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The weibull function as a diameter distribution model for mixed stands of Douglas-fir and Western hemlock Little, Susan Nancy 1980-12-31

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THE WEIBULL FUNCTION AS A DIAMETER DISTRIBUTION MODEL FOR MIXED STANDS OF DOUGLAS-FIR AND WESTERN HEMLOCK by SUSAN NANCY LITTLE FOR. SCI., PENNSYLVANIA STATE UNIVERSITY, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of FORESTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 1980 © Susan Nancy Little, 1980 In presenting this thesis in partial fulfiIment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis, for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of f> C-g^Vf The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 tt ABSTRACT The three-parameter Weibull function is a satisfactory model of the diameter distributions of mixed stands of western hemlock and Douglas-fir. Weibull distributions estimated by maximum likelihood (MLE) fit eighty of eighty three observed diameter distributions at the ct = .20 level of significance (Kolmogorov-Smirnov test). Weibull parameter predictors are derived by regressing stand characteristics of 42 stands against their MLE parameters. The Weibull diameter distributions predicted from stand age, mean diameter, mean height, site index and trees per acre fit 39 of 41 observed distributions in the test group at the oc = .20 level of significance. The results shown here compare favorably with those of other authors. The models given relating stand attributes to diameter distribution will prove useful in stand modeling and in updating forest inventories. TABLE OF CONTENTS Page TITLE PAGE i ABSTRACT ii TABLE OF CONTENTS iiLIST OF TABLES iv LIST OF FIGURES v ACKNOWLEDGEMENTS vi INTRODUCTION 1 LITERATURE REVIEW 4 WEIBULL PARAMETER ESTIMATION 9 DATA BASE 13 METHODS 22 RESULTS 4 DISCUSSION, APPLICATIONS AND FURTHER RESEARCH 4'4 LITERATURE CITED 4 8 APPENDIX I 5 5 APPENDIX II 8 APPENDIX III 5^ LIST OF TABLES Table Page I Descriptive Statistics of Estimation Plots 14 II Descriptive Statistics of Prediction Plots 18 III Comparison of Maximum Likelihood Estimated Weibull Distributions with Original Diameter Distributions 25 IV Regressed a Equations 7 V Regressed b Equations 28 VI Regressed c Equations 9 VII Regression Statistics 31 VIII Prediction Models which fit all MLE Distributions 32 IX MLE vs Predicted Distribution 35 LIST OF FIGURES V Figure Page 1 The relationship of goodness of fit to stand diameter 37 2 The relationship of goodness of fit to stand height 38 3 The relationship of goodness of fit to stand age 39 4 The relationship of goodness of fit to trees per acre 40 5 The relationship of goodness of fit to site index 41 6 Observed and predicted trees per acre by diameter class for plot 71041 42 7 Observed and predicted trees per acre by diameter class for plot 361216 42 8 Observed and predicted trees per acre by diameter class for plot 451266 43 9 Observed and predicted trees per acre by diameter class for plot 591354 43 ACKNOWLEDGMENTS The author is indebted to all individuals and agencies concerned with her studies and research. The author thanks Dr. Donald D. Munro who suggested the problem and whose constant advice, criticism, and enthusiasm were essential to the completion of this project. Data from the Regional Forest Nutrition Research Project were provided by Dr. William Atkinson and Mr. Al Becker of the University of Washington. Weibull parameter estimating subroutines were provided by Dr. Robert L. Bailey, Weyerhaeuser Co., and Dr. William G. Warren, Canadian Forest Service. Financial and computer support were supplied by the University of British Columbia and the United States Forest Service. The MacMillan Bloedel Ltd. Fellowship also gave financial assistance. The author extends her thanks to Drs. Jules P. Demaerschalk, James Arney, William G. Warren and Mr. Michael Staley for their help and useful feedback in the early stages of research. 1 INTRODUCTION The description of diameter (dbh) distributions poses a problem for the forest stand modeler. The distributional characteristics of the stand at any point in time may be required for the creation of stand tables, which delineate stand attributes such as volume or basal area per acre by diameter classes. Yet the computer storage of tree diameters may be cumbersome and costly in an already complicated model. Diameter distributions may be handled in either of two ways. The model may describe the initial stand in its entirety and "grow" individual trees or groups of trees over time (Mitchell, 1976; Leary, et al., 1977). In this case stand tables are readily available, as each tree or group of trees is stored with its diameter, height, crown ratio, etc.. The alternative approach is to generate the diameter distribution of the stand at the time it is needed via a mathematical function (Depta, 1974). The parameters of the function can be estimated from stand attributes, such as average diameter, trees per acre, and average height. Thus no individual tree data need be stored. Handling time is reduced, as stand tables can be generated directly by functions related to diameter classes. Another use for functional diameter relationships is in the area of aerial photo interpretation. Average height and number of stems per acre can be interpreted from aerial photographs of forest stands. Stand age and site index may be obtained from past records. Once a functional relationship has been developed between these stand characteristics and diameter 2 distribution, the photo interpreter can readily generate stand tables. This could be a valuable tool for periodically updating inventories on a large scale due to the simplicity of the method. In this thesis, the Weibull distribution will be investigated as a model for the diameter distributions of mixed stands of second growth Pseudotsuga  menziesii (Mirbel) Franco (Douglas-fir) and Tsuga heterophylla (Rafinesque) Sargent (western hemlock). The parameters, a, b, and c, of the Weibull function: where d represents diameter at breast height in inches; W(d), number of trees per acre having diameter d will be estimated from the observed diameter distribution of each plot, 0(d). The null hypothesis: will be rejected if for one plot maxl F(d) - 0(d)J exceeds the Kolmolgorov-Smirnov critical value doc(N). The acceptable probability level of committing a Type I error wi11 be = .20 throughout this thesis. If we cannot reject HQ, the parameters of F(d) will be regressed against the stand characteristics: arethmetic mean diameter, mean dominant and codominant height, stand age, number of trees per acre, and site index. F(d) = 1 - exp \- (d-a)c HN: F(d) = 0(d) 3 The distributions predicted from the resulting equations will be compared with F(d) and 0(d) for goodness of fit. LITERATURE REVIEW Stand tables, which list stand attributes (such as number of trees per acre) by diameter class, have proven Jto be valuable for forest managers. The construction of these tables has evolved around the biometrician's ability to summarize frequency distributions in mathematical terms. Meyer (1930) used Charlier curves to describe diameter distributions. Bliss and Reinker (1964) fitted Meyer's diameter distributions of even-aged stands of Douglas-fir with lognormal curves. They found that these lognormal curves could be adequately defined by the mean diameter and the variance of the diameters of each distribution. Nelson (1964) characterized diameter distributions of loblolly pine plantations with the gamma distribution. Leak (1965) used negative exponential curves to describe diameter distributions of uneven-aged stands. A summary of growth functions applied in forest biometry was given by Prodan (1968). Summarizing information by equations greatly reduces computer storage and calculation requirements. Depta (1974) developed a stand table generator for Weyerhaeuser Corp. Given sample data on basal area per acre, number of trees per acre, minimum, average, and maximum d.b.h., and average tree height, his model produced stand tables with diameter classes, stems per acre, basal area per acre, average total height, and cubic foot volume per acre for