 5245 5245 195 6112 9142 7681 270 6030 10294 8311 Total Observations 26 Species Group for BCFS-VAC C + Cmix & H + Hmix combined. Total Observations for BCFS-VAC 57 Note: H - Hemlock; other TABLE I I - DATA SUMMARY B C + Cmix - 266/POOR SALMO S.Y.U.-SC 180 MIN. MAX. MEAN N 0 220 110 3 872 1508 1190 2 1487 1487 1487 1 2391 2391 2391 1 3509 3509 3509 1 2467 7411 5175 6 5800 11374 7447 5 19 C + Cmix & H + Hmix combined 41 F - 968/MEDIUM NAKUSP & SL0CAN PSYU MIN. MAX. MEAN N 352 1216 640 3 466 2964 1747 10 1589 3595 2515 7 1958 5515 3736 2 3177 3966 3571 2 6185 6185 6185 1 25 F + Fmix 101 PI - 79/MEDIUM WESTLAKE P.W.C. MIN. MAX. MEAN N 982 982 982 1 553 2932 1939 14 344 4490 2189 4 1456 2787 2089 4 2703 2703 2703 1 24 PI + PI conif 62 same as for Table I. 17 VOLUME FUNCTIONS AND FITTING PROCEDURE Non-linear functions selected and f i t t e d to groups of data a f t e r con-s i d e r a t i o n of past studies on cumulative growth curves are numbered 1 - 7 below. As used below, the parameters a, b, c, d, U and G are to be estimated; a r e f e r s to the ultimate l i m i t i n g value; b, c, d describe the shape and pattern of curve; H i s the lower asymptote and G i s a v a r i a b l e f i x i n g the o r i g i n or determining the point of i n f l e c t i o n . X i s class age mid-point i n years, and V the net v o l -ume (above 7.1 inches diameter at breast height, dbh); e i s the exponential con-stant, which approximately equals 2.71828, and Ln i s nat u r a l logarithm (base e ) . Functions 1, 2, 4, 5, 6 and some of t h e i r properties were obtained from Grosenbaugh's (1965) generalized functions; Function 3 has been suggested and f i t t e d to volume/age curves of the BCFS by Mr. Steve Smith of the Faculty of Forestry, U.B.C. The points of i n f l e c t i o n have been proved and presented as Appendices 4 and 5. 1. MONOMOLECULAR FUNCTION V - a ( l - e " c ^ X - G ^ ) A. G = 0 B. G f i x e d or v a r i a b l e . Point of i n f l e c t i o n : none. 2. MULTIMOLECULAR FUNCTION V = a ( l - e " * c ^ X ~ G ^ ) d v J A. .0 B. G f i x e d or v a r i a b l e . Point of i n f l e c t i o n : X = G + iS^ . , V = a ( ^ i ) d c d 3. MULTIMOLECULAR (modified) V = a ( l - b e " " c i C ) d Point of i n f l e c t i o n : X = Ln(bd-b+1) v = a(bd-2b+l }d c bd-b+1 18 4. VON BERTALLANFY V = a ( l - e ~ c ^ X ~ G ^ ) A. G = 0 B. G fixe d or variable Point of i n f l e c t i o n : X = G + i g - 3 , V = 0.296 a 5. VERLHUST V = H + a/(l+e"" c ( X" G )) A. no lower parameter, H; G i n f l e c t i o n parameter B. lower parameter, H present; G i n f l e c t i o n parameter Point of i n f l e c t i o n : X = G , V = 0.5 a -c(X-G) 6. GOMPERTZ V = ae" e + H A. no lower parameter, H; G i n f l e c t i o n parameter B. H present; G i n f l e c t i o n parameter Point of i n f l e c t i o n : X = G , V = H + a/e 7. PEARLS REED (Grosenbaugh's new parameters) V ^ H + a / j l + ^ e " (b(X-G)+b 2(k-%)(X-G) 2+c(X-G) 3) ~J A. no lower parameter, H; G i n f l e c t i o n parameter B. lower parameter present; G i n f l e c t i o n parameter Point of i n f l e c t i o n : X = G , V = H + a k (Note: Function 7 reduces to 5 when k = h , c = 0) The non-linear parameters were estimated using the UBC BMD:X85 non-linea r computer program, an adaptation (by Jason Halm, revised September 1972) from University of C a l i f o r n i a , Los Angeles UCLA, BMD documentation. The program requires input of the p a r t i a l derivatives with respect to the parameters, arid uses the modified Gauss-Newton method of i t e r a t i v e solution (Hartley, 1961). I n i t i a l parameters were estimated by imposing several sketches on the 19 scattergram for each group, and suitable ranges calculated using the properties of the function in question. A convergence criterion of 0.00001 was used. The q2 c2 convergence criterion, q, is explained by the relationship n+1 - n / q , for S 2 five successive values of n, where S 2 n is the error mean n + l square at the beginning of the nth iteration. The maximum number of iterations was selected as tenv . After convergence using the i n i t i a l guessed values for the parameters, the estimates were then used for another set of iterations. The process was re-peated unt i l the attainment of optimal convergence. PARAMETER G AND DOUGLAS-FIR: The parameter G was used i n this study as a parameter defining the point of origin for Functions 1, 2, 3 and 4 or the point of inflection for Functions 5, 6 and 7. In selecting the parameter for the f i r s t case, three ap-proaches were used where possible. These were: (i) G fixed as zero or any other value suggested by the data, ( i i ) G as a variable taking on values from 0 to the minimum value for which data were available, and ( i i i ) G fixed as the origin considering the biological characteristics of the species. The proposition for ( i i i ) (J.H.G. Smith - personal communication) as briefly summarized in Table III applies only to Douglas-fir. The selection of the range of values for G was based on open and average growth stands (Table III, b. #7), and where possible the values for open and averaged stands were used separately (fixed). 20 LINEAR ESTIMATION: CONREG/MREG - multiple regression with/without intercept conditioned to zero f i t t e d to a l l the groups using a combination of the following transfor-mations of age, X : X 3, X 2, 1/X, 1/X2, 1/X3, X^, X"^, X 1 / 3, X" 1 / 3. The regressions were f i t t e d using the multiple regression program developed by Dr. A. Kozak of the Faculty of Forestry, U.B.C. (ROKA/MREG or ROKA/CONREG). 21 TABLE I I I - FIXING ORIGIN FOR INTERIOR DOUGLAS-FIR (a) Assumed f o r : open growth (0), dbh growth i n inches (gr. i n ins.) = 0,33 height growth, average growth (A), dbh gr. i n ins. = 0.20 ht. gr. f t . ** dense growth (D), dbh gr. i n i n s . = 0.05 ht. gr. f t . ** S I T E C L A S S GOOD MEDIUM POOR #1. Years to breast height (4.5 f t . ) 10.5 16.0 24.0 #2. Height as percentage of good s i t e 100 80 50 #3. Average ht. gr. f t . per year 1.0 0.8 0.5 #4. Maximum ht. gr. f t . per year (post juvenile) 2.0 1.6 1.0 #5. Maximum dbh gr. i n ins . per year * (0) 0.66 0.53 0.33 (A) 0.40 0.32 0.20 (D) 0.10 0.08 0.05 #6. Years from breast height to 7.1 i n . ' (0) 10.6 13.3 21.5 dbh (7.1 * //5) (A) 17.8 22.2 35.5 (D) 71.0 88.8 142.0 #7. Years from zero to 7.1 inches dbh (0) 21.1 29.3 45.5 (#1 + #6) (A) 28.3 38.2 59.5 (D) 81.5 104.8 166.0 * Calculated by multiplying (a) by #4. ** Height growth i n feet. 22 SELECTION OF THE BEST EQUATIONS Three non-linear functions were selected for each group of volume-age data on the basis of: (i ) the ov e r a l l standard error of estimate, ( i i ) the partitioned standard error of estimate, ( i i i ) the mean bias f o r each age, and (iv) the correlation index. Despite the fact that some functions describe a p a r t i c u l a r group better than another, a single function for the whole group, was selected using only the frequency of selec t i o n , an indication of f l e x i b i l i t y , as the c r i t e r i o n . The standard error of the estimate, SE £, which i s a measure of the spread of the data, i s an indication of the precision of the predicted response. This was computed by taking the square root of the sum of squares of deviations divided by the degrees of freedom. That i s , SE £ = ( £ ( v t " V t)VOl-p))**:where V t and V t are the actual and predicted response, N the number of observations and p, the number of parameters estimated for the non-linear functions. (For linea r models, the degrees of freedom i s given by N-p -1). For the graphic curves ( B r i t i s h Columbia Forest Service Volume/Age curves - BCFS VAC), the residuals were computed taking the predicted values from the curves, and a degree of free-dom of N-l. The range of data was then partitioned into classes, and the SEg for each class computed using the given formula. The number of classes varied ac-cording to the pa r t i c u l a r group of data. In some cases, the l a s t one or two sets of observations were l e f t out and the SEg computed as partitioned SEg. 23 The mean bias was calculated instead of the sum of biases because of the unequal and i n s u f f i c i e n t number of observations available for each age, X. The mean bias, MBj at age Xj with nj observations was computed by the r e l a t i o n -ship, MBj = 2 (Vj - Vj)/nJ. The mean biases for the classes were calculated but not used i n the selection of the functions because of the i n s u f f i c i e n t num-ber of classes to indicate suitable trends. Computation was si m i l a r to MBj. The correlation index, i d e n t i c a l to linear-regressions multiple, coeffi c i e n t of determination 100R2, except for the non-existence of "sum of squares due to regression", was computed using the id e n t i t y N A CI = 100(SS T O X A£ - £ (V t - V t ) 2 ) / S S T 0 T A L -t=l where the corrected t o t a l sum of squares (SS) ^^TOTAL i s given by N 9 » 2 t«l c t * l The co e f f i c i e n t of multiple determination, 10QR given by the r e l a t i o n 100R* = . . . - 1 - • 100 SSj f f iQ Rj ; S S I (^SS TQ T A E.where S 3 R E G R E S S I Q N = S S T O T A L " SSRESIDUAL. f * M u l t i p l i e d by 100, CI could be interpreted as the percentage v a r i a t i o n of the response accounted for by the function. The SEg i s expected to behave i n a manner opposite to CI, that i s , a large SEg for a small CI. Slight deviations should, however, be expected for functions which give mean square errors of about the same size but have d i f f e r -ent number of parameters'estimated. In such a case, for a pa r t i c u l a r function, a s l i g h t l y higher SEg for a s l i g h t l y higher CI than the other i s a p o s s i b i l i t y . Caution should be exercised i n comparing the BCFS-VAC>s and the func-tions' SEg and MB i n that:(i) by passing the curves through the means (BCFS 24 hand-fitted curves), the VACs should be expected to have the minimum mean bias at each age, and ( i i ) a smaller SE E, because of the larger degrees of freedom used for the BCFS-VACs, than those of the f i t t e d functions, especially where the sum of squares of the residuals are not very d i f f e r e n t . There i s room for argument as to the number of parameters which has to be considered i n deciding on the degrees of freedom of the hand-fitted curves. One school of thought i s of the opinion that since a certain number of means i s computed before the curve path i s determined, that number should be deducted from the t o t a l obser-vations to give the degrees of freedom. This i s worth considering especially for data where there are enough replications for each X. Another school i s of the opinion that since only one observation (mean) i s considered for each X when f i t t i n g the curve, a degree of freedom of N - l should be appropriate. Due to the few actual data available for the calculation of the means, the l a t t e r school of thought was used. An advantage of t h i s , i s that by giving the BCFS-VAC larger degrees of freedom, a large SE E would c l e a r l y expose the nature of the curve. What sets of combination of c r i t e r i a were used i n the selection of the best functions? Even though the selections were objective, some element of sub-j e c t i v i t y cannot be overlooked. Some few rules were adhered to. Functions that continually under- or over-estimated were not selected, even though they might give better SE E or CI. Where functions gave about the same trend i n biases, the extent of over- or under-estimation were considered. Another condition i s that the CIs for the functions to be selected for each group was not too f a r below the largest, say four per cent of less than, say f i v e per cent i n cases where the BCFS-VAC gave the highest CI. The SE Es - o v e r a l l and p a r t i a l - were the next s t a t i s t i c s considered and were used to ensure that functions 25 selected gave values which did not d i f f e r too much from a majority of the others not selected, p a r t i c u l a r l y the smallest SE £. I t should be recognized that s e l -ection based on the same s t a t i s t i c s but considered i n different order might re-s u l t i n the selection of different functions. Linear functions, MREG or CONREG, w i l l be included but not considered i n the selection except that th e i r s t a t i s t i c s w i l l be used as checks on those of the selected non-linear functions. The p o s s i b i l i t y that these or some other line a r functions could give better results than those selected, however, e x i s t s . 26 RESULTS The results are presented i n four parts. The f i r s t , as Appendices 7 to 12, show the functions with the estimated parameters and the respective points of i n f l e c t i o n , with the numbers of the selected "best" functions for each group underlined. The second part presented i n the subsequent pages (Tables IV - XI) i s made up of the s t a t i s t i c s used i n the selection of the most appropriate func-tions. These are the standard errors of estimate, the mean biases and the cor-r e l a t i o n indices for a l l functions and the BCFS-volume/age curves. Equivalent s t a t i s t i c s for the linea r regressions are also included. The mean biases for classes corresponding to those used for the p a r t i a l standard errors are also indicated but not used i n the selection. Table XII rep reseat is 1 the functions selected for each group and shows the points of i n f l e c t i o n (maximum current annual increment) and the age corres-ponding to maximum volume production per unit acre (culmination of the mean annual increment). Except for pine, VAC-79 (Table VII, and Appendix 3) which could not be appropriately described by a l l the functions due to u n s u i t a b i l i t y of the data, the groups were adequately f i t t e d by most of the functions. The Gompertz curve was selected as the function describing most of the data, especially where the data indicate a well-defined point of i n f l e c t i o n . Figures 3, 4 and 5 i l l u s t r a t e the Gompertz curve (6A) f i t t e d through a l l the data (including VAC-79 and 572 where i t was not one of the three selected); these represent the fourth part of the re s u l t s . These curves showing a l l the proper-27 ties, where possible, indicated in Figure 1, page 3, are presented in the next chapter - Discussion and Suggestions. TABLE IV - STATISTICS FOR FITTED FUNCTIONS - CEDAR, MEDIUM SITE (VAC 265) AGE STANDARD ERROR OF ESTIMATE M E A N B I A S CORRELATION FUNCTION OVERALL 30 - 130 195-270 30 - 130 195-270 30 50* 70* 90 130* 195 270 INDEX (Per Cent) 1A 1320.15 2152.63 1314.76 -1086.34 461.17 -983.12 -2232.75 -118.10 -1246.76 -633.38 519.84 308.64 72.81 IB 1388.18 1964.47 1291.69 -180.50 80.22 515.53 -1050.86 781.22 -599.10 -408.13 243.32 -343.83 78.27 3 1581.80 2273.79 1566.77 -540.90 763.53 -104.01 -1359.98 511.65 -892.21 -594.22 697.05 936.39 73.01 4A 1332.51 1974.94 1220.36 22.66 30.74 514.26 -605.93 1232.16 -325.09 -498.18 86.44 -114.08 79.11 4B 1361.17 1938.31 1260.39 22.95 30.88 514.35 -605.71 1232.50 -324.69 -497.78 86.63 -114.08 79.11 5A 1353.77 1922.81 1256.10 -39.30 5.52 -133.98 -913.25 1304.51 -38.56 -322.06 4.87 7.22 79.33 S 5B 1381.57 2138.61 1301.04 -29.15 12.96 78.20 -854.50 1248.50 -142.91 -354.90 . 