)V1531_} RETURNING MATERIALS: Place in book drop to LIBRARJES remove this checkout from .a-uncyunnL your record. FINES wiII be charged if book is returned after the date stamped below. 1-1; {.3 A SIMPLIFIED METHOD FOR QUANTIFYING DIAMETER DISTRIBUTIONS OF NORTHERN MICHIGAN UPLAND HARDWOODS WITH THE WEIBULL AND 83 FUNCTIONS BY Robert De Geus A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Forestry 1986 ABSTRACT A SIMPLIFIED METHOD FOR QUANTIFYING DIAMETER DISTRIBUTIONS OF NORTHERN MICHIGAN UPLAND HARDWOODS WITH THE WEIBULL AND SB FUNCTIONS by Robert De Geus This study presents a unified and simplified method for applying the Weibull or SB distribution to modelling of tree diameter distributions. This approach uses spreadsheet software and a microcomputer rather than a mainframe computer. The methodology is based on the use of point sample data and grouped frequencies Of observed data. Grouped frequencies, and maximum likelihood and percentile estimation are used to estimate distribution parameters. The Weibull and SB functions are compared for both estimation methods. Ten ecological groupings of 72 upland hardwood stands are used for diameter distribution modelling.‘The stands are composed of mixed species and age groups. The methods described are found to provide reasonably good models. The Weibull function*with maximum likelihood parameter estimation is found to perform best with these methods and data. Percentile parameter estimation for the Weibull is suggested as the starting point with these methods for obtaining the maximum likelihood estimates. ACKNOWLEDGEMENTS I would like to express my appreciation to Dr. Carl Ramm, graduate committee chairman, for his belief in a light hand on the reins. It is he, Dr. Kurt Pregitzer, and Dr. David Lusch that have made completion of this thesis possible. Recognition is due Niro Higuchi of INPA, Manaus, Brazil for providing perspective in an unfamiliar environment. The education represented by this thesis would. not have been complete without him. Dr. Phu Nguyen deserves many thanks for his help and friendship» Steve Westin and George Host were invaluable in many ways. The largest share of gratitude is reserved for my wife, Reenie, as well as my family. They must be wizards because I feel like the scarecrow. iii TABLE OF CONTENTS page LIST OF TABLES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Vi LIST OF FIGURES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Viii INTRODUCTION OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 1 Literature Review ..................................... 5 I. Background .................................... 5 II. The Distribution Models ..................... 11 A. The Weibull Distribution ................ 11 B. Johnson's SB Distribution ............... 18 III. Applications of Distribution Models to ForeStry OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 28 A. Stand Structure ......................... 28 B. Forest Growth and Yield Modelling ....... 29 C. Other Applications ...................... 31 Materials and MEthOds O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 34 I. The Sampling Method .......................... 34 II. Data PIOCESSing O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 40 A. Point Sample Data and Distribution “Odelling OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 41 B O calcu13tion T0018 O O O O O O O O O O O O O O O O O O O O O O O 42 III. Distribution Model Development ............. 43 A. weibUJ-l Distribution O O O O O O O O O O O O O O O O O O O O 43 Bo JOhflSOn'S SB DiStribUtion 00.000.000.000. 47 iv page IVO Data anaIYSis OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 51 A. GOOdflEBB’Of'Fit StatiStiCS 0000000000000. 51 Bo ELTP‘LEVEI ParamEterS 00000000000000.0000 54 C. Skewness and Kurtosis ...............R... 56 Results and Discussion ............................... 58 I. Modelling Results ............................ 58 II. Goodness-of—Fit Results ..................... 67 A. Percentile Estimation ................... 68 B. Maximum Likelihood Estimation ........... 71 III. Goodness-Of-Fit for Classes 11 Inches and Greater OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 75 IV. ELTP-Level Parameters ....................... 77 V. Skewness and Kurtosis ........................ 85 Summary and Conclusions ............................. 119 Appendix I: Ecological Land Type Phase Descriptions 125 Appendix II: SB Percentile Parameter Estimation Tables OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 131 Bibliography OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 137 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table 10: LIST OF TABLES page ECS Stand Assignments by ELTP ................ 38 Weibull Distribution Parameters .............. 59 SB Distribution Parameters ................... 60 Goodness-of-Fit Statistics for Predicted Diameter Distributions ....................... 69 Goodness-of-Fit Statistics for Predicted Diameter Distributions ....................... 72 Goodness-of-Fit Of Predicted Distributions for Diameter Classes 11" and Greater ......... 76 Observed and Predicted Averages for ELTP- Leve1 Parameters OOOOOOOOOOOOOOOOOOOOOOOOOOOOO 79 Observed and Predicted Averages for ELTP- Level Parameters OOOOOOOOOOOOOOOOOOOOOOOOOOOOO 80 Bias of Predicted ELTP-Level Parameters and Statistics OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 81 Observed and Predicted ELTP Kurtosis and Skewness OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 86 ELTP 1: Observed and Predicted Diameter Distributions OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 89 ELTP 10: Observed and Predicted Diameter Distributions OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 92 ELTP 12: Observed and Predicted Diameter Distributions OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 95 ELTP 20: Observed and Predicted Diameter DiStributiona OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 98 ELTP 21: Observed and Predicted Diameter Distributions OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 101 ELTP 35: Observed and Predicted Diameter Distributions OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 104 - ELTP 37: Observed and Predicted Diameter Distributions OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 107 vi Table Table Table Table 18: 19: 20: 21: vii page ELTP 40: Observed and Predicted Diameter Distributions O...0OOOOOOOOOOOOOOOOOOOOOO0.0 110 ELTP 43: Observed and Predicted Diameter Distributions 00......0000.00.00.00.00.00.... 113 ELTP 45: Observed and Predicted Diameter Distributions 0OOOOOOOOOOOIOOOOOOO0.......0. 116 Tables for percentile estimation of Johnson's distribution parameters .......... 132 Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Figure 12: Figure 13: LIST OF FIGURES page Skewness-Kurtosis space ...................... 7 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP l. OOOOOOOOOOOOOOOOOOOO...OOOOOOOOOOOOOO 90 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 1. OOOOOOOOOOIOOOO0.00000000000000000000 91 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 10. OOOOOOOOOOOOO...OOOOOOOOOOOOOOOOOOOO 93 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 10. 0.00000000000000000000000000000000CO 94 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 12. O00......O...OOOOOOOOOOOOOOOOOOOOOO. 96 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 12. O.C0.0000000000000000000000.......0. 97 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 20. 0.....0000...OOOOOOOOIOOOOOOOOOO0.0. 99 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 20. 00......COO...OOOOOOOOOOOOOOOOO...O. 100 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 21. OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOI 102 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 21. OOOOOOOOOOOOOOCOOOOOO0.0.0.0000... 103 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 35. I00.0.0.0...OOOOOOIOOOOOOOIOOOOOOO 105 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 35. 0.0.0.000...OOOOOOOOOOOOOOOOOOOOOO 106 viii Figure Figure Figure Figure Figure Figure Figure Figure 14: 15: 16: 17: 19: 20: 21: ix page Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 37. OOOOOOOOOOOOOOO0.00000000000000000 108 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 37. OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0.0. 109 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 4o. OOOOOOOIOOOOOOIOOOOOOOO00.0.0.0... 111 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 40. OOOOOOOOOOOOOOOOOOOOOOOOOIOOOOOOO. 112 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 43. OOOOOOOOOOOOIOIOOOO0.0000000000000 114 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 43. OOOOOOOOOOOOOOOOO00.000.000.000... 115 Observed and predicted (percentile parameter estimation) diameter distributions for ELTP 45. OOOOOOOOOOOOOOO...OOOOOOOOOOOOOOIO 117 Observed and predicted (maximum likelihood estimation) diameter distributions for ELTP 45. OOOOOOIOOOOOOIOOOOOOOOOOOOOOOOOOO. 118 INTRODUCTION Tree diameters are of interest to forest managers and researchers for a variety of reasons. First and foremost is the fact that tree diameter is well correlated with other important variables such as volume, value, conversion cost, and product specification. Secondly, from a silvicultural standpoint, diameter distributions are useful in understanding current and future stand structure. Obtaining a statistical or mathematical model of an observed diameter distribution can be an important step in developing growth and yield projection models. Recent projection models have incorporated diameter distribution models to obtain better projection accuracy. The improvement in projection is related to the capacity of a projection model to project stand attributes by diameter class. By predicting diameter distributions at a given time from stand variables, stand simulators using such models maintain some of the detail of individual tree simulators without the necessity for keeping track of individual trees throughout the simulated rotation (Little, 1983). This distributional information is useful in delineating yield by diameter class, identifying silvicultural opportunities, and projecting the impact of management decisions over a rotation. Distributional models have not been developed for forestry. Instead, general distribution functions that can generate a wide variety of curve shapes have been adapted for use in modelling tree diameter distributions..Any distribution model is dependent upon certain constants or parameters for use. These parameters are estimated from sample data. For the normal distribution, these parameters (mean and standard deviation) are familiar and well understood. The parameters of other distributions, such as those used for modelling diameter distributions, are not well understood. The result is that parameter estimation is often a difficult or vague process. One of the more important distribution models in tree diameter distribution modelling is the Weibull distribution. The Weibull distribution has been successfully applied to many diameter distribution modelling problems, principally in even-aged, single species plantations or forests. Another model that is relatively new to diameter distribution modelling is the SB distribution of ELL. Johnson (1949). The 58 distribution is more complex than the Weibull and is theoretically capable of producing a wider range of curve shapes (Hafley and Schreuder, 1977). The Weibull distribution is widely accepted in tree diameter distribution modelling and in growth and yield modelling based on distribution models. There is, however, 3 no biological basis for the use of this or any distribution model in forestry, and there remains some uncertainty about model selection and application (Bailey, 1980). Therefore, models such as the SB are of interest in the search for the best possible model. The scope of this study is two-fold. First, to address the problem of non-normal distribution parameter estimation, a simplified (and as explicit as possible) methodology for parameter estimation for the Weibull and SB distributions is described and applied. Second, the relative performance of the Weibull and SB distributions are assessed using observed diameter distributions from ten ecological forest groupings. The purpose of the first objective is to establish parameter estimation and distribution derivation methods that facilitate diameter distribution modelling while maintaining as much accuracy as possible. As such, three new facets of diameter distribution modelling are presented. First, data obtained from sampling with probability proportional to size (point sampling) are used to derive observed diameter distributions and estimate distribution parameters. Second, grouped frequency counts are used for parameter estimation. Third, mainframe computing is avoided in favor of more accessible (and cheaper) microcomputers with spreadsheet software. In addition, the data used comprise mixed species and age upland hardwoods. This latter point is a function of the data available as well as an interest in applying simplified methods to complicated data. The purpose of the second objective is to investigate the relative performance of two distribution models using two parameter estimation methods on observed data. This investigation assesses the different models as well as their underlying methodologies. The ultimate goal is the application of the appropriate model to growth and yield modelling with the observed data. This goal is not addressed in this study. However, the results of the study include distribution parameter estimates that may be appropriate for future studies. Finally, the methods presented may serve as a basis for future diameter distribution modelling either as the current data base develops or for an entirely new data set. Literature Review I. Background DeLiocourt first introduced mathematical and statis- tical models for tree diameter distributions to forestry in 1898. Since that time, a variety of models have been inves- tigated for application to tree diameter distributions such as the negative exponential, negative power function, gamma, beta, lognormal, Weibull, and Johnson's SB (Hafley and Schreuder, 1977). The goal in these studies was to find a model that could adequately describe diameter distribu- tions based on observed sample data. The desirable proper- ties of such a function were given by Hafley and Schreuder (1977): l. the model should have a mathematical form that allows ease of computation; 2. the parameters of the function should have properties that make their estimation relatively simple and exact; 3.the