even- and uneven-aged stands. Each stand was summarized by a series of coefficients relating to equations for each stand attribute. Individual tree and size class statistics were not stored in the computer, but were generated only when they were needed for final output. The •in capacity to describe distributions via mathematical functions enabled this program to handle large inventories and to fill gaps in information. The Weibull distribution was introduced to forest biometry by Bailey and Dell (1973). This function has special appeal because of its ability to take on - a variety of shapes and amounts of skewness. Since the parameters are directly related to shape (c) and scale (b), they should vary in a consistent manner with stand characteristics when applied to diameter distributions. Bailey and Dell gave an overview of parameter estimation methods and fitted four different diameter distributions with the Weibull. Stauffer (1977) gave a mathematical derivation of the Weibull function in order to give a precise interpretation for the parameters. The three-parameter Weibull function is: a, the location parameter, corresponds to the minimum diameter; b, the scale parameter, is inversely proportional to the number of allocations (i.e., number of trees); and c, the shape parameter, is an indication of the degree of nonrandomness. Bailey (1973) used maximum likelihood procedures to fit equations for 1095 Pinus radiata D. Don diameter distributions. Ninety percent of these equations had an adequate fit at the ©c = .10 level of significance by the Kolmogorov-Smirnov test for goodness of fit. He then attempted to regress age, number of stems per acre, and average height on the characteristics of the percentiles, Xp-. xp = b(-ln(l-p))1/c where x is the d.b.h. class of the p-th percentile. He came up with these relationships for the 24-th and 93-d percentiles: x.24 = B0 + B1A + B2/N + B3lo9lO(H) x.24 = C0 + C1a2 + C2/N + C3lo910(H) From these he calculated the parameters via Dubey's (1967) percentile estimation method. (See below.) At the = .05 level, 35% of these estimated curves had a satisfactory fit. The Weibull distribution has since been used in some growth models. Clutter and Allison (1973) used the Weibull distribution for diameters in their growth and yield model for Pinus radiata. If the user does not specify a diameter distribution, the model will approximate the distribution using stand age and number of stems as done by Bailey (1973). The Weibull was used by Schreuder and Swank (1974) to summarize diameter, basal area, surface area, biomass, and crown profile distributions for Pinus strobis and Pinus taeda. Rustagi (1977) modeled basal area distributions for even-aged Douglas-fir stands with the Weibull. Yang, Kozak and Smith (1978) described 7 previous attempts to fit volume growth and increment distributions with mathematical functions. They then developed a modified Weibull function: - the parameters of which can be estimated via nonlinear regression. This function had the best over-all fit for volume growth when compared with four other distributions found in the literature. Clutter and Belcher (1978) used the Weibull function to estimate number of trees per acre by one inch classes for slash pine plantations. Parameters of the diameter distributions of 487 plots were estimated via Harter and Moore's (1965) maximum likelihood algorithm. Standard multiple regression was then used to regress stand attributes on the parameter estimates, resulting in the following relationships: Y +YA+YH, +Y ln(H ) , r = .107 0 1 2d 3 d s = 1.044 a y •x b 6 + 6A+6/N + 6 ln(H,) ' , r = .357 0 12 3 d s = 1.091 yx c p + p /A , r2= .020 s = 1.091 ; 0 1 where A = stand age = mean height of dominants and codominants in feet and, N = number of trees per acre. The authors insisted that although the r-square values were low, these equations gave better results than using constant values for a, b, and c. 6 Observed mean diameter differed from predicted mean diameter by less than one inch in 99% of the observations (i.e., of the 487 plots that were used to derive the equations). Observed basal area differed by less than 2 25 ft from predicted basal area in 97.4% of the cases. Searching for a function more flexible than the Weibull, Hafley and Shreuder (1977) investigated the possibilities of Johnson's (1949) distribution and its modifications. The four-parameter function: fw-;= (*-E)Ux) ~pi^+ s '-fe)i23 for 6, X>0 ; - «»<Y<OO ; - e0<e<x<e + X<<» f(x) = 0 elsewhere. can take on a greater variety of shapes and is applicable to more cases than the Weibull. Schreuder and Hafley (1977) applied this curve to diameter-height relationships. They estimated parameters with maximum likelihood techniques, and achieved a satisfactory fit at the £* = .05 level with the Kolmogorov-Smirnov test. They did not compare the Sb directly with the Weibull function as a model of tree diameter distributions. WEIBULL PARAMETER ESTIMATION Weibull (1955) developed his equation in conjunction with his analysis of breaking strengths. It has been widely used in reliability and life testing analysis ever since. In life testing it is usually not feasible to test a large number of items, nor is it necessary to wait for all items to fail. Since most of the theoretical and applied work on Weibull parameter estimation has been done in the life testing field, the bulk of the algorithms for estimation are based on small, heavily censored samples. Graphical estimation procedures were developed by Kao (1959)?• Cohen (1965) defined the likelihood function for the Weibull distribution: Harter and Moore (1965) produced an iterative procedure for calculating maximum likelihood estimates. Warren (1976), in conjunction with his studies on breaking points of wood products, has further refined the maximum likelihood procedure. Seegrist and Arner (1978) discussed problems of maximum likelihood estimates when error components are correlated due to having repeated measurements on plots. Maximum likelihood estimation can be time consuming and complicated. Point estimation, although often not as accurate, is easier to use than maximum likelihood. Dubey (1967) devised a percentile estimator for the two parameter Weibull distribution, namely: M-lnO-p,)) -ln(-1n(l-p2)) C* ln(ypl) - lniyp2) b* =(yp^C* / In(l-P,) where is the i-th percentile and y ^ is the maximum value attributed to the percentile. He found that the percentiles which minimize the variance of the estimates are the 17-th and 97-th for c* and the 40-th and 82-d for b*. The percentile approach is appealing because it does not involve iterative calculations and is easily implemented. Englehardt and Bain wrote profusely on point estimates of Weibull parameters: (Bain and Antle (1967), Englehardt and Bain (1973, 1974), Bain (1972), and Englehardt (1975)). In 1977, they published simplified estimators of the parameters for complete samples and tolerance bounds for those estimates. Specifically: for z = 1/c: where s = [,84n^ and kn is a constant to remove bias. Ii for u = ln b: n u =2 Yj/n + Y z where "f is Euler's constant, 0.5772. This procedure, like the percentile approach, is easy to use. Although the debiasing constant, k, is difficult to calculate, Englehardt and Bain have solved for values for k of all sample sizes under 60 and for infinitely large samples. A complete discussion of the Weibull function, its history and applications, was given by Mann (1968). She discussed the applications and efficiency of several estimation procedures for censored and uncensored samples, as well as some methods of imposing confidence bounds on the various estimates. Mann, Shafer, and Singpurwalla (1974) published a text on statistical analysis of reliability and life testing data. They discussed the theory and mathematical derivation of estimation procedures and the applications thereof. They condensed the work of several previous authors, including d'Agostino (1971) (linear estimation) and Thomas and Wilson (1972) (point estimation). The tolerance bounds and confidence limits defined by Johns 12 and Leiberman (1966), Thoman, Bain, and Antle (1969), Bogdanoff and Pierce (1973), and Lawless (1975) were also discussed. The text is an excellent reference for anyone interested in the use of the Weibull distribution. 