37.72 -51.44 79.41 S 6A 1353.93 1915.68 1259.98 -39.67 24.94 207.06 -744.90 1219.71 -250.24 -425.54 78.82 -115.15 79.33 S 7A 1420.16 2466.87 1361.48 -122.76 -31.53 101.65 -947.10 1055.05 -313.46 -352.98 -125.45 212.65 79.23 MREG 1360.00 2055.05 1195.12 53.09 -23.58 -284.85 -982.84 1343.93 161.64 148.37 -46.12 35.01 79.14 (R2) BCFS-VAC 1513.16 1647.22 1499^69 201.87 917.39 218.00 -247.00 1718.00 4.33 -305.00 881.54 1010.60 71.93 * only one observation S selected oo TABLE V - STATISTICS FOR FITTED FUNCTIONS AGE STANDARD ERROR OF ESTIMATE FUNCTION OVERALL 30 - 130 195-270 30 - 130 195-270 30 50 1A 1684.24 1036.03 2285.37 -852.47 291.41 -1009.25 -604.93 IB 1580.88 437.44 2361.29 -65.75 34.60 23.00 266.96 2A 1589.44 506.82 2364.52 209.98 232.84 -78.77 618.80 2B 1663.07 543.52 2591.82 70.91 7.59 -53.21 572.82 4A 1606.90 473.58 2396.21 93.35 78.70 -69.98 577.23 AB 1612.93 582.45 2388.32 245.22 42.27 37.39 772.58 5A 1587.13 531.52 2357.10 -69.73 -54.42 -538.81 260.49 5B 1632.03 510.26 2546.61 77.07 37.40 -279.62 434.63 6A 1568.21 457.47 2339.17 50.49 -9.58 -306.46 429.84 7A 1929.46 365.26 3216.19 -43.32 0.39 -144.48 176.50 7B 2002.22 444.91 3595.82 -6.04 0.58 -106.34 203.96 MREG 1552.67 1370.06 2176.23 -72.73 52.90 77.41 -72.04 BCFS-VAC 1602.50 662.41 2077.32 -194.12 0.72 -65.00 440.00 CEDAR, POOR SITE M E A N B I A S 70* 110* -932.20 -1137.86 -213.41 -704.31 373.46 -56.54 198.13 -483.04 235.49 -396.28 460.83 -231.90 177.63 -23.90 269.43 -41.47 253.69 -97.89 -218.45 -21.79 -220.74 48.63 -421.81 -456.65 -38.00 -1209.00 I 266) 130* 195 -512.20 -202.03 -211.23 -288.93 361.66 -28.99 -133.81 -442.81 -36.92 -355.62 75.55 -372.73 383.91 -358.84 358.23 -271.63 307.81 -234;23 -25.94 -1032.57 34.91 -1032.40 208.51 182.55 -991.0 -705. CORRELATION INDEX 270 (Per Cent) 883.52 72.54 422.84 77.23 547.04 76.98 S 547.31 76.38 599.9 76.47 540.27 76.30 310.88 77.05 408.23 77.25 S 259.99 77.59 S 1239.95 70.32 1240.16 70.32 102.67 78.04 (R: 847.6 73.67 TABLE VI - STATISTICS FOR FITTED FUNCTIONS - PINE, GOOD SITE (VAC 578) AGE FUNCTION STANDARD ERROR OF ESTIMATE OVERALL 30-110 30-110 30* 50-M E A N 70 B I A S 90 110 130* 195* CORRELATION INDEX (Per Cent) 1A 851.47 902.55 -2.96 -1214.38 -296.43 200.72 382.84 -192.95 -166.35 -280.08 53.53 IB 850.02 889.44 368.63 0 320.47 443.83 566.35 56.39 212.33 609.38 53.69 2A 789.26 834.53 -12.82 -348.58 -8.69 45.24 165.87 -384.77 -146.46 491.94 62.43 S 2B 923.43 973.88 379.25 0 758.67 455.02 287.60 -233.94 0.79 604.46 48.56 4A 785.48 833.02 -23.80 -691.06 -146.75 141.85 193.40 -375.59 -253.86 123.51 60.46 4B 772.59 802.01 -70.75 0 161.42 -235.10 33.80 -286.48 41.24 705.17 61.75 S 5A 815.71 841.66 -112.53 -570.87 -160.77 -122.83 99.85 -180.72 164.58 837.50 59.86 6A 788.77 886.88 -37.40 -259.62 44.84 -7.05 50.82 -358.29 -74.98 559.90 62.47 S ' 7A 847.66 909.16 23.26 -410.65 -78.96 91.01 261.26 -234.57 -137.31 348.49 62.08 CONREG 788.64 845.77 4.42 -28.34 -97.43 105.72 164.82 -400.17 -280.91 211.10 62.48 (R: BCFS-VAC 761.48 775.52 18.70 -250.00 182.50 132.83 39.50 -558.50 -440.00 139.00 64.97 TABLE VII - STATISTICS FOR FITTED FUNCTIONS - PINE, MEDIUM SITE (VAC 79) AGE FUNCTION STANDARD ERROR OF ESTIMATE OVERALL 30-90 110-195 30-90 110-195 M E A N 30* B I A S 70 90 110 195 CORRELATION INDEX (Per Cent) 1A 939.50 1030.36 675.87 25.43 -101.40 -156.99 26.84 66.08 -176.19 197.74 7.35 IB 961.38 1060.06 845.52 23.06 -71.63 -60.22 18.91 58.39 -152.82 253.15 7.40 S 2A 961.61 1059.88 849.96 21.21 -80.09 -43.63 16.40 54.25 -171.35 284.96 7.35 3 984.59 1095.47 1177.92** 25.99 -38.20 112.45 2.07 88.11 -120.04 289.17 7.50 S 4A 940.93 1028.61 704.99 10.94 -28.00 140.05 -9.12 48.89 -137.75 410.98 7.07 S 4B 963.08 1060.27 863.43 10.61 -28.00 140.10 -9.57 48.90 -137.74 410.96 7.07 5A 962.67 1060.26 863.51 14.65 -58.36 -62.06 12.39 41.76 -165.14 368.76 7.15 6A 962.10 1059.96 854.98 19.27 -74.98 -68.58 16.80 49.90 -171.84 312.46 7.26 /A 1018.79 1136.55 *** -78.62 -133.54 -32.86 -69.67 -121.39 -256.21 357.14 5.91 MREG 938.04 1028.78 674.71 26.88 -101.93 -77.15 19.93 77.20 -160.04 130.50 7.64 (R2) BCFS-VAC 1011.22 1040.16 1005.50 114.84 -746.40 932.00 163.93 -261.25 -771.25 -647.00 NP ** degrees of freedom •» 1 *** degrees of freedom zero or negative NP not possible (negative index) TABLE VIII - STATISTICS FOR FITTED FUNCTIONS - PINE, POOR SITE (VAC 572) AGE FUNCTION 1A IB 2A 2B 3 4A 4B 5A 5B 6A 7A CONREG BCFS-VAC STANDARD ERROR OF ESTIMATE OVERALL 30-110 551.49 437.47 484.99 505.73 505.96 422.70 503.81 512.62 462.17 464.69 576.73 426.17 487.06 477.54 429.46 484.53 533.14 499.75 449.99 526.06 504.30 485.59 475.41 536.29 439.64 508.47 30-110 -191.91 -65.92 13.76 35.64 -2.36 -29.16 48.71 -131.92 -14.45 -28.96 -202.93 -19.44 -44.92 30* -413.32 -1.99 -45.58 -1.58 -42.88 -104.13 -0.02 -246.35 -59.81 -154.84 -238.54 56.74 0 M E 50 -406.05 -134.80 2.14 76.90 -21.54 -87.89 124.88 -194.06 -108.46 -108.14 -346.04 -80.25 -51.67 AN B 70 -84.36 55.51 193.35 231.19 161.30 132.31 266.67 50.88 125.79 141.32 -142.11 106.19 26.75 I A S 90 172.40 189.16 245.41 236.06 231.68 235.17 225.55 136.09 269.51 253.38 98.90 231.36 88.33 110 -521.58 -620.06 -645.80 -699.36 -631.71 -622.99 -742.38 -749.11 -557.17 -611.29 -544.29 -593.77 -400.50 195 785.49 275.94 363.15 148.29 422.50 233.59 183.41 446.58 89.96 244.85 638.02 132.86 994.00 CORRELATION INDEX (Per Cent) 58.98 74.19 S 70.72 70.81 70.79 73.57 S 68.40 67.29 75.63 S 73.12 65.49 75.50 (R2) 65.54 TABLE IX - STATISTICS FOR FITTED FUNCTIONS - DOUGLAS-FIR, GOOD SITE (VAC 757) AGE STANDARD ERROR OF ESTIMATE FUNCTION OVERALL 50-110 130-195 50-110 130-195 50 1A 2290.90 1985.16 3204.07 -47.19 -54.68 -1962.99 G=21.1 IB 2240.01 1900.46 3194.53 -51.30 -40.41 -1365.67 GVAR IB 2242.80 1830.87 3496.75 -15.70 33.36 29.22 G=28 IB 2220.08 1865.99 3191.99 -49.80 -25.64 -1033.85 2A 2252.50 1894.69 3503.30 -99.60 103.85 -471.48 G-21.1 2B 2247.36 1887.25 3500.01 -23.20 41.93 -142.11 G-28 2B 2246.36 1885.07 3500.47 -23.49 50.21 -17.16 3 2306.40 1967.99 3916.65 -50.47 -138.12 -181.50 4A 2218.22 1863.09 3191.26 -31.03 -42.73 -962.38 G=21.1 4B 2198.11 1822.98 3196.13 73.85 51.55 -70.85 G=28 4B 2203.12 1827.48 3202.63 -15.38 103.21 355.26 GVAR 4B 2301.06. 1914.90. 3609.75 103.10 -701.57 -49.75 5A 2266.61 1914.39 3513.31 -69.30 92.00 -468.63 53 2304.04 1963.73 3916.30 -20.19 42.99 -93.95 6A 2251.58 1892.88 3503.42 -19.70 44.21 -110.89 6B -2304.65 1964.43 3917.03 -19.23 44.57 -109.71 7B 2429.30 2143.00 5550.30 -91.48 39.74 276.37 7A 2366.91 2052.22 4527.89 -102.81 -13.22 132.68 MREG 2191.58 1814.27 2191.02 -10.28 20.80 80.40 BCFS-VAC 2141,14 1748.91 2954.00 -3.23 -0.13 0.50 M E A N B I A S 70* -1643.85 -1467.16 -1109.73 -1377.60 -1093.15 -943.48 -930.74 -861.23 -1227.30 -910.60 -741.47 -664.86 -784.42 -872.99 -804.50 -801.15 -555.22 -629.85 -1285.10 -959.00 90 533.02 469.47 344.94 434.69 337.09 383.52 368.46 392.58 474.03 376.54 336.58 592.36 441.47 405.75 411.70 412.46 274.09 290.69 314.68 332.78 110 -5.90 -179.76 -464.01 -262.71 -538.19 -540.68 -550.09 -633.41 -328.36 -561.15 -651.94 -562.84 -685.92 -586.84 -602.78 -603.70 -803.90 -799.91 -368.50 -418.40 130 371.16 178.85 -86.97 95.19 -163.40 -193.17 -192.89 -349.70 -17.66 -209.20 -267.88 -485.02 -366.40 -260.86 -271.33 -272.38 -397.71 -407.88 57.00 0 195 -196.63 -113.50 73.47 -65.92 192.93 120.29 131.24 -67.59 -51.09 138.47 226.90 -773.75 244.80 144.28 149.39 150.23 185.56 118.33 8.73 -0.17 CORRELATION INDEX (Per Cent) 