function should have the capacity to generate the widest possible range of curve shapes without sacrificing accuracy of fit to observed data. The last feature given was related to the fact that observed diameter distributions take on a variety of shapes. These shapes ranged from a reversed J-shaped curve in all-aged, undisturbed forests to a mound shaped, 6 negatively skewed curve in mature conifer plantations (Bailey and Dell, 1973; Lorimer and Krug, 1980). Bailey (1980) further suggested that a single function or distributional family is desirable for describing all possible distributional curves for tree diameters in a forest. The single function approach was shown to accomodate growth and yield projection. With regard to the models mentioned above, Hafley and Schreuder (1977) noted that the beta distribution and Johnson}s SB distribution yield the wider range of curve shapes. They based their conclusion on the plot of skewness (31) and kurtosis (82) values possible for each distribution function (Figure 1). Within this graph was an "impossible region" wherein no combinations of skewness and kurtosis could mathematically exist. According to their plot, the normal distribution occupied a single point, implying a single possible shape. The Weibull, gamma, lognormal, and exponential distributions each were represented by a line. The beta distribution and Johnsonfis SB distribution each occupied a region within the graph, implying more flexibility of shape than a line alone. Of those distributions that have been investigated, Hafley and Schreuder (1977) noted that only the Weibull and Johnsonfls SB functions meet all the criteria given for an appropriate distribution for modelling tree diameter distributions. The other functions (i.e., the gamma, Kurtosls Skewness 54 Figure I: Skewness-Q Kunosis space 8 lognormal, exponential, and beta distributions) showed problems with relative inflexibility, difficulty in parameter estimation, or difficulty in mathematical use of the function. In general, the mathematical form of a distribution was given by all authors as the probability density function (pdf) denoted as f(x). A further criterion for a pdf was given by Johnson (1949) as ease of integration of f(x) to obtain F(x), the cumulative probability density function (cdf). The cdf was given as a means of determining the proportion of observations that would occur below a certain threshold observation. In all studies citing criteria for distribution selection, the emphasis was on simplicity and/or ease of calculation in the use of a model. Most literature has not been concerned with this feature and Johnson (1949) claimed that this has held back the practical application of non- normal distributions.‘The result has been that the dominant distribution and theory in statistical applications has been the normal distribution (Johnson, 1949). One major reason given for this dominance was the ease with which moments of the distribution and, therefore, distribution parameter estimates can be obtained. Johnson (1949) noted that three basic approaches to distribution parameter estimation exist. These were given as maximum likelihood estimation, the method of moments, and percentile estimation. He also summarized two conceptual approaches to the use of distribution models in fitting observed frequencies. The first was the idea that there should exist a theoretical, statistical basis in applying a distribution function to observed data. This idea was further expanded to state that the parameters of a function must be estimated on the basis of statistical theory in order to ensure that these estimates accurately represent the underlying population distribution parameters. The second concept was that the most important criterion in using a distribution model to fit observed frequencies is nothing more than the goodness of the fit. Any underlying theory or statistical parameter estimation was said to be of little interest for this approach. Johnson (1949) concluded that subscribing to either approach is dependent on the individual's objectives in using a distribution model. He suggested that a desire to develop inferences about an underlying population based on the use of a distribution model would require some degree of theoretical basis. Maximum.likelihood parameter estimation was said by Cohen.(l965) to provide unique estimates of parameters with Ininimum bias in a statistically sound manner..He showed that maximum likelihood methods are based on the distribution function itself as applied to observed frequency data. Johnson (1949) stated that the method of 10 moments is most practically applied to the normal distribution since calculation of sample moments for non- normal distributions is often very difficult. As such, the method of moments was not given as a general procedure. Percentile estimation was presented as something of the pattern maker's approach to curve fitting. The generalized method was given by Dubey (1967) as one in which observations at particular percentile points of the observed distribution are used to estimate parameters of a curve identical to the observed curve. Percentile parameter estimation was noted to yield estimates that are not unique and may not have much theoretical relation to the underlying population distribution. Johnson (1949) related the methods of parameter estimation to the differing conceptual approaches to distribution function useuIHe wrote that from a theoretical standpoint, parameter estimates ought to be unique and should have some theoretical relationship to the underlying population distribution. From the standpoint of curve fitting alone, parameter estimates were not required to be unique or theoretically related to some underlying population distribution. In this latter case, Johnson (1949) concluded that any inference about a population using a distribution model would be entirely dependent upon the sampling method used and the depth of inference desired.‘The former conceptual approach was given as being ll appropriate for maximum likelihood parameter estimation. The latter case was given as being appropriate for either method of estimation. A generalized approach to the application of distribution models to observed data was given by Johnson anu Kotz (1970). They showed that observations on a continuous variable could be grouped into strata or classes, usually of equal width. They suggested that while the resulting variable was then discrete, this transformation was one merely of convenience. They wrote that inference or graphic presentation could be done on the basis of the original continuity of the data. II. The Distribution Models A. The Weibull Distribution Weibull (1951) introduced a distribution model that gained wide acceptance in materials strength testing. Since its introduction, many studies have explored other applications and the intricacies of the distributionfls properties. Despite these investigations, Dubey (1967) claimed that the Weibull distribution is not well understood. Bailey and Dell (1973) applied the Weibull distribution to the problem of modelling tree diameter distributions. Since their introduction, the Weibull distribution has gained a great deal of attention and, to 12 some degree, acceptance in forestry; There was no biological basis for their application; Bailey and Dell (1973) wrote that the Weibull distribution was chosen because of the relatively wide range of curve shapes it produces, the relative ease with which parameters may be estimated, and the ease by which the pdf can be integrated to obtain F1x). They found that the Weibull distribution effectively modelled even-aged conifer plantation diameter distributions. Bailey and Dell (1973) gave the three-parameter Weibull probability density function as f(x) = (C/b)(((x-a)/b)“(c-1)) eXPI-((x-a)/b)“c] where x = diameter at breast height (dbh) a = location parameter b = scale parameter c = shape parameter. The location parameter (a) was described by Zarnoch, Ramm, Rudolph, and Day (1980) as the smallest diameter in the population. In general, it was given by Dubey (1967) as the lower end-point of the distribution. Thus, in a population of trees, the location parameter (a) was noted to be theoretically zero. The scale parameter (b) was described by Little (1983) as the 63rd percentile of a population. In other words, b is the diameter below which 63% of all ordered observations would occur. 13 The shape parameter (c) was said by Bailey and Dell (1973) and Lorimer and Krug (1983) to be associated with the general degree of skewness of the distribution. They showed that for c3.6 the curve becomes negatively skewed..As c approaches infinity, the curve becomes a spike over a single point. Integration of the pdf resulted in the cumulative probability function (Bailey and Dell, 1973): F(x) = 1 - EXPI-((x-a)/b)cl with parameters as given above. This function was shown to provide the percentage of observations below a given diameter (x). Johnson and Kotz (1970) mentioned that in a strict sense it is necessary to use F(x) to obtain percentages for classes formed from a continuous variable. They further stated, though, that the probability density function f(x) may be used to obtain percentages for classes without any significant loss of accuracy in fit. In such a case they suggested that inferences based on such an approach could suffer some loss of precision because the continuous variable is treated altogether as a discrete variable. 14 The moments of the Weibull distribution were presented by Johnson and Kotz (1970). It was apparent that under any circumstance the moments present serious difficulty in calculation. Estimation of the parameters of the Weibull distribution was shown by Cohen (1965) and Bailey and Dell (1973) for maximum likelihood methods, by Zarnoch and Dell (1985) and Dubey (1967) for percentile methods. Other methods were found based on tables (Mann, 1967) and the sample coefficient of variation (Newby, 1980). Neither of the latter methods were shown to be a superior method for parameter estimation. Bailey and Dell (1973) noted‘that maximum likelihood estimators are generally the best but for the Weibull distribution the method requires iterative computations. They stressed that the easiest estimators to compute are based on percentiles. Dubey (1967) and Zarnoch and Dell (1985) presented methods for estimating the shape parameter using two percentile points: 0.17 and 0.97. These two percentiles were found by Dubey (1967) to have 82% asymptotic efficiency, the highest efficiency of the percentiles examined. For estimating the scale parameter b, Zarnoch and Dell (1985) suggested the 63rd quantile minus the estimate of the location parameter (a). For estimating the location parameter (a), they suggested setting it equal to the 15 smallest observed diameter. Dubey (1967), on the other hand, suggested a method for estimating a based on three percentiles. He wrote that his method yielded a value of the location parameter that would not exceed the smallest observation nor would it fall below some theoretical limit. Cohen (1965) developed the iterative form of the maximum.likelihood estimators for the Weibull distribution. The likelihood function of the Weibull pdf was manipulated to form two equations. In one, the shape parameter (c) appeared on both sides of the equation necessitating an initial estimate of c and an iterative solution. When the two sides of the equation converged to equality, the estimated value of c (3) that brought about the equality was the maximum likelihood estimate of the shape parameter. In the second equation the estimate of c was used to estimate the scale parameter u». He offered no method for estimating the location parameter (a) in conjunction with b and c. Cohen (1965) only dealt with the two-parameter Weibull pdf which does not include a location parameter. It seemed possible that a likelihood function could be derived for the three- parameter Weibull pdf and that a, b, and c could be estimated more or less simultaneously. Zarnoch and Dell (1985) presented maximum likelihood estimation as a difficult and costly method. However, Cohen (1965) pointed out that solving for the shape parameter (c) by using the iterative formula could be done effectively 16 using trial and error, giving a simpler and cheaper method. He wrote that it is only necessary to obtain two estimates of c that bracket the final value of 6. The bracketing was explained as the situation when two values of the left hand side of the iterative formula for c are above and below the value of the right hand side. The final step was given as linear interpolation between the two incomplete estimates of c to obtain the actual estimate of c. The vague proviso given was that the two bracketing values of 6 must be within a "sufficiently narrow interval”. The functional forms of these estimation methods were given as follows: From Cohen (1965) and Zarnoch and Dell (1985) for maximum likelihood estimation, the following methods applied: 1. c is estimated iteratively from [(ina 1n xi)/(2xie)1 - 1/8 = [(1/n) (21a xi)] 2. b is estimated from E = [(l/n) (ine)11/5 From Zarnoch and Dell (1985) for percentile estimation, these methods were given: 1. a is estimated from a = (x1 xn.- x22 )/(x1 + xn - 2x2 ) where xi = the ith ordered value (ascending) in the sample, 17 n = the sample size, x1 = the smallest ordered value in the sample. 2. b is estimated from B = ‘3 + x.63n where x.63n = the 63rd quantile of the sample. 