13 DATA BASE The data for this thesis came from the Regional Forest Nutrition Research Project (Univ. Wash., 1976) (a cooperative fertilization study with forest industry and several government resource agencies). The study was coordinated by the University of Washington under Dr. William Atkinson. Eighty three of the control plots in natural stands were selected for my study based on the following criteria: 1. Plot elevation between 0 and 2,500 ft.-'' above sea level. 2. Douglas-fir site index (King, 50 ) total height of site trees 77-143 feet at 50 years from seed. 3. Age at breast height 12-42 yrs. 4. All plots located west of the Cascades. 5. Mixed stands of western hemlock and Douglas-fir. Plot size ranged from 1/10 acre to 1/5 acre. Tree height was measured to the nearest foot and dbh to the 0.1 inch. Tables I and II list these plots along with some of their stand statistics. 1/ Because the data were all reported in English units, all measurements and results reported here are likewise in English units. iH TABLE Is ST^KD DATA - ESTIMATION SET Plot Species 11006 71037 81048 131076 141082 141084 TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock TPA 460 447 13 8.3 8.5 2.0 205 14.3 200 14.3 5 14.1 8D 3.0 3.4 161095 201118 211123 211125 % in 1.8 7.6 TOTAL Douglas-fir Big leaf Maple 327 320 7 11.2 11.4 3.5 4.1 3.5 Dmax 19.8 21.8 TOTAL 950 5.8 2.2 1.7 10.4 Douglas-fir 950 5.8 TOTAL 1460 4.1 1.8 1.7 10.0 Douglas-fir 1360 4.1 W. Hemlock 50 4.2 W. Redcedar 10 1.8 Other 40 2.5 TOTAL 470 8.8 3.6 2.0 16.1 Douglas-fir 340 9.9 W. Hemlock 110 6.2 W. Redcedar 15 2.6 Red Alder 5 8.5 TOTAL 480 8.4 5.5 1.7 25.8 Douglas-fir 215 12.7 W. Hemlock 145 5.4 W. Redcedar 95 2.9 Sitka Spruce 5 2.7 Red Alder 20 10.2 TOTAL 640 7.8 4.2 2.0 18.3 Douglas-fir 334 10.7 W. Hemlock 280 4.9 W. Redcedar 13 4.2 Other 13 2.5 TOTAL 400 8.4 3.3 2.8 15.5 Douglas-fir 393 8.5 Other 7 5.0 TOTAL 380 11.4 3.3 5.6 18.2 Douglas-fir 373 11.4 Other 7 6.9 21.1 Stand age 35 39 21 36 52 52 35 26 39 39 Site index 113 133 130 80 118 118 129 143 130 130 Average height 82 112 67 62 108 114 98 85 110 102 D — Mean diameter (inches) sD -- Standard deviation of diameters (inches) D . ~ Minimum diameter (inches) mm D — Maximum diameter (inches) max TPA — Trees per acre Stand Age — (years) Site Index — King, Douglas-fir (feet at 50 yrs) Average Height — Mean dominant, codominant ht. (feet] \5 TABLE I: STAND DATA - ESTIMATION SET Plot Species TPA Dmin Dma Stand age Site index Average height 211128 TOTAL 407 9.6 3.8 2.1 18.8 45 124 105 Douglas-fir 354 9.9 W. Hemlock 40 6.1 Red Alder 13 10.0 221132 TOTAL 420 10.3 5.0 2.3 20.7 45 124 113 Douglas-fir 293 12.5 W. Hemlock 107 5.6 Red Alder 20 3.6 33119 3 TOTAL 880 4.2 2.3 1.6 12.2 19 123 51 Douglas-fir 450 5.4 W. Hemlock 430 2.8 341202 TOTAL 610 8.0 3.1 3.8 18.1 27 137 87 Douglas-fir 580 8.1 W. Hemlock 30 7.2 361215 TOTAL 305 10.2 5.7 1.7 21.9 39 138 110 Douglas-fir 135 14.8 W. Hemlock 145 6.4 Red Alder 15 11.2 Other 10 1.7 371221 TOTAL 320 11.3 3.7 3.0 19.7 42 127 109 Douglas-fir 180 13.4 W. Hemlock 130 8.7 Red Alder 10 7.3 411244 TOTAL 387 8.9 3.3 1.7 17.2 28 140 92 Douglas-fir 354 9.6 W. Hemlock 13 2.3 Other 20 2.6 431253 TOTAL 1610 5.1 2.2 1.7 11.2 37 98 67 Douglas-fir 580 5.7 W. Hemlock 640 5.2 W. Redcedar 390 4.1 451270 TOTAL 620 7.5 4.2 2.0 23.1 33 131 94 Douglas-fir 167 12.3 W. Hemlock 313 6.2 W. Redcedar 80 4.7 Big leaf Maple 20 5.7 Red Alder 7 6.2 Other 33 4.7 16 TABLE IJ STAND DATA - ESTIMATION SET Plot Species TPA X> 8D Pnin °max Stand Site Average age index height 531313 TOTAL 493 7.5 2.0 3.8 13.0 21 125 65 Douglas-fir 493 7.5 531317 TOTAL 527 7.0 3.4 1.5 15.0 21 125 58 Douglas-fir 447 7.9 Other 80 1.7 541319 TOTAL 1100 4.6 2.3 1.6 12.0 35 106 72 Douglas-fir 1100 4.6 551327 TOTAL 640 7.4 2.9 2.0 13.4 37 94 77 Douglas-f ir 640 7.4 571341 TOTAL 400 9.9 3.5 2.1 17.4 29 137 88 Douglas-fir 333 10.6 W. Hemlock 20 6.7 H. Redcedar 27 4.5 Red Alder 13 7.4 Other 7 7.0 601355 TOTAL 500 7.6 4.0 1.6 17.1 42 105 97 Douglas-fir 400 9.0 Other 100 1.9 601360 TOTAL 840 6.4 2.2 1.6 15.3 42 105 77 Douglas-fir 820 6.5 Other 20 1.7 681404 TOTAL 570 6.7 4.7 1.6 15.3 30 135 91 Douglas-fir 340 9.9 Other 230 1.9 761453 TOTAL 420 6.7 3.6 1.5 16.4 32 98 73 Douglas-fir 360 7.0 Other 60 4.9 771462 TOTAL 1280 3.7 1.8 1.6 8.9 15 77 43 Douglas-fir 1270 3.7 Sitka Spruce 10 2.1 791473 TOTAL 630 7.3 3.3 1.5 15.4 30 121 87 Douglas-fir 420 8.5 W. Hemlock 40 3.6 W. Redcedar 50 1.9 Big leaf Maple 20 2.5 Red Alder 100 7.3 811485 TOTAL 600 8.2 2.4 4.3 14.6 31 109 69 Douglas-fir 600 8.2 iT TABLE Is STAND DATA - ESTIMATION SET Plot Species TPA Dmln Dmax stand site Average age index height 821490 TOTAL 1110 5.2 2.6 Douglas-fir 220 8.0 W. Hemlock 720 4.9 Red Alder 10 8.1 Other 160 2.4 831494 TOTAL 1560 5.1 2.4 Douglas-fir 520 6.5 W. Hemlock 950 4.5 Other 90 2.9 951568 TOTAL 1340 4.5 1.7 Douglas-fir 1340 4.5 961576 TOTAL 280 11.3 . 3.4 Douglas-fir 280 11.3 971582 TOTAL 520 7.2 3.6 Douglas-fir 420 8.3 Other 100 2.8 981588 TOTAL 374 10.2 3.4 Douglas-fir 347 10.7 Big leaf Maple 20 3.7 Other 7 2.5 991591 TOTAL 380 6.8 5.4 Douglas-fir 313 7.7 W. Hemlock 47 3.0 Other 20 1.9 1011602 TOTAL 730 6.2 2.4 Douglas-fir 730 6.2 1011605 TOTAL 1060 4.9 2.0 Douglas-fir 1060 4.9 1101660 TOTAL 830 3.7 2.0 Douglas-fir 400 4.2 H. Hemlock 340 2.3 W. Redcedar 10 1.7 Other 80 2.8 1.7 12.8 1.5 12.2 21 122 60 3.0 18.6 1.5 14.6 2.5 18.4 1.6 28.5 2.8 12.7 1.2 9.4 1.7 6.8 22 1.5 9.1 19 39 30 38 34 24 24 12 130 100 112 108 125 121 118 118 132 68 44 94 72 100 79 71 59 36 1131675 TOTAL 1300 4.0 Douglas-fir 1080 4.0 Other 220 4.0 2.0 1.6 9.9 23 117 59 IS TABLE II: STAND DATA - PREDICTION SET Plot Species TPA D 8D °fliin Dm ax Stand age Site index Average height 11004 TOTAL 407 9.3 2.9 4.3 15.2 35 113 81 Douglas-fir 407 9.3 51025 TOTAL 593 6.4 2.4 1.7 11.7 30 101 67 Douglas-fir 553 6.6 W. Hemlock 40 4.1 51027 TOTAL 626 7.0 2.4 2.2 17.0 30 101 65 Douglas-f ir 613 7.1 W. Hemlock 13 4.0 71041 TOTAL 220 13.0 4.0 5.8 20.2 39 133 106 Douglas-fir 205 13.4 W. Hemlock 15 7.4 B1043 TOTAL. 1180 5.3 2.0 2.0 11.4 21 130 66 Douglas-fir 1170 5.3 Other 10 2.4 131074 TOTAL 1600 4.4 2.1 1.6 9.4 36 80 55 Douglas-fir 1580 4.3 Other 20 . 5.1 161091 TOTAL 659 7.7 3.7 2.2 16.8 35 129 98 Douglas-fir 373 9.9 W. Hemlock 253 4.9 W. Redcedar 20 4.0 Other 13 5.3 171098 TOTAL 1530 4.9 2.0 1.8 12.3 20 122 63 Douglas-fir 980 5.4 W. Hemlock 550 4.1 171100 TOTAL 2280 3.8 1.7 1.6 9.9 20 122 59 Douglas-fir 1550 4.1 W. Hemlock 730 3.1 191111 TOTAL 314 10.7 4.2 1.5 18.5 45 126 117 Douglas-fir 280 11.7 W. Hemlock 7 2.3 W. Redcedar 27 2.2 191114 TOTAL 267 11.7 3.5 7.0 20.0 45 126 117 Douglas-fir 267 11.7 201119 TOTAL 687 6.4 3.3 1.8 17.9 26 143 84 Douglas-fir 680 6.4 Other 7 2.6 X - Mean diameter (inches) -- Standard deviation of diameters (inches) — Minimum^djameter (inches) Maximum diameter (inches) TPA — Trees per acre Stand Age — (years) Site Index — King, Douglas-fir (feet at 50 yrs) Average Height — Mean dominant, codosrinant ht. (feet) TABLE lis STAND DATA - PREDICTION SET Plot Species TPA 8D Drain ^max stand Site Average age index height 331194 341204 361216 371222 411246 431254 451266 541324 551330 571342 591353 TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock Other TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir Other TOTAL Douglas-fir W. Hemlock W. Redcedar TOTAL Douglas-fir W. Hemlock W. Redcedar Other TOTAL Douglas-fir TOTAL Douglas-fir TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock W. Redcedar 1150 900 250 680 480 160 40 340 185 155 430 240 190 373 360 13 1050 340 480 230 473 193 100 73 107 790 790 930 930 360 327 33 813 520 173 120 4.7 5.1 3.2 6.8 7.9 3.7 5.5 10.9 14.2 7.0 10.3 12.9 7.0 9.8 10.0 4.8 6.0 7.2 5.8 4.5 7.0 12.8 4.2 3.0 2.0 6.0 6.0 5.8 5.8 9.7 9.8 9.3 1.9 3.2 5.3 4.4 3.0 3.0 5.4 3.0 2.2 3.8 3.5 1.8 8.9 19 1.8 15.0 27 2.0 3.1 3.1 1.5 1.5 23.3 18.9 16.8 13.5 19.6 39 42 28 37 33 1.6 14.6 35 2.4 13.3 37 2.1 17.2 29 1.4 17.1 34 123 137 138 127 140 98 131 106 94 137 99 52 86 109 108 91 83 95 80 73 82 72 20 TABLE II: STAND DATA - PREDICTION SET Plot Species TPA D 8D Pmin Dmax Stand age Site index Average height 591354 TOTAL 993 5.2 3.5 1.4 23.0 34 99 71 Douglas-fir 633 6.2 W. Hemlock 73 2.7 \ W. Redcedar 280 3.6 Other 7 3.6 681407 TOTAL 420 8.5 3.9 1.5 16.8 30 135 90 Douglas-fir 350 9.9 Other 70 1.7 691410 TOTAL 260 9.3 4.5 2.0 17.5 29 130 84 Douglas-fir 247 9.6 Other 13 3.8 691412 TOTAL 327 8.4 5.3 1.7 20.2 29 130 87 Douglas-fir 220 11.4 W. Redcedar 20 2.2 Red Alder 7 9.7 Other 80 1.9 7614 54 TOTAL 1774 4.0 2.0 1.6 11.4 32 98 61 Douglas-fir 1687 3.8 Other 87 6.8 771458 TOTAL 860 3.5 1.7 1.4 9.2 15 77 39 Douglas-fir 790 3.6 Other 70 2.5 781468 TOTAL 1013 5.7 2.4 2.2 16.7 42 108 85 Douglas-fir 960 5.7 W. Hemlock 13 8.2 Red Alder 33 7.0 Other 7 2.3 791472 TOTAL 690 6.8 3.1 1.6 13.7 30 121 85 Douglas-fir 350 8.5 W. Hemlock 10 1.6 W. Redcedar 100 2.5 Big leaf Maple 140 . 