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1 FUNDAMENTAL RESEARCH For. Sc. 60(4): Copyrght 2014 Socety of Amercan Foresters bometrcs Dervng a Dameter Dstrbuton from St Table Data Quang V. Cao Dean W. Coble St tables lst numbers of trees per unt area for each dameter class. A st table projecton model, whch projects a st table through tme, requres knowledge of the dstrbuton of tree dameters n each class so that t can predct the movements of trees from one dameter class nto another. In ths study, we evaluated dfferent statstcal dstrbutons for modelng st table dameter data. The dstrbutons were unform, truncated segmented Webull, truncated segmented S B. Results ndcated that the truncated Webull wth parameters obtaned from the least-squares method was overall the best method. Keywords: truncated dstrbuton, segmented dstrbuton, unform dstrbuton, Webull, S B Forest management decsons are based on nformaton provded by growth--yeld models, whch ranged from smple whole-st models, to sze-class dstrbuton models, fnally to complcated ndvdual-tree smulaton models. Sze-class dstrbuton models can be based on a probablty densty functon such as the beta (Clutter Bennett 1965), the Webull (Baley Dell 1973), Johnson s S B dstrbuton (Hafley Schreuder 1977), or a mxture of more than one dstrbuton (Zhang et al. 2003, Podlask 2010). Qn et al. (2007) dscussed dfferent approaches to project a dameter dstrbuton through tme. Because nventory data are typcally summarzed nto st tables, whch lst the number of trees per unt area for each dameter class, another type of sze-class dstrbuton method based on st tables s often used. Ths method, called the st table projecton method, predcts a future st table based on the current st table. When projected nto the future, trees n a dameter class wll ncrease n sze, as a result, span more than one dameter class. Some trees n the current dameter class wll stay n that class, whereas others wll move up one or more dameter classes. Knowledge of the dstrbuton of tree dameters n each class should enable us to model these movements of trees from one class nto another. Tradtonal st table projecton methods assume a unform dstrbuton wthn each dameter class (Penaar Harrson 1988, Avery Burkhart 2002, Husch et al. 2003). Nepal Somers (1992) ntroduced a method n whch a truncated Webull was used to descrbe the dstrbuton of trees n each dameter class. Ths approach was later adopted by other researchers (Cao Baldwn 1999a, 1999b, Cao 2007, Allen et al. 2011). To date, no study has been conducted to determne whether the unform, truncated Webull, or other dstrbutons would be most approprate for fttng the st table data. The objectves of ths study were to nvestgate the use of other statstcal functons for characterzng dameter dstrbuton, gven number of trees n each dameter class, to evaluate the dfferent methods for characterzng dameter dstrbutons from st table data contanng 2-4-cm dameter classes, to evaluate the dfferent methods n projectng the st tables nto the future. Data Loblolly pne (Pnus taeda L.) data from the Southwde Seed Source Study (Wells Wakeley 1966) were used n ths study. A sample of 100 plots was romly selected from the orgnal data. Tree dameter at breast heght (dbh) n each plot was measured at ages 10, 15 (or 16), 20 (or 22), 25 (or 27) years (Table 1). Numbers of trees n each 2-cm class were totaled for the begnnng end of each of the 300 growth perods. The process was repeated for 4-cm dameter classes. Slash pne (Pnus ellott Engelm.) data from the East Texas Pne Plantaton Research Project (Allen et al. 2011) were also used n ths study. Dbh was measured on 78 plots that ranged n age from 9 to 40 years (Table 1). Smlar to the loblolly pne data, the numbers of trees n each 2-4-cm class were totaled for the begnnng end of each of the 387 growth perods. Manuscrpt receved November 21, 2012; accepted July 19, 2013; publshed onlne September 19, Afflatons: Quang V. Cao (qcao@lsu.edu), Lousana State Unversty, Agrcultural Center, School of Renewable Natural Resources, Baton Rouge, LA. Dean W. Coble (dcoble@sfasu.edu), Stephen F. Austn State Unversty, Arthur Temple College of Forestry Agrculture. Acknowledgments: Fundng for ths project was provded n part by the McIntre-Stenns funds. Fundng for the slash pne data was provded by the East Texas Pne Plantaton Research Project. 628 Forest Scence August 2014

