A comparison of diameter distribution models for

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Jun 7, 2018 - To cite this article: Rafaella Carvalho Mayrinck, Antonio Carlos Ferraz Filho, Andressa Ribeiro,. Ximena Mendes de Oliveira & Renato Ribeiro ...
Southern Forests: a Journal of Forest Science

ISSN: 2070-2620 (Print) 2070-2639 (Online) Journal homepage: http://www.tandfonline.com/loi/tsfs20

A comparison of diameter distribution models for Khaya ivorensis A.Chev. plantations in Brazil Rafaella Carvalho Mayrinck, Antonio Carlos Ferraz Filho, Andressa Ribeiro, Ximena Mendes de Oliveira & Renato Ribeiro de Lima To cite this article: Rafaella Carvalho Mayrinck, Antonio Carlos Ferraz Filho, Andressa Ribeiro, Ximena Mendes de Oliveira & Renato Ribeiro de Lima (2018): A comparison of diameter distribution models for Khaya ivorensis A.Chev. plantations in Brazil, Southern Forests: a Journal of Forest Science, DOI: 10.2989/20702620.2018.1463189 To link to this article: https://doi.org/10.2989/20702620.2018.1463189

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SOUTHERN FORESTS

This is the final version of the article that is published ahead of the print and online issue

ISSN 2070-2620 EISSN 2070-2639 https://doi.org/10.2989/20702620.2018.1463189

A comparison of diameter distribution models for Khaya ivorensis A.Chev. plantations in Brazil Rafaella Carvalho Mayrinck1*  , Antonio Carlos Ferraz Filho2  , Andressa Ribeiro2  , Ximena Mendes de Oliveira3  and Renato Ribeiro de Lima4 School of Environment and Sustainability, University of Saskatchewan, Saskatoon, Canada Department of Engineering, Federal University of Piauí, Bom Jesus, Piauí, Brazil 3 Department of Forest Science, Federal University of Lavras, Minas Gerais, Brazil 4 Department of Statistic, Federal University of Lavras, Minas Gerais, Brazil * Corresponding author, email: [email protected] 1

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The purpose of this study was to compare Beta, Gamma, Johnson’s SB and Weibull functions fitted by different methods for describing the horizontal structure of Khaya ivorensis (African mahogany) plantations in Brazil. The database comprised 128 plots from six plantations at varying ages. The function fits were compared using the Kolmogoroff–Smirnoff test, mean bias and mean absolute error for the number of trees and basal area per hectare per diameter class. Johnson’s SB outperformed the other functions, although all functions provided an adequate fit. The best methods were method of moments and maximum likelihood fitted using 25% of the minimum observed diameter as the location parameter for the Johnson’s SB function. The errors were greater in diameter classes with higher frequencies. Location and scale parameters were highly correlated with mean diameter and age for the Weibull and Johnson’s SB functions, respectively, which is convenient for diameter prediction. Gamma’s scale parameter had medium correlation with age. Beta’s parameters had low correlation with stand attributes assessed. Keywords: African mahogany, diameter distributions, horizontal structure, probability density functions

Introduction Khaya ivorensis A.Chev., known as African mahogany, produces valuable hardwood used commercially in the international market. Air-dried and oven-dried lumber exported from Ghana reaches prices of €650 and €1 049 m−3, respectively (ITTO 2017). It is a tropical forest tree, with medium growth rate and, when grown in plantations, can be a suitable alternative for wood from tropical native forests. It is strong enough for structural use and adequate for inner decoration and furniture (Nordahlia et al. 2013; ITTO 2017; Moura et al. 2017). Given its good wood properties, countries such as Brazil, Australia and some countries in Asia started to establish plantations using K. ivorensis and other species of African mahogany, such as K. senegalensis. However, silvicultural practices are at an initial stage and there is little information about management and silvicultural treatments for African mahogany (Ribeiro et al. 2016, 2017), especially regarding growth and yield studies. Probability density functions (pdf) are especially important in managing plantations for structural use, where not just the total stand volume matters, but also individual tree size. In this regard, forest structure can be assessed by stand density (number of trees per hectare) or by basal area, where basal area gives more weight to the larger, higher-priced trees in a stand (Maltamo et al. 1995; Palahí et al. 2007). Stand structure information is required in multipurpose forestry in order to determine a priori the effect of

management and silvicultural practices on wood quality and dimension, as well as to scale harvesting, trading and transport. A variety of probability density functions, e.g. Johnson’s SB, Weibull, Normal, Beta and Gamma, have been successfully used in forest studies by several authors for describing theoretical distributions based on observed data (Nokoe and Okojie 1984; Cao 2004; Dalla Corte et al. 2004; Arce 2005; Palahí et al. 2007; GorgosoVarela and Rojo-Alboreca 2014; Ribeiro et al. 2014). Many studies have shown that species, age, site index, spacing, along with other dendrometric characteristics, influence the probability density function which yields the best fit for a stand (Nokoe and Okojie 1984; Barra et al. 2004; Cao 2004; Arce 2005; Palahí et al. 2007; Gorgoso-Varela and Rojo-Alboreca 2014). The main objective of this study was to compare the efficacy and accuracy of Beta, Gamma, Johnson’s SB and Weibull functions, fitted by different methods (three methods for Weibull distribution: maximum likelihood, method of moments and percentiles; five methods for Johnson’s SB distribution: maximum likelihood, method of moments, mode [Hafley and Buford 1985], linear regression [Zhou and McTague 1996] and Knoebel–Burkhart’s method [Knoebel and Burkhart 1991] and, lastly, the method of moments for Beta and Gamma distributions). We also tested different values for the location parameter for Weibull and Johnson’s SB pdf. In addition, the correlation of

Southern Forests is co-published by NISC (Pty) Ltd and Informa UK Limited (trading as Taylor & Francis Group)

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Mayrinck, Ferraz Filho, Ribeiro, Mendes de Oliveira and Lima α −1

the function’s parameters to stand attributes was assessed, because this correlation is key to predict the development of a stand for different ages and the effect of thinning on growth, for example. Materials and methods Database The database used in this study was obtained from 128 permanent sample plots established in six African mahogany plantations in Brazil measured at different ages. Table 1 summarises the database used in this study. Plot sizes varied from 573 to 4 372 m2. Ages ranged from 1.1 to 14 years and the tree densities ranged from 70 to 400 trees ha−1.