5.5 Red Alder 90 7.1 811481 TOTAL 550 8.5 3.1 1.8 17.5 31 109 75 Douglas-fir 530 8.8 Other 20 2.0 21 TABLE lis STAND DATA - PREDICTION SET Plot Species TPA Dfliin °max Stand Site Average age index height 821492 TOTAL Douglas-fir W. Hemlock Red Alder Other 880 400 420 30 30 6.7 B.l 5.4 8.0 2.7 2.7 2.3 12.1 21 122 59 831498 TOTAL Douglas-fir W. Hemlock 1270 610 660 5.7 7.2 4.4 2.4 1.7 12.8 22 130 62 951565 TOTAL Douglas-fir Red Alder Other 1930 1850 20 60 1.7 1.5 9.0 19 100 49 961572 TOTAL Douglas-fir W. Hemlock Other 380 340 27 13 9.8 10.5 4.5 3.1 3.3 1.9 15.4 39 112 93 971580 981584 TOTAL Douglas-fir Big leaf Maple TOTAL Douglas-fir Other 470 440 30 327 320 7 8.5 8.7 5.5 11.3 11.5 2.3 3.0 2.9 3.4 2.3 14.8 17.0 30 38 108 125 74 94 991592 TOTAL Douglas-fir W. Hemlock Sitka Spruce Other 599 253 153 33 160 6.9 10.4 6.1 6.0 2.4 4.4 1.5 16.5 34 121 79 1101658 TOTAL 670 3.6 Douglas-fir 410 4.2 W. Hemlock 110 1.8 Red Alder 20 4.6 Other 130 1.9 1.9 1.7 7.7 12 132 34 1131678 TOTAL 850 5.2 Douglas-fir 550 5.7 W. Hemlock 20 3.5 Red Alder 100 4.9 Other 180 3.8 2.7 1.6 11.5 23 117 65 22 METHODS The data were split by DUPLEX, ,an algorithm developed by R. W. Kennard (Snee, 1977), into an estimation set and a prediction set. Each set contained a comparable range of sites, densities, and age groups. The estimation set was used to find a suitable relationship between stand characteristics and the Weibull diameter parameters a, b and c. The prediction set was used to test that relationship. The parameters of the Weibull distribution for each plot were estimated from the given individual tree diameter data. This estimation relied on the availability of an adequate estimating routine. Three algorithms found in the literature were tested: Dubey's (1967) percentile method, and two maximum likelihood routines, Warren (1976) and Bailey (1973) (see Appendix I). Bailey's FITTER routine was selected because it gave a better fit than the percentile method and, unlike Warren's routine, it gave unbiased estimates. After the Weibull parameters were estimated for the diameter distributions of all plots, stand characteristics from the data were regressed* on the estimated parameters of the estimation plots. The characteristics considered were: stand total age, mean dominant and codominant height, 1/ Regression done via BMD:2R routine. £3 arithmetic mean diameter, site index, number of stems ( >0.1 inch d.b.h.) per acre, and the inverses and squares thereof. The best overall models were then used to predict the Weibull parameters for the prediction plots. The resulting distributions were compared with the corresponding estimates from the observed diameter data. (See Appendix I). To be a useful model, the Weibull distribution should describe the diameter distributions of mixed species stands as well as, if not better than simpler models such as the normal distribution. Here, a regressed relationship was deemed satisfactory if all of the predicted distributions satisfied the Kolmogorov-Smirnov (K-S) goodness of fit test at the «- = 0.20 level (Massey, 1951). This level of significance was thought to be an acceptable compromise between the goodness of fit obtained for Weibull distributions on even-aged, single species stands (Bliss & Rienker, 1964;•<* = 0.15; Bailey, 1973. 35% fit at oc = 0.05) and the fits obtainable with simpler one- and two-parameter functions. RESULTS The Weibull parameters a, b, and c were estimated by Bailey's FITTER routine for all 83 stands. The cumulative distributions defined by the maximum likelihood estimated parameters were compared with the observed diameter distributions. All of the estimated distributions passed the Kolmogorov-Smirnov (K-S) test at the <*.= 0.20 level of significance; d 2(30) = .131-/ (Table III). The stand characteristics mean diameter, mean dominant and codominant height, stand age, trees per acre, and site index were regressed against the estimated parameters of the Weibull diameter distributions of the estimation set. The best resulting predictors for the parameters based on correlation coefficients and F ratios are listed in Tables IV, V, and VI. The diameter distributions of the prediction set as defined by all combinations of a, b, and c predictors were compared with the distributions which were estimated from the observed diameter distributions via maximum likelihood. The only combinations to fail the K-S test at the <*-= 0.20 level were those involving predictor a (8) or b (8). 1/ Lilliefors (1967) critical value for d 2(30) was used here because the parameters of the distribution were estimated from "the sample. X 1? TABLE III: Comparison of Maximum Likelihood Estimated Weibull Distribution with Original Diameter Distribution >(N)i/ Plot *w b* TW 5.9216 1.8617 .068 .159 8.1952 2.4969 .056 .153 6.0071 2.3716 .062 .131 6.1935 2.4098 .063 .130 8.7014 2.3690 .053 .183 11.2241 2.6992 .072 .177 4.2696 1.9549 .041 .166 5.4440 2.3902 .061 .130 3.7726 1.6709 .095 .100 3.4661 1.7869 .093 .105 9.2849 2.4701 .058 .130 8.0054 1.3355 .071 .130 6.9514 1.7192 .054 .126 6.9613 1.4752 .041 .130 3.8235 1.7479 .046 .103 3.1728 1.8152 .080 .084 11.9021 2.7535 .094 .173 5.7082 1.4494 .087 .183 7.1932 1.9827 .051 .160 5.4706 1.5645 .046 .124 7.1396 1.9922 .063 .163 9.8310 2.2681 .092 .169 9.0550 2.2264 .055 .159 10.4275 1.9175 .076 .157 3.4860 1.4431 .081 .131 3.5633 1.7218 .089 .118 4.9760 1.4900 .049 .159 5.8880 1.6470 .051 .153 10.1425 1.5651 .063 .159 12.2880 2.1683 .075 .153 12.5558 3.2579 .056 .206 9.3328 1.9554 .051 .177 9.9678 2.8635 .066 .160 10.8782 3.6038 .057 .163 4.6327 1.9442 .046 .100 5.5194 1.6684 .047 .124 6.1243 1.0381 .098 .150 6.7211 1.5060 .095 .132 4.4572 2.0177 .075 .149 7.8732 2.1918 .083 .142 4.0349 1.5962 .055 .121 5.5978 1.7028 .043 .142 11004 11006 51025 51027 71037 71041 81043 81048 131074 131076 141082 141084 161091 161095 171098 171100 191111 191114 201118 201119 211123 211125 221128 221132 331193 331194 341202 341204 361215 361216 371221 371222 411244 411246 431253 4 31254 451266 451270 531313 531317 541319 541324 4.0000 0000 0000 5000 5000 .0000 5000 1.0000 1.0000 1.0000 0.5000 1.0000 1.5000 1.5000 1.5000 1.0000 0.0000 6.5000 2.0000 1.5000 5.0000 2.5000 5000 0000 0000 5000 5000 5000 0000 0000 0000 2.0000 0.0000 0.0000 1.0000 1.0000 1.0000 1.5000 3.5000 0.0000 1.0000 1.0000 1. 1. 1. 1. 3. 1. 1. 0. 