Evaluation of three methods for estimating the Weibull distribution parameters of Chinese pine (Pinus tabulaeformis)

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1 JOURNAL OF FOREST SCIENCE, 54, 2008 (12): Evaluato of three methods for estmatg the Weull dstruto parameters of Chese pe (Pus taulaeforms) Y. Le Researh Isttute of Resoure Iformato ad Tehques, Chese Aademy of Forestry, Bejg, Cha ABSTRACT: Weull dstruto was used to ft tree dameter data olleted from 86 sample plots loated Chese pe stad Bejg. To estmate the Weull dstruto parameters, three methods [amely maxmum lkelhood estmato method (MLE), method of momet (MOM) ad least-squares regresso method (LSM)] were ompared ad evaluated o the ass of the mea square error (MSE) ad sample sze. For these sample plots, the momet method was superor for estmatg the parameters of Weull dstruto for tree dameter dstruto. Keywords: Weull dstruto; dameter dstruto; parameter estmato Tree dameter dstrutos play a mportat role stad modellg. A umer of dfferet dstruto futos have ee used to model dameter dstrutos, ludg Beta, Logormal, Johso s S, ad Weull oes. The Weull dstruto, trodued y Baley ad Dell (1973) as a model for dameter dstrutos, has ee appled extesvely forestry due to (1) ts alty to desre a wde rage of umodal dstrutos ludg reversed-j shaped, expoetal, ad ormal frequey dstrutos, (2) the relatve smplty of parameter estmato, ad (3) ts losed umulatve desty futoal form (e.g. Baley, Dell 1973; Shreuder, Swak 1974; Shreuder et al. 1979; Lttle 1983; Reolls et al. 1985; Mavurra et al. 2002), ad (4) ts prevous suess desrg dameter frequey dstrutos wth oreal stad types (e.g. Baley, Dell 1973; Lttle 1983; Klkk et al. 1989; Lu et al. 2004; Newto et al. 2004, 2005). It s mportat that dfferet estmato methods are ompared to ft parameters of the Weull proalty desty futo (PDF) from gve tree dameter reast heght (dh) data forest vetory eause the estmate parameters play a major role developg a stad-level dameter dstruto yeld model ased o stad varales employg the parameter predto method,.e. expressg the parameters of a proalty desty futo (PDF) haraterzg the dameter frequey dstruto as a futo of stad-level varales (Hyk, Moser 1983). Therefore, may other methods have ee proposed to estmate the parameters of Weull PDF dstruto forestry, suh as the maxmum lkelhood estmato (MLE), the peretle estmato (PCT), ad the method of momet (MOM) estmato. MLE s geerally osdered the est as t s asymptotally the most effet method, ad thus t s the most frequetly used method to estmate parameters of dstrutos. However, the MLE does ot exst ases where the lkelhood futo a e made artrarly large. Ths ours, for example, to dstrutos whose rage depeds o ther parameters, suh as the three-parameter Weull dstruto as we foud our smulato study. Some other methods have ee proposed to estmate the parameters of the Weull dstruto, suh as the ME, the PCT ad the least-squares method (LSM). Zaroh ad Dell (1985) om- The author s very grateful to MOST for ts support of ths work through Projet 2006BAD23B02 ad to the Ivetory Isttute of Bejg Forestry for ts data. 566 J. FOR. SCI., 54, 2008 (12):

2 pared the Weull dstruto estmato methods of oth PCT ad MLE ased o the mea square error (MSE) whh there s a dfferee etwee the estmate ad the true value of the parameter. They foud that the MLE s superor auray ad has a smaller MSE ompared wth the PCT. Shver (1988) evaluated three-parameter estmate methods (MLE, PCT ad MOM) of the Weull dstruto uthed slash pe platatos ased o the MSE ad the oluso supports the results of Zaroh ad Dell (1985). The LSM has osstetly ee foud to e superor for estmatg the parameters of S dstruto (Zhou, MTague 1996; Kamzah et al. 1999; Zhag et al. 2003) forestry applatos, ut the LSM s used very lttle for estmatg the parameters of Weull dstruto forestry applatos. The LSM provdes alteratves to the MLE ad MOM. Addtoally, ths method has a advatage omputato that most of the statstal software pakages urretly avalale (S-Plus, SAS, SPSS, ) support