2 Table 1. Age Range of tree varables, by age. No. of plots No. of trees/plot Dbh (cm) Total heght (m) Loblolly pne 10 yr yr yr yr yr yr yr Slash pne 7 yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr yr Method Suppose that x x 1 are the upper lower lmts, respectvely, of the th dameter class F F 1 are correspondng emprcal cumulatve dstrbuton functons (cdfs). The goal was to connect the F values by use of a smooth monotonc nondecreasng functon. Cdfs are a natural choce for ths task. The statstcal dstrbutons nvestgated n ths study nclude the unform, truncated segmented Webull, truncated segmented S B dstrbutons. Unform Dstrbuton When a hstogram s used to dsplay a dameter dstrbuton, the mplct assumpton s that tree dameters n each class follow a unform (or rectangular) dstrbuton. In other words, all trees n the same dameter class have the same probablty of occurrence. The cumulatve dstrbuton functon of X, a dameter rom varable, for the th dameter class s Fˆ x F 1 F F 1 x x x x 1, x 1 x x, (1) 1 where x 1 x are the lower upper lmts of the th dameter class, respectvely. The correspondng probablty densty functon (pdf) of X for the th dameter class s fˆx F F 1 x x x 1 x x. (2) 1, Truncated Webull Dstrbuton Nepal Somers (1992) assumed that all tree dameters n each plot follow a general Webull dstrbuton. The Webull functon (Baley Dell 1973) has been extensvely used to characterze dstrbuton of tree dameters n forest sts (McTague Baley 1987, Kangas Maltamo 2000, Zhang et al. 2003, Newton et al. 2005). The cdf of ths Webull functon s Gx 1 exp x a b c, x a, (3) ts correspondng pdf s gx b c x a b c1exp x a c b, x a, (4) where a x 0 s the Webull locaton parameter, x 0 s the lower lmt of the smallest dameter class, b c are the scale shape parameters, respectvely. Nepal Somers (1992) further assumed that tree dameters n each class follow a dfferent truncated verson of the general Webull dstrbuton. The cdf of ths truncated Webull dstrbuton for the th dameter class s F Fˆ x F 1 F 1 G x G x G x 1 Gx 1, ts correspondng pdf s F fˆx F 1 G x G x 1 x 1 x x, (5) gx, x 1 x x. (6) The scale shape parameters (b c, respectvely) of the general Webull dstrbuton are unknown can be estmated by means of the method of moments,.e., recovered from the arthmetc quadratc mean dameters (Nepal Somers 1992). These parameters can also be estmated by use of the maxmum lkelhood estmaton (MLE) method to maxmze the lkelhood functon or the least-squares (LS) method to mnmze the squared dfference between the observed predcted cdfs. The LS method s smlar to the cdf regresson technque proposed by Cao (2004). All three methods are ncluded n the evaluaton. Truncated S B Dstrbuton Ths approach s smlar to the one above, except that the S B dstrbuton replaces the Webull as the general dameter dstrbuton. Johnson s S B dstrbuton (Johnson 1949) has recently been used to model dameter dstrbutons (Splehto 1999, Zhang et al. 2003, Rennolls Wang 2005, Palahí et al. 2007). The S B cdf s Gx ln x 1 2 x, 1 x 2, (7) where () s the stard normal cdf, 1 x 0 s the mnmum value of x, 2 x m s the maxmum value of x, x m s the upper lmt of the largest dameter class, m s the number of dameter classes n the Forest Scence August