0. y a*, b*, c* are the estimated Weibull parameters V d - max/Sn (x) - FD (x) 3/ a.2(N) - Kolmogoror-Smirnov limit for d for fit at 20% level of significance Z.6 TABLE III (Continued) Plot b* c* d V d.2(N)i/ 551327 1.0000 7.1695 2.3550 .056 .157 551330 2.0000 4.2236 1.7714 .047 .132 571341 0.0000 10.9818 3.0761 .038 .160 571342 0.0000 10.9145 2.8211 .073 .166 591353 1.0000 5.8602 1.5572 .049 .155 591354 1.0000 4.5779 1.3143 .047 .104 601355 0.0000 8.4775 1.9472 .124 .169 601360 1.0000 6.0191 2.5392 .062 .139 681404 1.0000 5.8259 1.0608 .202 .163 681407 0.0000 9.5229 2.2623 .138 .183 691410 1.0000 9.2792 1.8733 .084 .190 691412 1.0000 7.8292 1.1820 .162 .169 761453 1.0000 6.2550 1.5107 .076 .159 761454 1.0000 3.3434 1.6222 .080 .078 771458 1.0000 2.7473 1.5137 .099 .137 771462 1.0000 3.0129 1.6326 .078 .112 781468 1.5000 4.7896 1.9342 .067 .103 791472 0.5000 7.0273 2.0676 .060 .153 791473 0.0000 8.1590 2.3065 .055 .159 811481 0.0000 9.4805 2.9009 .079 .163 8114 85 4.0000 4.7198 1.8031 .033 .160 821490 1.0000 4.6869 1.6929 .064 .121 821492 2.0000 5.1969 1.7361 .063 .131 831494 1.0000 4.6034 1.7963 .040 .102 831498 1.0000 5.3542 2.0770 .056 .112 951565 1.0000 2.9076 1.6261 .098 .091 951568 1.0000 3.9121 2.1489 .067 .110 961572 0.0000 10.9154 3.4388 .084 .163 961576 0.0000 12.4302 3.6758 .056 .183 971580 2.0000 7.3454 2.2804 .097 .173 971582 0.5000 7.5120 1.9103 .086 .166 981584 0.0000 12.3532 4.5658 .071 .169 981588 0.0000 11.3259 3.3332 .054 .163 991591 1.0000 6.1411 1.1413 .061 .163 991592 1.0000 6.3467 1.2426 .123 .131 1011602 2.5000 4.1552 1.6311 .093 .150 1011605 0.5000 4.9292 2.3247 .061 .123 1101658 1.0000 2.9.53 1.5847 .089 .157 1101660 1.0000 2.9956 1.7325 .107 .149 1131675 1.0000 3.3690 1.6392 .081 .111 1131678 1.0000 4.6670 1.6160 .113 .137 TABLE IV; a Equations a(l) • 0.4737 + 448.4/TPA a(2) = 3.086 - 0.6655(D) + 0.08021(D2) - 27.58(A)/TPA a(3) =• -413.3 - 43.22(D) + 640.1/D - 574.1/TPA + 1.032(D2) + 296.6(ln D) - 21.83(A)/TPA a(4) = -444.1 - 46.31(D) + 689.5/D - 1275.0/TPA + 1.093(D2) + 318.9(ln D) - 2.872(A)/SI a(5) = -410.7 - 43.33(D) + 635.0/D +0.00005556(SI2) + 295.3(ln D) + 1.0 4 8 ( 02) - 14.86(HT)/TPA a(6) = -466.0 - 47.99(D) + 724.6/D - 1347.0/TPA + 1.126(D2) + 333.3(ln D) - 2.544(HT)/SI a(7) = -325.8 - 35.84(D) +496.8/D + 0.8543(D2) + 0.00003266(SI2) + 238.6(ln D) + 0.01514(TPA)/A a(8) = 372.5 -39.37(D) + 0.008466(SI) + 567.8/D - 0.0002378(A2) + 0.91931D2) + 268.2(ln D) + 0.06373(TPA)/HT a(9) = -332.4 - 35.63(D) + 509.5/D - 0.0004968(A2) + 0.8411(D2) + 241.2(ln D) + 0.0726(TPA)/SI ,a(10) " -386.3 - 40.60(D) + 598.1/D - 0.0006213 (A2) + 0.965KD2) + 278.1(ln D) - 7.318 (SI)/TPA A = D = HT stand age (total years) SI = site index (King, Douglas fir, feet at 50 yrs) arithmetic mean diameter (inches) TPA = trees per acre (dbh 0.1 in.) = ave. dominant & codominant height (feet) TABLE V; b Equations b(l) = 2.648 + 0.06581(HT) - 0.04242(TPA)/A b(2) = 6.297 + 0.001277(A2) + 0.00007376(SI2) - 0.1870(TPA)/HT b(3) = -0.2065 + 0.05548{HT) + 1.159(In A) - 0.2231(TPA)/SI b(4) = 17.75 + 0.05619(HT) - 2.412(ln TPA) b(5) - 13.10 + 3.914(ln A) - 2.600(ln TPA) - 7.488(A)/HT b(6) = 3.497 + 3.188(ln HT) - 1.791(ln TPA) + 14.72 (A)/TPA b(7) = 6.364 + 0.03952(A) - 29.30/A - 236.9/SI + 13.84(HT)/TPA b(8) " 23.20 + 5.665(HT)/SI - 3.156(ln TPA) b(9) = 18.90 + 2.617(ln A) - 3.276(ln TPA) b(10) - 4.011 + 0.0007134(A2) + 4.451(ln D) - 1.038(ln TPA) 00' TABLE VI: c Equations c(l) - 1.562 + 255.6/TPA c(2) • 0.8321 + 0.2908(D) - 0.0004183(A2) - 0.0002045(HT2) c(3) = 0.6100 + 0.4136(D) - 311.65/TPA + 0.0004033(A2) - 0.0002179(HT2) c(4) » 165.4 + 16.77(D) - 253.8/D - 0.3649(E)2) - 0.0001321 (HT2) - 0.00003068 (SI2) - 115.9(ln D) c(5) = 38.96 - 0.1473(A) + 0.3883(D) + 0.4969(HT) + 0.002466(A2) - 0.002059(HT2) - 14.72(ln HT) + 0.0674(TPA)/SI c(6) » -0.1859 + 0.6313(D) + 0.1174(HT) + 10.28/D + 0.001338(A2) - 0.0008732(HT2) - 2.415(ln A) - 2.940 (SD/TPA 3o Several combinations of predicting equations passed the K-S test at the <* = 0.20 level with 100% fit of all plots. These superior combinations, (or distribution models), can be found in Table VIII with their respective d statistics. As a class, the combination of equations a(2) and b(10) was a superior model, having the lowest average d statistic and the least amount of variance in fit regardless of the c equation used. The best model was: a(2)=3.086 - 0.6655(D) + 0.08021 (D2) - 27.58 (A)/TPA b(10) = 4.011 + 0.0007134 (A2) + 4.451 (ln D) - 1.038 (ln TPA) c(4) = 165.4 + 16.77(D) - 253.8/D - .3649 (D2) - .0001321 (HT2) - .00003068 (SI2) - 115.9 (ln D) where D = mean stand diameter (inches) A = stand age (total years) SI = site index (King, Douglas-fir, height at 50 yrs) HT = mean dominant and codominant height (feet) TPA = trees per acre with an average d = .0587, sH = .0317, and max d = .1441 . 31 TABLE VII: Regression Statistics Equation F(n,v) R SEE (inches) a(l) 3.896(1,40) .2979 1.336 a(2) 6.717(3,38) .5887 1.161 a(3) 5.737(6,35) .7042 1.063 a(4) 5.480 (6,35) .6960 , 1.075 a(5) 5.674(6,35) .7022 1.066 a(6) 5.736(6,35) .7041 1.063 a(7) 3.852(6,35) .6306 1.162 a(8) 3.432(7,34) .6434 1.163 a(9) 4.088(6,35) .6419 1.148 a(10) 4.820 (6,35) .6727 1.108 b(l) 37.58(2,39) .8114 1.597 b(2) 22.62(3,38) .8006 1.659 b(3) 26.95(3,38) .8248 1.565 b(4) 51.29(2,39) .8512 1.434 b(5) 33.97(3,38) .8535 1.443 b(6) 34.45(3,38) .8551 1.445 b(7) 20.57(4,37) .8306 1.562 b(8) 48.64(2,39) .8449 1.462 b(9) 48.42(2,39) .8442 1.464 b(10) 39.18(3,38) .8693 1.368 c(l) 7.719(1,40) .4022 .5412 c(2) 8.863 (3,38) .6416 .4652 c(3) 7.485(4,37) .6588 .4570 c(4) 5.827(6,35) .7069 .4470 c(5) 5.068(7,34) .7146 .4485 c(6) 5.039(7,34) .7136 .4492 3£ Table VIII — Prediction Models which fit all MLE Distributions EQUATION # d sd max d abc 2 5 2 .0852 .0408 .1625 2 5 3 .0830 .0405 .1746 2 5 4 .0843 .0371 .1694 2 5 6 .0839 .0426 .1890 2 6 2 .0780 .0407 .1732 2 6 3 .0750 .0408 .1663 2 6 4 .0650 .0389 .1672 2 6 5 .0827 .0439 .1868 2 6 6 .0767 .0421 .1868 2 11 1 .0735 .0572 .1705 2 11 2 .0591 .0346 .1549 2 11 3 .0563 .0346 .1739 2 11 4 .0587 .0317 .1441 2 11 6 .0566 .0384 .1883 7 11 2 .08 21 .0459 .1666 7 11 3 .0795 .0465 .1775 7 11 4 .0789 .0422 .1733 7 11 6 .0813 .0446 .1794 1. d = average K-S statistic over all prediction plots s^ = standard deviation of d max d = maximum value of d found over all plots 33 A plot by plot comparison of the distributions predicted by the model: with the observed stand diameter distributions can be found in Table IX. The K-S statistic d was less than d ^(N) for 95% of the plots. According to the Kolmogorov - Smirnov test, the predicted Weibull diameter distributions are satisfactory models of the observed distributions. The quality of fit, as reflected by the K-S statistic d, was not dependent on the value of any one stand attribute (Figures 1-5). The model predicts consistantly over the range of mean diameter, height, site, stocking, and stand age exhibited in the prediction set. The figures in Appendix III show all three distributions (MLE, predicted, and observed) for 18 of the prediction set The predicted distributions did not fit the observed diameter distributions at the oc = .20 level of significance for plots 691412 and 991592. The trees in these plots divided easily into distinct species - size groups. Stand 691412 (fit at d = .01) was 67% Douglas-fir (mean diameter 11.4 inches) and 33% cedar and other species (mean diameter 1.9 inches). Stand 991592 (fit at oc = .10) was 42% Douglas-fir (10.4 in. dbh) 31% hemlock and spruce (6.0 in. dbh) and 27% other species (2.4 in. dbh). The polymodal nature of these two diameter density distributions accounts for the lack of fit with the unimodal Weibull model. F(d) = 1.0 - exp plots. 34 Although the difference between the observed and predicted distributions was not significant at the oc = .20 probability level, plots 71041, 361216, 451266, and 591354 showed a tendency in the model towards bias. The predicted distributions for these plots underestimated the number of trees in the small diameter classes and overestimated the number of trees in the middle diameter classes (Figures 6-9). TABLE IX: MLE vs Predicted Distribution (a(2) b(10) c(4)) d.20OO) = .190 Plot a y b y c y dMLE y d<)3 11004 4.0000 5.9126 1.8617 .0470 .0683 1.4432 8.5643 2.7774 51025 1.0000 6.0771 2.3716 .0594 .0651 0.7155 6.2829 2.0695 51027 1.5000 6.1935 2.4098 .0441 .1027 1.0467 6.6419 2.2585 71041 3.0000 11.2241 2.6992 .0301 .1006 3.1164 10.9179 2.9267 81043 1.5000 4.2696 1.9549 .0495 .0580 1.3184 4.3939 1.7620 131074 1.0000 3.7726 1.6709 .0881 .1825 1.0847 3.8379 2.1205 161091 1.5000 6.9514 1.7192 .0379 .0469 1.2519 7.2286 1.5712 171098 1.5000 3.8235 1.7479 .0327 .0388 1.3898 3.7538 1.8461 171100 1.0000 3.1728 1.8152 .1082 .0792 1.4731 2.2186 1.4335 191111 0.0000 11.9021 2.7535 .0578 .1295 1.2053 10.0470 2.3373 191114 6.5000 5.7082 1.4494 .1226 .1355 1.6340 10.6077 2.6085 201119 1.5000 5.4706 1.5645 .0580 .0949 1.0683 5.9763 1.4188 33119 4 1.5000 3.5633 1.7218 .0190 .0821 1.2734 3.8310 1.9692 341204 1.5000 5.8884 1.6467 .0441 .0800 1.1693 6.2855 1.5151 361216 0.0000 12.2880 2.1683 .1061 .1588 2.2319 9.6907 2.5561 371222 2.0000 9.3328 1.9554 .0700 .0988 2.0727 9.3668 2.4634 411246 0.0000 10.8782 3.6038 .0199 .0682 2.2281 8.5962 2.6055 1/ The first line contains the maximum likelihood estimate, the second line contains the predicted parameters. 