40.11 42.74 45.09' 43.75 44.62 44.87 94.92 44.57 43.85 44.86 44.61 42.20 43.92 44.69 44.66 44.66 44.37 44.41 45.19 45.40 (R ) TABLE X - STATISTICS FOR FITTED FUNCTIONS - DOUGLAS-FIR, MEDIUM SITE (VAC-968) AGE FUNCTION STANDARD ERROR OF ESTIMATE OVERALL 30-110 30-110 30 50 M E A N B 70 I A S 90 110 195* CORRELATION INDEX (Per Cent) 1A 1252.82 1242.33 -116.30 -629.35 -214.92 -38.71 677.25 81.30 1464.83 44.85 2A 1246.95 1258.06 50.99 198.30 251.55 -146.40 133.86 -684.79 985.21 47.74 2B 1443.27 1493.87 24.79 639.96 628.51 -611.69 -515.43 -1148.73 1188.74 33.17 3 1291.32 1284.11 -55.24 -490.38 -101.54 -7.54 628.44 -21.64 1427.90 46.50 AA 1200.19 1220.08 110.88 60.04 200.37 202.76 199.15 -670.11 617.53 49.38 4B 1353.62 1312.66 65.52 639.35 395.06 -483.77 -9.43 -445.46 2031.21 38.41 5A 1230.04 1255.00 36.03 -432.43 100.25 133.81 533.94 -422.48 458.77 49.15 5B 1240.02 1269.99 -7.55 -164.88 62.15 -23.94 402.70 -472.98 180.55 50.67 S 6A 1217.23 1245.09 -16.10 -301.17 75.60 5.19 390.56 -528.11 202.55 50.20 S 7A 1269.78 1300.41 -47.40 -107.01 47.99 -157.98 315.17 -410.42 341.48 50.73 S MREG 1199.41 1172.00 19.73 -418.76 11.70 102.57 645.01 -197.57 -463.77 . 49.45 (R2) BCFS-VAC 1228.79 1243.81 -54.08 250.00 97.70 -234.43 86.50 -778.50 810.00 44.63 TABLE XI - STATISTICS FOR FITTED FUNCTIONS - DOUGLAS-FIR, POOR SITE (VAC 575) AGE FUNCTION STANDARD ERROR OF ESTIMATE OVERALL 30-110 130-195 30-110 130-195 30 M E 50 AN B I 70 A S 90 110 130 195 290 CORRELATION INDEX (Per Cent) 1A 886.75 524.49 1523.98 -234.83 298.67 -273.72 -445.02 -269.60 61.25 82.95 891.18 180.83 -S92.93 47.11 2A 841.20 486.36 1528.12 -176.00 64.62 93.77 -154.10 -263.14 -207.28 -348.77 411.73 -21.27 -464.97 53.48 3 835.91 451.06 1614.21 -0.65 83.66 153.69 53.77 -26.98 -43.58 -261.79 451.58 -10.50 -446.40 55.14 S 4A 833.15 427.37 1484.41 -88.33 132.14 17.02 -185.49 -135.61 51.03 -47.13 682.99 -7.21 -677.81 53.31 5A 821.47 433.79 1524.19 -15.89 71.20 51.66 -26.30 28.09 18.93 -301.95 362.07 1.07 -390.91 . 55.64'S 6A 825.23 441.87 1530.96 -1.35 91.45 141.65 42.21 -17.12 -34.51 -262.51 447.22 1.07 -427.79 55.23 S 6B 834.61 447.12 1612.99 -6.40 42.02 101.01 25.65 -5.37 -19.22 -263.58 425.78 -58.04 -492.45 55.27 7A 854.60 449.95 1695.03 159.72 -39.27 165.20 255.26 239.48 36.25 -196.08 166.52 -79.63 -458.77 54.22 7B 853.37 472.11 1815.79 60.48 -73.23 26.90 120.00 146.61 -33.67 -192.28 166.30 -125.88 -504.88 55.47 MREG 832.55 444.74 1535.37 -29.40 94.53 -18.62 60.55 -60.92 -27.21 -215.81 477.42 -61.06 118.88 55.50 (R2) BCFS-VAC 793.90 429.33 1338.17 -2.66 102.22 165.20 -0.11 -4.80 -0.20 -287.00 383.33 0 0 55.67 /36,' TABLE XII - SOME PROPERTIES OF THE SELECTED FUNCTIONS SPECIES GROUP SITE FUNCTION MAXIMUM CAI (years) MAXIMUM MAI (years) C + C mix Medium (H + H mix) C + C mix Poor (H + H mix) PI conif Good PI Medium (+ PI conif) PI conif Poor Good F Medium (+ F mix) F + F mix Poor 5A 5B 6A BCFS 2A 5B 6A BCFS 2A 4B 6A BCFS IB 3 4A BCFS IB 4A 5B BCFS VERHULST VERHULST GOMPERTZ VAC-265 MULTIMOL. VERHULST GOMPERTZ VAC-266 MULTIMOL. VON BERT GOMPERTZ VAC-578 MONOMOL. MULTIMOL. VON BERT VAC-79 MONOMOL. VON BERT VERHULST VAC-572 GVAR IB MONOMOL. G21.1 4B VON BERT 6A BCFS 5B 6A 7A BCFS 3 5A 6A BCFS GOMPERTZ VAC-757 VERHULST GOMPERTZ PEARL/REED VAC-968 MULTIMOL. VERHULST'; GOMPERTZ VAC-575 103.92 94.69 81.75 88.00 114.71 142.07 133.04 90.00 46.28 46.81 48.14 53.00 0 20.50 26.18 54.00 30.00 83.47 90.00 58.00 44.50 58.10 64.20 65.00 85.11 68.88 49.98 24.90 74.00 89.15 76.72 90.00 142.0 140.0 130.0 117.0 180.0 200.0 210.0 132.0 74.0 74.0 76.0 78.0 35.0 40.0 44.0 88.5 110.0 140.0 158.0 92.0 93.0 94.0 98.0 90.0 92.0 98.0 97.5 89.0 120.0 120.0 118.0 120.0 37 DISCUSSION AND SUGGESTIONS Examination of the scattergrams (Appendix 3) revealed that the data obtained for VAC-79 were unsuitable for analysis. This was confirmed by the low correlation indices obtained for a l l the functions f i t t e d . A l l the other groups indicated p a r t i c u l a r sigmoid trends. However, for a l l the groups, absence of equal number of replications and equal spaced data, especially between ages 130 and 195 pose serious threats as to the v a l i d i t y of the functions f i t t e d and the conclusions to be drawn. The mean biases showed that some functions give very high values — over- or under-estimation — whereas the respective standard errors of estimate are low or CI high. Unequal number of observations for each age i s a possible cause. For instance, the function might pass through or close to the means of some age class having the greatest number of o v e r a l l observations, thereby mini-mizing the o v e r a l l error but not the biases for those ages which do not have enough observations to exert such influence. The mean biases for the classes, corresponding to the same used for the p a r t i a l SE E, presented among others i n Tables IV - XI, i l l u s t r a t e the d i f f i c u l t y i n deciding on the best functions where there are not enough equally spaced data to warrant grouping into classes. Consideration of the SE £ alone could also lead to misleading r e s u l t s , due to the differences i n the number of parameters estimated and the closeness of the sum of squares of the errors. The sequence used i n the selection could also be varied but possible differences i n the choice of the functions should be expected. Another major factor that caused serious problems i n the selection of the most appropriate functions was that of deciding how large or small an under-38 estimation or over-estimation could be considered acceptable. No s p e c i f i c l i m i t s could be set up, except that functions that gave "reasonable" estimates were selected. Certain other properties of the functions including the maximum current and mean annual increments could be taken into consideration i n the s e l -ection of the functions but t h i s should be done only where there i s enough e v i -dence to suggest such p r e f e r e n t i a l treatment. The f i t t i n g of the curves to each group i n d i r e c t l y involved consider-ation of the l e v e l , shape, steepness, complexity, sampling v a r i a t i o n i n the data c o l l e c t i o n , and the years taken for. the average trees i n the stand to grow into the 7.1 inches diameter at breast height ('origin'). The l e v e l indicates how high or low the curves are, with respect to variation i n s i t e . The results i n d i -cate that good s i t e s generally have higher levels than poor ones. In the second column of Table X I I I , page 39, the ultimate l i m i t i n g value "a" could be consid-ered as an indicator of the l e v e l , even though this i s not always true from the results obtained (compare Cedar poor and medium s i t e s for Gompertz). The para-meter also t e l l s us that good s i t e s have the highest ultimate value. Similar conclusions could be drawn by considering other functions not presented i n Table XIII but provided i n Appendices 7 - 14. The shape and steepness are indica t i v e of s i t e quality and stand man-agement. The parameter "c", shown i n the t h i r d column of Table XIII i s an i n d i -cator of the shape of the curve. Generally, the Verhulst and Gompertz functions indicate that steeper curves have higher "c" than gradual ones. For the m u l t i -molecular functions, "c" (and "b" for modified) depends on the shape and steepness. Because some estimates of parameters are dependent on others, i t might not always be possible to i s o l a t e the effect of one parameter. This explains the lowest value of "c" obtained for the medium