3 Fgure 1. Probablty densty functons derved from tree dameters from a sample plot. The dstrbutons are unform (A), truncated Webull (moments) (B), truncated Webull (LS) (C), truncated S B (D), segmented Webull (E), segmented S B (F). The graph for the truncated Webull (MLE) s smlar to those n B C. plot, are unknown shape parameters. The S B probablty densty functon s gx 2 1 x 1 2 x2 exp 1 2 ln x x, 1 x 2. (8) Tree dameters n the th dameter class follow the followng truncated S B cdf F Fˆ x F 1 F 1 G x G x G x 1 Gx 1, ts correspondng pdf s F fˆx F 1 G x G x 1 x 1 x x, (9) gx, x 1 x x. (10) The two shape parameters,, can be obtaned from the percentle method that nvolves the 50th 95th percentles (Knoebel Burkhart 1991), the MLE method, or the LS method, whch mnmzes the squared dfference between the observed predcted cdfs. Prelmnary analyss revealed that the performance of the percentle MLE methods was much worse than that of the LS method; therefore, these two methods were dropped from the evaluaton. Segmented Webull Dstrbuton Cao Burkhart (1984) proposed the jonng of dfferent functons, each n the form of a modfed Webull cdf, to form a segmented cdf to approxmate dameter dstrbutons of forest sts, especally those that were rregular as n thnned or mxed sts. A smlar approach was appled n ths study to use a segment of dfferent Webull cdfs for each dameter class then jon them together to form a segmented Webull dstrbuton. The cdf for the th dameter class s Fˆ x 1 exp x a ts correspondng pdf s fˆx c x a c c 1exp x a ; x 1 x x, (11) c, x 1 x x. (12) Note that the Webull locaton parameter, a, s kept constant at x 0 (the lower lmt of the smallest dameter class), whereas the scale shape parameters for the th dameter class, c, vary from 630 Forest Scence August 2014

4 Table 2. Evaluaton statstcs of the seven methods for the current sts ther relatve ranks (n parentheses) for 2-4-cm dameter classes, by speces. Method Kolmogorov-Smrnov Anderson-Darlng R 2 ln(l) Loblolly pne 2-cm dameter classes (n 300) Unform (3.74) (1.00) (2.01) 13,902 (1.81) Truncated Webull (moments) (5.51) (1.42) (1.34) 13,816 (1.00) Truncated Webull (MLE) (1.00) (3.02) (1.00) 13,893 (1.71) Truncated Webull (LS) (4.18) (2.34) (1.13) 13,885 (1.65) Truncated S B (LS) (4.84) (1.96) (3.99) 13,912 (1.89) Segmented Webull (7.00) (3.20) (5.27) 13,912 (1.90) Segmented S B (5.79) (7.00) (7.00) 14,461 (7.00) 4-cm dameter classes (n 300) Unform (5.96) (1.88) (3.42) 14,944 (6.69) Truncated Webull (moments) (7.00) (1.00) (2.17) 14,670 (1.98) Truncated Webull (MLE) (1.00) (7.00) (7.00) 14,820 (4.56) Truncated Webull (LS) (2.50) (2.02) (1.06) 14,621 (1.13) Truncated S B (LS) (4.34) (2.15) (2.31) 14,683 (2.21) Segmented Webull (3.77) (1.50) (1.00) 14,613 (1.00) Segmented S B (3.88) (5.52) (1.69) 14,962 (7.00) Slash pne 2-cm dameter classes (n 387) Unform (5.33) (2.50) (7.00) 68,767 (4.09) Truncated Webull (moments) (7.00) (2.32) (5.30) 68,530 (1.91) Truncated Webull (MLE) (1.00) (5.50) (4.52) 68,733 (3.77) Truncated Webull (LS) (1.80) (3.21) (1.26) 68,610 (2.64) Truncated S B (LS) (1.23) (1.00) (1.00) 68,432 (1.00) Segmented Webull (4.76) (3.64) (2.17) 68,511 (1.72) Segmented S B (3.26) (7.00) (2.37) 69,083 (7.00) 4-cm dameter classes (n 387) Unform (7.00) (6.22) (6.03) 71,450 (7.00) Truncated Webull (moments) (5.59) (3.92) (4.21) 70,459 (3.71) Truncated Webull (MLE) (5.80) (7.00) (7.00) 70,703 (4.52) Truncated Webull (LS) (1.00) (1.24) (1.00) 69,793 (1.50) Truncated S B (LS) (1.25) (1.00) (1.17) 69,642 (1.00) Segmented Webull (1.58) (1.44) (1.13) 69,778 (1.45) Segmented S B (1.61) (2.31) (1.38) 70,522 (3.92) one dameter class to the next. The constrant Fˆ 1 (x 1 ) Fˆ (x 1 ) F 1 s mposed so that the resultng cdf s contnuous at all jon ponts. Consequently, c are computed as follows ln ln1 F ln1 F 1 c lnx a lnx 1 a, (13) x a ln1 F 1/c. (14) A drawback of ths method s that a mnmum of three dameter classes s requred. Segmented S B Dstrbuton Smlar to the segmented Webull dstrbuton, a segment of dfferent S B cdfs s used to descrbe