2/ K-S statistic for predicted distribution against MLE. 3/ K~S statistic for predicted distribution against observed. TABLE IX: continued. Plot a b c dMLE a0 431254 1.0000 5.5479 1.6959 .0220 .0580 1.0048 5.7322 1.7233 451266 1.0000 6.1243 1.0381 .0637 .1600 0.4506 7.0768 1.4097 541324 1.0000 5.5978 1.7028 .0138 .0475 0.7587 5.9345 1.7206 551330 2.0000 4.2236 1.7714 .0727 .1000 0.8161 5.6846 1.9221 571342 0.0000 10.9145 2.8211 .0573 .1335 1.9970 8.6355 2.7846 5913 53 1.0000 5.8602 1.5572 .0628 .1090 0.9106 6.0446 1.9707 591354 1.0000 4.5779 1.3143 .1010 .1552 0.8491 5.0046 1.8804 681407 0.0000 9.5229 2.2623 .0332 .1571 1.2825 7.9296 2.0860 691410 1.0000 9.2792 1.8733 .1067 .1883 0.7497 8.7600 2.5967 691412 1.0000 7.8292 1.1820 .1399 .2809 0.7406 8.1007 2.1373 761454 1.0000 3.3434 1.6222 .0428 .1099 1.2103 3.1180 1.7182 771458 1.0000 2.7473 1.5137 .0578 .1485 1.2616 2.6943 1.4734 781468 1.5000 4.7896 1.9342 .0942 .1407 0.7623 5.8533 1.5856 791472 0.5000 7.0273 2.0676 .0361 .0810 1.0497 6.3680 1.6580 811481 0.0000 9.4805 2.9009 .0364 .0907 1.6804 7.6802 2.5801 821492 2.0000 5.1969 1.7361 .0305 .0946 1.5498 5.7217 2.1237 831498 1.0000 5.3542 2.0770 .0320 .0483 1.4287 4.7089 1.8574 951565 1.0000 2.9076 1.6261 .0806 .0686 1.4584 2.1003 1.4339 961572 0.0000 10.9154 3.4388 .0422 .1058 1.4 815 9.1113 2.7657 971580 2.0000 7.3454 2.2804 .019 2 .1235 1.4864 7.8087 2.6171 981584 0.0000 12.3532 4.5658 .0389 .0956 2.6398 9.8388 3.1718 991592 1.0000 6.3467 1.2426 .1000 .2172 0.7664 6.8167 1.8280 1101658 1.0000 2.9122 1.5748 .0780 .1499 1.2042 3.1270 1.3974 1131678 1.0000 4.6665 1.6158 .0462 .1632 1.0407 4.6872 1.8621 3T Figure 1. The relationship of goodness of fit to stand diameter. 4.10 5.35 6.60 7.85 9.11 10.36 11.61 12.37 14.11 . + i + « + + + + 1 + + e (• + 4 + 1  .25 • 20 .17 -o .15 o •r— -t-> to •r— +-> <0 +-> oo CO 1 .10 .02 * • 0 + . -t + + 1——+ + +—-+—-+ • + •( + , + + +- ^ + _ + .3.47 _ 5.98 7.23 8.48 9.73 10.99 12.24 <3.49 14.75 16.00 Diameter (inches) STATISTICS.. R SQUARED .00848 . SLOPE (B) - .00160 STD FRRGfi OF P -Figure 2. The relationship of goodness of fit to stand height. 39.00 49.00 59.00 69.00 79.00 89.00 99.00 109.00 119.00 129.00 . + r + r + + H (- + + + + 1 r + -t r- + + + T .25 + .22 .20 .17 ,15 .12 +•> «/> T-•I » rO 4-> CO CO ^ .10 .07 .05 .02 it * 4 ft * * * + ^ + + + i + t + + + + + + <—-+ +-—+ 1-77—+-.00 44.00 54.00 64.00 74.00 84.00 94.00 104.00 "114.00 124.00 1 Height (feet) STATISTICS.. R SQUARED .00360 SLOPE (B) .00013 STD ERROR OF B -33 Figure 3. The relationship of goodness of fit to stand age. 14.00 13.00 22.00 26.00 30.00 34.00 3B.00 42.00 46.00 50.00 .+ -f + 1 (• + i (• + + + r 1 + r r + + + + +. .25 + • * .22 .20 .17 •° .15 4-> «/» +-> fO +J i/"> CO i .12 .10 .07 .05 0 + « * . + + + + + r 1- 1 +: 1 * + + + + 1—--•t- + + +-12.00 . 16.00 20.00 24.00 23.00 32.00 36.00 40.00 44.00 48.00 'STATISTICS.. R SQUARED Stand Age (total years) .02950 SLOPE lb) - .00092 STD ERROR OF B -+, 2.00 .OOuf; Figure 4. The relationship of goodness of fit to trees per acre. C" 323.00 529.00 735.00 941.00 1147.00 1353.00 1559.00 1765.00 1971.00 2177.00 * 1- 1 i i 1 i i 1 1 •+ ^ <• + * + < + +—-+ +• .25 *** .22 + .20 .17 u 5 -12 -t-> » JO .07 .05 .02, » • * * * * « * * * 0' * ft + t -+ 4 - • -+ - / /22o!« ' 4.26.00 632.00 + 838.00 1044.00 ~T250.00 ^1456.00 ,1662.00 1868T.0O 2074.00 2288.00 •STATISTICS.. R SQUARED Trees per Acre .07018 SLOPE <B) - -.00002 " STD ERROR OF B - .00001 Figure 5. The relationship of goodness of fit to site index. 81.00 89.00 97.00 105.00 , 113.00 121.00 129.00 137.00 ' 145.00 153.00 • + < < ' 1 < + 1- i i r- H 1 1 + ( 1 1 , e— .22 .20 ,17 .15 4-> t/> •r~ % .12 +-> CO CO .10 .07 .05 0 + .+--77.00 i i « ~H f + e f. + + H i < + + ^ -+ + +-85.00 93.00 101.00 109.00 117.00 125.00 133.'00 ' 141.'00 ' 14y.'oo""+~^LOO •STATISTICS.. R SQUARED Site Index (feet at 50 years),'• .00299 SLOPE (B) - -.00015 STD ERROR OF B .00043 Figure 6. Observed and predicted trees per acre by diameter class for plot 710^1. 80-60 H HQ] Zo *f-8 8-12. 12.-16 Diameter Class Observed I—I Predicted Figure 7. Observed and predicted trees per acre by diameter class for plot 361216. IbOA 12,0-&• SO 5 5"-'o 10-/5 Diameter Glass l^>l Observed I J Predicted Figure 8. Observed and predicted trees per acre by diameter class for plot 451266 Diameter Glass Figure 9. Observed and predicted trees per acre by diameter class for plot 591354. 5 Hoo < 6 Zoo-zoo looi \^ZA. Observed [ f Predicted 2 O-M N-8 8M2 Diameter Glass DISCUSSION, APPLICATIONS, and FUTURE RESEARCH All of the 83 MLE diameter distributions fit the observed distribution at the probability level <X = .20. The null hypothesis, F(d) = 0(d), cannot be rejected. The model cumulative diameter distribution d = dbh F(d) = cumulative frequency a = f(mean diameter, stand age, trees per acre) b = f(mean diameter, stand age, trees per acre) c = f(mean diameter, height, site index) successfully describes the MLE diameter distributions of the 41 prediction plots. This does not guarentee that the model is sound. When tested against the observed distributions, the model did not fit for two plots. In other cases, bias was observed towards underestimation of the number of small trees. Previous applications of the Weibull to diameter distributions are difficult to compare due to the different statistical tests and the criteria for fit chosen. Shreuder and Swank (1974) compared four distributions by log likelihood (ln L) statistics. Although the Weibull has larger ln L than the other models for six of seven samples all of the ln L are extremely small (ln L — e"^^). No criteria of fit was given by the authors to test the strength of the Weibull as a model for diameter distributions. Clutter and c F(d) = 1.0 - expi-H5 Belcher (1978) gave the coefficients of determination for their prediction equations (a = f(age,height), r = .107; b = f(age,height,trees per acre), 2 2 r = .357; c = f(age), r = .200). They compared predicted and observed mean diameter and basal area per acre. This choice of test shows a concern for average stand descriptors rather than an interest in distributional qualities. Bailey (1973) predicted the percentiles of the two parameter Weibull from age, height, and trees per acre ( r > .95) and then obtained the Weibull parameters from the percentiles. Sixty five percent of the predicted distributions fit the observed diameter distributions at the c< = .05 level (K-S). The level of significance chosen by Bailey is not as exacting as the one used in the present study ( d Q^(N) = 1.36/NTIT ; d 20^) = 1«07//~N). Although the results of this study are not directly comparable with those of other authors, the fits demonstrated here for mixed species stands appear as good if not better than those found for plantations of pine. The primary application for the Weibull diameter distribution model will be in computer simulation of forest stands. It is not necessary for the simulator to "grow" individual trees in order to maintain distributional information at each time interval. The model presented here treats diameter distribution as independent of stand history. If the simulator predicts trees per acre, mean diameter, and mean height for a stand given site index, the diameter distribution of the stand can be generated at any age. This greatly reduces the computation time and storage requirements of the simulator. HQ> The model developed in this thesis describes the diameter distribution of the entire stand. Modeling the diameter distribution of individual species within a mixed species stand may eliminate the bias evident in the model presented in this thesis. Mean diameter, height, and trees per acre may be all that is needed to predict individual distributions. However, it may prove necessary to track the parameters of the Weibull over time for each species in the stand. These questions should be answered through future research. The results reported here are based on data from untreated second growth stands. Silvicultural treatment is intended to have a positive impact on the diameter distribution of the stand. Fertilization may increase diameter growth through an increase in site quality. Thinning directly alters the distribution through selective removal of trees. Fertilization may induce a shift in the population mean diameter; thinning will tend to skew the diameter