the least-squares estmato ut may ot support the MLE ad MOM, therefore t s worth trodug the LSM for fttg the Weull dstruto ad omparg ther performaes wth the MLE ad MOM. The ojetve of ths researh s to assess ad ompare the auray of the aove three estmators of two-parameter Weull dstruto. Computer smulato tehques are used to geerate Weull populatos wth kow parameters ad the estmators are aalyzed ad evaluated from Chese pe (Pus taulaeforms) data ad smulato data usg approprate statstal proedures. MATERIALS AND METHODS Feld data desrpto The data were provded y the Ivetory Isttute of Bejg Forestry. They osst of a systemat sample of permaet plots wth a 5-year re-measuremet terval. From the vetory plots over the whole of Bejg, all plots wth 10 trees at least were used ths study (see Tale 1),.e. eghty-sx ha permaet sample plots (PSPs) loated platatos stuated o uplad stes throughout orth-wester Bejg. The PSPs data ossted of 256 measuremets otaed the followg years: 1987, 1991, 1996 ad All 256 measuremet data of 86 sample plots were seleted to estmate the two-parameter Weull futo usg MLE, MOM ad LSM methods order to osstetly ompare the three dfferet estmators. Methods of estmato The proalty ad umulatve dstruto futos of the three-parameter Weull dstruto for a radom varale D are D a 1 D a ƒ(d;a,,) = ( ) exp ( ( ) ) = 0 (a D ) (1) (D < a) D a F(D;a,,) = 1 exp ( ( ) ) (2) D dameter at reast heght ( m), a loato parameter, sale parameter, shape parameter. The parameters of Equato (1) were estmated from the dvdual tree dameter data of eah set of dameter data y maxmum lkelhood estmato. I some plots the proedure of maxmum lkelhood estmates a result a egatve value for the loato parameter a. The parameter a a e osdered as the smallest possle dameter the stad ad thus t should e etwee 0 ad the m- Tale 1. Desrptve statsts of stad ad tree varales Stad varale (86 plots) Tree varale ( = 15,676 trees) dh (m) age (years) N (trees/ha) H (m) BA (m 2 /ha) dh (m) BA (m 2 /tree) Mea Stadard devato M Max , dh dameter at reast heght; N stad desty; H average heght of domat ad odomat trees; BA asal area; Mea, M., Max. mea, mmum ad maxmum dameter at reast heght respetvely J. FOR. SCI., 54, 2008 (12):

3 mum oserved value some ases (Baley, Dell 1973). A approxmato to ths smallest possle dameter s gve y mmum dameter at reast heght (Dmm), whh s the mmum oserved dameter o the sample plots. By artrarly settg a to 0.5 Dmm some studes ad the estmatg parameters ad, three-parameter Weull futo a e otaed (Klkk et al. 1989). Thus, the two-parameter Weull dstruto was osdered ths study as follows D F(D;,) = 1 exp ( ( ) ) (3) Three methods (MLE, MOM ad LSM) metoed aove were used to estmate the Weull dstruto ths study. Maxmum lkelhood estmator (MLE) The method of maxmum lkelhood s a ommoly used proedure for the Weull dstruto forestry eause t has very desrale propertes. Estmato of the parameters y maxmum lkelhood has ee foud to produe osstetly etter goodess-of-ft statsts ompared to the prevous methods, ut t also puts the greatest demads o the omputatoal resoures (Cao, MCarty 2005). Cosder the Weull PDF gve (1), the the lkelhood futo (L) wll e L(D 1,..., D ;,,) = Π ( ) exp ( ( ) ) (4) =1 O takg the logarthms of (4), dfferetatg wth respet to ad respetvely, ad satsfyg the equatos = [ 1 D ] 1/ (5) =1 D = [( D ld ) ( D) 1 1 ld ] 1 (6) =1 =1 The value of has to e otaed from (6) y the use of stadard teratve proedures (.e. Newto- Raphso method) ad the used (5) to ota. 1 =1 Methods of momets (MOM) The method of momets s aother tehque ommoly used the feld of parameter estmato. I the Weull dstruto, the k momet readly follows from (1) as 1 k m k = ( ) k/ Г ( 1 + ) (7) Г gamma futo, Г(s) = 0 xs 1 e x dx, (s > 0). D The from (7), we a fd the frst ad the seod momet as follows 1 1 m 1 = µ = ( ) 1/ Г (1 + ) (8) m 2 = µ 2 + σ 2 = ( ) 2/ {Г (1 + ) [Г (1 + )] 2 } (9) σ 2 varae of tree dameters a plot, m 1, m arthmet mea dameter ad quadrat mea 2 dameter a plot, respetvely. Whe m 2 s dvded y the square