a dameter dstrbuton for each dameter class. These segments are then joned together to form a segmented S B dstrbuton. The cdf for the th dameter class s Fˆ x ln x 1 2 x, x 1 x x, (15) ts correspondng pdf s fˆx 2 1 x 1 2 x2 exp 1 2 ln x x, x 1 x x. (16) The parameters 1 2 are set at the mnmum maxmum dameters (x 0 x m, respectvely) kept constant over the segments. The shape parameters for the th dameter class,, are computed as follows z z 1 ln x 1 2 x ln x x 1, (17) z ln x 1 2 x, (18) where z ln[(x )/( 2 x )] z 1 ln[(x 1 1 )/( 2 x 1 )]. Smlar to the segmented Webull, the segmented S B requres a mnmum of three dameter classes. Evaluaton The seven methods were ftted to the st table (2-4-cm classes) of the current sts but were evaluated on the orgnal dameter measurements. Four goodness-of-ft statstcs were used; Forest Scence August

5 the lower the value of these statstcs, the better the ft. The four evaluaton statstcs were the followng 1. The average one-sample Kolmogorov-Smrnov (KS) statstc (Massey 1951) KS 1 N 1 N KS, (19) where KS max{max 1jn (j/n u j ), max 1jn [u j (j 1)/n ], n s the number of trees n the th plot-age combnaton, u j Fˆ (x j ), N s the total number of plot-age combnatons. 2. The average Anderson-Darlng (AD) statstc (Anderson Darlng 1954) AD 1 N 1 N AD, (20) n where AD n j1 (2j 1)[ln(u j ) ln(1u n j1]/n. 3. The R 2 value N R j1 N 1 n F x j Fˆ x j 2, (21) n F x j F 2 j1 where F (x j ) Fˆ (x j ) are the observed predcted cdf, respectvely, of the jth dameter n the th class, N n 1 F x j j1 F. N n 1 4. The negatve log-lkelhood statstc lnl N n lnf ˆxj. (22) 1 j1 In addton, the seven methods were ncorporated nto the st table projecton system developed by Nepal Somers (1992). The methods were then evaluated on the bass of ther ablty to predct dameters of the future sts. Two forms of error ndex (EI) proposed by Reynolds et al. (1988) were used to quantfy goodness of ft m EI N nk nˆk, (23) k1 m EI B bk bˆk, (24) k1 where EI N EI B are error ndces based on number of trees/ha basal area/ha for the th plot, respectvely, n k nˆk are the observed predcted numbers of trees/ha of the kth dameter class n the th plot, respectvely, k bˆk are the observed predcted numbers of trees/ha of the kth dameter class n the th plot, respectvely, m s the number of dameter classes n the th plot. Poudel Cao (2013) stated that the tradtonal ordnal ranks for m methods (1, 2,, m) show the order of the methods but fal to depct the exact postons of the methods compared wth one Table 3. Mean error ndces of the seven methods for the future sts ther relatve ranks (n parentheses) for 2-4-cm dameter classes, by speces. Mean error ndex based on Basal area Method No. trees/ha (m 2 /ha) Loblolly pne 2-cm dameter classes (n 300) Unform 499 (2.67) (4.09) Truncated Webull (moments) 490 (1.00) (1.41) Truncated Webull (MLE) 492 (1.39) (1.00) Truncated Webull (LS) 491 (1.31) (1.05) Truncated S B (LS) 516 (5.53) (5.88) Segmented Webull 524 (7.00) (7.00) Segmented S B 506 (3.78) (6.13) 4-cm dameter classes (n 300) Unform 336 (3.10) (3.33) Truncated Webull (moments) 309 (1.00) 9.73 (1.00) Truncated Webull (MLE) 324 (2.18) (1.99) Truncated Webull (LS) 317 (1.61) 9.78 (1.19) Truncated S B (LS) 388 (7.00) (7.00) Segmented Webull 350 (4.14) (3.90) Segmented S B 330 (2.60) (2.56) Slash pne 2-cm dameter classes (n 387) Unform 171 (3.25) 4.82 (5.72) Truncated Webull (moments) 167 (1.84) 4.74 (3.22) Truncated Webull (MLE) 167 (1.67) 4.70 (1.99) Truncated Webull (LS) 165 (1.00) 4.66 (1.00) Truncated S B (LS) 179 (6.18) 4.87 (7.00) Segmented Webull 182 (7.00) 4.83 (5.88) Segmented S B 167 (1.80) 4.70 (1.98) 4-cm dameter classes (n 300) Unform 149 (5.50) 4.03 (5.85) Truncated Webull (moments) 129 (3.35) 3.59 (3.58) Truncated Webull (MLE) 114 (1.67) 3.19 (1.52) Truncated Webull (LS) 108 (1.00) 3.09 (1.00) Truncated S B (LS) 162 (7.00) 4.25 (7.00) Segmented