distribution. The Weibull function is flexible enough to handle such variation. Whether or not the model presented here can reflect the changes induced through silviculture, either by changing input values (site index, mean diameter, trees per acre), or by calibrating the coefficients will have to be answered by further research. HI The diameter distribution model may be useful in inventory updating. Through aerial photograph interpretation, mean height, mean diameter, and trees per acre can be estimated. Site index and age may be obtained from past records. With these data, diameter distributions can be predicted for each stand. This will speed the process of inventory updating and reduce the number of costly field plots needed. The Weibull function can model the diameter distributions of mixed stands of Douglas-fir and western hemlock. Its future will depend on the development of models which accurately predict small diameter classes and which predict separate distributions for individual species groups within a mixed stand. HQ LITERATURE CITED Bailey, Robert L., 1973. Weibull model for Pinus radiata diameter distribution. In Statistics in Forestry Research, I.U.F.R.O. Proc. subgroup S6.02; Vancouver, B.C. pp. 51-19. Bailey, Robert L. and T. R. Dell, 1973. Quantifying diameter distributions with the Weibull function. Forest Science 19:97-104. Bain, Lee J. and Charles E. Antle, 1967. Estimation of parameters in the Weibull distribution. Technometrics 9:621-627. Bliss, C. I. and K. A. Reinker, 1964. A lognormal approach to diameter distributions in even-aged stands. Forest Science 10:350-360. Bogdanoff, David A. and Donald A. Pierce, 1973. Bayes fiducial inference for the Weibull distribution. Journal of American Statistical Association 68:659-664. Clutter, Jerome L. and B. J. Allison, 1973. A growth and yield model for Pinus radiata in New Zealand. In Growth Models for Tree and Stand  Simulation. Proc. I.U.F.R.O. working party S4, 01-4; Ed. J. Fries. Inst Skogsprod., Stockholm. Rapp, Uppsatser No. 30, pp. 136-160. 19 Clutter, Jerome L. and David M. Belcher, 1978. Yield of site prepared slash pine plantations in the lower coase plain of Georgia and Florida. In Growth Models for Long Term Forcasting of Timber Yields, Ed. Joran Fries, et al. Publ. FWS-1-78 School of Forest and Wildlife Resources, VPI and State Univ.; pp. 53-70. Cohen, Clifford A., 1965. Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples. Technometrics 13:171-182. D'Agostino, Ralph B., 1971. Linear estimation of the Weibull parameters. Technometrics 13:171-182. Depta, David J., 1974. Large in-place inventories based upon stand tables In Inventory Design and Analysis, Ed. W. E. Frayer, G. B. Hartman, D. R. Bower. Proc. Workshop of Inventory Working Group, S.A.F. pp. 275-288. Downton, F., 1965. Linear estimates of parameters in the extreme value distribution. Technometrics 8:3-17. Dubey, Satya D., 1967. Some percentile estimates for Weibull parameters. Technometrics 9:119-129. Ek, A. R., J. N. Issos and R. L. Bailey, 1975. Solving for Weibull diameter distribution parameters to obtain specified mean diameters. Forest Science 21:290-292. 50 Englehardt, Max and Lee J. Bain, 1974. Some results on point estimation for the two parameter Weibull or extreme value distribution. Technometrics 16:49-56. Englehardt, Max and Lee J. Bain, 1977. Simplified statistical procedures for the Weibull or extreme value distribution. Technometrics 19:323-331. Finkelstein, J. M., 1976. Confidence bounds on the parameters of the Weibull process. Technometrics 18:115-117. Freund, John E., 1971. Mathematical Statistics, Prentice Hall, Inc. Englewood Cliffs, N.J. pp. 125+ Hafley, W. L. and H. T. Schreuder, 1977. Statistical distributions for fitting diameter and height data in even-aged stands. Canadian Journal of Forest Research 7:481-487. Harter, H. Leon and Albert H. Moore, 1965. Maximum likelihood estimation of the parameters of Gamma and Weibull populations from complete and from censored samples. Technometrics 7:639-643. Johns, M. V. and G. J. Lieberman, 1966. An exact asymptotical efficient confidence bound for reliability in the case of the Weibull distribution. Technometrics 8:135-175. 51 Kao, J. H. K., 1959. A graphical estimation of mixed Weibull parameters in life testing of electron tubes. Technometrics 1:389-407. Lawless, J. F., 1975. Construction of tolerance bounds for the extreme-value and Weibull distributions. Technometrics 17:255-261. Leak, William B., 1965. The J-shaped probability distribution. Forest Science 11:405-411. Leary, Rolph A., Jerold T. Hahn, and Roland G. Buchman, 1977. A generalized forest growth projection system - preliminary research findings. Phase I of Forest Resources Evaluation Program (FREP). Manuscript, North Central Forest Experiment Station, USDA (300 pp.) Lilliefors, Hubert W., 1967. On the Kolmogorov-Smirnov test for normality with mean and variance unknown. American Statistical Assoc. Journal 62:399-402. Mann, Nancy R., 1968. Point and interval estimation procedures for the two-parameter Weibull and extreme-value distributions. Technometrics 10(2):231-256. Mann, Nancy R., Schafer, Ray E., and Singpurwalla, Nozer D.,1974. Methods  of Statistical Analysis for Life Testing. John Wiley, New York. Massey, Frank J., Jr., 1951. The Kolmogorov-Smirnov test for goodness of fit. American Statistical Assoc. Journal 46:68-78. 52 Meyer, Walter H., 1930. Diameter distribution series in even-aged forest stands.Yale Univ. Sch. For. Bull. 28, 105 pp. Mitchell, Kenneth J., 1975. Dynamics and simulated yield of Douglas-fir. Forest Science Monograpoh No. 17, 39 pp. Nelson, Thomas C, 1964. Diameter distribution and growth of loblolly pine. Forest Science 10:105-114. Prodan, Michail, 1968. Forest Biometrics. Pergamon Press. N.Y. pp 39-56. Rustagi, Krishna P., 1978. Predicting stand structure in even-aged stands. In Growth Models for Long Term Forecasting of Timber Yields, Ed. Joran Fries, et aj. Publ. FWS-1-78. School of Forest and Wildlife Resources, VPI and State Univ.; pp. 193-208. Schmidt, J. W. and R. E. Taylor, 1970. Simulation and Analysis of Industrial  Systems. Richard D. Irwin, Inc., Homewood, 111. 644 pp. Schreuder, Hans T. and W. L. Hafley, 1977. A useful bivariate distribution for describing stand structure of tree heights and diameters. Biometrics 33:471-478. Schreuder, Hans T. and Wayne T. Swank, 1974. Coniferous stands characterized with the Weibull distribution. Canadian Journal of Forest Research 4:518-523. 53 Seegrist, Donald W. and Stanford L. Arner, 1978. Statistical analysis of linear growth and yield models with correlated observations from permanent plots remeasured at fixed intervals. In Growth Models for  Long Term Forecasting of Timber Yields. Ed. Joran Fries, et jH. Pub 1. FWS-1-78 School of Forest and Wildlife Resources, VPI; pp. 209-223. Snee, Ronald D., 1977. Validation of regression models: methods and examples. Technometrics 19:415-428. Stauffer, 1977. A derivation for the Weibull Distribution. Univ. of B.C., Faculty of Forestry. Unpublished manuscript. Thoman, D.R., L.J. Bain and C.E. Antle, 1969. Inferences on the parameters of the Weibull distribution. Technometrics 11:445-460. Thomas, David R. and Wanda M. Wilson, 1972. Linear order statistic estimation for the two-parameter Weibull and extreme-value distributions from Type II progressively censored samples. Technometrics 14:679-692. University of Washington, College of Forest Resources, 1976. Regional Forest Nutrition Research Project. Biennial Report 1974-1976. 67 pp. Warren, William G., 1976. Maximum likelihood estimation of the three-parameter Weibull. Forintek, Vancouver, B.C., unpublished. Weibull W., 1955. A statistical distribution function of wide applicability. Journal of Applied Mechanics 18:293-297. 