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Diameter Structure Analysis of Forest Stand and Selection of Suitable Model

Tsogt Khongor1, Chinsu Lin1* and Zandraabal Tsogt2
1Department of Forestry and Natural Resources, National Chiayi University, 300 University Road, Chiayi,
60004, Taiwan, *corresponding author: e-mail: chinsu@mail.ncyu.edu.tw
2Department of Forest Resources, Institute of Botany, Mongolian Academy of Sciences, Jukov Avenue 77,
Ulaanbaatar 210351, Mongolia


Ecologically and economically it is important to understand how many tree stems are in each diameter class. The purpose of this study was to fi nd larch forest (Larix sibirica) diameter distribution model among Weibull, Burr and Johnson SB distributions. Inventory was conducted near Gachuurt village, Ulaanbaatar, Mongolia. The goodness of fi t test were accompanied with Kolmogorov- Smirnov, Anderson-Darling and Chi-Squared tests for distribution models. Study result shows Johnson SB distribution gave the best performance in terms of quality of fi t to the diameter distribution of larch forest.

Keyword: diameter distribution model,Weibull,Burr,Johnson SB,larch


Detailed information of forest stand is crucial for forest research and planning. This information used for input of ecosystem modeling and/or forest growth and yield models. In the analysis of stand dynamics, detailed data for all trees on a plot is often lacking. In such case, we may generate missing data using various theoretical diameter (D) distributions. For many years there were various activity and interest in describing the frequency distribution of D measurements in forest stands using probability density functions. First study of D distribution mathematical description was negative exponential (DeLiocourt 1898), and since then, researchers used various distributions. All distribution models have their advantage and sensitive in specifi c shape. Weibull distribution able to describe Exponential, Normal and Lognormal distribution shapes (Bailey & Dell, 1973; Lin et al., 2007), while Burr distribution cover much larger area of skewness and kurtosis plane than the Weibull distribution (Lindsay et al., 1996). Moreover, it is closely approximate with above mentioned distributions plus Gamma, Logistic and several Pearson type distributions. Johnson SB distribution cover different region of skewness and kurtosis plane than the Burr (Johnson, 1949; Hafl ey & Schreuder, 1977), and it is closely approximate Beta and generalized Weibull distributions. In case of Mongolian forests, Khongor et al. (2011a) published the birch forest D study using Weibull and Lognormal distributions and compared the accurateness of these models. For larch forest D distribution, Khongor et al., (2011b) used Exponential, Lognormal and Gaussian (or Normal) distributions, but they did not used Weibull, Burr and Johnson SB before. The purpose of this study is to investigate the suitability of the Weibull, Burr and Johnson SB distributions for modeling D distribution of larch forest (Larix sibirica).
Weibull distribution One of the most popular models is the Weibull distribution, fi rst introduced to the forestry research fi eld by Bailey and Dell (1973). The popularity of the Weibull distribution depends largely on its simplicity and yet relatively good fl exibility. It describes the inverse J shape for $<1 and the exponential distribution for $ =1. For 1< $ <3.6 the density function is mound shaped and positively skewed and for $ =3.6 the density function becomes approximately normal. If $ >3.6 the density function becomes increasingly negatively skewed. With the support random variable x: the Probability Density Function (pdf) of Weibull 3 parameter distribution is given as:

where, $ > 0 - shape parameter, ' > 0 - scale parameter, * - location parameter if *=0, then the distribution is 2 parameter. Burr distribution The Burr distribution was introduced to the forestry research by Lindsay et al. (1996). This distribution is inherently more fl exible, because it covers a much larger area of the skewnesskurtosis plane than the Weibull distribution (Lindsay et al., 1996; Rodriguez, 1977; Tadikamalla, 1980). The Burr (Zimmer & Burr, 1963) distribution has a fl exible shape, controllable scale and location, which makes it appealing to fi t to data. It is sometimes considered as an alternative to a Normal distribution when data show slight positive skewness. With the support random variable x: , the pdf of Burr 4 parameter distribution is given as:
where, k, $ > 0 - two shape parameters, ' > 0 - scale parameter, * - location parameter if *=0, then the distribution is Burr 3 parameter. Johnson SB distribution The Johnson SB (1949) have been much commonly used in forest distributional studies (Hafl ey & Schreuder, 1977), because of its fl exibility of distributional form and its ability to represent equally well positive and negative skewed distributions. The pdf of SB distribution transforms a bounded random variable by subtracting the minimum and dividing by the range. The logit of this transformation is then distributed as a standard normal variable. Following Johnson, consider this transform z on the random variable x:
where, [ - minimum value of x, - maximum value of x.
Within our context x is a D measurement. Then the pdf of D is defi ned as
where, * and + shape (+ > 0), ? scale (? > 0) and [ location parameter.