s i t e of Douglas-fir for both the Gcmpertz and the TABLE X I I I - BEHAVIOUR OF PARAMETER ESTIMATES -c(X-G) GOMPERTZ (6A) V = ae""e Group S i c e a c G Cedar Medium 8759 0.017262 81.75 Poor 9000 0.010896 133.04 Pine Good 3605.0 0.053324 48.14 Medium 2399.3 0.032873 24.18 Poor 2800.0 0.016986 92.58 F i r Good 8733.1 0.034154 64.20 Medium 6693.9 0.017333 68.88 Poor 2480.0 0.032931 76.5J2 VERHULST, (5A) V = a / ( l + e ~ c ( X ~ G ) ) Group S i t e a c G Cedar Medium 8421.3 0.025615 103.92 Poor 7785.5 0.019982 150.00 Pine Good 3326.5 0.078713 50.00 Medium 2336.3 0.044031 34.84 Poor 2204.5 0.034550 90.00 F i r Good 8571.1 0.046153 74.67 Medium 6000.0 0.027669 85.11 Poor 2441.0 0.051060 89.15 40 Verhulst functions. The maximum CAI and maximum MAI provided in Table XII show that for a l l the functions and the BCFS-VAC, good sites attain their maximum CAI earlier and have shorter rotation (maximum MAI) than poor ones. The only abnormalities were the results obtained for medium site lodgepole pine (VAC-79), the data for which were unsuitable. Description of volume-age curves with the non-linear par-ameters could be applied to the level of management of the stands. For instance, i t is expected that intensive management could result in shorter rotations than those in less intensively managed stands. Species groups differ from each other in their rotation ages and inflection points of the fitted functions. The indi-cated trend (Table XII) in changes in parameter with respect to species groups and sites, could be effectively used in the formulation of a generalized growth function for a l l species and sites. The idea of fixing the origin (the years to grow trees of 7.1 inches diameter at breast height) should be further investigated. Ordinary fitting without any biological considerations, were considered as well as the proposition for Douglas-fir (Table III), Most of the results obtained were from the former technique, except that for the Verhulst function, the introduction of an arbitrary lower asymptote (5B) resulted in the lowering of the inflection point. Fixing the origin for Douglas-fir gave interesting results. The method produced one of the three functions selected for good site, Dpuglas-rfir (757>;, Function 4B, age =» 21.1 years; Table IX). For the medium site, i t gave the worst results (2B and 4B; Table X), while the introduction of the suggested origin for poor site could not bring about convergence of the non-linear functions to solutions. This might be due to the nature of the data for the group, for there were observations re-corded below the postulated minimum origin. This is expected because trees of 41 better s i t e , older age and superior microsite and genetic endowments within any age class are the f i r s t to grow into the measurable volume class. The s t a t i s t i c s presented i n Tables IV - XI could be used as strong indicators of the functions l i k e l y to be most suitable for use to describe volume/age relationships of certain species and species groups. The Von Bertallanfy function, which i s a s p e c i a l form of the multimolecular function, describes a l l the pine groups; the same could be said of the Verhulst and Gompertz for cedar, and Gompertz for Douglas-fir. I f , however, we-base our c r i -t e r i a of "best" on the function describing most of the groups (that i s , f l e x i -b i l i t y ) and s i m p l i c i t y , the Gompertz function could be considered as the "best". The Gompertz curve described s i x out of the seven groups which were appropriately described by a l l the other functions. None of the functions described the pine-medium s i t e appropriately. Figures 3, 4 and 5 on the next pages i l l u s t r a t e the curve shapes and other properties of the Gompertz function f i t t e d to a l l the eight groups studied; the f i r s t part of Table XIII gives a summary of the behaviour of the parameters. Clearly, the Gompertz curve as a volume/age function takes care of a l l the v a r i -ation i n curve shapes expected for different s i t e s and groups. The parameters shown (Table XIII) for the Gompertz also show how meaningful they could be. Consideration of the performance of a l l the functions and i n p a r t i c u l a r the Gompertz curve, i n comparison with the BCFS-volume/age cruves i s possible for four out of the eight groups (Tables VI, V I I I , IX, XI) since only these have the same set of data from which the BCFS-VACs were constructed. Tables IX and XI show the functions do not d i f f e r much from each other and from BCFS-VACs 757 and 575 i n their correlation indices. However, as expected and FIGURE 3 PINE G O M P E R T Z F U N C T I O N S / C U R V E S 6000-5000-AGE IN YEARS - AGE CLASS MID- POINTS 43 F I G U R E 4 CEDAR G O M P E R T Z F U N C T I O N S / C U R V E S FIGURE 5 DOUGLAS-FIR G O M P E R T Z F U N C T I O N S / C U R V E S 45 explained e a r l i e r , the BCFS-VACs have smaller mean biases for each age and lover o v e r a l l and p a r t i a l standard errors of estimate than the f i t t e d functions. In Table VI, the BCFS-VAC 578 i s shown as being s l i g h t l y better than the func-tions f i t t e d (CI about 2 per cent more). The f i t t e d functions, except f o r the monomolecular 1A, and p a r t i c u l a r l y the selected ones are shown i n Table VIII as being able to describe the data better than the BCFS-VAC 572. (CI about 10 per cent more and smaller standard errors of estimate and mean biases.) I t i s cer-tain from the results that the differences between the functions selected and the BCFS-VACs f o r the four groups considered are not enough to warrant the con-clusion that the functions are better than the BCFS-VACs. This could imply that i t should be possible to describe the e x i s t i n g VACs by non-linear functions assuming them to be correct. This should be simpler and more reasonable than f i t t i n g to actual data which are unreplicated, not equally spaced and are s t a t i s -t i c a l l y not suitable for analyses. Using the Gompertz curve, for instance, the i n f l e c t i o n point could be fixed by G, reasonably selecting a range for the u l t i -mate l i m i t i n g value "a", and leaving only "c" to be estimated. By examining a wide variety of curve shapes, a range of values could be selected for the para-meter "c", irrespective of the growth type. If volume/age data are available by stand densities (average basal areas) and s i t e indices, these could be used to evolve a common: function. This w i l l involve f i t t i n g non-linear functions to each stand density and s i t e class. The parameters obtained from the best function can then be regressed on suitable variables (example, transformations of stand density and s i t e index) using the linear least squares procedure. The two sets of equations can then be combined to give the required function. For p r a c t i c a l purposes, the d i f f i c u l t y , cost, a v a i l a b i l i t y of non-linear 46 least squares estimation computer programs, the time involved in fitting the functions and in using them thereafter, are important. Computer time and cost for the non-linear program used are dependent on the number of iterations, and to some extent on the number of observations and the variations. For observa-tions of less than one hundred, only the number of iterations appears to be important. Where i n i t i a l parameters are selected carefully, convergence could be attained after a few iterations. In this study, the number of iterations was set at ten, and where i