Γ (α + β α) −1 d − dmin  β −1 d − dmin   Γ (α +fβ( x) )= d − dmin   d − dmin    1 − 1 dmax − d(1) − − dmin    f (x) = 1max  Γ (α ) Γ ( β  ) d min  Γ (α ) Γ ( β )  dmax − dmin 



dmax − dmin 

β −1

dmax − dmin

1 dmax − d

 1

1  1 1 f (d ) = ) )α −1.e − β .( d −dmin ) (2)   ( d − dmin )α −1.αeα−−1 (βd.(d−−ddminmin α β −1 α β Γ α β Γ ( ) ( ) Γ (α + β )  d − dmin   d − dmin  1 f (x) =   1 −  Γ (α ) Γ ( β )  dmax − dmin   dmax − dmin  dmax − dmin 2 For both Equations 1 and 2, α is the shape parameter, δ λ  1  d − ε    δ dmin 1 β −1 d ) =scale parameter, expand γ−1max ln  the + (δare  minimum  d −ε ) − λαd β is f (the (dε+))(αβ=−λ1.)+e −ε 1−d d ) − +ε−− d γ + (δ ) ln1 λ 2π (Γdf(α    on  2 1 −d dexp −ddmin )  .( d−  min mind2 is diameters found the plot, the d ε λ ε d − + − ( )( ) β λ +ε −d 2 π f (dand d d − ) = maximum  ( ) f ( xα) =     1 −  min Γ (α ) βof Γ (α dmin dminΓ is the dmax − dmin ) Γ ( β )  dclass midpoint the diameter be estimated, max − to  dmax −and

f (d ) =

   d − α γ      d − α γ   gamma function.  f (d ) = exp  −  γ    − exp  −  γ     1βe−1 .(d−d )  d−dε − α 2        d −α  δ   β1λ α −1. − γβ+− (δexp −   f (d ) = f (d )=f (d ) = exp − exp ln    d d − ) ( )   (3) α ε − d ) minβ 2     λ +ε −βd  +    2π ( d Γ− (εα)()λ  β          min

(D

− Dˆ ij

)