3. c is estimated from a = lnlln(1-Pk )/1n(1'Pi )1 1nl0 of a standard normal variate. Based on this value of 2 they proposed setting four points as +/-z and +/-32. For the SB' the distances between each of the outer and inner end-points would be smaller than the 25 distance between the two inner points. They maintained that this relationship would be the result of the bounding on the SB. In detail, Slifker and Shapiro (1980) asked to let the sample quantiles x32, x2, x_z, and x_3z correspond to 32, z, -z, and -32. Three relationships were then defined as m = x3z ' xz n = x--z ' x-3z These three measures of distance within the sample distribution were used to establish the following criteria - for selection of one of Johnson's models: if mn/p2 >1 then fit so; if mn/p2 = 1 then fit SL' the lognormal; if mn/p2 < 1 then fit 33' Using m, n, and p and hypergeometric trigonometric functions they defined parameter estimators for 30' SL' and SB..For SB these estimators were given as 3 = z/{cosh‘1(1/2t(1+p/m)(1+p/n)11/2)} A . _ ((p/n)-(p/m)){1+p/n)(1+p/m,-4}1/2 9 = d sinh l i((P7n)(P/m)-l) P[{(1+p/n)(1+p/m)-2}2-4]1/2 i = ((p/n)(p/m))-1 . x2 + x-z i + P((P/n)-(P/m)) w = - i ‘7' -§((p/n)(p/m)-1) 26 Slifker and Shapiro (1980) based the initial choice of 2 on sample size. They suggested that for a moderately sized sample 2 should be set at less than one: as sample size increases, the largest practical choice of 2 also increases. For determining actual percentiles from the data, Slifker and Shapiro (1980) showed that percentages P- are 3 obtained from tabulated 2 values with j = 32, z, -z, -Bz. The next step given was to obtain the percentile x‘i) from i = nPj + 1/2. The final step given was to obtain the value of the observations corresponding to the four observed percentile points and using these to compute m, n, and p. Mage (1980) followed a somewhat similar line of development but made allowance for the use of four equidistant 2 values that need not be symmetric. He maintained that the use of four symmetric and equidistant normal variates leads to gross simplification. He also stated that Slifker and Shapiro's (1980) use of hyperbolic trigonometric functions offered more difficulty but was more powerful than Mage's use of natural log functions with 2 values that need not be symmetrical. In addition, Mage (1980) wrote that the use of symmetric normal variates provides for maximum efficiencies in SB parameter estimation. 27 Mage (1980) pointed out a major difficulty in both percentile estimation methods. He wrote that the choice of an initial 2 value dictates the resulting parameter estimates. He established that different 2 values yield different parameter estimates. He stressed that the ambiguity of obtaining different parameter estimates by different percentile choices may be unacceptable in some applications. However, from examination of Johnson's (1949) numerical examples as well as those of Slifker and Shapiro (1980) and Mage (1980), it appeared as though a particular range of 2 values could apply to particular types of populations. Mage (1980) made a superficial comparison between Johnson's moment estimators, Slifker and Shapiro's percentile estimators, and Mage's percentile estimators. He reached no conclusion but from visual inspection it appeared that there was better agreement between the moment estimators and Slifker and Shapiro's estimators than with Mage's estimators. Johnson (1949) noted that the SB was capable of producing two unique curve shapes. He showed that a bimodal or dish shaped curve and a flat topped curve were possible. In general, these curves were taken to be somewhat trivial. However, their existence seemed to further imply the difficulty in using the current, seemingly vague percentile estimation methods for Johnson's SB' 28 III. Applications of Distribution Models to Forestry A. Stand structure Lorimer and Krug (1983) wrote that the most promising method of indirectly assessing forest age structure has been the interpretation of diameter distributions. They proposed using distribution function parameter estimates and distribution curve shapes as indices for forest age structure. Lorimer and Krug (1983) found that even-aged stands typically have unimodal diameter distributions.‘They noted that these stands exhibit varying degrees of positive skewness at a young age but approach a more symmetric distribution with time. All-aged stands were shown to have steeply descending, monotonic diameter distributions that can be represented by the negative exponential distribution function. Multi-aged stands were said to consist of several age classes that may or may not have equal prominence. These stands varied from near normal to irregular negative exponential distributions. The variation depended in part on the proportion of shade tolerant species in the stand. Shade tolerant species were found by Lorimer and Krug (1983) to have the greatest variety of curve forms, frequently deviating from a symmetric, unimodal shape. The lack of symmetry was attributed to the large number of suppressed trees. 29 Lorimer and Krug (1983) concluded that for all-aged stands, the Weibull shape parameter (c) is less than 1.0 in most cases and is therefore distinct from the shape parameters of even-aged stands.‘They decided that it was difficult to distinguish even-aged stands from multi-aged stands through the use of diameter distributions. Lorimer and Krug (1983) also noted that the diameter distribution of overstory trees alone was near-normal for all species. This was the case even when the total distribution was highly skewed. Lorimer and Frelich (1984) wrote that the negative exponential distribution has been used with all-aged forest diameter distributions.‘They stated that a semi-log plot of this distribution produces a straight line which implies an invariant rate of attrition from one size class to the next. This rate was shown to be inappropriate since mortality rates decline precipitously as trees progress from saplings to dominant trees. They provided evidence that it is possible to demand properties of a distribution model that agree with biological conditions associated with a population to be modelled. 8. Forest growth and yield modelling Bailey (1980) noted that recent growth and yield models have incorporated techniques for predicting changes in diameter distribution with stand age. He described these models as using an assumed distribution that is fitted to 30 stand data. The models were further described as using estimates of distribution parameters to develop least squares (regression) equations to predict the parameters, and the distributions, from stand age. Bailey (1980) set as a requirement for these models that a diameter distribution remain in a given family over the projection time. This requirement was easily justified in that it would be very awkward to respecify a distribution function, and use parameter predictions, at any step in the projection. From this point Bailey (1980) set out to show that the Weibull distribution exhibits a key property that provides some tentative biological justification for its use. It was shown by Bailey (1980) that the Weibull distribution allows the assumption of a nonconstant relative growth rate in diameter at any two ages. On the other hand, he showed that Johnson's 38 distribution forces the assumption of constant relative growth rate in diameter at two ages. He concluded, based on his data, that relative diameter growth rate cannot be assumed to be constant over all ages for a given density. Hyink and Moser (1983) acknowledged that the diameter distribution method for predicting yields and stand structure in even-aged forests was firmly established.‘They went on to present two approaches to the use of diameter distribution models in growth.and yield modelling. One was 31 the parameter prediction model, defined as the process of predicting the future values of distribution function parameters. This step was given to lead to computation of stand average attributes such as volume, basal area per acre, trees per acre, and quadratic mean diameter. The other method was the parameter recovery model, defined as the process of predicting future values of stand attributes and then computing the distribution parameter estimates of the underlying diameter distribution. In general, Hyink and Moser (1983) stated that the parameter prediction models are viewed as being somewhat more informative than the parameter recovery models. Their reasoning was that parameter prediction models allow computation of total stand attribute values as well as their distribution by diameter class. . Hyink and Moser (1983) concluded with a remark that tends to sum up the state of knowledge of distribution models and their use in forestry. They stated that there is poor, if any, understanding of the biological relationships between specific distribution function parameters, the forest populations they characterize, and the characteristics of the site upon which they reside. C. Other applications Quang and Burkhardt (1984) presented a method for modelling irregular diameter distribution curves.‘They focused on the Weibull distribution but showed the general 32 applicability of the method to distribution functions. In essence, they used the cumulative probability function of a distribution and different sets of parameter estimates to model segments of a sample distribution. The segment cumulative functions were then joined to create an overall model of the irregular curve. Ek, Issos, and Bailey (1975) discussed how to estimate Weibull distribution parameters so as to obtain a particular result beyond that of fitting an observed distribution. They showed that a distribution could be modelled in such a way as to have the model produce a specific quadratic mean diameter. They defined the expected value of the quadratic mean as a quadratic equation involving gamma functions. The distribution parameters were defined as the positive roots of the quadratic equation. No explicit solution of parameters was found possible when the shape parameter (c) is unknown. Little (1983) has provided the only research to date on fitting a distribution model to tree diameter distributions in a mixed species forest. She developed parameter prediction equations based on stand attributes but pointed out that individual species distributions were still needed to complete the work. She found, however, that the Weibull distribution fit the observed diameter distribution of the mixed stands quite well. The major 33 implication of her work was that the use of distribution models need not be restricted to even-aged, single-species forests or stands. Zarnoch, et al. (1980) used the Weibull distribution to model changes in red pine diameter distributions under different thinning treatments. They looked at proportions of trees per acre and basal area per acre estimated by the Weibull function for the various thinnings. They found that the Weibull provided an adequate fit to observed diameter distributions. Materials and Methods I. The Sampling Method During the summers of 1983-1985, eighty forest stands were selected and sampled as part of the Ecological Classification System (ECS) study - a cooperative agreement between Michigan State University and the 0.8. Forest Service. The 80 stands are composed of upland hardwoods and many are in "late successional" stages.‘The sampling was done on the western unit of the Huron-Manistee National Forest located in the northwestern portion of the Lower Penninsula of Michigan. This area includes parts of Newaygo, Lake, Wexford, and Manistee counties. Stands were selected at random from a list of stands which exhibited the appropriate overstory characteristics. The list was compiled by air photo interpretation and ground reconnaisance. The critical overstory characteristics for the ECS study were defined as composition, degree of disturbance, age, basal area, and degree of aspen presence. The overstory was limited to well-stocked upland hardwoods of at least forty years of age. The aspen component was restricted to 20 % of total basal area or less. Disturbance was defined as evidence of harvesting (or other cutting) or fire in the last forty years. Evidence of grazing, insect attack or disease were also cause for 34 35 rejecting a stand. Stand size was defined as a minimum of 2.5 acres. Other criteria for stand selection were that the stand have some consistency of slope and aspect and that it be on Forest Service land. Prior to identifying possible sample stands, the sample region was stratified by landform. Stands were then located randomly within the landform strata. A minimum number of sample stands was identified for the strata and for overstory community types. These types were defined as upland oak, mixed oak-red maple, and northern hardwoods. Once a stand was located on the ground and selected, simple random sampling was used. Permanent markers were placed at point center of all sample points as well as at a reference point that was described in stand summary notes. Four sample points were used in ECS stands 8-80 (stands numbered consecutively by chronology)‘while 6 sample points were used in stands 1-7. The initially larger sample size was used to assess the level of variability within stands. Results of this assessment allowed reduction of the sample size. The main sample point was randomly located from the reference point. The three or five remaining points were located as satellites at random azimuths and distances from the main point. Sample points were rejected if they fell outside the stand boundary. In such cases, a new random azimuth and distance was chosen and followed from the main point. 36 The overstory data which were collected included tree species, diameter, height, age, ten—year diameter increment, crown ratio, and crown class. Understory species abundance and cover were also sampled as part of the inventory. Additionally, soil profile was described and textural samples taken for laboratory analysis. At each sample point, variable radius plot sampling (point sampling) was performed. A basal area factor (BAF) of 10 (English) was used on all sample points in all stands except stands 60-66. Stands 61-66 were sampled using a BAF of 5 and stand 60 was sampled with a BAF of 5 for two points and a BAF of 10 for two points. The variable of interest in this study, diameter at breast height (dbh), was measured to 1/10 inch on all tally trees at a sample point. The minimum dbh was 3.5 inches, the lower limit of tree merchantability. Data was recorded on 0.5. Forest Service-style tally sheets, one sheet per point. The intent of stratifying and selecting stands as described was to eventually segregate stands based on landform overstory community type, ground flora species cover and abundance, and soil characteristics. Via this post-stratification, stands were grouped and their overstory productivity levels determined from per acre stand averages. The goal of this process was to identify strata characteristics by which land units could be 37 identified with regard to their potential productivity. The potentiality was derived from the "late successional" nature of many of the stands sampled. The groups of stands were termed Ecological Land Type Phases (ELTP).‘Of the eighty stands sampled, 72 were classed as one of eleven ELTP's. The remaining 8 stands were not classified due to irregularities in overstory composition and/or other characteristics that did not conform to ELTP definitions. The current ELTP classifications (Table l) were not given as final but only as an initial step in the process of defining the classification scheme and system. These unofficial classifications were issued in March of 1986. Additional samples were expected to be added to the ECS data base and these stands and further investigation could alter the current ELTP‘s. This study utilized these current ELTP's to establish an initial basis for ELTP diameter distributions and a methodology by which distribution modelling may be accomplished as ELTP's develop. Each ELTP was given a numeric code.