Webull 161 (6.85) 3.85 (4.96) Segmented S B 111 (1.30) 3.16 (1.37) another. They proposed a new method of rankng, n whch method s gven the followng relatve rank R 1 m 1S S mn, (25) S max S mn where R s the relatve rank of method ( 1, 2,, m), m s the number of methods (equal to 7 for ths study), S s the goodnessof-ft statstc produced by method, S mn s the mnmum value of S, S max s the maxmum value of S. In ths relatve rankng system, the best the worst methods have relatve ranks of 1 m, respectvely. Ranks of the remanng methods are expressed as real numbers between 1 m, thus the magntude not only the order of the S values s taken nto consderaton. Results Dscusson Fgure 1 shows dfferent probablty densty functons derved from a st table from a sample plot. For ths example, the shapes of the truncated Webull S B are smlar (except for the largest dameter class), whereas the shapes of the segmented Webull S B are smlar (except for the largest dameter class). Evaluaton statstcs of the seven methods ther relatve ranks for 2-4-cm dameter classes are shown for the loblolly slash 632 Forest Scence August 2014

6 Table 4. Overall ranks for the seven methods. Loblolly pne Slash pne 2-cm classes 4-cm classes 2-cm classes 4-cm classes Overall Current Future Current Future Current Future Current Future Total rank Unform Truncated Webull (moments) Truncated Webull (MLE) Truncated Webull (LS) Truncated S B Segmented Webull Segmented S B pne data sets, based on dameters of the current sts (Table 2) future sts (Table 3). The sum of the relatve ranks for each method was used to determne the overall relatve rank for that method (Table 4). Loblolly Pne Data Table 4 shows that the truncated Webull (MLE) was the best performer for the 2-cm class, wth ranks of 1.00 (current sts) 1.01 (future sts). It was closely followed by the truncated Webull (LS), whch ranked for the current future sts, respectvely. The unform dstrbuton dd relatvely well, rankng The bottom two methods were the segmented Webull S B. For the 4-cm dameter classes, the truncated Webull (MLE) was last n predctng current dameters, whereas the truncated S B was last n predctng future dameters. The best overall for ths group was the truncated Webull (LS), wth relatve ranks of for the current future sts, respectvely. The unform dstrbuton dd not perform well for the current sts (rankng 6.25) but was better for predctng dameters of the future sts (rankng 3.21). Slash Pne Data For the 2-cm classes, the truncated Webull (LS) was the most consstent method wth relatve ranks of for the current futures sts, respectvely (Table 4). The unform dstrbuton dd poorly ranked last for ths dameter wdth. The truncated Webull (LS) was the clear wnner for the 4-cm dameter classes, rankng n predctng dameters of the current future sts, respectvely. The segmented S B was second, the unform dstrbuton agan was last. Predctng Dameters of Current Future Sts Common sense dctates that f a method does well n representng dameter dstrbutons of the current sts, t should do equally well n predctng dameter dstrbutons of the future sts. Results from Table 4 shows that the rankngs for the current future sts dd not necessarly go h-n-h. The truncated S B the segmented Webull predcted dameters of the current sts better than those of the future sts. The reverse was true for the truncated Webull (moments) the truncated Webull (MLE). Results were mxed for the rest of the dstrbutons. Even though the same st table projecton system (Nepal Somers 1992) was used to project dameters nto the future, the dfference n the relatve rankngs before after the growth perod ndcates that the projecton system s a complcated process that ts outcome can be unpredctable. Dameter Class Wdth When the class wdth ncreases from 2 to 4 cm, the change n relatve rankngs was generally neglgble (Table 4). The exceptons for the loblolly pne data were