n i t i a l estimates were selected accurately, convergence was attained before the eighth iteration; the cost involved on a "normal priority job" being only about one U.B.C. Computing Centre dollar (CC $1.00), which is comparable to the cost of fitting polynomials or transformations of age using linear least squares estimation. For non-convergence on the tenth iteration, the cost involved was about CC $3.00 or less. No difficulty should be envisaged in fitting most of the non-linear functions, especially where biological inter-pretation could be given to the parameters. Many computer programs are available using the non-linear least squares estimation technique in one form or the other. Some of the programs require that the first partial derivatives of the response with respect to the parameters be given (UBC BMD:X85 was used in this study), and others have sub-programs which compute these derivatives. Forest mensurationists can certainly develop non-linear fitting rou-tines appropriate to forestry problems, given the encouragement by forest man-agers. Foresters must discard the notion that non-linear functions are difficult to f i t . Mention should be made of the work currently being done by Jensen (1973) and with Homeyer (1970, 1971) in the form of production of "matchacurves" which could be used conveniently in the selection of some of the i n i t i a l estimates of non-linear parameters. 47 In terms of applying a function like the Gompertz, foresters will only have to rely on the use of the exponential and logarithms, both of which are available in most mathematical tables. 48 CONCLUSIONS In the report submitted to the Productivity Committee of the B r i t i s h Columbia Forest Service (PC 006), Smith (1973) noted the need for further con-sideration of methods of f i t t i n g volume/age relationships, at least on a zonal; basis. Such studies might involve the f i t t i n g of functions to respective species and s i t e s , and then the pooling of a l l the functions. In doing t h i s , the fact that some functions might describe a p a r t i c u l a r species group or s i t e better than others should be recognized. Use of some non-linear functions might be appropriate i n this case since the parameters are usually f l e x i b l e to changes i n various curve forms. The Gompertz curve (Winsor, 1932), V = ae~ e f i t t e d to the eight groups of data considered, demonstrated s u f f i c i e n t f l e x i b i l i t y . The parameters a, c and G could be given b i o l o g i c a l interpretation, thereby making i t suitable for use as an appropriate volume/age function. The l e v e l ( s i t e q u a l i t y ) , curve shape (growth pattern), steepness (stand management) and sampling variations were cl e a r l y i l l u s t r a t e d by the parameters when the function was f i t t e d to a l l the groups. Comparison of the non-linear functions with the BCFS-VACs. indicated that results from both approaches do not d i f f e r much from each other, thus show-ing the p o s s i b i l i t y of replacing the exi s t i n g curves by suitable functions. The f i t t i n g of functions to the BCFS-curves should be investigated for the other functions, since some of them might be more appropriate than the Gompertz func-tion which was found to be the best when applied to the actual groups of data used herein. 49 REFERENCES Beck, Donald E, (1971). Height-growth patterns and s i t e index of white pine i n the Southern Appalachians. Forest S c i . 17:252-260. Bertallanfy, L. Von, (1938). A quantitative theory of organic growth. I I Inquiries on growth laws. Human B i o l . 10:181-213. — — — — — • ( 1 9 5 7 ) . Quantitative laws on metabolism and growth. Quart. Rev. B i o l . 32:217-231. Box, G. E. P. (1960). F i t t i n g empirical data. Annals N. Y. Acad. S c i . 86: 792-816. Box, G. E. P. and Lucas, L. H. (1959). Design of experiment i n non-linear s i t u -ations. Biometrika 46:77-90. B r i c k e l l , James E. (1966). Site index curves for Engelman Spruce i n the nor-thern and central Rocky mountains. Inter-mountain Forest and Range Exp. Sta. U.S. Forest Service Res. Note INT-42, 8 pp. — — — — ( 1 9 6 8 ) • A method for constructing s i t e index curves from measurement of tree age and height - i t s application to inland Douglas-fir. Inter-mountain Forest and Range Exp. St. U.S. Forest Service Paper INT-47, 23 p. Chapman, D. G. (1961). S t a t i s t i c a l problems i n pupulation dynamics. Proc. Fourth Berkeley Sump. Math. Stat, and Prob. Univ. C a l i f . Press, Berkeley and Los Angeles, pp. 153-168. «(1967). Stochastic models i n animal population ecology. Proc. F i f t h Berkeley Symp. Math Stat, and Prob. Univ. C a l i f . Press, Berkeley, and Los Angeles, pp. 147-162. Cooper, Charles F. (1961). Equations for the description of past growth i n even-aged stands of ponderosa pine. Forest S c i . 7:72-80. Cornell, R. G. (1965). Spearman estimation for a simple exponential model. Biometrics 21:858-864. Curtis, R. 0., DeMars, D. J . and Herman, F. R. (1974), Which dependent variable i n s i t e index - height - age regressions? Forest S c i . 20:74-87. Demaerschalk, J u l i e n P. (1973). Derivation and analysis of compatible tree taper and volume estimating systems. Unpublished Univ. of B r i t i s h Columbia Ph.D. Thesis. 131 pp. Draper, N. R. and Smith, H. (1966). Applied Regression Analysis, John Wiley and Sons Inc., New York. 407 pp. Evert, F. (1969). Use of form factor i n tree volume estimation. Jour. For. 67:126-128. 50 Ez e k i e l , M. and Fox, K. A. (1967). Methods of correlation and Regression Analysis. 3rd Edition. Wiley and Sons, New York. 548 pp. Freese, E. (1960). Linear Regression methods for forest research. USDA Forest Service Research Paper NFPL-17. 136 pp. Furnival, George M. (1961). An index for comparing equations used i n construc-ting volume tables. Forest S c i . 7 (4):337-341. Gregory, R. A. and Haack, P. M. (1964). Equations and tables for estimating cubic-foot volume of i n t e r i o r Alaska tree species. USDA Forest Ser-vice Res. Note NDR-6. 21 pp. Grosenbaugh, L. R. (1965). Generalization and reparameterization of some sigmoid and other non-linear functions. Biometrics 21 (3):708-714. Haack, P. M. (1963). Volume tables for Hemlock and Sitka Spruce on the Ehngach National Forest, Alaska. USDA For. Service, South For. Exp. Sta. Occasional Paper No. 134. 32 pp. Hartley, H. 0. (1948). The estimation of non-linear parameters by " i n t e r n a l least squares". Biometrika 35:32-45. -(1961). The modified Gauss-Newton method for the f i t t i n g of non-lin e a r and regression functions by least squares. Technometrics 3:269-280. Hejjas, J . (1967). Comparison of absolute and r e l a t i v e standard errors and estimates of tree volumes. Univ. of B r i t i s h Columbia, Forest Faculty M.F. Thesis, Unpublished. 58 pp. Husch, B. (1963). Forest mensuration and s t a t i s t i c s . Ronald Press, New York. 474 pp. Jensen, Chester E. (1973). Matchacurve-3: multiple-component and multidimensional mathematical models for Natural Resource Studies. USDA Forest Service, Inter-mountain For. and Range Exp. Sta. Res. Paper INT-146. Jensen, C. E. and Homeyer, J . W. (1970). Matchacurve-1 for Algebraic transforms to describe sigmoid or bell-shaped curves. USDA Fdrest Service, Inter-mountain For. and Range Exp. Sta. Res. Paper. 22 pp. •(1971). Matchacurve-2 for algebraic transforms to describe curves of the class X™. USDA For. Service, Inter-mountain For. and Range Exp. Sta. Paper INT-106. Johnson, N. 0. (1935). A trend l i n e for growth series. Journ. Amer. Stat. Assoc. 