‘The first digit of the code indicated the potential late successional overstory community; The first number was also strongly related to soil development within sandy soils. ELTP 1: pin oak - white oak ELTP l_: black oak - white oak ELTP 2_: mixed oak - red maple 38 Table 1 : ECS Stand Assignments by ELTP ELTP ECS STANDS l: 48, 49, 50, 53, 54, 55, 65, 78, 79 10: 1, 3, 28, 33, 38, 61, 63 12: 29, 30, 34, 51, 59, 66, 71, 75 20: 15, 18, 39, 45, 46, 76 21: 8, 11, 47, 52, 58, 67, 68, 70, 80 35: 14, 16, 17, 20, 60, 64, 69, 74, 77 37: 4, 5, 10, 44, 72 40: 7, 36, 40, 41, 43, 56, 57 43: 21, 23, 25, 26, 31, 35 45: 6, 22, 24, 27, 32, 37 47: 2 NO ELTP: 9, 12, 13, 19, 42, 62, 73 39 ELTP 3_: red oak - red maple ELTP 4_: northern hardwoods. The second digit in the code described the soil substrata characteristics relative to plant associations. 0: no textural substrata 1: bands of sandy loam or coarser material 2: subirrigation 3: bands of sandy clay loam or finer materials in ELTP's undifferentiated by ground flora 5: sandy clay loam or finer bands beneath ELTP's with diagnostic ground flora. A brief description of each ELTP can be found in Appendix I. 40 II. Data Processing All overstory sample data collected were entered into computer files as well as retained as original and photocopies of original tally sheets. Individual files were created for each stand. Species codes and dbh data were extracted from these files to create stand files in a statistical analysis program. The data in these files remained as species and dbh of sample tally trees. Stand files of species and dbh were grouped according to ELTP groupings. The ELTP files then contained total tally trees for all stands in an ELTP. The statistical program was used to produce frequency distributions of tally trees by one inch dbh classes. The dbh classes were defined as being centered on the inch with width of plus 0.4 inch and minus 0.5 inch. For example, the 4 inch class had width 3.5-4.4 inches. The sampling design truncated the diameter samples at 3.5 inches. Samples of trees less than 3.5 inches in diameter, though available, were found to be inappropriate for this study. Therefore, the population of interest was defined as all trees in ECS stands of diameter 3.5 inches and greater. Once the frequency distributions of tally trees within an ELTP were obtained, the number of tally trees in each diameter class in an ELTP were transformed to per acre 41 values by the function: I tally trees * [{(baf/ba)/# points}/#stands in an ELTP] where baf basal area factor ba basal area = 0.00545415 * mm2 as derived from Husch, Miller and Beers (1982). This conversion factor resulted in unbiased estimates of trees per acre for point sampling. All stands with the same BAF over all points and the same number of points were processed as a group to the point of obtaining per acre frequencies by diameter class. There were four such groups: 10 BAF and 4 points, 10 BAF and 6 points, 5 BAF and 4 points, and one stand (60) with two points at 10 BAF and two points at 5 BAF. Frequency distributions for each ELTP were then compiled by combining per acre frequencies from appropriate groups. A" Point Sample Data and Distribution Modelling The per acre frequency distributions of tree dbh by ELTP were used to develop modelled dbh distributions using the Weibull distribution model and Johnson's SB distribution model. This methodology agreed with the theoretical foundation developed by Van Deusen (1986). He used relative frequency - percentage of total trees in a diameter class - to develop the theory behind the use of point sample data to model dbh distributions. That 42 development parallelled the use of absolute frequencies - the actual number of trees in a diameter class - in modelling dbh distributions, as was done in this study. The contrast in methodology was in using observed tally frequencies as the basis for modelling and then expanding the predicted tally frequencies to per acre values. Van Deusen's point was that point sampling, or sampling with probability proportional to size, has dbh2 as the variable in the proportional function. The dbh2 term was applied directly in transforming observed tally frequencies to modelled per acre frequencies. In the contrasting, incorrect methodology, the dbh2 proportional transformation would have to be applied after the modelling portion of the process. Because the method used here modelled the per acre frequencies, the dbh2 term was incorporated in the modelling, following exactly the theory developed by Van Deusen (1986). B. Calculation Tools The criteria set forth in the objectives of this study included that the calculations be kept relatively simple. The goal was to provide methodology making the modelling of diameter distributions readily accessible, concise, and yet reasonably accurate. To that end, all calculation was done using a simple spreadsheet program and/or a programmable calculator. 43 A simple spreadsheet program was used in carrying out the preceeding and following methods. The program was considered to be a standard sort of spreadsheet that would be available to most foresters interested in diameter distributions. More complex spreadsheets were noted to be capable of allowing an individual to perform the calculations more quickly and easily. A programmable calculator was also found to a capable tool for use in carrying out the necessary calculations. Obviously, the time required in the use of a calculator was greater than for a spreadsheet. The key element in the use of the spreadsheet was defining a series of ”macros": individual keys or short sequences of keys that stored longer typed formulas or a series of commands to the program. The use of macros eliminated the need to type in long formulas and command sequences repeatedlyu The macros were saved to diskette and loaded with the spreadsheet program. III. Distribution Model Development A. Weibull Distribution The Weibull probability density function was given earlier as f(x) (c/b) ((x-a)/b)°‘1 expl-(«x-a)/b)°1 dbh where x smallest dbh in distribution a 44 b 63rd percentile of the distribution c shape parameter. In order to apply this function to modelling diameter distributions, the parameters a, b, and c first had to be estimated.'The conceptual approaches to parameter estimation for the Weibull distribution have been presented. The method for estimation from percentiles was given by Zarnoch and Dell (1985). Estimation of a, the location parameter, was done according to the definition of the population of interest. That is, since the samples were truncated at a diameter of 3.5 inches, the smallest possible diameter, 3 was set at 3.5 inches. This value then became the estimate of a. Estimation of b, the scale parameter, was given by Zarnoch and Dell (1985) as B = -3 + x.63n where n = total number of trees per acre 3 = the estimate of a, the location parameter x.63n = the 63rd percentile of the distribution. They also defined the estimator of c, the shape parameter, as 6 = {1n t1n(1-pk)/1n(1-pi)11/{1nt(xnpk-S)/(xnpi-a)1} 0.97366 where Pk Pi = 0.16733. 45 Cohen (1965) and Zarnoch and Dell (1985) gave the maximum likelihood estimators of b and c as [(22 x15 1n ximz x16)! - 1/6 = [(l/nHZln xiH from which c is derived through iteration, and 8 = («i/annienl/e A In both formulas xi = xi - a. Percentile estimation was given by Cohen (1965) and Zarnoch and Dell (1985) as a good way to obtain an initial value of 3 upon which to base the iterative solutions of the first equation for maximum.likelihood estimation. They noted that when the two sides of the equation are equal, the value of 6 used to obtain this equality is the maximum likelihood estimate of c. For both estimation methods, grouped frequency counts were the basis for determining parameter estimates. Though this approach ran the risk of suppressing possibly important distributional information as would be available in a complete data list, the simpler nature of this approach was in accordance with the objectives of the study. For percentile estimation a list of observed cumulative per-acre frequencies and corresponding diameters were obtained for each ELTP. Total number of trees per acre (n) was then multiplied by the two values of p given by Zarnoch and Dell (1985). This provided two cumulative 46 frequency values, x k and xnpi' The diameter classes nP corresponding to those two cumulative frequencies were then used to calculate c. The same n for the ELTP was then multiplied by 0.63 to gain one more cumulative frequency. The corresponding diameter class for this value was then used to estimate b..All calculations were easily done on a programmable calculator. For maximum likelihood estimation, diameter class and frequency data for an ELTP were entered into a spreadsheet. Diameter class was necessarily transformed by subtracting 3.5 from all classes. Three other values (columns) were A calculated: frequency * In x, frequency * x°, and frequency * In x * x6. Columns were summed to obtain the elements of the maximum likelihood estimation formula. The percentile estimate of c was obtained first. That estimate of c was used to solve for the maximum likelihood estimate of c through the iterative formula. In most cases it was necessary to complete only five iterations. At that point values for each side of the equation were sufficiently close (identical in value up to the fourth significant digit) that linear interpolation, as suggested by Cohen (1965), yielded the correct result. The estimate of b was calculated in the same spreadsheet. The parameter estimates from each method for each ELTP were then used to calculate f(x) for the Weibull distribution. Number of trees per acre by diameter class 47 (absolute frequencies) were obtained by multiplying f(x) by the total observed number of trees per acre. Tables of observed and predicted relative and absolute frequencies were compiled for each ELTP. Chi-squared and Kolmogorov- Smirnoff one sample goodness of fit test statistics were computed for each parameter estimation method in each ELTP. In addition, ELTP level attributes were calculated to assess the predictive power of the parameter estimation methods and the model beyond simple curve fitting. B. Johnson's 38 Distribution The SB distribution probability density function was given earlier as f(x) = (CM/271) (1/[(x-w)(W+1-x)l}exp{-1/2[g+d 1n((x-w)/(W+1-x))12} where x = dbh ‘ d shape parameter 9 = shape parameter 1 = scale parameter w location parameter (smallest diameter). The parameters to be estimated were discussed earlier. The method for percentile estimation was given by Slifker and Shapiro (1980). Estimation of the parameters was noted as being dependent upon the selection of an initial z-valu-e, where z is a unit normal variate as given in tabular form by Steele and Torrey (1980). Sliker and Shapiro (1980) let this initial 2 be the basis of four 48 symmetrical values of z : +/-z and +/-3z. The probabilities associated with each of these values, designated Pj, were then used to determine the distribution percentile x‘i) from i = nP- + 1/2, where n equals the total 3 number of trees per acre. Using the four percentiles corresponding to the four z-values, Slifker and Shapiro (1980) developed three relationships m=x3z-xz n - x-z "" X_32 p = x - x 2 '2' .These three relationships were used in the hyperbolic trigonometric functions to estimate the four parameters: 3 = z/{cosh'1(.5[(1+p/m)(1+p/n)11/2)} A . . -1 ((p/n)-(p/m)){(1+p/m)(1+p/n)-4}1/2 9 = d Sinh v- 2((p/n)(p/m)-1) 1 _ Pil(1+p/n)(l+p/m)-2]2-4}1/2 ((p/n)(p/m))-1 A x2 + x_z i + 9((p/n)_(p/m)) W = _, 5 ‘7' ‘§((p/n)(p/m)-1) Slifker and Shapiro (1980) suggested an initial 2- value less than one. In the process of using this method of parameter estimation, a set of tables were developed giving a range of z-values from 0.50 to 0.90 in increments of 0.10 (Appendix II). The tables included sample size and z-value, 49 and gave the four appropriate cumulative frequencies for the four corresponding z-values. Parameter estimation was then simply a matter of choosing a z-value within 0.50 through 0.90 for a particular ELTP, obtaining values for m, n, and p, and calculating the estimates. The method.for maximum.likelihood estimation was given by Johnson (1949). It was based on the distribution transformation function and so reduces the problem to fitting a normal distribution. Johnson (1949) defined the transformation function as ti = ln[(xi - a)/(a + l - xi)], the moments of which yielded estimates of d and g, the shape parameters. The moments were given as E = Zfi/n and sfz = 2(fi - f)2/n and the estimators as ‘E/Sf <3 and 0» ll l/sf. This method required that values for a and 1, the location and scale parameters, be estimated or known prior to estimating d and g..As in the case of the Weibull distribution, the most straight forward approach was to define the range of diameters given by the samples as the population range of diameters. Therefore, the location 50 parameter, w, was set at 3.5 and the scale parameter, 1, was set at the maximum observed diameter class for an ELTP minus 3.5. Because w+l defines the upper end of the 58 distribution, this method of setting 93 and ’1 gave the observed range. In addition, because the SB distribution has high contact at both ends of its curve, the upper distribution end was defined as the upper bound of the maximum diameter class plus 0.5. In the case of each parameter estimation method, the grouped frequencies of the observed distributions were used in computation. Again, some specificity in the complete data list may have been lost due to this approach but the conciseness of calculation and data handling was considered to be an important aspect of the study. Maximum likelihood parameter estimation for the SB distribution was carried out with the use of a spreadsheet. Diameter classes and frequencies were entered, the transformation fi was calculated for each class and summed, the moments were obtained- from these sums, and the parameter estimates calculated. Parameter estimates from each method were used to calculate»f(x) for each ELTP. Absolute frequencies were obtained by multiplying f(x) by total observed trees per acre for an ELTP. As in the case of the Weibull, goodness of fit statistics and ELTP level attributes were calculated to assess the quality of prediction for the model and each parameter estimation method. 