worse rankngs produced by dstrbutons such as the unform, truncated Webull (MLE), truncated S B. For the slash pne data, the two segmented dstrbutons (Webull S B ) mproved ther rankngs as the class wdth ncreases from 2 to 4 cm. Dfferent Results from Data Sets For the most part, results from the loblolly the slash pne data sets were smlar. The dfference n relatve rankngs for some methods mght be caused by the varatons among data sets rather than by the dfference n speces. Overall, the truncated Webull (LS) was the best performer for both loblolly slash pne data. The segmented S B the unform dstrbuton were the worst methods for the loblolly pne slash pne data, respectvely. Whch Dstrbuton to Use? The unform dstrbuton s the smplest method the mplct dstrbuton used n straghtforward st table projecton models (Avery Burkhart 2002, Husch et al. 2003). However, ts overall performance ranked last compared wth those for the rest of the methods, ndcatng that ths smple dstrbuton was not adequate to model dameter dstrbutons from st table data. The S B dstrbuton has never been used n a st table projecton model. In ths study, the truncated segmented S B receved overall rankngs of , respectvely, whch were near the bottom of the seven methods (Table 4). The poor performance of the S B dstrbuton mght be because ts lower upper lmts were arbtrarly set at the smallest largest dameter classes, respectvely. The segmented Webull dstrbuton receved an overall relatve rank of 6.18, next to last among the seven methods. Because a dfferent set of Webull parameters are used for each segment (dameter class), the segmented Webull should be flexble to hle dstrbutons wth varatons among dameter classes. However, ts projectons of dameters of the future sts were poorer than those from the remanng methods, even though t predcted the current sts relatvely well. The verson of the truncated Webull dstrbuton used n many recent st projecton models (Nepal Somers 1992, Cao Baldwn 1999a, 1999b, Cao 2007, Allen et al. 2011) was based on the method of moments. It was found to be just a medocre performer, rankng 3.79 overall (Table 4). The MLE verson receved an overall relatve rank of The LS verson of the truncated Webull dstrbuton was the best most consstent among the Forest Scence August

7 seven methods. Whereas the moments approach s based on pont estmates,.e., the dameter moments, the am of the LS approach s to optmze the entre dstrbuton by mnmzng the squared dfference between the observed predcted cumulatve probabltes. Table 4 shows three dstnct groups of methods, based on ther overall relatve ranks. The truncated Webull (LS) was clearly the wnner, performng well n predctng dameters of both current future sts for both data sets. Group 2 ncluded the moments MLE versons of the truncated Webull dstrbuton. The last group, made up by the segmented Webull S B, truncated S B, unform dstrbutons, performed poorly should not be used for st projecton. Lterature Cted ALLEN, M.G. II, D.W. COBLE, Q.V. CAO, J. YEISER, AND I. HUNG A modfed st table projecton growth model for unmanaged loblolly slash pne plantatons n east Texas. South. J. Appl. For. 35: ANDERSON, T.W., AND D.A. DARLING A test of goodness of ft. J. Am. Stat. Assoc. 49: AVERY, T.E., AND H.E. BURKHART Forest measurements, 5th ed. McGraw-Hll, New York. 456 p. BAILEY, R.L., AND T.R. DELL Quantfyng dameter dstrbutons wth the Webull functon. For. Sc. 19: CAO, Q.V Predctng parameters of a Webull functon for modelng dameter dstrbuton. For. Sc. 50: CAO, Q.V Incorporatng whole-st ndvdual-tree models n a st-table projecton system. For. Sc. 53: CAO, Q.V., AND V.C. BALDWIN JR. 1999a. A new algorthm for st table projecton models. For. Sc. 45: CAO, Q.V., AND V.C. BALDWIN JR. 1999b. Projectng a st table through tme. P n Proc. of the Southern slvculture research conference. USDA For. Serv., Gen. Tech. Rep. SRS-30, Ashevlle, NC. 