1

Bias = ε∑and Probability density function fitting where λ γarex the γ   scale parameters, 2   location  j =d1 − NC   d ˆ− α and  α  j δ NP    D λ Dij parameters  d − βε−1    1 NP  δ−are  −ij −  exp −α −11 γ(asymmetry ) = exp  − f (dγ) and exp Diameter distribution data was fitted using four pdf: Beta, f (drespectively, the shape    = + (δd) ln     Γ + α β   ( ) − − d d d    Bias x = β β min   )min  + ε −d dis α2−1 1 − dj =1− ε)( λ + ε−β −d1    1 f (x )2=π∑(Γ Gamma, Johnson’s SB and Weibull, represented by the and kurtosis, respectively), themidpoint  α) Γ+NC ββλand 1 ( ) −−ddmin dof−−dthe j d dNP min Γ α d d d ˆ ( ( ) min  min  max − d  max  1 − max Dijto D)ij =estimated. f −( xbe 1 NP diameter class Equations 1, 2, 3 and 4, respectively. Johnson’s SB was dmax − MAE = ∑ j =1 x Γ (α ) Γ ( β )  dmax − dmin   dmax − dmin  NC NP fitted using five fitting methods: maximum likelihood, method D − Dˆ j γ   γ  NP ( ij  ij )  1  ˆ  ij − Dij 1 α −1.  ed−−1α. d −d  of moments, Knoebel–Burkhart (Knoebel and Burkhart Bias = ∑fj =(1d ) = expx −  d −NP 1α D ( (4) −−dexp MAE ) NC == ∑βj =1 α (d1.89 x1min ) α−−1.e −ββ1.d −d1.43 j f (d )NP 1991), mode and linear regression. Beta and Gamma −  1    (   ) −1  1.43 ) = Γd(α−)dβminNC  (dj −dmin ) d − dβmin  Γ (1.89 + NP  f d ( 1  functions were fitted by only the method of moments. The f (d ) =  (α ) β α  1 −  Γ (1.89 ) Γ (1.43 )  dΓmax − dmin  − d d d max min  max − dmin  Weibull function was fitted using three methods: percentile, where α, Dβ −and Dˆ ij γ 1are the location, scale and shape ij NP ˆmidpoint of the diameter 1.89 −1class to 1.43 MAE = ∑ j =1 and d x isDthe method of moments and the maximum likelihood. parameters, ij − Dδ 1 λ +ij 1.43  d − ε   1  d − dmin (1.89 NC j NPΓ NP Bias x  ) 1 .( dd−d − )dmin exp γ + (δ ) ln  d − ε = 6.6 For Weibull and Johnson’s S B functions, we also be estimated. − 11− d )∑= j =f1(d )NC 1f (= −1.e −    δ λ 0.81 f (x) = dmin)j)Γ d − λ + ε− −dmin d ) exp −2  γ d+ (δ )−lnd λ +ε −d (Γd(−1.89 (NP min 2π(1.43 ) ε )(dmax  max  tested for the best values of the location parameter (ε for Γ ( 6.6 ) 0.816.6 f (d ) = ( d−1− ε )( λ + ε − d )  1.43−12  λ +ε − 2π 1.89 Performance of the probability density functions Johnson’s SB and α for Weibull). For this, we tested Γ (1.89 + 1.43 )  d − dmin   d − dmin  1 )=  methodˆ was   1γ−  1 different fitting evaluated by analysing bias different percentage values of the observed minimum f (dEach γ −d D   dmax Dij−−   Γ (1.89 − dmin d d    ) Γ (−1.43 ij  d 1 1.57)  d 18.19 NP max max min 1   − − − e d d 6.6 1. . ) min −d d − (α  min  absolute  (MAE)   0.81 =5)f (and f ( x )MAE (Equation error diameter to represent the location parameter. For Johnson’s − γdmin x=)f (=   ( α2.44 d )mean =1 2.44 − x6.6 ∑ exp)(18.19 d−))×exp −(Equation 6) γ  + −   α α d − d − 2π ( dΓj =−  NC NP β  and 0.81 )per j  −hectare  −diameter  − class  β  per exp (dtrees ) (=6.6 for the number fof SB, values between 5% and 95%, with 5% intervals of the    exp 2   1β      1area  β et  al. e − ddiameter d −d  ) class 6.6 − 1.per .(2.44      1 − basal per hectare (Palahí observed minimum diameter in each plot, were tested. f ( xfor         ) = exp − 2.13 ( d − dmin )) ln   0.81  6.6 + ( −1.57   Γ ( 6.6 0.81 1.89 −1 1.43 −1 ) 2 18.19 2.44 + − d 2007; Gorgoso-Varela and Rojo-Alboreca 2014): For the Weibull function we tested values between 0%        Γ (1.89 + 1.43  18.19 −1.57 )  d −ˆ d d − dmin  1 × f ( x ) = NP Dij − Dij min 1  f (d ) = 1− and 100% of the observed minimum diameter, with 10%  ˆ Bias) Γ=(∑ x Γ (1.89 − 1.43 d d d d 2.44 18.19 2.44 − + − d d D D − ( ij min )  max min max −  NC   1)π jNP =2 ijmax 1)( intervals. Details on the probability density functions’ 18.19 j (5) Bias xNP6.8   6.8  = ∑ f ( x ) = −1.57    1 j = 2     d j− 4.88NP ×  different parameter estimations methods can be found in f ( x ) = exp  − d − 4.88 −NC exp +   1)(−18.19     2π ( d −2.44 2.44 − d )     d − 2.44 1.57 ln  1−  2.13 +( −10.40   1  ) Scolforo et al. (2003), Gorgoso-Varela and Rojo-Alboreca      10.4exp 6.6 −1.e −2 18.19 ) − d    1f (x ) =  .( d −d+ 2.44 dd−D−dij 2.44  26.6 − Dˆ)ij  (NP    0.81 min (2014) and Mayrinck (2017).    1 exp − 2.13Γ+((6.6 −1.57 ) ln= ∑ D − Dˆ x   (6) ) 0.81 MAE jNP =1 +ij 2.44ij− d 1 We tested the following for each pair of percentiles:  18.19  2   NC NP  j MAE = ∑ j =1 x 24 and 93, 25 and 75, and 30 and 90. For each pair of NC 6.8j   NP 6.8    4.88  of18.19 4.88 per  d −number  d −area    trees  orbasal percentiles, we also tested the value of the location the observed where Dfij (is − 1.57 = − − − x ) exp exp   6.8 6.8 ( ) = × f x    −1 1.43  1.89     dfor   dij1.43 − 4.88 − 4.88  i in Γ  estimated plot parameter using the same methodology previouslyf ( x ) = hectare class is the 10.4 10.40 d (1.89 )18.19  − number ( dj,−−+Dˆ2.44 )( )   ofd − dmin  1.4 min 1.89 −1 1 − exp  −  f(d ) = −2πexp   d−+d2.44    10.4 10.40 Γ 1.89 + 1.43     mentioned. Thus, a total of 41 combinations of different trees or basal area per hectare for class i in plot j, NC is the (   )  d −−ddmin   j2 d d −−ddmin    min  min  f (d1)= Γ(1.89 ) Γ (1.43 ) dmax d − 2.44   1 − inmax fitting methods and functions were tested to choose the number of exp classes (considering the data) found Γ (1.89 ) Γ (1.43 + ( −1.57  dmax − dmin    dmax − dmin  ) ln )observed − 2.13 2 18.19 + 2.44 − d    best-fitting method for each plot. To fit and compare plot j, and NP is  the number of plots.   1  1 of fit of each fitting − 6.6 −1.emethod, distributions, the observed diameters of each plot were To evaluate the goodness ) .( d −dthe 0.81 − f (x) = d d   ( ) 1   min 6.6 −1.e − 6.6 the Kolmogoroff–Smirnoff classified into 1 cm classes. mean and standard  0.81 .( d −d ) Γ ( 6.6 ) 10.81of − f ( x ) = deviation d d   (  min ) 6.8  d −Γ   6.6 ( 6.66.8 ) 0.81 4.88  d − 4.88     f ( x ) = exp  −    − exp  −       10.4       10.40     = Minas 1.57 Table 1: Characteristics of African mahogany data used to fit different distributions. MG Gerais, − GO = Goiás,PA = 18.19 Pará × f ( x ) = −1.57 18.19 f ( x ) = 2π ( d − 2.44 )(18.19 + 2.44 − d ) × − d) 2 2π ( d − 2.44 )(18.19 + 2.44 Plot No. of Stand Plot Ages (years) Location (city/state)* Coordinates N ha−1 d − 2.44  1    2 shape plots area (ha) size (m2) exp − 12.13 + ( −1.57 ) ln    2.44 − d   18.19 + 2.44 São Roque de Minas – MG 20.24° S, 46.36° W 56.5 286 800 Circle 1.1;+ 1.9; 3.2;) ln 4;5.2 – − d    exp−2 2.13 ( −1.57 2 1.3; 2.3; 3.1; 4.4; 5.2;  18.19 Piumhi – MG 20.46° S, 46.95° W 9.2 303/250 843 Circle 6.3 + 2.44 14 − d      NP