Material and Methods

Field measurement. Study plot was selected near the Gachuurt village in the vicinity of Ulaanbaatar city, Mongolia, located at 48°00’18.9’’N and 107°13’23.1’ E with altitudinal elevation 1607-1627 m above sea level. The forest consisted of natural stands and any management activity had taken previously. Inventory was conducted in summer of 2009. Composition of the stands is pure larch. Plot size was 0.2 ha, i.e. 40 x 50 m in area. D were measured for all trees >1.3 m, and totally 275 stems were counted. Average D of tree stands in plot was 14.6 cm with standard error mean 0.497 cm. Diameter of tree stems ranges from 2 to 32 cm. D distribution skewness value was 0.38 indicating that the tail on the right side of the pdf is longer than the left side and kurtosis value - 0.97 indicating statistically fl attered peak.
Data analysis. The goodness of fi t of empirical D distribution was tested using three theoretical distributions: Johnson SB, Weibull and Burr. The distribution parameters were estimated using the EasyFit 5.5 Professional distribution fi tting software (Table 2). To calculate goodness-of-fi t of the actual D and height distributions with theoretical distributions, the KS (Kolmogorov-Smirnov) test, 2 test and AD (Anderson-Darling) test (Anderson & Darling, 1952) were used. KS test is distribution free and based on empirical distribution. It is used for continuous distributions and compares curves maximum distance. It is more sensitive near the center of distribution than at the tails. The 2 test divides the range of the data into a set of equiprobable classes.

AD test is a statistical test of whether there is evidence that a given sample of data did not arise from a given probability distribution. In its basic form, the test assumes that there are no parameters to be estimated in the distribution being tested, in which case the test and its set of critical values is distribution-free.


The parameter estimates of the three models are given in Table 1. The predictions from each model were compared with observed frequencies. The KS, AD and 2 tests and P value for KS and 2 tests were computed for each model (Table 2). All tested distribution models Table 2. Summary of empirical diameter distribution for larch forest ($=0.05) Distribution Kolmogorov Smirnov (critical value 0.08189) Anderson Darling (critical value 2.5018) Chi-Squared (critical value 15.507) statistic P-value statistic statistic P-value Johnson SB 0.03106 0.94589 0.18795 7.6044 0.47303 Weibull 0.06891 0.14004 1.8136 10.535 0.22946 Burr 0.07506 0.08567 1.909 13.562 0.09392 were statistically fi tted with observed diameter distribution and among them Johnson SB distribution was more fl exible than Weibull and Burr distributions. By the defi nition, the area under the pdf graph must equal 1, so the theoretical pdf values have to be multiplied by the total number of stems to match the histogram and the D coverage of bin width to calculate the number of stems in each D class. Tree stems are smoothly distributed in diameter classes and it is statistically unimodal.

It is easy to fi t such distribution, but here fl attered peak is problem that causes under/ over prediction. Though all models passed on goodness of fi t test, Weibull and Burr models over-predict D classes around 10-16 cm and lower-predict 4-6 cm and 24-30 cm classes. It is evident that the Johnson SB model was more fl exible in fi tting fl attered D distribution of larch forest stand (Fig. 1).


Weibull and Burr theoretical distributions fi t the best for right tailed D distributions whilst Johnson SB distribution has ability to represent equally well right and left tailed distributions. With this reason we have chosen the Weibull, Burr and Johnson SB distributions to test their suitability and fl exibility for larch forest D distribution. Then, our study result suggest Johnson SB distribution for larch forest and that is would not be necessary to believe about the Johnson SB distribution is the best for all over larch forest in Mongolia. Every forest stand D distributions are different depending on the site quality, climatic condition and history of natural or human disturbances. Supposedly, empirical distribution models would accurately work in big scale if the geographic and climate conditions are same. But, random disturbances, such as forest fi re, insect invasion or selective logging are change the forest structure and shape in different forms. Specially, every forest ever infl uenced with forest fi re in Mongolia and near urbanized areas all forests under danger of illegal timber logging. However, it is still important that stand specifi c forest structure information for model development and research or management planning in small scale forest area. If we needed bigger scale as regional forest D structure, we have to collect more stand D data to fi t general D distribution. The required amount of stem numbers or sample plots for regional D distribution study would be defi ned by stability of a chosen model. If the one fails we need to collect more stand samples and do it again until it become statistically stable. Westphal (2006) suggested that for the regional scale diameter distribution is reverse J shaped because of many small stems and relatively fewer big stems. Strong intensity disturbances or high intensity regeneration may change D structure as bimodal. If disturbance happened repeatedly in same forest, then the D distribution would forms multimodal shape. In this study, we used unimodal D distribution. However, it may not be suffi cient when a frequency distribution is reverse J with hump, bimodal or multimodal, and therefore, irregular shaped D distributions should have tested by mixture distribution (Zhang & Liu, 2006).


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