51 IV. Data Analysis A. Goodness-of-Fit Statistics Two tests were used to statistically determine the goodness-of-fit of the predicted diameter distributions to the observed distributions. The first was the Chi-squared test and the second was the Kolmogorov-Smirnoff one sample test. Steele and Torrie (1980) and Conover (1980) defined the Chi-squared test statistic as x2 = 1((Observed - Expected)2)/Expected with degrees of freedom equal to the number of classes minus one, minus the number of parameters estimated from the data. They specified that the test statistic be compared to tabulated values of Chi-squared based on.a predetermined alpha level and degrees of freedom. Steele and Torrie (1980) noted difficulties with the test statistic when there are class frequencies less than one. They suggested that consecutive classes with frequencies less than one be combined in both observed and expected distributions. However, Steele and Torrie (1980) and Conover (1980) agreed that there is no accepted protocol for handling frequencies less than one. For testing the predicted or "expected" diameter distributions against the observed ELTP distributions, consecutive classes with frequencies less than one were combined. The Chi-squared test statistic was then 52 calculated. Degrees of freedom were determined by the number of classes after combining. Two degrees of freedom were subtracted for parameters estimated for the Weibull distributions, both parameter estimation methods, and for the SB distribution, maximum likelihood estimation. Four degrees of freedom were subtracted for the SB distribution obtained through percentile estimation. Alpha level was set a priori at 0.05. The precedent found in the literature for diameter distribution modelling was 0.10.4Almost all such studies were conducted on even- aged, single species forests using fractional area sampling. The stands in this study were mixed species, mixed age stands sampled using probability proportional to size. Under these conditions, it was decided that acceptance or rejection criteria of the goodness-of-fit hypotheses could be relaxed somewhat. By decreasing the value of alpha, the rejection region for the hypothesis test was made smaller thereby decreasing the possibility of incorrect rejection of the null hypothesis. That is, the chance of a Type I error was thereby reduced. The second test used, the Kolmogorov-Smirnoff one sample test, was discussed by Conover (1980). He defined the test statistic as D = sup | Fn(x) - Fo(x) | x . where Fn(x) = cumulative relative frequency of the predicted distribution 53 and Fo(x) = cumulative relative frequency of the observed distribution. The test was presented as one which is concerned with the cumulative distributions: observed and expected. Conover (1980) described it as looking at the absolute value of the differences between the cumulative relative frequencies of the observed and predicted distributions. The largest of these differences was given as the value of the test statistic. Tabulated values of D were referred to where comparison is based on alpha level and sample size. Sample size in this case was defined as the total trees per acre for an ELTP.‘The large sample approximation, for n>40, was given as D = 1.36/(n +frVI'0)1/2. The Chi-square test gave a class by class assessment of goodness of fit. The Kolmogorov-Smirnoff test was used to provide secondary goodness of fit information. The Kolmogorov-Smirnoff test was rather sensitive to distributional differences in the smaller diameter classes but rather insensitive to differences in the upper tails. Conversely, the Chi-Squared test was overly sensitive to differences in the upper tails and, in one case, gave an unreliable indication of goodness of fit. The Kolmogorov- Smirnoff test was used to provide further goodness of fit discrimination when the Chi-squared test did not provide reliable results. 54 The Chi-squared test was also applied to that portion of each distribution greater than or equal to 11 inches in diameter. Again, classes with frequencies less than one were combined. The trees greater than or equal to 11 inches in diameter were defined as sawtimber size. Because of size and potential for producing a higher value product than smaller trees, the sawtimber portion of the distributions was considered critical in modelling.‘The Chi-square test on this portion of the distributions was intended to provide further evidence on the quality of the performance of each model and parameter estimation method. B. ELTP-Level Parameters Bailey (1980), Hyink and Moser (1983), and Little (1983) made the point that stand level attributes are of real interest in growth.and yield modelling.‘These attributes included basal area per acre (BA/a), number of trees per acre (not/a),.arithmetic mean diameter at breast height (amdbh), and quadratic mean diameter at breast height (qmdbh). The ability of a distribution model to accurately predict frequencies of trees per acre by diameter class was said to be of basic importance to growth and yield modelling. The further ability of a model to predict a distribution that also yields accurate predictions of stand level attributes that are based on easily obtained measurements and are closely correlated with volume was noted to be of equal significance. 55 Therefore, the second level of model and parameter estimation method assessment was to determine how well the predicted distributions agreed with the observed distributions on ELTP level attributes. Not/a was calculated by summing the observed and predicted per acre frequencies respectively for each model and parameter estimation method in each ELTP. BA/a was calculated by obtaining the basal area of each diameter class, multiplying by class per acre frequencies, and summing. debh was calculated by dividing the BA/a by not/a, then dividing that quotient by 0.00545415 and taking the square root of the result..Amdbh was calculated by multiplying diameter class by frequency, summing the products, and dividing by not/a. No conclusive means of assessing attribute prediction for an individual ELTP was found. Instead, predictions were assessed by model and parameter estimation method through the use of ATEST. Rauscher (1986) developed ATEST, a computer program written in BASIC, to determine the bias of a predictor based on observed values. This bias was described as the difference between observed and predicted values given in units of measure and as a percentage of the observed. The program was described as giving a value for bias of a prediction based on normally distributed differences or non-normally distributed differences. Normally distributed differences were given to lead to use 56 of Studentls t in order to establish a confidence interval about the bias. Non-normally distributed differences were given to lead to the use of a trimmed mean and a jackknifed estimate of variance to obtain a confidence interval about the bias. An alpha level of 0.05 was used in constructing the confidence intervals. Using this program meant that bias and accuracy would be reported for a model and parameter estimation method over all ELTP's. Therefore, attribute prediction was assessed on an overall level. The main interest in using ATEST was to establish whether the prediction was significantly biased and to what degree it was accurate. Whether or not the 95 % confidence interval about the bias contained zero was the criterion for determining if the bias was significantly different from zero. C. Skewness and Kurtosis The final level of analysis was the comparison of observed and predicted values of skewness and kurtosis. ATEST was also used in this comparison. Steele and Torrie (1980) defined the coefficients of skewness and kurtosis as incorporating the second, third, and fourth moments of the mean as calculated from sample data. These moments were defined as -1 m2 = n (x1 - i)2 m3 = n"1 (xi - i)3 57 -1 (Xi - i)4 and the coefficients were defined as J31 = m3/(m23/2) 32 = M4/III22. skewness kurtosis They defined skewness as a measure of displacement of the mode of a distribution from centrality. Kurtosis was defined as a measure of the ”peakedness” of the distribution. The three moments were calculated in a spreadsheet by first entering the diameter classes and frequencies for an ELTP. Arithmetic mean diameter was calculated as given earlier. Each diameter was deviated from the mean and raised to the second, third, and fourth power in turn and multiplied by the class frequency. The deviations were then summed and divided by the total trees per acre. One of the key features of each model was given as flexibility in generating a variety of curve shapes. The SB distribution was said to be more flexible than the Weibull, i.e., that it could generate a wider variety of curve shapes. The intent of this comparison was to bring to light information of an ancillary nature regarding any questions arising from differences in model performance. Results and Discussion The results of this study may be put into three categories: model results, goodness-of-fit results, and ELTP-level parameter predictions. Results from the application of the distribution models and parameter estimation methods include stand tables of observed and predicted trees per acre and accompanying distribution graphs. This section also includes discussion of the models and methods used from the standpoint of application mechanics. Results of goodness-of-fit tests include test statistics and outcomes and a discussion of the relative performance of models and methods with respect to observed values. Prediction results include predicted ELTP-level parameter values, relative measures of bias with respect to observed values, and discussion of the significance of the parameters and their prediction through diameter distribution modelling. I. Modelling Results Tables 2 and 3 present the results of parameter estimation for the Weibull and SB distributions, respectively. The non-unique nature of estimates obtained by percentile estimation is apparent in the Weibull distribution. ELTP's l and 10 have identical shape parameter estimates as do ELTP's 20 and 35. 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On the whole, the Weibull shape parameter estimates show a range of curve shapes from reverse J-shape to a somewhat strongly right skewed mound shape (0.8876 - 1.53595). The maximum likelihood estimates (MLE).areIdifferent from the percentile estimates (PCTE) for both models. The MLE are noticeably more conservative in that they exhibit a narrower range of values. For the Weibull, three of the shape parameter estimates differ in a significant way between estimation methods for the same ELTP. The PCTE give values of c less than 1.0 for ELTP's 12, 40, and 43 whereas corresponding MLE are greater than 1.0. There is a basic change in curve shape as c differs from 1.0; when c is less than 1.0 the curve becomes a reverse J-shaped distribution; when c is greater than 1.0 the curve becomes mound-shaped. For ELTP 12 the PCTE does not correspond to the class of curve of the observed distribution (Figure 6). The reverse is true for ELTP's 40 and 43 (Figures 16 and 18). Far less can be discerned by examining the parameter estimates for the SB distribution (Table 3). The main observation that can be offered is that relative magnitudes of parameter estimates are similar between parameter estimation methods. This at least suggests some degree of internal consistency in the model. 62 The SB percentile parameter estimation process appears to offer more difficulty in practice than in theory. Initial estimates for ELTP's 1, 10, 12, 20, 21, 43, and 45 are those given in Table 3. These estimates are the direct result of the estimation method described.‘The estimates for ELTP's 35, 37, and 40 in Table 3 are the result of some modification. Estimates obtained directly by the given method do not produce a distribution of sufficient range in application to this data set. The scale parameter in particular is too small. This problem is corrected by rejecting the underestimates of the scale parameter.'The corrected estimates are values of the scale and location parameters that produce frequencies in a range of diameters identical to that of the appropriate ELTP..For this problem. the sum of a and l is set to equal the maximum diameter class and the solutions for 4 and 1 are obtained by two or three iterations of f(x), when the proper range is obtained. The correction of percentile estimates is permissible since percentile estimation is a non-deterministic method. Percentile estimation of parameters is simply a systematic and repeatable method for fitting a curve to an observed frequency curve. For curve fitting only, one may obtain parameter estimates by trial and error alone. The appropriate estimates are those that provide the best curve fit. 63 Correcting parameter estimates creates two problems. First, the process is time consuming and ill-defined. It is probable that the use of a complete data list instead of grouped frequencies would provide sufficient distributional detail to overcome the problem. Second, when the scale and location parameter estimates are corrected, the shape of the curve is altered. In effect, the curve becomes less kurtotic as the location parameter is decreased and the scale parameter is increased. The fit to the complete observed distribution may then be better or worse. If the shape parameter(s) need to be corrected, estimate correction becomes increasingly complex. The method of maximum likelihood estimation.for the SB is straight forward, as given. The use of the moments of the transformed variable (diameter) is simple and precise. The only difficulty with this method is that a and 1 must be taken from the sample data directly. Since the SB curve approaches zero very quickly at the bounds the values of a and 1 must be set so as to allow sufficient frequencies to occur near the bounds. The method described is adequate for accommodating this feature. Both estimation methods for the Weibull distribution are easy to apply as described. The MLE method requires somewhat more time than the PCTE method and about the same as the SB MLE method. Weibull PCTE are the result of about three hours of work, including the production of the 64 frequency tables. The Weibull MLE method currently requires about six hours including obtaining the distribution frequencies. Time savings for this method are possible because of the piecemeal setup of calculations, the limited nature of the spreadsheet used, and the speed limitations on the computer used. The