ij

(

)

min

min

(

)

min

min

( (

) )

min

min

min

Iporá – GO 16.44° S, 51.11° W Iraí de Minas – MG 18.98° S, 47.46° W Santo Antônio do Tauá – PA 1.18° S, 48.13° W Pirapora – MG 17.71°S, 44.91°W Pirapora – MG 17.71°S, 44.91°W

64.3 178.1 39.3 121.3 1.3

416 416 70 277 238/400/100

573 Circle 4.9; 6 20 6.8   4.7  787 27 6.8     d − 4.882.7;   d − 4.88  6.83.4; Square   6.8 − exp −  d − 4.88 f ( x ) = Square exp −  d − 4.88 14   4 372 4    10.4         −exp − 10.40 f ( x ) = Square exp − 1.3; 4.4; 1 074 2.2;3.3; 5.3;6.3 30    10.4 10.40               4.4;    Square  5.3 2 205.3 1.3; 2.2; 3.3; 6

Southern Forests 2018: 1–8

3

(KS) test value (Dn) was used, as in Cao (2004) and Gorgoso-Varela and Rojo-Alboreca (2014). The Dn is obtained from the greatest difference between observed and estimated data as follows: Dn = max|F(x) – s(x)|, where F(x) is the cumulative value for estimated data and s(x) is the cumulative value for observed data. The percentage of plots where the estimated distribution was not statistically different to the observed data (according to KS with 95% confidence) was also evaluated. In order to observe the behaviour of bias and MAE for the number of trees per hectare and basal area, the errors of the best tested functions were plotted by diameter class. Correlation of the pdf parameters with stand attributes A correlation matrix was obtained in order to analyse correlations between parameters of the functions and the stand attributes. Stand attributes evaluated were: age, number of trees per hectare, basal area, mean diameter, mean height and dominant height (mean height of the 30 largest trees per hectare, as in Ribeiro et al. 2016). Results Probability density function selection Before comparing the behaviour of all the different tested functions and fitting methods, we first selected the best minimum diameter values for the location parameters of the Johnson’s SB and Weibull distributions. Table 2 compares the location parameter by the mean value of the KS statistic for the Weibull and Johnson’s S B functions. The best location value for each function fitted by each methodology is highlighted in bold in Table 2. The best values for the location parameter varied between 0 and 60% of the minimum observed diameter

(Table 2). For Johnson’s SB moments and maximum likelihood methods, the best values for the location parameter was 25% of the observed minimum diameter value. For Weibull method of moments and maximum likelihood, the best values were 0 and 10% of the minimum diameter, respectively. For Weibull percentile, when using the percentile pairs of 24–93 and 25–75, the best values were 40% and 60%, respectively. For the percentile of 25–97, values of 50% and 60% of minimum diameter were the best, equally adequate. Methods using the best location parameter, highlighted in bold in Table 2, were then compared with all the other fitting methods (Table 3). Table 3 compares the functions fitted by different methods using mean values of MAE, bias, the percentage of plots where the fitted distribution was not statistically different from the observed data (according to the KS test), and the mean value for the KS test. All the probability density functions were ranked considering each evaluation criterion previously described (ranked in parentheses in Table 3), and the sum of the ranked values was calculated to select the best fitting method (last column in Table 3). The lower the rank, the better the fit. Given that the database is diverse in age and density, we stratified the database into a total of six classes of age and density (N). The statistics presented in Table 3 were calculated again for each class, to make sure the results did not change with age and density variation. The stand age classes (in years) were as follows: Class 1: 1 to 4.4; Class 2: 4.5 to 6.3; and Class 3: 14. The stand density classes (in trees per hectare) were as follows: Class 1: 238 to 286; Class 2: 416; and Class 3: 70 to 100. The statistics for each class were calculated as in Table 3 and the values were also ranked. To simplify, Table 4 presents the sum of the ranking values only (similar to the last column in

Table 2: Kolmogoroff–Smirnoff mean value for the Johnson’s SB and Weibull distributions fitted by moments, maximum likelihood and percentile methods. Highlighted values were the best in each method evaluated

Parameter 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Johnson’s SB moments – 0.0990 0.0924 0.0893 0.0880 0.0877 0.0881 0.0891 0.0903 0.0919 0.0938 0.0960 0.0983 0.1009 0.1040 0.1077 0.1121 0.1176 0.1246 0.1336 –

Johnson’s SB maximum likelihood – 0.1104 0.0979 0.0935 0.0921 0.0921 0.0929 0.0943 0.0960 0.0982 0.1007 0.1035 0.1067 0.1103 0.1145 0.1196 0.1262 0.1350 0.1478 0.1701 –

Weibull moments 0.1063 – 0.1095 – 0.1155 – 0.1250 – 0.1370 – 0.1515 – 0.1690 – 0.1911 – 0.2168 – 0.2470 – 0.2830

Weibull maximum likelihood 0.1250 – 0.1230 – 0.1273 – 0.1364 – 0.1482 – 0.1612 – 0.1782 – 0.2009 – 0.2339 – 0.2750 – 0.3137

Weibull percentille (24–93) 0.1155 – 0.1159 – 0.1158 – 0.1157 – 0.1154 – 0.1157 – 0.1158 – 0.1189 – 0.1233 – 0.1316 – 0.1459