5H Yang, R. C, Kozak, A. and J. H. G. Smith, 1978. The potential of Weibul1-type functions as flexible growth curves. Canadian Journal of Forest Research 8:424-431. APPENDIX I The strength of the regression models depended on the accuracy of the inputs. It was necessary, therefore, to estimate the Weibull parameters of the individual plot diameter distributions from the tree diameter data as accurately as possible. The parameter estimation algorithms developed in the life-testing research were not considered, as they were developed for heavily censored samples, and in most cases were time consuming and inadequate when using complete samples of one hundred or more observations. (For a more thorough discussion, see Mann, et aj_., 1974.) The Englehardt and Bain (1977) algorithm for simplified point estimates was abandoned because the debiasing constant, k, would have to be calculated for each sample, thus rendering the technique more cumbersome than the more accurate maximum likelihood algorithms. Three estimation algorithms, Dubey's (1967) percentile method, Bailey's (1973) maximum likelihood estimator, and Warren's (1977) maximum likelihood method were considered for the estimation of the diameter distributions. These routines, DUBEY (Dubey), FITTER (Bailey), and WINWAR (Warren) were tested and compared in the following manner: Using a random number generator, samples were taken from cumulative Weibull distributions having b and c parameters within the range of those expected for the data set. A number, y (0^y<l), corresponding in this case to a cumulative frequency, was randomly selected from a uniform 5.6 distribution. From the Weibull formula it follows that x = bC-lnd-y))^. • Given y, b, and c, the diameter, x, was calculated. This was repeated until a sample of 100 diameters was obtained. (See Freund (1971) for more detail.) The sample was then run through each of the three-parameter estimating routines, yielding three estimated diameter distributions. The Kolmogorov-Smirnov test for goodness of fit was used to compare these distributions with the original distribution. Critical values, d^N) were given by Massey 1951) such that Pr j™*x|sN(x)-Fo(x)| > d (N)j = oc where F (X) is the theoretical cumulative distribution o S^(X) is an observed cumulative distribution for a sample of N. The difference, d = |SN(X) - F (X)( was calculated for twenty observations. The average maximum d, d, was calculiated for ten samples from each distribution. The d and corresponding standard deviations, s^, are listed in Appendix II. According to Chebyshev's theorem: PR(^x- ker <x<vi+ ko-) ^ 1- 1/k2 -If k = 1/7*, Pr (x < )X +(r/fai., In this case, jx and cr were estimated by d and s^, respectively, so that Pr(d<d + Sj -Ala) > 1- <*• If for (l-a)% of the samples, d<da(N), the estimated distribution has a good fit at the OLlevel of probability. In other words, if d + sg -AR <6JH) the estimated distribution fits the original distrubution at the level. According to the results in Appendix II, all of the curves estimated by FITTER fit at theoc= 0.01 level, (d 01(20) = 0.356). WINWAR estimates for distributions where b = 10 with 3.5 < c 5.0 fit at the oc= .05 level. The remaining WINWAR estimates fit at the ot= 0.01 level. The DUBEY estimates for (b = 10, 3.5 < c < 5.0) did not fit at the<X= 0.20 level of probability. The remainder fit at the 0.05 level. Bailey's FITTER routine was chosen for this study because it gave the best overall fit for distributions within the expected data range, and, unlike WINWAR, it is an unbiased estimator. APPENDIX Hi Comparison of Dubey, Warren (WINWAR), and Bailey (FITTER) estimates for b and c over a range of true values for b and c. with a - 1.0. For each combination of b and c, each estimator used the same random sample (N - 100) from the given population. Twenty observations from the estimated curve were used to calculate d. The Kolmogoror-Smirnov statistic, d, was averaged over ten estimations. 10.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 8d 8d ad sd Sd 8d 8d DUBEY .0806 WINWAR .0334 FITTER .0335 .0337 .0232 .0229 .0906 .0330 .0333 .0392 .0233 .0230 .1034 .0330 .0333 .0424 .0234 .0230 .1179 .0394 .0324 .0440 .0324 .0231 .1334 .0390 .0324 .0450 .0319 .0231 .1497 .0457 .0330 .0458 .0340 .2300 .1664 .0451 .0327 .0456 .0336 .0229 b - 20.0 DUBEY .0640 WINWAR .0345 FITTER .0351 .0290 .0244 .0242 .0666 .0339 .0346 .0337 .0234 .0239 .0709 .0374 .0363 .0379 .0305 .0268 .0753 .0373 .0363 .0419 .0297 .0267 .0813 .0385 .0365 .0446 .0280 .0268 .0877 .0381 .0363 .0476 .0284 .0270 .0943 .0384 .0363 .0504 .0281 .0271 b - 30.0 DUBEY .0410 WINWAR .0381 FITTER .0377 .0296 .0278 .0270 .0418 .0377 .0379 .0327 .0270 .0264 .0435 .0388 .0372 .0353 .0279 .0267 .0462 .0317 .0369 .0377 .0287 .0263 .0498 .0378 .0368 .0393 .0298 .0258 .0545 .0383 .0372 .0396 .0306 .0258 .0597 .0388 .0368 .0402 .0299 .0262 b » 40.0 DUBEY .0432 WINWAR .0386 FITTER .0371 .0275 .0228 .0236 .0430 .0376 .0363 .0259 .0238 .0241 .0436 .0377 .0362 .0248 .0242 .0243 .0477 .0341 .0327 .0237 .0236 .0241 .0460 .0350 .0329 .0234 .0236 .0212 .0439 .0384 .0369 .0366 .0303 .0261 .0461 .0386 .0371 .0384 .0300 .0261 03> APPENDIX III Graphical illustration of the observed, estimated, and predicted diameter distributions of some typical stands in the prediction set. Figure 1. Observed, estimated, and predicted diameter distributions of stand noo4 O • 5 . LO IS SO Diameter (inches) Figure 2. Observed, estimated, and predicted diameter distributions of stand 71041 Diameter (inches) 1/ estimated from observed distribution by maximum likelihood 2/ predicted from stand characteristics by model a(2) b(10) c(4) Figure 3. Observed, estimated, and predicted diameter distributions of stand 161091 Diameter (inches) Figure 4. Observed, estimated, and predicted diameter distributions of stand 191114 Diameter (inches) 1/ estimated from observed distribution by maximum likelihood 2/ predicted from stand characteristics by model a(2) b(10) c(4) Figure 5. Observed, estimated, and predicted diameter distributions of stand 331194 Diameter (inches) Figure 6. Observed, estimated, and predicted diameter distributions of stand 361216 Diameter (inches) 1/ estimated from observed distribution by maximum likelihood 2/ predicted from stand characteristics by model a(2) b(10) c(4) Figure 7. Observed, estimated, and predicted diameter distributions of stand 431254 O S 1 CD IS 20 25 Diameter (inches) Figure 8. Observed, estimated, and predicted diameter distributions of stand 451266 o s LO is 20 Diameter (inches) 1/ estimated from observed distribution by maximum likelihood dj predicted from stand characteristics by model a(2) b(10) c(4) Figure 9. Observed, estimated, and predicted diameter distributions of stand 541324 Diameter (inches) Figure 10. Observed, estimated, and predicted diameter distributions of stand 541324 i. o r— Diameter (inches) 1/ estimated from observed distribution by maximum likelihood 2/ predicted from stand characteristics by model a(2) b(10) c(4) <o5 Figure 11. Observed, estimated, and predicted diameter distributions of stand 551330 Diameter (inches) Figure 12. Observed, estimated, and predicted diameter distributions of stand 591354 i. O i— o c CT 0) > o.e J— (D.S. (— o.o 1 o is Diameter (inches) 20 1/ estimated from observed distribution by ^i^.Ji^^ood 1/ predicted from stand characteristics by model a(2) b(10). c(4} &6 Figure 13. Observed, estimated, and predicted diameter distributions of stand 691410 Diameter (inches) Figure 14. Observed, estimated, and predicted diameter distributions of stand 791472 >-o o.e QJ CT <D > ro O. •<* 3 0.2 O.O observed , , estimated^ predicted-• • • i i i —i——i—i i o i s Diameter (inches) 1/ estimated from observed distribution by ""i1^."1*}.1!!?0^-, il predicted from stand characteristics by model a(2) b(10) d4) 67 Figure 15. Observed, estimated, and predicted diameter distributions of stand 821492 O S . 1 O 15 20 25 Diameter (inches) Figure 16 . Observed, estimated, and predicted diameter distributions of stand 831498 Diameter (inches) 1/ estimated from observed distribution by maximum likelihood J/ predicted from stand characteristics by model a(2) b(10) c(4) Figure 17. Observed, estimated, and predicted diameter distributions of stand 991592 Diameter (inches) Figure 18. Observed, estimated, and predicted diameter distributions of stand 1101658 o o. c at cr QJ S- o. u. > •r-4-> _ ro O. O. 2 O.O I i i_ IO IS Diameter (inches) observed 1/ estimated^ predicted— _i i i_ 20 1/ estimated from observed distribution by maximum likelihood 27 predicted from stand characteristics by model a(2) b(10) c(t) 

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