SB MLE are the result of approximately four hours of work including obtaining frequencies. The SB PCTE require about six to seven hours of work for the initial estimates. The correction of estimates adds an undetermined amount of time to the process. The Weibull PCTE method is clearly quicker and is also easier than the other methods. The SB.MLE method is easier and quicker than the Weibull MLE method because the latter is done iteratively. The use of spreadsheet and calculator makes obtaining parameter estimates rather easy, quick, and immediate in the sense that no mainframe computer time is required. Tables 11 - 20 and accompanying Figures 2 - 21 detail the observed and predicted distributions and their curves. Parameter estimation method results are paired in the graphs. Pairing methods results is preferred because of the inherent differences in the methods. Both relative frequency (RF) and absolute frequency (AF) are given in the tables for each ELTP. 65 Some aspects of curve behavior can now be linked to parameter estimate values. Although ELTP's l and 10 (Tables 11 and 12; Figures 2 and 4) have the same shape parameter estimate via percentile estimation, the difference in their scale parameter estimates causes rather different curves. This difference gives an idea of what effect changing the scale parameter has on the distribution. The disagreement between percentile and maximum likelihood estimates of the Weibull shape parameter for ELTP's 12, 40, and 43 can be seen in Tables l3, l8, and 19 and Figures 6, 7, 16, 17, 18, and 19. The discrepancy is in fact small and may be related to the truncation of the distribution. Because of this truncation it cannot necessarily be said that one or the other estimate is incorrect. The behavior of the distribution around c = 1.0 is interesting in spite of the truncation. It can be seen that small changes in c around 1.0 definately change the general class of curve that results. At c equal to about one the distribution develops a shoulder on the left tail. When c becomes even slightly less than one the distribution becomes a definite reverse J-shaped curve (Tables 13 and 19, Figures 6 and 19). In general, visual inspection reveals a reasonable approximation of the observed distributions by the predicted in all cases except ELTP's 35 and 37. These ELTP's will be discussed later. 66 Closer inspection reveals that the left tail of the observed curves can be highly variable. This variability makes fitting a curve to the observations overall somewhat difficult. Where the observed curves are relatively regular, the predicted curves can be seen to fit much better (Figures 2 - 21). The variability of frequencies in the lower diameter classes is a product of sample size, sampling method, and the stands sampled. A larger sample size would provide greater opportunity for observing smaller diameter trees. The expansion of point sample frequencies to per acre frequencies results in small diameter trees being weighted much more heavily than larger diameter trees. The outcome is that the absence or presence in a sample of a single 4 inch tree greatly affects the expanded frequencies. Finally, in many stands, the inconsistent presence of multiple stemmed red maples of small diameter (< 7”) ends up creating a sawtoothed left tail in ELTP's 35 and greater. Though trees less than 3.5 inches in diameter are not included in the observed frequencies, there is no indication that this truncation presents any problems in modelling. The location parameter for each model provides the means by which a lower bound is placed on the distribution. Frequencies beyond that point are then allocated according to the observed frequencies by either parameter estimation method. 67 II. Goodness-of-Fit Results The null hypothesis of the Chi-squared goodness-of- fit test states that there is no difference between observed and predicted distributions. Failure to reject the null hypothesis leads to the conclusion that the predicted distribution is as representative of the underlying population distribution as is the observed distribution. The same null hypothesis and conclusion are applicable to the observed and predicted ELTP diameter distributions generated in this study. The Chi-squared test examines individual class absolute frequencies to assess goodness-of-fit. The Kolmogorov-Smirnoff test works with cumulative relative frequencies of the distributions. The differences between these tests present different aspects of goodness of fit. The Chi-squared test is used in this study as the primary indicator of goodness-of—fit because it shows how well individual class absolute frequencies are predicted. These class frequencies are of interest in assessing the accuracy of a prediction on the dependent variable of the study: frequencies of trees per acre by diameter class. The Kolmogorov-Smirnoff test is used as an indicator of the accuracy of the distribution in general. As such it can be expected to be less discriminating and more related to the distributions as continuous rather than discrete as they are treated here. 68 A. Percentile Estimation Table 4 provides the calculated test statistics for the PCTE models. According to the Chi-squared test, both models produced significantly different distributions (alpha = 0.05) for six ELTP's. That only 40% of the observed distributions for each model are non-significant indicates that the PCTE models did not do an adequate job of modelling the observed distributions. This is especially so considering that the alpha level was set to allow greater latitude by reducing the size of the rejection region. However, the hypothesis tests are not the final word on how well the models perform, as is shown in the sections to follow. The PCTE SB model produces non-significant results for the so-called low site ELTP's: l, 10, and 12. The PCTE Weibull model produces no such pattern of non-significance. Both PCTE models produce significant results for the higher site ELTP's - 35, 37, 40, 43, and 45 - with the exception of the Weibull PCTE model for ELTP 45. Over all ELTP's the SB PCTE model produces somewhat lower Chi-squared scores. Considering that three of the SB distributions have corrected parameter estimates, the lower scores over all ELTP's is a point of interest. However, because of the increased difficulty in parameter estimation, the small improvement over the Weibull PCTE model over all ELTP's may not be worth noting. 69 :m =fi§fi§§iéfi 1% .O m mag .0 3 8—8 .0 a Becca .O N— 03 .o u n la n u u .a On g .6 O— 83 .0 Cu u "'0 .o a g .b a max wt 3:“... a .AB .VN lea .au .5— .2 180.2. an .3 uh .3 I; .8 a .N— a .5 ha .a— «(2.5 03.33. an: mugs-micramozs. . mnu soc-3&8... 58.2. .. EN 508 .o .2 3.0 Mg 5.6 on 018.9 3 3.6 an «08.0 Nu 09.0 Nu 8.0 Na «N666 an :6 3 a0! 5.. Bran an! 86 n 0:30 NM .Nu 5m .VN Rah .aN .05 .a 3N0 .Nm 3.0” 3 .hu .8 .N mad— .Al .8 Negro ”Batu—aha:— guo Eng 8.» annug h: 8 mag u v oafih 70 As a final note on the SB PCTE Chi-squared scores, the degrees of freedom are two fewer than for the Weibull. The reduction in degrees of freedom increases the size of the rejection region for the goodness-of-fit test from that of the Weibull. The fewer degrees of freedom therefore makes statistical fitting of the SB PCTE model more difficult than for the Weibull. Rejection of the null hypothesis in the Chi-squared test is of prime concern. As such, the Weibull and SB PCTE distributions model the observed distributions about equally well. This result indicates that both models are about equally accurate in predicting individual class frequencies for ELTP's, though the models do not model the same ELTP's equally well. Rejection of the null hypothesis in the Kolmogorov- Smirnoff test is of concern for it indicates a lack of cumulative distributional accuracy. For this study, this result is considered less important than the results of the Chi-squared test. Howevery the Kolmogorov-Smirnoff test provides information on the general suitability of the models to this application which can lead to general comments on model selection. The two tests do not agree in all cases (Table 4). The Kolmogorov-Smirnoff test is more sensitive to differences in the small diameter classes. However, in many cases the cumulative distributions approach each other reasonably well to produce better agreement than indicated 71 by the Chi-squared test.'The Komogorov-Smirnoff test indicates that although the two models produce about equal accuracy in class frequency prediction, the Weibull PCTE model provides a better overall fit than the 58° This is likely due to the corrections made to the SB parameter estimates in order to obtain a suitable diameter range.‘The resulting distributions may be less accurate overall than if corrections were not necessary. The Weibull PCTE model must be preferred over the SB PCTE model in this study based on goodness-of-fit results. Both goodness-of-fit tests indicate that the Weibull PCTE model performs at least as well as the SB PCTE model. This is the telling point. The greater difficulty of estimating parameters for the SB PCTE model requires that it perform better than the Weibull in order for its use to be justified. Since it performs only about as well at best, nothing appears to be gained by the increased complexity. B. Maximum Likelihood Estimation The distributions derived through maximum.likelihood estimation of parameters do a much better job of fitting the observed data (Table 5). Though this method of parameter estimation is more difficult in general than the percentile estimation methods, the improvement in.accuracy justifies its consideration. 03.33. an: “sorta-mitomozs. u m-» 8:36:83 .3365 x 36 u i.- 72 2... £686 : .366 2.... $66 : 9.: 9 «R 5686 n. 5.6». «8 E36 3 3.2 T. :6 $36 6. 8.: :6 336 S 5.: 9 2.. .926 3 .866 2.— 55.6 3 .966 am can :86 n. 19.8 36 886 m. .0666 8 NM 886 «a and 8» 836 u. 8.6 a... «m... «~86 «a .96... .9 236 w. :6. 8 an 836 m. and. a... 836 «a 8.: a. .8 3.86 a. #6. an 986 2 2.. 2 an k8... : 66.: «an 386 3 E6 . c ml. .5 «:5 c mu". .6 «.56 Ed an u... 433.. guhaunhm~0 g3 Bro—DUE... 8.... manhunt—hm h: 3 g n u Quin. 73 Both the Chi-squared and Kolmogorov-Smirnoff tests show that the MLE models perform decidedly better than the PCTE models. The improvement in performance can be determined by the relative magnitudes of the test . statistics. Only in the case of ELTP's 35 and 37 is there no improvement. Both of those distributions are so irregular that curve fitting of any regular sort would prove difficult or impossible. Examination of the distribution data and curves in Tables 11 - 20 and Figures 2 - 21 gives an idea of how close in agreement the MLE models come to most of the observed distributions. 0n the whole, the Weibull MLE model does somewhat better than the SB MLE model. The differences could be explained by noting that two of the 58 parameters were taken directly from the sample data while the Weibull required only one parameter to be treated in that way.‘The setting of parameter estimates from sample data may introduce additional error into the estimation process. In making a choice between these two models, it should be acknowledged that differences in accuracy are not substantial. Instead, the basis for choice should be that although the iterative calculations for estimating the Weibull shape parameter are somewhat time consuming, that method is more exact than relying on setting sample observations as parameter estimates. Therefore the Weibull MLE model is indicated as superior to the SB MLE model on the basis of goodness-of-fit tests on the whole distribution. 74 Goodness-of-fit tests on the whole distribution are, however, not the final word in determining the adequacy of {a model. Hafley and Schreuder (1977) use goodness-of-fit criteria as a relative measure of distribution adequacy. Still other studies (Little, 1983; Zarnoch, 33 al,, 1980; Johnson, 1949) do not base their findings completely on goodness-of-fit tests, or also are interested in how well predicted distributions estimate certain aspects of the data. It is typical that distribution modelling is a step in dealing with a larger estimation or modelling problem. It is the nature of the data itself that dictates what is important in modelling. In this case, diameter distributions are presented as possibly leading to forest growth and yield modelling. Other criteria in addition to goodness-of-fit to the entire distribution are of interest for assessing the relative worth of a distribution model for growth and yield modelling. These criteria include how well the model predicts frequencies of larger diameter, higher value trees and how well the model predicts forest level attributes. Both of these criteria are not mutually exclusive of the overall goodness-of-fit tests but need not follow the pattern of accuracy established by the tests. 75 III. Goodness-of-Fit for Classes 11 Inches and Greater Trees 11 inches in diameter and greater are defined as sawtimber size and as such demand attention as potentially higher value trees than those less than ll inches. Table 6 presents Chi-squared goodness-of-fit scores for the two models and two parameter estimation methods. The Kolmogorov-Smirnoff test is not included because the concern is with accuracy of class frequency prediction alone. Results are much improved over the full distributions“ The percentile estimation models produce seven non-significant distributions. Because degrees of freedom serve to scale the rejection region, the results between the full distributions and the sawtimber segments may be compared.‘Visual inspection of the curves indicates that the sawtimber segments are more regular than the smaller size class segments. The more regular curve would be easier to model on its own..As it is, both models predict the sawtimber segment quite well as a part of a total distribution. The MLE models are again superior in prediction to the PCTE models. For this segment, there is no real difference between models for a parameter estimation method.‘Therefore, the choice of model must go to the Weibull since it provides the easier and more exact means for parameter estimation. 76 0 h v 0 h 0 0 0 v Q 0‘ «am.- 00.0 v0.v la .8 3.0 8.” 05.0— 0N.0 0v.— 3.? N<~xu Mia *Q'OO‘DKOQKW 0.! «0.0 8.0 an; «N‘— . 3.0 «N6 0— .0 0 .v 0v.0 a; 9.06 an”. oucuuaezcmau nos-oauca a guad‘oxhounuv no.0 u era—- m .500" 0 “5.0 0v 0 he .0 s. 8 .0 0* N 09.0 v INN .N— 0.. 0 .00.“ 0 5.0" 50 0 13.3 s 2 .N a v 60.0 0 RN .0 am 0 .0..: 0 0.0 a v 05.0 0 “0.: N— N 0— .u v '04 0a N 00.” v 9.00 a mu N320 st 9.28 E...” 0.3.80 00 04:”! an! E 9.! 63 00000.5 E~0 g g~§~0h0~0 aha—0000 .3 S... 3 g u 0 .3”... 