30:717. Lipton, S. and McGilchrist, C. A. (1964). The derivations of methods for f i t t i n g expoential curves. Biometrika 51:504-508. 51 Lundgren, A. L. and Dolid, W. A. (1970). B i o l o g i c a l growth functions describe published s i t e index curves for Lake States timber species. USDA Forest Service, North Central For. Exp. Sta. Research Paper NC-36, 8 pp. Moser, J . W. and H a l l , 0. F. (1969). Deriving growth and y i e l d functions for uneven-aged forest stands. Forest S c i . 15:183-188. Muench, H. (1959). Catalytic models i n epidemiology. Harvard Univ. Press, Cambridge. Munro, D. D. (1964). Weighted least squares solutions improve precision of tree volume estimates. For. Chron. 40:400-401. Nelder, J . A. (1961). The f i t t i n g of a generalization of the l o g i s t i c curve. B i o -metrics 18:614-616. Pearl , R. P. and Reed, L. J. (1923). On the mathematical theory of population growth. Metron. 3:6-19. Pienaar, L. V. and Turnbull, K. J. (1973). The Chapman-Richards Generalization of Von Bertallanfy's growth model for basal area growth and y i e l d i n even-aged stands. Forest S c i . 19:2-22. Richards, F. J . (1959). A f l e x i b l e growth function for empirical use. Journ. of Exp. Botany 10 (29):290-300. Schumacher, F. X. (1939). A new growth curve and i t s r e l a t i o n to timber y i e l d studies. Journ. Forestry 37:819-820. Smalley, G. W. and Beck, D. E. (1971). Cubic-foot volume table and point-sampling factors for white pine plantations i n the Southern Appala-chians. USDA Forest Service Research Note S0-118. 2 pp. Smith, J . H. G. (1973). Report to the Productivity Committee of the B r i t i s h Columbia Forest Service on PC 006, Univ. of B. C , Fac. of For. 76 pp. Speckman, J. A. and Cornell, R. G. (1965). Estimation for a one parameter expoential model. Journ. of American Stat. Assoc. 60:560-572. Spurr, Stephen H. (1952). Forest Inventory, Ronald Press Co., New York. 476 pp. Winsor, C. P. (1932). The Gompertz curve as a growth curve. Proc. Nat. Acad. Sciences, Washington, D. C. 18:1-8. 52 APPENDIX 1 TESTS ON EMPIRICAL DATA NORMALITY: CHI-SQUARE DISPERSION TEST n Based on V (Oi - e ^ ) ^ -w 2 ^ —eT ' y n-t-1 where Oi and e± are the observed andnexpected values estimated from hand drawn sketches, respectively and o the number of observations. For normal observation (each age class and whole c l a s s e s ) , t = 2 since two parameters - the mean and the variance - are required. An a l t e r n a t i v e t e s t f o r normality: i s by considering the residuals and te s t i n g f o r the assumption that i f Y — N ( b Q + b>X,<T2) then N (0,o~ 2) 2 where Y i s the response and g the er r o r ; N refe r s to normality, the popula-t i o n variance; b Q , b , are the c o e f f i c i e n t s estimated. To use t h i s t e s t , the residuals must be tested for unbiasedness. References: Basic mathematical s t a t i s t i c s texts. Freund, J . E. (1971). Mathematical S t a t i s t i c s . P r e n t i c e - H a l l Inc., New Jersey. 463 pp. Snedecor, G. W. a l . (1967). S t a t i s t i c a l Methods. Iowa State Univ. Press, Ames, Iowa. 593 pp. Wetheril, G. B. (1967). Elementary S t a t i s t i c a l Methods, Methuen Co., London. 329 pp. 53 APPENDIX 2 TESTS ON EMPIRICAL DATA HOMOGENEITY OF VARIANCE: BARTLETT'S TEST Applicable to cases of equal and unequal sample sizes (for X's) and i s based on a s t a t i s t i c whose sampling d i s t r i b u t i o n i s approximated very closely by the - d i s t r i b u t i o n when the k random samples are drawn from independent normal populations. Hypothesis of equal variances rejected when at <* l e v e l , b = 2.3026 q/h (k - 1 ) , * 2 k 9 q i s computed by q = (N-k) Log S - Y ( n - - l ) Log Sr P |=i 1 1 k and h = 1 + 1 ( ]> _1 - __1__ ) 3(k-1) i = l n i - l N-k 2 2 and S i s the variance of X^  with n.i observations, S the i k * ' 1 P pooled variance, S = 2_ (nj - 1) Sf and N, the t o t a l number of observations. p ' i = l N-k 1 Log i s common logarithm (base 10); i f natural Log, l n , i s used, the constant 2.3026 used for computation of b i s eliminated. The test i s less r e l i a b l e i f n^< 5. An alternative test could be done by examining the plot of residuals versus the response. Reference: Walpole, R. E. & Myers, R. (1972). Probability & S t a t i s -t i c s for Engineers & S c i e n t i s t s . Macmillan Co., New York. 506 pp. 54-z o 5 o Ul o CC o Ul z O CC < a. ui o, ui 2 D _1 O > 13 ooo -j 10 ooo 9ooo 8ooo 7ooo • 6000 • 5ooo • 4000 3ooo 2ooo looo DOUGLAS-F IR GOOD SITE VAC 757 •2 •X-•X-. 12 472 • 11 100 • 10 653 APPENDIX 3 (!) 10 30 50 70 AGE IN YEARS 90 110 130 150 - AGE CLASS MID-POINTS 170 190 210 6ooo Ul Ul u. 5ooo o GO 4000 z o , >-Ul < CC o O Ul < a 3ooo CC Ul CC o. O u. Ul 2ooo 5 K-=> Ul > looo > < 0 DOUGLAS-F IR MEDIUM SITE VAC 968 - 1 — 10 • 2 * APPENDIX 3 (II) - r i 1 1 1 — 30 50 70 90 110 130 150 170 AGE IN YEARS - AGE CLASS MID - POINTS —I 130 190 5ooo -J o CO ( J >-5 z rr UJ UJ 4 0 0 0 H 3000 2ooo H 13 J~ - J UJ O 2 > looo D O U G L A S - F I R POOR SITE VAC 575 A P P E N D I X 3 ( II 4 ? — p — i -•X-•X-0 , 0 3 0 50 70 90 110 130 150 AGE IN YEARS - AGE CLASS MID - POINTS 170 190 195 210 230 lo 000 z 9ooo < uj 8ooo a cc O 7ooo z I ui IL o m 5ooo y U) 4<>00 cc ( J < P= 3ooo H 5 „ 3 2 0 0 0 _J o > < looo CEDAR CEDAR and MIXTURES M E D I U M S I T E VAC 265 A P P E N D I X 3 ( IV 10 30 50 70 AGE IN YEARS I 1 1 1 °0 110 130 150 AGE CLASS MID- POINTS 170 — I — 190 210 230 — i 250 — i — 270 — i 290 56 10 ooo A CEDAR and CEDAR MIXTURE POOR SITE VAC 266 APPENDIX 3 ( V ) 9ooo 8ooo 7ooo 6ooo 5ooo 4ooo H 3ooo 2ooo H looo i— 10 —i— 70 —i— 90 —i— 110 1 1 1 1 — 130 150 170 190 AGE CLASS MID- POINTS —i 230 —i 250 30 50 210 270 AGE IN YEARS 5ooo ui UJ u. O ta 3 4ooo S 5 3 0 0 0 < O rr ^ CC o Ui o. cc £2 z> i--I Ul o Z looo 5 2ooo H PINE APPENDIX 3 ( VI ) GOOD SITE VAC 578 0 10 30 50 70 AGE IN YEARS 90 110 130 150 AGE CLASS MID- POINTS 170 190 5? PINE M E D I U M S I T E APPENDIX 3 ( VII ) VAC 79 6ooo 5ooo ui ui o m 3 ° >- 4000 o ui j . 5 , i 3 ooo < ui cc Q CL CC O 2ooo UJ u. " j u i O • > > looo 0 10 30 50 70 AGE IN YEARS 90 110 130 150 AGE CLASS MID - POINTS 170 190 APPENDIX 3 (VIII to 3 O PINE P O O R S I T E VAC 572 3ooo 5 1 3 ° "j u-0,1-> ui < 2ooo looo H 0 10 30 50 70 90 110 130 150 AGE IN YEARS - AGE CLASS MID-POINTS 170 190 58 APPENDIX 4 INFLECTION POINTS MULTIMOLECULAR (NUMBERS 2 & 3) form V = a(l-be- c( x l- G>) d or V = a(l-be~ c X) d first derivative, vl(x.) = ad (l-be- c X) d _ 1.bc e" c X second derivative, V u(3e.) = ad(d-l) (l fbe~ c X) d" 2.bc e~cX.bc e~ c X - ad(l-be- c X) d- 1.bc 2e _ c X for inflection, V n(x.) = 0 (and V m ( X ) 4 0) adb 2c 2 (d-1) ( l - e - c X ) d - 2 e - 2 c X = adbc 2(l-e - c X ) d T 1e- c X b(d-l)e" c X = l - e - c X e-cX(bd-b+1) = 1 - cX Ln e = - Ln(bd-b+1) r cX = - Ln(bd-b+1) X = Ln(bd-b+1) c For function #2, b=l and X = Ln d. Substituting the values in the respective c functions give for function #2, V = a (d-l) d and for function #3, V = a (bd-2b+l)d. d bb-b+1 Points for Von Bertallanfy (#4) obtained by substituting 3 for d. form APPENDIX 5 INFLECTION POINTS GOMPERTZ CURVE (FUNCTION NUMBER 6) -c(x-G) V = H + a e~e -c(X-G) V1(PC) = -a e~ e (-)c e~ c< x-«> = ac e-e" C ( X" G ) e-c(X-G)  c .e . ' VH(X) = ac e-e~ C ( X~ G ).c e-c<X-G>.e-c<X-G> -c(X-G) + e"e (-c)e-c(x-G> for inflection e-c(X-G) = i C(X-G) = X = G and V-H = a/e , e = 2.71828 VERLHUST FUNCTION (FUNCTION NUMBER 5) form V = H + a(l + e" 0^" 6^)"" 1 V l ( x ) = ac(l+e-c<x-G>r2.e-c<X-G> V u ( x ) = 2ac 2e- c( X- G>.