618 p. CAO, Q.V., AND H.E. BURKHART A segmented dstrbuton approach for modelng dameter frequency data. For. Sc. 30: CLUTTER, J.L., AND F.A. BENNETT Dameter dstrbuton n old-feld slash pne plantatons. Georga For. Res. Councl, Rep. 13, Athens, GA. 9p. HAFLEY, W.L., AND H.T. SCHREUDER Statstcal dstrbutons for fttng dameter heght data n even- aged sts. Can. J. For. Res. 7: HUSCH, B., T.W. BEERS, AND J.A. KERSHAW JR Forest mensuraton, 4th ed. John Wley & Sons, Hoboken, NJ. 443 p. JOHNSON, N.L Systems of frequency curves generated by methods of translaton. Bometrka 36: KANGAS, A., AND M. MALTAMO Performance of percentle based dameter dstrbuton predcton Webull method n ndependent data sets. Slva Fenn. 34: KNOEBEL, B.R., AND H.E. BURKHART Bvarate dstrbuton approach to modelng forest dameter dstrbutons at two ponts n tme. Bometrcs 47: MASSEY, F.G. JR The Kolmogorov-Smrnov test for goodness of ft. J. Am. Stat. Assoc. 46: MCTAGUE, J.P., AND R.L. BAILEY Compatble basal area dameter dstrbuton models for thnned loblolly pne plantatons n Santa Catarna. Braz. For. Sc. 33: NEPAL, S.K., AND G.L. SOMERS A generalzed approach to st table projecton. For. Sc. 38: NEWTON, P.F., Y. LEIB, AND S.Y. ZHANG St-level dameter dstrbuton yeld model for black spruce plantatons. For. Ecol. Manage. 209: PALAHÍ, M., T. PUKKALA, E. BLASCO, AND A. TRASOBARES Comparson of beta, Johnson s S B, Webull truncated Webull functons for modelng the dameter dstrbuton of forest sts n Catalona (north-east of Span). Eur. J. For. Res. 126: PIENAAR, L.V., AND W.M. HARRISON A st table projecton n unthnned even-aged sts. For. Sc. 34: PODLASKI, R Two-component mxture models for dameter dstrbutons n mxed-speces, two-age cohort sts. For. Sc. 56: POUDEL, K.P., AND Q.V. CAO Evaluaton of methods to predct Webull parameters for characterzng dameter dstrbutons. For. Sc. 59: QIN, J., Q.V. CAO, AND D.C. BLOUIN Projecton of a dameter dstrbuton through tme. Can. J. For. Res. 37: RENNOLLS, K., AND M. WANG A new parameterzaton of Johnson s S B dstrbuton wth applcaton to fttng forest tree dameter data. Can. J. For. Res. 35: REYNOLDS, M.R., T.E. BURK, AND W.C. HUANG Goodness-of-ft tests model selecton procedures for dameter dstrbuton models. For. Sc. 34: SIIPILEHTO, J Improvng the accuracy of predcted basal-area dameter dstrbuton n advanced sts by determnng stem number. Slva Fenn. 33: WELLS, O.O., AND P.C. WAKELEY Geographc varaton n survval, growth, fusform rust nfecton of planted loblolly pne. Socety of Amercan Foresters, For. Sc. Monogr. 11, Bethesda, MD. 40 p. ZHANG, L., C.P. KEVIN, AND L. CHUANMIN A comparson of estmaton methods for fttng Webull Johnson s S B dstrbutons to mxed spruce-fr sts n northeastern North Amerca. Can. J. For. Res. 33: Appendx Below are the dervatons for estmatng parameters of the segmented Webull S B dstrbutons. Segmented Webull Dstrbuton The cdf for the th dameter class passes through F, resultng n whch can be rewrtten as F Fˆ x 1 exp x a x a c, (A1) c ln1 F. (A2) Smlarly, the followng equaton shows that the cdf for the th dameter class also passes through F 1 x 1 a c ln1 F 1. (A3) Dvdng A2 by A3 gves x c a x 1 a ln1 F ln1 F 1. The parameters c are then computed as follows ln ln1 F ln1 F 1 c lnx a lnx 1 a, (A4) (A5) 634 Forest Scence August 2014

8 x a ln1 F 1/c. (A6) Smlarly, the same cdf for the th dameter class also passes through F 1, leadng to Segmented S B Dstrbuton The S B cdf for the th dameter class (Equaton 15) s based on a segment of the stard normal cdf. Let z ln[(x 1 )/ ( 2 x)] be the correspondng stard normal rom varable. The segmented S B cdf for the th dameter class passes through F, resultng n Therefore, F Fˆ x ln x 1 2 x z. (A7) z 1 1 F 1 ln x x 1. (A9) The shape parameters for the th dameter class,, are obtaned from A8 A9 as follows z z 1 ln x 1 2 x ln x x 1, (A10) z 1 F ln x 1 2 x. (A8) z ln x 1 2 x. (A11) Forest Scence August