77 ELTP 37 stands out as the most intractable in all cases. The observed distribution is bimodal and reaches its maximum diameter class somewhat suddenly (Table 17 and Figures 14 and 15). In addition, both modes appear quite leptokutotic, making the curve even more unbalanced. The reasons for this are not entirely clear. A breakdown of species distributions reveals that the modes are not especially associated with individual species or particular groups of species..Age data is incomplete so it is unknown whether the modes coincide with different age groups. A breakdown by stands shows that three of the five stands exhibit the identical modes of the ELTP distribution, one at 4 inches and another at 10 inches. The other two stands have no trees in the 4 inch class and have their respective modes at 10 inches. Given the shape of the observed curve, there is almost certainly no way to adequately model it using the methods described in this study. IV. ELTP-Level Parameters- The final level of assessment looks at prediction of the parameters basal area per acre (ba/a), total trees per acre (not/a), arithmetic mean diameter (amdbh), and quadratic mean diameter (qmdbh). All of these parameters are directly related to the distributions from which they are calculated. However, they are summaries of the distributions and as such can be different in accuracy than the predicted curves from which they are derived. 78 These parameters are considered important for what their prediction accuracy indicates about the respective .models and parameter estimation methods. The goodness-of- fit tests deal with the details of the distributions. Consideration of ELTP-level parameter predictions leads to statements about how well the modelled distribution as an aggregate characterizes an ELTP in a standard manner of summary. The ability to predict and therefore project forest level parameters is considered a key to obtaining the best results from growth and yield modelling (Little, 1983). The observed and predicted estimates of the ELTP- level parameters are given in Tables 7 and 8. The error in prediction (bias) and the confidence limits about that error are presented in Table 9. Predictions are by ELTP and distribution parameter estimation method by model. Errors and confidence intervals are for ELTP-level parameter predictions over all ELTP's. By and large, the predictions are adequate in accuracy, the most extreme deviation being for ba/a for the SB PCTE model. The most extreme value of the confidence interval for that prediction is 17% of the mean of the observed ba/a. Though accuracy is generally good, several of the predictions are slightly to clearly biased, generally upwards. 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Ga.-.3.2..v .96.. .fl.-.k.:& .36.. 33 “8.7.8.“; :26: 8.17.2.m... 8a.? 39.59%; 3.? “86.84.-. 9. 3:! 69:35 8... “8:8; :8. “8:85 8.- 26...»... ..~... in... 81.84 5.. 32.8.. .8. $1.85 2... 29.9..-. on... .8! 5 x8 ES 8 um. .23 8 a man B .6... man 26:5 N... am u... 44!! 3E... am 3:8 44!! an; Burma: 2 E gnu Bro—Dunn a a: u m ouih 82 occurs for the MLE models. Biased prediction is determined by the fact that the 95% confidence interval does not include zero. Arithmetic mean diameter (amdbh) and total trees per acre (not/a) are used here as direct summaries of the distributions. Their values are given in Table 7. Inspection of observed and predicted values shows that the MLE distributions are very close in prediction of the observed values. The bias and 95% confidence intervals in Table 9 confirms the better accuracy of the MLE distribution predictions compared to the PCTE model predictions. This result is no surprise since the accuracy in MLE model curve fitting is superior. Yet for their lesser accuracy, the PCTE predictions are unbiased for both models. The Weibull and SB MLE predictions are biased, however slightly, for both amdbh and not/a. The sign on the bias indicates the direction of the bias. Where the 95% confidence intervals include zero this sign is of no consequence as the bias is not significantly different from zero (alpha = 0.05). Where the confidence interval does not include zero the sign of the bias is of interest. Both MLE models underestimate amdbh, indicated by the positive sign on the bias (as given by the ATEST program). Both MLE models overestimate not/a, as indicated by the negative bias. The implication is that the MLE models tend 83 to overestimate the frequencies of trees in the small diameter classes and/or underestimate the frequencies in the larger diameter classes. The amount of bias in the MLE models predictions is small: no more than 3% of the parameter mean for the extreme value of the confidence interval. However, the result is unexpected in view of the precedents set in the literature. Percentile estimation is generally expected to produce distributions with greater bias than maximum likelihood estimation (Zarnoch and Dell, 1985). The predictions considered.here are somewhat removed from distribution estimation so that this expectation may not apply. Yet these ELTP-level parameters are directly related to the distributions as summaries. Quadratic mean diameter (qmdbh) and basal area per acre (ba/a) are more sophisticated summaries than amdbh and not/a and are easier to relate to volume. Observed and predicted values for these ELTP-level parameters are included in Table 8. The errors in prediction and 95% confidence intervals about the errors are in Table 9. Only the Weibull MLE predictions are biased for qmdbh. As before, the magnitude of the bias is small: the extreme value of the confidence interval is only 1% of the ELTP mean qmdbh. The SB MLE prediction of qmdbh is the most accurate over all ELTP's., There is less to indicate this result in the distributions and their curves. 84 All models produce biased predictions of ba/a for all ELTP's. The PCTE predictions are two to three times less accurate than the least accurate MLE predictions. The errors for these predictions are much higher than for any other predictions. As such, the PCTE predictions overestimate ba/a by as much as 17% of the ELTP mean ba/a. The 53 MLE predictions are also overestimates of ba/a.‘The Weibull MLE predictions underestimate ba/a. The magnitude of bias for both MLE models is relatively small. In general, the Weibull MLE predictions of ELTP-level parameters are the most accurate. However, all of those predictions are biased. The SB MLE predictions are only slightly less accurate than the Weibull MLE predictions, in general. These predictions tend to be biased as well, except for qmdbh. The PCTE predictions are generally less accurate than the MLE predictions. Except for the ba/a predictions, they are all unbiased. From a statistical standpoint, unbiased results are preferred.‘Therefore, though the MLE models produce more accurate curves and predictions of ELTP-level attributes, the biasedness of their summary predictions makes their superiority less certain. This result cannot be applied to the models in general. Observations on the biasedness of predictions are limited only to the data used in this study. 85 V. Skewness and Kurtosis The coefficients of skewness and kurtosis are normally used to assess the normality of a distribution or curve.‘They are overall measures of curve shape in two dimensions. In this study, these coefficients are used to assess the relative curve shapes of observed and predicted curves. The observed and predicted coefficients are in Table 10. The bias of the predictions and the 95% confidence intervals about the bias are presented in Table 9. For skewness predictions, the SB models produce biased results. The biasedness in both cases is the result of overestimating skewness. This overestimation of skewness is seen in the distributions (Tables 11 - 20) and graphs (Figures 2 - 21) as reflected in the overestimation of frequencies in the small diameter classes.‘The overestimation of skewness may be the result of an underestimation of the distribution scale parameter. All predictions for kurtosis are unbiased for both models. The most accurate predictor of overall curve shape as described by skewness and kurtosis is the Weibull PCTE distribution. This is somewhat surprising given that the SB distribution was noted to be the more flexible model. Two reasons may be given for this result. First, the range of curve shapes given by the observed data is not very wide. Other than the two irregular distributions, ELTP's 35 and 08— .u 9—0.6 an .v 851d 80* .t a.“ «a .n saw; :on .Q g; mug scam; “~84 nmNNJ 3.” Cuba." 34 3104 :3.— Ron.— man.— ”:26 Mann.” vs.“ ‘8‘... 836 33 ”in.“ £8 .0 $13.. g3 33.! g 285 “a; ”fit.— 9.84 ~03.— Nana.“ g.— 0864 3..— an; «Nun.— NJ. in“. Ed .38 .v 08— .v an .m an": .o. a .5 than .0 kn"; g; :8 .5 0mm»; enhvé Saw.— $06 “8.— a; 03.6 '83.” in." 38.— szba 3m 88.» 3.5.... 38d 8h.» 38... 3:... Rad 88... 12.. 88... a mg 3.3 mag 3:8 g.— a." man»...— 8'6." '8..— scan.— mi.” 3.0 Nana." an“. 58.x 83... thin 82 .n Skd 88.» 336 BR... 33... 986 vs; «RN.— 83.0 03.0.6 000'." '86 swan.— Duo—.— 3.— 25d g 2 mug and Eng 8 g u on Si.— 87 37, only two general curve types appear. Second, the difficulties encountered in the percentile estimation of the SB parameters is certain to have given a less than satisfactory indication of that distribution's potential performance. Allied to this point is the fact that maximum likelihood estimation for the SB distribution is less explicit than for the Weibull. In being less explicit, the method may be subject to additional error in estimation. The result in both cases could be parameter estimates that are not sufficiently accurate. The parameter estimation methods are at fault for the SB and that places them in a position inferior to the Weibull distribution in this study. The implications of accurately predicting skewness and kurtosis may be carried further. The correlation between the Weibull shape parameter estimates (both methods) and the skewness and kurtosis of the predicted curves is -0.80 and -0.39, respectively. The two tailed 95% critical correlation value is -0.44, n = 20. The weibull shape parameter is clearly correlated with skewness. 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I III! 6 p /I 6a ,/'z 20' no. :06 soon, who; um I .l _ who; :2...) I l I :2. 92:02.0 118 .9 Ed .3“. 9333333.. .3136 A8333: 39.23.: «2.2a: “.30th us not; SN 8.39“. 3.0520 on Q0 N0 on ma 0N VN NN ON up or VP NP 0.. m 0 v P[ b I I-i. "IIILHIIWI b [D n L b P o , /./ ./ .0w 0” /.// .ON “- . , O C / rOm" 8 \ d . ‘ u v Q / ~ o .. /\ n / ram 0 50 m4, am ll.l| WA! 230;; .III.. .Oh coinage rOm Summary and Conclusions In general, the nature of the data analyzed in this study presents a complex modelling problem. First, the ELTP's are composed of mixed upland hardwood species and, to some extent, mixed age groups. This sort of forest as a whole has not yet been considered in diameter distribution modelling in the literature. Second, the ELTP”s are composed of stands with disparate diameter distributions. The expected and actual result of these two conditions is a series of somewhat regular to very irregular diameter distributions. The use of point sample data is suitable for modelling ELTP diameter distributions. The methods given are based on sampling and distribution theory taken from sampling with probability proportional to size~.A larger number of sample points per stand may lead to improved accuracy of small diameter class (< 8") frequencies. Overall, the MLE models especially provide good fits to the observed data except in cases where the observed distributions are irregular to very irregular (ELTP”s 35 and 37). ELTP 37 may present a case where the compound distribution modelling method of Cao and Burkhart (1984) *would be appropriate. Otherwise, the good fits to observed data indicate that point sample data are useful in 119 120 modelling observed diameter distributions. This point is especially important since a great deal of forest inventory is done with point sampling. The grouping of frequency data before estimating parameters may result in diminished accuracy. However, the MLE models in particular provide results that are sufficiently good to state that the possible loss of accuracy may not be great enough to offset the ease of the method. The use of the grouped frequency counts should include an acknowledgement of this possible shortcoming. The fact that any of the models used in this study produce a good overall fit to the observed data may be taken as evidence that these methods are worth consideration. In these methods, three new facets of tree diameter distribution modelling are given: modelling of mixed species and mixed age upland hardwood diameter distributions; the use of grouped frequency counts for distribution parameter estimation; and the use of point sample data for modelling diameter distributions. Using the grouped frequencies facilitates the use of a microcomputer and spreadsheet software for calculations. The chief advantages in this are that a mainframe computer is not necessary and manipulation of data is reduced. As to the models themselves, the Weibull distribution provides the best overall accuracy with these data with the least effort. The SB distribution is theoretically more 121 flexible and therefore more interesting than the Weibull, but its estimation accuracy is not commensurate with its greater complexity. Therefore, the Weibull distribution is preferred in this study. The relatively poorer performance of the SB with respect to the Weibull may be related to the parameter estimation methods chosen for the SB. The simplest and most explicit methods available are described and used in this study but these methods are not as simple or explicit as those for the Weibull distribution. This weakness may be the cause of the unacceptable level of accuracy of the 38 as compared to the Weibull. In addition, the SB may be more sensitive than the Weibull to the use of grouped frequencies for parameter estimation.‘The result of inaccurate parameter estimates is an inaccurate fit of predicted to observed data. These speculations serve to point out that the most significant impediment to the common use of the SB is the difficulty in parameter estimation previously described. Both parameter estimation methods for the Weibull distribution are explicit and straight forward. The percentile estimation method as given in this study is the easier of the two methods for parameter estimation. The Weibull PCTE model is not as accurate in predicting class frequencies as the Weibull MLE model, but for estimation of a somewhat generalized nature, the PCTE model is .A .