(l+e- c( X- G ))- 3.e- c ( X- G ) + aC2(l+e-c(X-G))-2>e-c(X-G) = a c2 e-c(X-G) ( 1 + e-c(X-G) }-3 2 e-c(X-G)_ ( 1 + e-c(X-G) } Inflection, V 1 : L ( - X - ) = 0 2 e-c(X-G)_1_e-c(X-G) „ Q exp-c(X-G) X = G and V-H = a (2)" 1 = 0.5a 60 APPENDIX 6 PEARL & REED FUNCTION (Grosenbaugh's suggested parameters, 1965) Y - H + a ( l e-CbU+b^k-^U^cU3))-! k and X-G = Y ;_G U = 0 ") -H = ak f a t P o : L n t °^ inflection W - Y^ H - (1 + h± e-CbU+b^k^^+cU 3))-! , a n d a k 2 1 for monotoniclty, h Z (.k-h) < 3c but up to 3 inflections may occur for 2 b* effective asymptotes, both b and c must be positive within negative parens. dY = Y 1 ^ ) = W(l-W).a(b+2b2(k-yu+3cU2) dX and d2Y = Y1;L(x.) = W(l-W)a (b+2b2(k-%)U+3cU2)2(l-2W)+2b2(k-%)+6cU * dX? * multiple inflection may occur unless h Z Oa-H)^*A^^ Z 0 and b^ only one asymptote exists unless b and c are ^positive. Source: Table 4, pg. 713 Biometrics 21 (2), 1965. APPENDIX 7 PREDICTED FUNCTIONS + CEDAR GROUP (MEDIUM SITE - VAC 265) Function 1A V = IB V = -.0081452X 9000 (1-e ) 3 V = 4A V 4B V = 5A V = -.0075736(X-24.04) 10245 (1-e ) 5.8652 -.019105X) 7500 (1-0.50073 e ) \l- '. 8 8 2 L 2 (i~' 0l5502X 3 U e } ) 8^3/a+e'' 0256^(X-103.92) ) -.02245(X-94.69) 5B V - -901.44+9444.4/(1+e ) -.017262(X-81.75) -e 6A V = 8759 e Inflection: none none 64.59 , 2976.75 70.29 , 2611.08 70.29 , 2611.08 103.92 , 4210.65 94.69 , 3820.76 81.75 , 3222.23 7A* a - 8098.3 k = .31756 b = .029306 G = 66.765 c - 1.3620 x 10""6 66.77 , 2571.70 MREG V - 937.887+0.381117 X2-.00103875 X3 BCFS-VAC 122.30 , 4738.20 90 , 3400 + Best functions underlined * See Appendix 6; applicable to a l l other appendices on "Predicted Functions". APPENDIX 8 PREDICTED FUNCTIONS CEDAR GROUP (POOR SITE - VAC 266) -.0039567X Function 1A V = 10,000 ( l - e ) -.0036382(X-28.0) IB V - 12000 ( l - e ) 2.5539 -.0081741X 2A V = 9293.4 ( l - e ) 2.7525 -.011133(X-4.976) 2B V = 8000 ( l - e ) -.011057X 3 4A V = 8000 ( l - e ) -.011697(x-10.0) 3 4B V = 8000 ( l - e ) .019982(X-150) 5A V = 7785.5/(1+e ) -.017586(X-142.07) 5B V = -650+8500/(1+e ) -.010896(X-133.04) -e 6A V = 9000 e In f l e c t i o n : none none 114.71 , 2602.15 96.17 , 2308.80 99.36 , 2368.00 103.92 , 2368.00 150.00 , 3892.75 142.07 , 3600.00 133.04, 3310.92 7A a = 6207.7 k = .32341 b = .009267 G = 88.398 c = 9.1455 x 1 0 - 6 88.40 , 2007.63 7B H a b c BCFS-VAC = .00003 - 6207.4 = .0076496 = 1.0110 x 10 k = .32103 G = 89.069 -5 MREG V = 2763.62 + .069919 X z - 83818.7/X 89.05 , 1992.76 106.23 , 2763.61 90 , 2500 63 APPENDIX 9 PREDICTED FUNCTIONS PINE GROUP (GOOD SITE - VAC 578) •.0081964X) Function 1A V = 5570.7 (1-e -.023036(X-30) IB V - 3635.9 (1-e ) 7.5001 -.043533X 2A V - 3727.8 (1-e ) 3.4597 -,05051(X-30) 2B V - 3562.5 (1-e ) -.026772X 3 4A V = 4106.7 (1-e ) -.065152(X-29.95) 3 4B V - 3459.1 (1-e ) -.078713(X-50) 5A V - 3326.5/(l+e ) -.053324(X-48.141) -e 6A V - 3605 e In f l e c t i o n : none none 46.28 , 1274.91 54.57 , 1094.40 41.04 , 1215.58 46.81 , 1023.89 50.00 , 1663.25 48.14 , 1326.39 7A a = 3815.5 b - .077309 c = 7.3558 x 10 -6 k = .32749 G = 44.816 CONREF V = 46.5017X+111.4X^-144921/X1/3 44.82 , 1249.54 none BCFS-VAC 53 , 1500 APPENDIX 10 PREDICTED FUNCTIONS PINE GROUP (MEDIUM SITE - VAC 79) -.019609X) Function 1A V = 2561.2 (1-e -,023598(X-6.887) IB V - 2479.1 (1-e ) -.027500X 2A V » 2435.1 (1-e ) 1.5 -.013174X 0 , 3 3_ V - 2500.0 (l-1.4408e ) -.041957X 3 4A V - 2293.9 (1-e ) -041955(X-l.5 * 10~5) 4B V = 2294.0 (1-e ) -.04403KX-34.843) 5A V - 2336.3/(1+e ) -.032873(X-24.178) -e 6A V - 2399.3 e Inflection: none none 14.74 , 468.51 none 26.18 , 678.99 26.19 , 679.02 34.84 , 1168.15 24.18 , 882.65 7A a = 2345.9 k - .35022 b = .009635 G = 4.3645 c - 6.4717 * 10~6 MREG V - 3549.16 - 13638.3/X3* 4.36 , 821.58 none BCFS-VAC 52 , 900 65 APPENDIX 11 PREDICTED FUNCTIONS PINE GROUP (POOR SITE - VAC 572) -.0053239X Function 1A V = 2800 (1-e ) -.004532(X-29.90) IB V = 4400 (1-e ) 4.9851 -.019845X 2A V =» 2478.1 (1-e ) 2.4989 -.017154(X-27.014) 2B V - 2824.8 (1-e ) 3.0383 -.017715X 3 V - 2455 (l-1.2524e ) -.013161X 4A V - 3000 (1-e ) -.02129(X-28.999) 4B V = 2634.6 (1-e ) -;»03455(X-90) 5A V = 2204.5/(l+e ) -.0077725(X-90) 5B V = -3000 + 7937.6/(l+e ) -.016986(X-92.579) -e 6A V - 2800e I n f l e c t i o n : none none 80.95 , 811.58 80.43 , 787.55 71.56 , 654.99 83.47 , 888.00 80.60 , 779.84 90.00 , 1102.25 90.00 , 968.80 92.58 , 1030.06 7A a • 2076.3 k = .25759 b = .048096 G = 46.855 c = 2.7769 * 1 0 - 6 CONREG V - 685.945X^1227.40X1/3 46.86 , 534.83 none BCFS-VAC 60 , 550 66 APPENDIX 12 PREDICTED FUNCTIONS DOUGLAS-FIR (GOOD SITE - VAC 757) Function 1A G=21.1 IB G VAR. IB G=28.0 IB 2A G=21.1 2B G=28.0 2B G=21.1 4B 0=28.0 4B G VAR. 4B 5A 5B 6A -.0070658X V - 12007 (l-e ) -.011998(X-21.1) V - 10157 (l-e ) -.02103(X-40.885) V - 9063.8 (l-e ) -.014429(X-280) V = 9722.3 (l-e ) 5.5270 -.29512X V = 8741.8 (l-e ) 2.8195 -.028684(X-21.1) V = 8831.3 (l-e ) 2.20 -.028333(X-28.0) V - 8821.7 (l-e ) 9.7357 -.030952X V = 8999.8 (l-0.71800e ) 4A V = 9280.0 (l-e -.02115X ) -.029696(X-21.1) 3 V = 8794.4 (l-e ) -.033894(X-28.0) 3 V - 8645.9 (l-e ) V = 10,000 (l-e -,023460(X-16.0) ) -.046153(X-74.671) V = 8571.1/(l+e ) -.034731(X-55.0) V = 4207.8 + 12946/(1+e ) -.034154(X-64.197) -e V - 8733.1 e -.034238(X-64.34) -e 6B V - 20.00 + 8717.4e Inflection: none none none none 57.93 , 2900.53 57.24 , 2566.82 55.83 , 2365.98 64.10 , 3266.93 52.03 , 2746.88 58.10 , 2603.14 60.41 , 2599.19 62.83 , 2960.00 74.67 , 4285.55 55.00 , 2265.20 64.20 , 3212.73 64.34 , 3224.75 67 APPENDIX 12 (cont'd.) Function 7A a = 8756.00 k = .41649 b = .057590 G - 69.804 c = 1.0288 * 10" 6 69.80 , 3646.79 7B H = 200.00 k = .39926 a - 8416.3 G = 69.723 b = .064353 c - 2.5360 * 10~ 6 69.72 , 3560.29 MREG V = 11265.1 - 485750/X none BCFS-VAC 70 , 4000 68 APPENDIX 13 PREDICTED FUNCTIONS DOUGLAS-FIR (MEDIUM SITE - VAC 968) -.0079233X Function 1A V = 6000 (1-e ) ( -.027037X) 2A V = 5314.3 (1-e ) 4.2331 3.4998 -.050998(X-29.30) 2B V = 5000.0 (1-e ) -.014306X 3 V - 5303.1 (1-0.75321 e, ) 2.2932 -.020674X 4A V = 5874.8 (1-e ) -.055452(X-29) 3 4B V = 4155.0 (1-e ) -.027669(X-85.112) 5A V = 6000.0/(l+e ) -.01472(X-35.625) 5B V = -4946.9 + 12000/(1+e ) -.017333(X-68.879) -e 6A V = 6693.9 e I n f l e c t i o n : none 53.86 , 1698.45 53.38 , 1540.00 47.56 , 1763.28 53.14 , 1738.94 48.81 , 1229.88 85.11 , 2000.00 35.63 , 1053.10 68.88 , 2462.55 7A a = 6000.0 k = .28311 b = .042448 G = 49.98 c = 2.1680 * 10~ 5 49.98 , 1698.66 MREG V = 42.3911 + 33.8789X none BCFS-VAC none 69 APPENDIX 14 PREDICTED FUNCTIONS DOUGLAS-FIR (POOR SITE - VAC 575) 8.3587 -.0030626X Function 1A V = 5000 (l-e ) 6.2344 -.027719X 2A V = 2520 (l - e ) -.030877X 2 V = 2499.6 (l-1.1986e ) -.015650X 3 4A V - 2817.2 (l-i ) -.051060(X-89.154) 5A V = 2441.0/(1+e ) -.032931(X-76.715) -e 6A V = 2480 e -.03233(X-79.025) -e 6B V - 45.176 + 2500e I n f l e c t i o n : none 66.03 , 847.22 74.00 , 841.62 70.20 , 833.89 89.15 , 1220.50 76.72 , 912.34 79.03 , 964.88 7A a = 2508.8 k = .55873 b - .034866 G = 97.531 c = 4.0035 * 1 0 - 5 97.53 , 1401.74 7B H a 138.30 2416.6 .0300 k = .55053 G - 98.596 c = 4.1451 * 10" 5 98.60 , 1468.71 MREG V - -23746.4 - 56.7503X + 6269.54X 1 / 3 + 184555/X 65.60 , 629.17 BCFS-VAC 90 , 1250 APPENDIX 15 MAXIMUM MAI FOR THE GOMPERTZ FUNCTION -c(X-S) -e V = ae By definition, MAI - -JL Age Maximum occurs at dMAI = Q dX d(MAI) -e(X-G) -e -c(X-G) -1 d x = ace e X -c(X-G) -e -2 - ae X -c(X-G) -1 For maximum, ce -X =0 -c(X-G) * c e = 1 Ln c + Ln X - c(X-G) = 0 c, G known; solve for X by any iterative procedure or otherwise. 

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