Weibull percentille (25–75) 0.1651 – 0.1672 – 0.1665 – 0.1659 – 0.1651 – 0.1642 – 0.1633 – 0.1653 – 0.1681 – 0.1745 – 0.1899

Weibull percentille (25–97) 0.0974 – 0.0981 – 0.0980 – 0.0978 – 0.0977 – 0.0976 – 0.0976 – 0.1009 – 0.1053 – 0.1141 – 0.1315

4

Mayrinck, Ferraz Filho, Ribeiro, Mendes de Oliveira and Lima

Table 3). The lower the sum of the ranking value related to the method, the better the method performs. Values in bold highlight the three best methods for each class. The last column ranks the methods considering all classes, in order to observe the general tendency again, as in Table 3. Results in Table 3 are similar to those in Table 4. For most of the classes, the methods Johnson’s SB method of moments (dmin*0.25), Johnson’s SB maximum likelihood (dmin*0.25) and Weibull percentile (25–97) (dmin*0.5) were the best. There were some exceptions. For the age class 14, the best methods were Johnson’s SB Knoebel–Burkhart, Weibull percentile (24–93) (dmin*0.4) and Johnson’s SB method of moments (dmin*0.25), in that order. For the density of 416 trees per hectare, the best methods were Johnson’s SB method of moments (dmin*0.25), Weibull percentile (24–93) (dmin*0.4) and Johnson’s SB Knoebel–Burkhart. The age class 5.5 to 6.6 fitted best using Weibull maximum likelihood (dmin*0.1), instead of the Johnson’s SB maximum likelihood method (dmin*0.25). As a general rule, the worstperforming methods were Johnson’s SB regression and mode. For Johnson’s SB regression, only 22.2% of plots had statistically equal observed and estimated values, according to the KS test with 95% confidence limits.

On average, Johnson’s SB method of moments overestimated 16.7 trees per hectare per diameter class and overestimated 0.360 m2 per hectare per diameter class, whilst the Johnson’s SB maximum likelihood method overSestimated 16.8 trees per hectare per diameter class and overestimated the basal area by 0.358 m2 per hectare per diameter class. Both methods resulted in quite similar errors, and SB method of moments had a higher rate of conformance between fitted and observed values (99.64% vs 99.1%). The Weibull percentile method, using the percentiles 25–97 pairs and the location value of 50% for the minimum diameter, had smaller errors for number of trees per hectare per diameter class (16.1 trees) and for basal area (0.358 m2), and the second-highest percentage of conformance (99.28%). Beta was the third-best distribution, underestimating the trees by 19.7 and the basal area by 0.389 m2 per hectare per diameter class. The percentage of plots where the fitted distribution was statistically equal to the observed data was 97.11% for the Beta function. Regarding the Gamma function, the percentage of plots where estimated and observed distributions were not statistically different according to the KS test was 95.67%. In addition, Gamma underestimated by 22.6 trees per

Table 3: Mean bias and mean absolute error (MAE) for number of trees per hectare (N) and basal area per hectare (G) (ranked position), percentage of plots where theoretical and observed frequencies were similar according to the Kolmogoroff–Smirnoff (KS) test, and the mean value of the KS test. Sum of rank values is the sum of all ranked values from preceding columns. Best values for each category are highlighted in bold Method Beta moments Gamma moments Johnson’s SB Knoebel–Burkhart Johnson’s SB mode Johnson’s SB moments (dmin*0.25) Johnson’s SB maximum likelihood (dmin*0.25) Johnson’s SB regression Weibull moments (dmin*0) Weibull maximum likelihood (dmin*0.1) Weibull percentille (24–93) (dmin*0.4) Weibull percentille (25–75) (dmin*0.5) Weibull percentille (25–97) (dmin*0.5)

Bias N −0.171 (5) −0.241 (8) 0.433 (9) 1.938 (12) 0.136 (2) 0.109 (1) 0.164 (3) 0.942 (11) 0.819 (10) 0.169 (4) 0.233 (7) 0.211 (6)

MAE N 19.7 (8) 22.6 (10) 19.3 (7) 24.6 (11) 16.7 (2) 16.8 (3) 50.9 (12) 18.2 (5) 18.7 (6) 17.6 (4) 20.3 (9) 16.1 (1)

Bias G −0.047 (2) −0.054 (6) −0.059 (9) −0.050 (4) −0.056 (7) −0.053 (5) −0.170 (12) −0.048 (3) -0.038 (1) −0.066 (10) −0.111 (11) −0.057 (8)

MAE G 0.389 (6) 0.439 (9) 0.382 (4) 0.452 (10) 0.360 (2) 0.358 (1) 0.771 (11) 0.388 (5) 0.399 (7) 0.379 (3) 0.438 (8) 0.358 (1)

Adhered plots (%) 97.11 (7) 95.67 (8) 97.29 (6) 73.47 (11) 99.64 (1) 99.10 (4) 22.20 (12) 98.19 (5) 93.68 (10) 99.28 (2) 94.58 (9) 99.28 (2)

Mean Sum of rank KS values 0.028 (7) 35 (6) 0.043 (8) 49 (9) 0.027 (6) 41 (7) 0.265 (11) 59 (11) 0.003 (1) 15 (1) 0.009 (4) 18 (2) 0.778 (12) 62 (12) 0.018 (5) 34 (5) 0.063 (10) 44 (8) 0.007 (2) 25 (4) 0.054 (9) 53 (10) 0.007 (2) 20 (3)