of m 1, the expresso of otag oly s 2 1 σ 2 Г(1 + ) Г 2 (1 + ) = (10) µ 2 Г 2 (1 + 1 ) O takg the square roots of (10), the oeffet of varato (CV) s 2 1 Г(1 + ) Г 2 (1 + ) CV = (11) Г 2 (1 + 1 ) I order to estmate ad, we eed to alulate the CV of tree dameters plots ad get the estmator of (11). The sale parameter () a the e estmated usg the followg equato ˆ = { x / Г [(1/ĉ) + 1]} ĉ (12) x mea of the tree dameters. Least squares method (LSM) For the estmato of Weull parameters, the least-squares method (LSM) s extesvely used egeerg ad mathemats prolems. We a get a lear relato etwee the two parameters takg the logarthms of (3) as follows 1 l l [ ] = l D l (13) 1 F(D) Y = l{ l[1 F(D)]} X = ld λ = l. Let D 1, D 2,..., D e a radom sample of D ad F(D) s estmated ad replaed y the meda rak method as follows: F(D) =( 0.3)/( + 0.4) (D, = 1, 2,, ad D 1 < D 2 < < D ) (14) 568 J. FOR. SCI., 54, 2008 (12):

4 Tale 2. Numer of tmes mmzg MSE for 256 dameter frequey dstruto measuremets y method Method No. of tmes the method gves the est estmate mea SD MOM MLE LSM eause F(D) of the mea rak method [F(D) = /( + 1)] may e a larger value for smaller ad a smaller value for larger. Thus, equato (13) s a lear equato ad s expressed as Y = X + λ (15) Computg ad λ y smple lear regresso (15) ad the parameters ad a e estmated as: = [ xy 1/( X Y]/[ X 2 1/( X) 2 ] (16) λ = 1/( Y / X (17) = exp( λ/) (18) Statstal rtera For quattatve omparso of dfferet estmators, mea square error (MSE) was used to test the estmators of the three methods y the 256 dameter frequey dstruto measuremets (oservatos) from 86 sample plots for feld data ths study. MSE s a measure of the auray of the estmator. MSE a e alulated as elow MSE = {Fˆ (D ) F(D )} 2 (19) Fˆ (D ) = 1 exp( D /ˆ ) ĉ value of the umulatve dstruto futo (CDF) of the Weull dstruto evaluated at dh of tree a plot y usg dfferet estmatos, F(D ) oserved umulatve proalty of tree a plot, umer of trees a plot. I ths study, testg ad evaluato omputatos were ompleted usg the Forstat statstal pakage (Tag et al. 2006). RESULTS AND DISCUSSION The 256 dameter frequey dstruto measuremets (oservatos) from 86 sample plots were used to estmate the two-parameter Weull futo ased o the MLE, LSM ad MOM. The est estmated method was evaluated aordg to mmum MSE, mea ad SD MSE. Tale 2 dsplays the summares of the MSE dator for 256 dameter frequey dstruto measuremets. From Tale 2, the MOM produed the est estmate 152 tmes out of 256 dameter frequey dstruto measuremets, whh s approxmately 59.3%, followed y the LSM 69 tmes (27.0%) ad the MLE 35 tmes (13.7%), respetvely. The mea MSEs from 152 tmes MOM, 69 tmes LSM ad 35 tmes MLE are , ad m, respetvely. The Weull parameters ad were estmated y the maxmum lkelhood method (MLE) for 35 dameter frequey dstruto measuremets. The parameter values of the MLE raged as follows: , ; the LSM for 69 dameter frequey dstruto measuremets, the parameter values raged as follows: , ; the MOM for 152 dameter frequey dstruto measuremets, the parameter values raged as follows: , The MOM aheved good estmated results eause t volved more alulatos ad requred more omputato tme tha the LSM or the MLE (Al-Fawza 2000). Although the results from the LSM ad the MLS estmated methods were feror to the MOM ased o the MSE rtero ths study, the LSM ad the MLE amed at fttg the etre dameter dstruto tself (rather tha just the average dameter or plot-level dameter attruted suh as dameter momets). Therefore, t seemed reasoale to expet the LSM or the MLE method estmatg the Weull dstruto futo. Atually, Cao ad MCarty (2005) reported that the umulatve dstruto futo (CDF) regresso method produed etter results tha those from the MOM ased o the h-square statst for lololly J. FOR. SCI., 54, 2008 (12):