‘I 122 appropriate. If more accuracy is required, the PCTE estimates of c are a good starting point for the iterative solution of the maximum likelihood estimate of c. The Weibull MLE parameter estimates are suitable for inferences about stand structure. ELTP-level parameters are predicted most accurately from the MLE models, particularly the Weibull. However, the increase in accuracy is marred by a slight bias in prediction. The PCTE model predictions yield unbiased results, except for predictions of basal area per acre. It should be noted, however, that the 95% confidence intervals about the errors are wider for the PCTE models than for the MLE models. This wider confidence interval indicates an overall lower reliability in individual predictions. Based on the extremes of the error confidence intervals and consideration of the very small size of the biasedness in most biased predictions, both methods produce acceptable accuracy for all ELTP-level parameters except for the PCTE model predictions of basal area per acre. Acceptable accuracy in these cases is defined as error less than 10% of the observed parameter mean across ELTP's. The ability of the Weibull model to accurately predict ELTP-level parameters means that it provides an accurate summary of the ELTP's. An accurate summary is desirable for growth and yield modelling. One feature of growth and yield modelling based on a distribution model is 123 that it allows diameter class specificity without the necessity of modelling individual trees (Little, 1983). A distribution model that is accurate in class frequencies and summary at the outset will be less subject to error through projection cycles. The Weibull PCTE model is most accurate in predicting observed distribution skewness and kurtosis. In doing so, this model does best in modelling observed curve shape as measured by skewness and kurtosis. However, the difference in accuracy between the Weibull and SB models in predicting skewness and kurtosis is not great except that the SB models produce biased estimates of skewness. In general, prediction of observed skewness and kurtosis gives a much more generalized look at curve fitting than the goodness- of-fit tests. The results of this study suggest that the Weibull model and its parameter estimation methods provide unbiased estimates of observed curve shape. In general, the use of spreadsheet software for estimating distribution parameters and obtaining distribution frequencies provides a quick and easy alternative to mainframe computing.‘This method may bring diameter distribution modelling within the reach of those who are interested in diameter distribution modelling but who don't have access to more expensive and powerful hardware systems. 124 To summarize, this study examines the relative performance of the Weibull distribution and the SB distribution in modelling ELTP diameter distributions. The ELTP's are composed of upland hardwoods of somewhat mixed age groups. Conventional distribution parameter estimation methods are described and applied to grouped frequency counts of trees per acre by diameter class as obtained from point sample data. The Weibull distribution is preferred for modelling the observed ELTP diameter distributionsw The Weibull PCTE model provides an adequate approximation to observed data on its own. In addition, the Weibull PCTE model provides a good starting point for maximum likelihood parameter estimation for the Weibull distribution. APPENDICES Appendix I. Ecological Land Type Phase Descriptions The following ELTP descriptions are from Cleland, Hart, Pregitzer, Host, and Padley (March, 1986; unpublished). The descriptions are not in any way complete but are intended to provide general classification background as may be desired for further understanding of the diameter distributions developed in this study. The classifications are considered unofficial at this time since the study is still in progress. Therefore, ELTP as presented in this study is not a definitive classification. Species names: Black oak : Quercus velutina White oak : Quercus alba (Northern) red oak : Quercus rubra (Upland) pin oak : Quercus ellipsoidalis Red maple : Acer rubrum Sugar maple : Acer saccharum White ash : Fraxinus americana (Bigtooth and quaking) aspen : Populus grandidentata Populus tremuloides Basswood : Tilia americana Beech : Fagus grandifolia 125 126 ELTP 1: Pin oak-white oak-Deschampsia plant association on excessively well drained sands of outwash plainsm<0verstory composition is upland pin oak and white oak. The canopy is relatively open and ground flora coverage is very low. Red maple does not occur in the understory. Stand averages for the overstory: species ' E512 Gross Vol SI pin oak 53 1033 56 white oak 34 335 49 81 1438 age = 72 MAI = 20.6 cu.ft./a/yr. ELTP 10: Black oak-white oak-Vaccinium plant association on excessively well drained sands of outwash plains. Similar to ELTP 1 except in having a more closed canopy and in the presence of red maple and bracken fern in the understory. Overstory composition is black oak, white oak, and northern red oak. species E512 Gross vol SI black oak 32 767 50 white oak 32 606 42 red oak 18 388 54 85 1817 age = 81 MAI = 23.1 cu.ft./a/yr. 127 BEER 13: Black oak-white oak-Vaccinium plant association on sub—irrigated, excessively well drained sands of outwash plains. Similar to ELTP 10 except for the presence of sub- irrigation and the wider presence of witch hazel as a shrub. Sub-irrigation refers to the presence of a water table within tree rooting depth for an extended period of time. Overstory composition is the same as ELTP 10. species E512 Gross vol SI black oak 35 828 56 white oak 40 685 49 red oak Z l_gy 54 90 1832 age = 73 MAI = 25.3 cu.ft./a/yr. ELTP 20: Mixed oak-red maple-low Viburnum plant association on well to excessively well drained sands on overwashed moraines, kame terraces, spillways, and outwash plains. The overstory of this ELTP is made up of northern red oak, white oak, black oak, and red maple. Red maple and witch hazel are well represented in the 1-3" classes. species E513 Gross vol SI red oak 30 819 61 white oak 30 689 52 black oak 35 865 60 red maple 5 ' 11 102 2533 age = 81 MAI = 31.2 cu.ft./a/yr. 128 ELTP II: Mixed oak-red maple-low Viburnum plant association. Similar to ELTP 20 except for the absence of sub-irrigation and the presence of sandy loam textural bands beneath well to somewhat excessively well drained sands. Overstory composition is northern red oak, white oak, black oak, and red maple. species S513 Gross vol SI red oak 40 967 65 white oak 30 682 53 black oak 21 534 66 red maple II III 107 2460 age = 72 MAI = 34.4 cu.ft./a/yr. ELTP SS: Red oak-red maple-high Viburnum plant association on well drained sands with fine loamy substrata on moraines and overwashed lake bedsm Overstory composition is northern red oak, white oak, and red maple. species E512 Gross vol SI red oak 73 2166 77 white oak 14 372 63 red maple II III 74 117 3169 age = 72 MAI = 44.3 cu.ft./a/yr. 129 ELTP SI: Red oak-red maple-Desmodium plant association on well to moderately well drained sandy loams over loamy substrata on ground moraines and fine textured glacial lakebeds. Overstory composition is northern red oak, white oak, and red maple. species Eéli Gross vol SI red oak 62 2192 85 white oak 20 498 63 red maple IS SIS 69 113 3561 age = 73 MAI = 48.9 cu.ft./a/yr. ELTP SS: Sugar maple-beech-Lycopodium plant association on well drained morainal sands. Overstory composition is sugar maple, beech, northern red oak, and red maple. species S513 Gross vol SI sugar maple 27 589 65 red oak 32 973 76 red maple 1 SS 65 104 2717 age = 62 MAI = 43.6 cu.ft./a/yr. ELTP 43: Sugar maple-northern red oak-Lycopodium plant association on well drained morainal sands with fine textured substrata. Overstory composition is sugar maple, beech, northern red oak, and red maple. 130 species 5513 Gross vol SI sugar maple 40 969 73 red oak 50 1840 . 88 red maple S SS 73 119 3649 age = 67 MAI = 54.3 cu.ft./a/yr. ELTP 45: Sugar maple-white ash-Osmorhiza plant association on well to moderately well drained morainal sands over fine substrata. Overstory composition is sugar maple, white ash, northern red oak, and red maple. species 5513 Gross vol SI sugar maple 52 1388 76 red oak 9 343 86 white ash I5 SZI 85 126 3868 age = 66 MAI = 59.5 cu.ft./a/yr. Appendix II SB Percentile Parameter Estimation Tables The tables in this appendix may be used to facilitate estimation of the parameters of the SB distribution by percentiles. Use of these tables is mentioned in the materials and methods section (Chapter 3). The tables are based on the percentile estimation methods of Shapiro and Slifker (1980). These methods are described in Chapter 3. The sample size (n) appears in the first column on the left. The second column contains the values of z, the tabulated standard normal variate (Steele and Torrie, 1980). The remaining four columns contain the four symmetric percentiles according to the sample size and z-value. Given a sample size and an initial value of z, the corresponding percentiles are read from the table. These percentiles identify individual observations from a list of ordered observations (in this case, diameter). The four appropriate observations are then used to calculate m, n, and p.‘The parameter estimates can then be obtained accordingly via the hyperbolic trigonometric functions given in Chapter 3. The sample sizes include 50 and 75 and otherwise are given by tens from 100 through 400. Interpolation of percentiles between sample sizes is possible. Likewise, interpolation between z-values is possible. 131 Teble 21: 132 Table. for percentile eetieetion o! Johnson's distribution per-meters 3 343333 83388 120 120 130 130 130 130 130 140 140 140 140 140 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 3.84 2.30 1.40 0.91 0.68 5.51 3.19 1.84 1.12 0.76 7.18 4.09 2.” 1.32 0.85 7.85 4.45 2.47 1.40 0.89 8.52 4.81 2.65 1.48 0.92 9.18 5.17 2.83 1.57 0.96 9.85 5.53 3.01 1.65 0.99 15.93 14.22 12.60 11.10 9.71 23.64 21.07 18.65 16.39 14.31 31.35 27.93 24.70 21.69 18.91 34.44 30.67 27.12 23.81 20.75 37.52 33.42 29.54 25.93 22.59 40.61 36.16 31.96 28.05 24.43 43.69 38.90 34.38 30.17 26.27 3 35.” 36.79 38.40 39.91 41.30 52.36 54.93 57.35 59.61 61.69 69.65 73.07 76.30 79.31 82.09 76.57 80.33 83.88 87.19 90.25 83.48 87.58 91.46 95.07 98.41 90.40 9‘. 84 99.04 102.95 106.57 97.31 102.10 106.62 110.83 114.73 4 47.16 48.71 49.61 50.09 . 50.33 70.49 72.81 74.16 74.89 75.24 93.82 96.91 98.71 99.68 100.15 103.15 100.55 100.53, 109.60 110.12 112.48 116.19 1‘3.” 119.52 120.08 121.82 125.83 128.17 129.43 130.05 131.15 135.47 137.99 139.35 140.01 150 150 150 150 150 160 160 160 160 160 170 170 170 170 170 180 180 180 180 180 190 190 190 190 190 200 200 200 210 210 210 210 210 220 220 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.0 0.7 0.0 0.9 10.52 5.89 3.19 1.73 1.03 11.19 6.24 3.36 1.81 1.06 11.86 6.60 3.54 1.89 1.10 12.52 6.96 3.72 1.98 1.13 13.19 7.32 3.90 2.06 1.17 13.86 7.68 4.08 2.14 1.20 ‘4.53 8.04 4.26 2.22 1.24 15.20 8.40 4.44 2.30 1.27 133 46.78 41.65 36.80 32.29 28.12 49.86 44.39 39.22 34.40 29.96 52.95 47.13 41.64 36.52 31.80 56.03 49.87 44.06 38.64 33.64 59.12 52.62 46.48 40.76 35.48 62.20 55.36 48.90 42.88 37.32 65.29 58.10 51.32 45.00 39.16 68.37 60.85 53.74 47.12 41.00 3 104.23 109.36 114.20 118.72 122.89 111.14 116.61 121.78 126.60 131.04 118.06 123.87 129.36 134.48 139.20 124.97 131.13 136.94 142.36 147.36 131.89 138.38 144.52 150.24 155.52 138.80 145.64 152.10 158.12 ‘63.“ 145.72 152.90 159.68 166.00 171.84 152.63 160.15 167.26 173.88 180.00 4 140.48 145.11 147.82 149.27 149.98 149.81 154.76 157.64 159.19 159.94 159.14 164.40 167.46 169.11 169.91 150.40 174.04 177.29 179.02 179.07 177.81 183.68 187.10 ‘m.9‘ 189.84 187.14 193.32 196.92 198.86 199.80 196.47 202.96 206.74 208.78 209.77 205.80 212.60 216.56 218.70 219.73 230 230 230 230 230 240 240 240 §§§§§ “£3333 §§§ 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.5 0.7 0.9 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 15.86 8.76 4.62 2.39 1.31 16.53 9.12 4.80 2.47 1.34 17.20 9.48 4.98 2.55 1.38 17.87 9. B3 5.15 2.63 1.41 18.54 10.19 5.33 2.71 1.45 19.20 10.55 5.51 2.80 1.48 19.87 10.91 5.69 2.88 1.52 20.54 11.27 5.87 2.96 1.55 134 71.46 63.59 56.16 49.24 42.84 74.54 66.33 58.58 51.36 44.68 77.63 69.08 61.00 53.48 46.53 80.71 71.82 63.42 55.59 48.37 83.80 74.56 65.84 57.71 50.21 86.88 77.30 68.26 59.83 52.05 89.97 80.05 70.68 61.95 53.89 93.05 82.79 73.10 64.07 55.73 3 159.55 167.41 174.84 181.76 188.16 166.46 174.67 182.42 189.64 196.32 173.38 181.93 190.00 197.53 204.48 180.29 189.18 197.58 205.41 212.63 187.21 196.44 M. 16 213.29 220.79 194.12 203.70 212.74 221.17 228.95 2°‘.o4 210.95 220.32 229.05 237.11 207.95 218.21 236.93 245.27 4 215.14 222.24 226.38 228.61 229.70 224.47 231.88 236.20 238.53 239.66 233.80 241.53 246.03 ‘248.45 249.63 243.13 251.17 255.85 258.37 259.59 252.46 260.81 265.67 268.29 269.56 261.80 270.45 275.49 278.20 279.52 271.13 280.09 285.31 288.12 289.49 280.46 289.73 295.13 298.04 299.45 310 310 310 310 310 320 320 320 320 320 330 330 340 340 340 340 340 350 350 350 350 370 370 370 370 370 0.5 0.6 0.7 0.9 089 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.9 0.9 21.21 11.63 6.05 3.04 1.59 21.88 11.99 6.23 3.12 1.62 22.54 12.35 6.41 3.21 1.66 23.21 12.71 6.59 3.29 1.69 23.88 13.07 6.77 3.37 1.73 24.55 13.42 6.94 3.45 1.76 25.22 13.78 7.12 3.53 1.80 25.88 14.14 7.30 3.62 1.83 135 2 96.14 85.53 75.52 66. 19 57.57 99.22 88.28 77.94 68.31 59.41 102.31 91.02 80.36 70.43 61.25 105.39 93.76 82.78 72.55 63.09 108.48 96.51 85.20 74.67 64.94 111.56 99.25 87.62 76.78 66.78 114.65 101.99 90.04 78. W 68.62 117.73 104.73 92.46 81.02 70.46 3 214.87 225.47 235.48 244.81 253.43 221.78 232.72 243.06 252.69 261.59 228.70 239.98 250.64 260.57 269.75 235.61 247.24 258.22 268.45 277.91 254.50 265.80 276.34 286.07 249.44 261.75 273.38 284.22 294.22 256.36 269.01 280.96 292.10 302.38 263.27 276.27 288.54 299.98 310.54 4 289.79 299.37 304.95 307.96 309.42 299.12 309.01 314.77 317.88 319.38 308.46 318.65 324.59 327.79 329.35 317.79 328.29 334.41 337.71 339.31 327.12 337.94 344.24 347.63 349.28 336.45 347.58 354.06 357.55 359.24 345.78 357.22 363.88 367.47 369.21 355.12 366.66 373.70 377.38 379.17 390 390 390 390 390 400 400 400 400 o. 0.6 0.7 0.8 0.9 0.5 0.6 0.7 .0.8 0.9 26.55 14.50 7.48 3.70 1.87 27.22 14.86 7.66 3.78 1.90 136 120.82 107.48 94.88 83.14 72.30 123.90 110.22 97.30 85.26 74.14 3 270.19 283.52 296.12 307.86 318.70 277.10 290.78 303.70 315.74 326.86 4 364.45 376.50 383.52 387.30 389.14 373.78 386.14 393.34 397.22 399.10 BIBLIOGRAPHY 1. 5. 6. 10. 11. 12. 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