Table 4: Rank values summarising all the statistics (bias and mean absolute error) for number of trees per hectare (N) and basal area per hectare (G), percentage of adherence and mean value of the Kolmogoroff–Smirnoff test for the age and density classes. Best values for each category are highlighted in bold Method Beta moments Gamma moments Johnson’s SB Knoebel–Burkhart Johnson’s SB mode Johnson’s SB moments (dmin*0.25) Johnson’s SB maximum likelihood (dmin*0.25) Johnson’s SB regression Weibull moments (dmin*0) Weibull maximum likelihood (dmin*0.1) Weibull percentille (24–93)(dmin*0.4) Weibull percentille (25–75)(dmin*0.5) Weibull percentille (25–97)(dmin*0.5)

1 to 4.5 47 58 47 64 18 22 63 26 28 26 53 16

Age class 5.5 to 6.6 48 47 30 59 21 28 62 44 27 29 51 21

14 39 43 21 35 25 34 36 40 35 21 47 27

N ha−1 class 238 to 286 416 35 44 39 46 47 41 58 50 16 17 21 23 51 54 22 24 35 32 24 20 43 47 17 10

70 to 100 42 44 43 58 23 20 62 30 28 32 54 22

Rank 8 9 7 11 2 3 12 6 5 4 10 1



dmax − dmin

 1  1 6.6 −1.e − .( d −dmin )  1 0.81   f (x) = ( d − dmin 1 6.6)−1.e −  .( d −dmin ) 6.6 0.81 Γ ( 6.6 ) 0.81 − f (x) = d d   ( ) min Γ ( 6.6 ) 0.816.6

Southern Forests 2018: 1–8

hectare per diameter class and underestimated basal area by 0.439 m2 per hectare. Pdf parameters’ correlation with stand attributes Equations 7 to 10 show the fit for Beta, Gamma, Johnson’s SB method of moments (dmin*0.25) and Weibull percentile (25–97)(dmin*0.5) functions, respectively, where d refers to the midpoint of the diameter class to be estimated, dmin and dmax are the minimum and maximum diameters on each plot. Since each plot was fitted individually, the parameters obtained on each plot were averaged to yield equations (7) to (10). Table 5 presents stratified average values for parameters and their standard deviation for the best fittings obtained for each distribution, as previously described. Classes were made based on age and stand density. 1.89 −1

Γ (1.89 + 1.43 )  d − dmin  1.89 −1 Γ (1.89 + 1.43 )  d − d  × f (d ) = f (d Γ ) =(1.89 ) Γ (1.43 )  d  − d min  × min  ) max  d d Γ (1.89 ) Γ (1.43 − min   max 1.43 −1

  d −ddmin  1.43 −1 1 − dmin  1  1 − 1 −  − dmin − d−min   dmaxdmax   dmax dmax dmin − dmin



dmax − dmin 

(7)

 1 

1 1 1.e−−1.e − 1.( d−.(ddmin ) )  0.81  −dmin (8) f ( x )f (=x ) =  0.81 ( d − dmindmin )6.6)−6.6    6.6 6.6 ( d − Γ ( 6.6 0.81 ) Γ ( 6.6 ) 0.81

5

−1.57 18.19 × f ( x )−=1.57 18.19 d) 2π ( d − 2.44 )(18.19 + 2.44 − × f (x) = 2π ( d − 2.44 )(18.19 + 2.44 − d ) 2  1  d − 2.44 2     1− 2.13 + ( −1.57 )ln  d − 2.44  exp    2  + ( −1.57 ) ln   18.19 + 2.44 −d  exp −  2.13   (9)  18.19 + 2.44 − d      2     d − 4.88 6.8      d − 4.88 6.8    d  d − − 4.88 6.8    − exp − − 4.88 6.8    (10) f ( x )= exp 10.4 − − − f ( x ) = exp exp       10.40            10.4         10.40          All the tested functions had their parameters correlated with stand attributes using a matrix of correlation. The highest correlation between each parameter of each function and a stand attribute is given in bold in Table 6. For those functions with more than one fitting method (Johnson’s SB and Weibull), the best one was chosen for the analysis (Johnson’s S B fitted by the method of moments and Weibull percentile [25–97] [dmin*0.5]). The highest correlation between the stand attribute and the parameter of the function is highlighted in bold. For the Beta function, parameter α showed the highest correlation with mean dominant height, whereas parameter β was highly correlated with mean diameter. For the Gamma function, parameter α was best correlated with the inverse of mean height and parameter β correlated well with age.

Table 5: Average parameter values and (standard deviation) for each distribution fitted using the best fitting method. Best values for each −1.57 18.19 18.19 category are highlighted bold ×× f ( x )f (=x ) = in−1.57

+ 2.44 − d−)d ) ( d −( d2.44 )(18.19 2π 2π − 2.44 + 2.44 )(18.19

2 2 Age class  1  1 Parameter − 2.44       d −d2.44 − + − expexp 2.13 1.57 ln − + − 2.13 1.57 ln ( ) ( ) 1 to 4.5 5.5 to 6.6       + 2.44 − d− d    18.19 + 2.44  18.19  2  2 α Beta moments  1.89 (0.81) 1.89 (0.81) 

Distribution

β 1.43 (0.57) 1.43 (0.57) α 6.30 (3.58) 6.32 (3.59) 6.8 6.8 6.86.8    d −d4.88  0.83  (0.56)     d−d4.88 β− 4.88 −3.75 4.88       (1.85)  = exp f (=x) exp exp −  −  ε    − −exp −  −  f ( x )         (1.20) 10.40 2.19 (1.20) Johnson’s SB  10.4   2.19  10.40      10.4                 17.03  (8.20)  λ moments 17.12 (8.21) δ (dmin*0.25) −1.46 (0.57) −1.46 (0.57) γ 1.96 (0.59) 1.96 (0.59) α Weibull percentille 4.38 (2.41) 4.39 (2.41) β (25–97) (dmin*0.5) 9.95 (5.02) 10.02 (5.05) γ 6.71 (2.16) 6.72 (2.16)