5 pe platatos the souther Uted States eause the CDF regresso amed at fttg the CDF of dameter dstruto. Also, the LSM mproves the fttg of the dstruto eause more formato s used tha the MOM. CONCLUSION I ths study, the good results of the MOM terms of the umer of tmes for the lowest values of MSE dated that the MOM was a superor method to estmate the dameter dstruto of Weull futo for Chese pe stad Bejg. However, from the aspet of estmated performae, the LSM ad the MLE of fttg Weull futo were also good methods eause the methods are easy ad quk estmates well as there exsts a lot of software to estmate the parameters of Weull dstruto. Speally, the LSM method mproves the fttg of tree dameter dstrutos eause more formato s used tha the MOM method. Se the regresso method uses smple lear regresso to estmate the parameters ad of the Weull futo, t may e a approprate method for predtg a future stad. Akowledgemets The author would lke to thak Dr. Mohammad Al-Fawza for provdg hs formato to ths paper. R e f e r e e s BAILEY R.L., DELL T.R., Quatfyg dameter dstrutos wth the Weull futo. Forest See, 19: CAO Q.V., MCARTY S.M., Preseted at the Thrteeth Beal Souther Slvultural Researh Coferee. Memphs, TN. HYINK D.M., MOSER J.W., A geeralzed framework for projetg forest yeld ad stad struture usg dameter dstrutos. Forest See, 29: KAMZIAH A.K., AHMAD M.I., LAPONGAN J., Nolear regresso approah to estmatg Johso SB parameters for dameter data. Caada Joural of Forest Researh, 29: KILKKI P., MALTAMO M., MYKKANEN R., PAIVINEN R., Use of the Weull futo estmatg the asal area dh-dstruto. Slva Fea, 23: LITTLE S.N., Weull dameter dstrutos for mxed stads of wester ofers. Caada Joural of Forest Researh, 13: LIU C.M., ZHANG S.Y., LEI Y., ZHANG L.J., Comparso of three methods for predtg dameter dstrutos of lak sprue (Pea maraa) platatos easter Caada. Caada Joural of Forest Researh, 34: MABVURIRA D., MALTAMO M., KANGAS A., Predtg ad alratg dameter dstrutos of Eualyptus grads (Hll) Made platatos Zmawe. New Forests, 23: NEWTON P.F., LEI Y., ZHANG S.Y., A parameter reovery model for estmatg lak sprue dameter dstruto wth the otext of a stad desty maagemet dagram. The Forestry Chrole, 3: NEWTON P.F., LEI Y., ZHANG S.Y., Stad-level dstae-depedet dameter dstruto model for lak sprue platatos. Forest Eology ad Maagemet, 209: RENNOLLS K., GEARY D.N., ROLLINSON T.J.D., Charaterzg dameter dstrutos y the use of the Weull dstruto. Forestry, 58: SCHREUDER H.T., SWANK W.T., Coferous stads haraterzed wth the Weull dstruto. Caada Joural of Forest Researh, 4: SCHREUDER H.T., HAFLEY W.L., BENNETT F.A., Yeld predto for uthed atural slash pe stads. Forest See, 25: SHIVER B.D., Sample sze ad estmato methods for the Weull dstruto for uthed slash pe platato dameter dstruto. Forest See, 34: TANG S., LAN K.J., LI Y., Gude of ForStat.2.0. (Upulsh.) ZARNOCH S.J., DELL T.R., A evaluato of peretle ad maxmum lkelhood estmators of Weull parameters. Forest See, 31: ZHANG L., PACKARD K.C., LIU C., A omparso of estmato methods for fttg Weull ad Johso s SB dstrutos to mxed sprue-fr stads ortheaster North Amera. Caada Joural of Forest Researh, 33: ZHOU BAILIN, MTAGUE J.P., Comparso ad evaluato of fve method of estmato of the Johso system parameters. Caada Joural of Forest Researh, 26: AL-FAWZAN MOHAMMAD, Method for estmatg the parameters of the Weull dstruto. (Upulsh.) Reeved for pulato July 8, 2008 Aepted after orretos Septemer 1, J. FOR. SCI., 54, 2008 (12):

6 Hodotee troh metód a určee parametrov Weullového rozdelea čískej orove (Pus taulaeforms) ABSTRAKT: Na vyrovae hrúok stromov zozeraýh z 86 výskumýh plôh čískej orove v Pekgu sa použlo Weullove rozdelee. Pr určovaí parametrov Weullového rozdelea sa prostredítvom stredej kvadratkej hyy a počtu prípadov porovával a hodotl tr metódy, meovte metóda maxmálej verohodost MLE, mometová metóda MOM a regresá metóda ajmešíh štvorov LSM. Na určee parametrov Weullového rozdelea hrúok stromov výerovýh plôh ola ajlepša mometová metóda. Kľúčové slová: Weullove rozdelee; rozdelee hrúok; určee parametrov Correspodg author: Prof. Dr. Yuaa Le, Researh Isttute of Resoure Iformato ad Tehques, Chese Aademy of Forestry, Bejg , Cha, P. R. tel.: , fax: , e-mal: [email protected]., [email protected] J. FOR. SCI., 54, 2008 (12):