Gamma moments

14 1.68 (0.40) 1.41 (0.35) 5.03 (1.62) 3.85 (1.64) 8.00 (1.22) 65.58 (2.57) −1.13 (0.39) 1.79 (0.29) 16.01 (2.44) 37.20 (2.58) 5.94 (1.65)

Stand density class (stems ha−1) 70 to 100 238 to 286 416 2.14 (1.31) 1.89 (0.81) 2.07 (0.88) 1.64 (0.69) 1.43 (0.57) 1.59 (0.58) 5.03 (3.85) 6.30 (3.58) 6.60 (3.81) 3.85 (1.64) 0.83 (0.56) 0.81 (0.53) 8.00 (1.22) 2.19 (1.20) 2.44 (1.16) 65.58 (2.57) 17.03 (8.20) 18.19 (8.05) −1.13 (0.39) −1.46 (0.57) −1.57 (0.51) 1.79 (0.29) 1.96 (0.59) 2.13 (0.54) 5.69 (3.25) 4.38 (2.41) 4.88 (2.33) 14.27 (6.63) 9.95 (5.02) 10.40 (4.72) 6.03 (1.75) 6.71 (2.16) 6.80 (1.83)

Table 6: Correlation of stand attributes with parameters of the probability density functions tested in fitting diameter distributions of African mahogany plantations in Brazil. Best values for each category are highlighted in bold

Function Beta moments Gamma moments Johnson’s SB moments (dmin*0.25) Weibull percentille (25–97) (dmin*0.5) * p < 0.05, ** p < 0.01

Parameter α β α β ε λ γ δ α β γ

Mean diameter 0.00 0.21* −0.14* 0.52* 0.90* 0.81* −0.39* 0.50* 0.90* 0.68* 0.13*

Age 0.04 0.00 0.04 0.76* 0.76* 0.94* 0.12* 0.07 0.76* 0.92* 0.09

Stand attribute Mean dominant Mean height height 0.12* −0.01 0.07* 0.18* 0.09* −0.14* 0.71* 0.51* 0.11 0.86* 0.11* 0.78* 0.02* −0.36* 0.10 −0.44* 0.76* 0.87* 0.91* 0.66* 0.16* 0.08

Number of trees ha−1 0.08 0.07 0.03 −0.15* −0.11* −0.14* −0.14* −0.12** −0.06 −0.15* 0.09*

Basal area (m2 ha−1) 0.00 0.10* −0.12* 0.55* 0.79* 0.77* −0.19* −0.31 0.79* 0.65* 0.00

6

Mayrinck, Ferraz Filho, Ribeiro, Mendes de Oliveira and Lima

forest modelling by several researchers in fitting a variety of natural and planted stands (Cao 2004; Palahí et al. 2006; Palahí et al. 2007; Gorgoso et al. 2008; GorgosoVarela and Rojo-Alboreca 2014; Ribeiro et al. 2014; Madi et al. 2017). Gorgoso-Varela and Rojo-Alboreca (2014) compared the Weibull function fitted by three methods (maximum likelihood, moments and percentiles) and Johnson’s SB function fitted by four methods (conditional maximum likelihood, moments, mode and Knoebel–Burkhart) for Betula pubescens (birch) stands and found that the best fits were made using percentiles and maximum likelihood methods for this function. Scolforo et al. (2003) verified that Johnson’s SB distribution accurately describes the diameter distribution of Pinus taeda. These authors also found that the moments and maximum likelihood methods provided the best fits. Palahí et al. (2007), comparing the Beta, Johnson’s SB, Weibull and truncated Weibull functions to describe diameter distributions of forest stands in Catalonia, found that the truncated Weibull was the best function, followed by the Beta function. Gorgoso-Varela et al. (2008), when fitting the Beta distribution to model the diameter distribution for Betula alba L. (birch) and Quercus robur (pedunculate oak) dominated stands, had satisfactory results, with percentages of rejection by the KS test of just 0.8% for the total number of cases in birch stands and 1.2% in pedunculate oak stands, at a significance level of 5%. In this study, we found the best values for the location parameter to vary between 0 and 60% of the minimum observed diameter. Similarly, Scolforo et al. (2003),

Johnson’s SB parameters ε, γ and δ were highly correlated with mean diameter, and λ was best correlated with age. Weibull’s parameters α and γ were best correlated with mean diameter, whereas β was best correlated with age. Error by diameter class Figure 1 shows MAE and bias by number of trees and basal area per hectare per diameter class (a, b, c and d, respectively) for the fitting methods highlighted in Table 4. Greater MAEs were concentrated around the class of 15.5 cm for both stand density and basal area. For bias, the errors were concentrated around the 12.5 and 22.2 cm classes. The function with the larger errors was Gamma, especially for MAE in number of trees per hectare. Johnson’s SB, Weibull and Beta had similar errors for MAE. Discussion Selecting the best probability density function Based on bias, MAE and the percentage of conformance of fitted and observed values according to the KS test, all functions performed well. The exceptions were the Johnson’s SB fitted by the linear regression and mode methods. The best functions and fitting methods were, in top-down order, Johnson’s S B method of moments and maximum likelihood both using the location parameter as 25% of the minimum observed diameter, the Weibull 25–97 percentile method using 50% of minimum observed diameter, and the Beta function fitted with the method of moments. These functions have been successfully used in (a)

Beta Gamma Johnson’s S B Weibull

10 8 6 4

2 1 0.5 0 −0.5 −1 −1.5 −2

2

−2.5

0 (c)

(d)

0.25

0.05

0.2

BIAS (m2 ha−1)

MAE (m2 ha–1)

(b)

1.5

BIAS (stems ha−1)

MAE (stems ha–1)

12

0.15

0

−0.05

0.1

0.05

−0.1 −0.15

0 0

10

20 30 DIAMETER CLASS (cm)

0

10

20 30 DIAMETER CLASS (cm)

Figure 1: Mean absolute error (MAE) and bias for error in stand density (stems ha−1) (a and b, respectively) and MAE and bias for error in basal area (c and d, respectively) by function

Southern Forests 2018: 1–8

7

studying Pinus taeda (loblolly pine) plantations, found that values of up to 45% of observed minimum diameter can be safely used in fitting Johnson’s SB function. GorgosoVarela and Rojo-Alboreca (2014), studying pedunculate oak (Quercus robur) and birch (Betula pubescens) stands, observed that 10% of minimum observed diameter was the best value for the location parameter for the Weibull and Johnson’s SB functions. Figure 2 illustrates the fit of the different functions at ages 1.3, 3, 6.3 and 14 years, on random plots. The theoretical and observed diameter distributions became left asymmetric with age because trees were getting larger over time (Figure 2). In these cases, there were no poor fits and all the theoretical distributions showed no significant differences with the observed values according to the KS test. This was expected because even the poorest distribution (Gamma) showed high conformance between observed and predicted values (95.67%). Correlation of pdf parameters with stand attributes Parameters of the Beta function (α and β, shape and scale, respectively) showed low correlation with stand attributes. The parameter α was best correlated with mean dominant height (r  2 = 0.12) and the parameter β was highest correlated with mean diameter (r 2 = 0.21). For the Gamma function, the shape parameter (a) was highest correlated with the inverse of mean dominant height, but it was low (r  2  = 0.14). The scale parameter (β) was most highly correlated with age (r 2 = 0.76). In this study, as the stand aged, the parameters α and β decreased (Table 5). The

70

(a)

same trend was observed by Ribeiro et al. (2014) studying Eremanthus erythropappus stands in Brazil. Weibull parameters α, β and γ are location, scale and shape parameters, respectively. In he present study, α and β were highly correlated with mean diameter (r 2 = 0.90) and age (r 2= 0.92), respectively. Parameter γ had the highest correlation with mean dominant height (r 2 = 0.13). In this study, α and β increased with age, whereas γ decreased. Ribeiro et al. (2014) also found an increasing trend for α and β with time. Similarly, Nokoe and Okojie (1984), studying African mahogany plantations in Nigeria, found that location and scale parameters for the Weibull function increased with increasing stand age, and the shape parameter decreased. Johnson’s SB parameters of location (ε) and scale (λ) were highly correlated with mean diameter (r 2 = 0.9) and age (r 2 = 0.94), respectively. Shape parameters δ and γ (kurtosis and symmetry) were most highly correlated with mean diameter (r 2 = 0.50) and the inverse of it (r 2 = −0.39). Scolforo et al. (2003) also found that the scale parameter (λ) was highly correlated with age. It is desirable that parameters be highly correlated with stand attributes, especially those easily measured and estimated. Thus, diameter distributions can be estimated by estimating stand attributes. Some authors have taken this approach such as Cao (2004), Leite et al. (2013), Palahí et al. (2006) and Schneider et al. (2008). Errors by diameter class All functions had similar errors in distribution by diameter class (Figure 1), except for the Gamma distribution, whose

Observed frequency Beta moments Gamma moments S B moments (dmin*0.25) Weibull percentille (25−97)(dmin*0.5)

60 50

(b)

STAND DENSITY (stems ha−1)

40 30 20 10 0 70

(c)

(d)

60 50 40 30 20 10 0

5

10

15

20 25 30 35 40 45 50 55 60 65 70

5

10

15

DIAMETER CLASS (cm)

20 25 30 35 40 45 50 55 60 65 70

Figure 2: Observed and theoretical diameter distribution fitted by Beta moments, Gamma moments, Johnson’s SB maximum likelihood and the Weibull percentile methods for plots at ages 1.3 (a), 3 (b), 6.3 (c) and 14 (d) years

8

Mayrinck, Ferraz Filho, Ribeiro, Mendes de Oliveira and Lima

errors were the higher, compared with those of the other pdfs. Similarly, Maltamo et al. (1995) compared the Beta and Weibull distributions in describing basal area distributions in Scots pine (Pinus sylvestris) and Norway spruce (Picea abies) stands and concluded that the Weibull and Beta functions presented similar errors. In the present study the errors were greater between diameter classes of 12 and 19 cm, where the frequency of trees were higher. The results are in accordance with Gorgoso-Varela et al. (2008), who found greater errors in the diameter class of 15.5 cm, which was the diameter class with the highest number of trees. Conclusion All tested functions yielded good fits for predicting diameter distributions for African mahogany plantations. The best fitting methods were Johnson’s SB moments, Johnson’s S B maximum likelihood, and the Weibull percentile methods. The best value for the location parameter for Johnson’s SB function was 25% of the minimum diameter of observed data, and for the Weibull function 50% of the minimum diameter of observed data. The worst method was Johnson’s SB regression. For both Johnson’s SB and Weibull functions, location and scale parameters were highly correlated with mean diameter and age, respectively. This is positive, because it can ease the prediction of diameter distributions by using stand attributes. ORCID Rafaella Mayrinck https://orcid.org/0000-0001-7772-6502 Antonio Ferraz Filho https://orcid.org/0000-0001-9178-918X Andressa Ribeiro https://orcid.org/0000-0002-8923-1395 Ximena Mendes https://orcid.org/0000-0002-9887-7075

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Received 10 September 2017, revised 7 February 2018, accepted 18 March 2018