Probability density functions for description . . . PROBABILITY DENSITY FUNCTIONS FOR DESCRIPTION OF DIAMETER DISTRIBUTION IN THINNED STANDS OF Tectona grandis

ABSTRACT: The objective of this study was to evaluate the effectiveness of fatigue life, Frechet, Gamma, Generalized Gamma, Generalized Logistic, Log-logistic, Nakagami, Beta, Burr, Dagum, Weibull and Hyperbolic distributions in describing diameter distribution in teak stands subjected to thinning at different ages. Data used in this study originated from 238 rectangular permanent plots 490 m2 in size, installed in stands of Tectona grandis L. f. in Mato Grosso state, Brazil. The plots were measured at ages 34, 43, 55, 68, 81, 82, 92, 104, 105, 120, 134 and 145 months on average. Thinning was done in two occasions: the first was systematic at age 81months, with a basal area intensity of 36%, while the second was selective at age 104 months on average and removed poorer trees, reducing basal area by 30%. Fittings were assessed by the Kolmogorov-Smirnov goodness-of-fit test. The Log-logistic (3P), Burr (3P), Hyperbolic (3P), Burr (4P), Weibull (3P), Hyperbolic (2P), Fatigue Life (3P) and Nakagami functions provided more satisfactory values for the k-s test than the more commonly used Weibull function.


INTRODUCTION
Wood from Tectona grandis L. f. ranks high in the global market, particularly for the main intended purpose of furniture making, it being necessary to adopt thinning procedures in order to boost tree diameter for multiple product applications, along with artificial pruning in order to obtain better quality wood in the final cutting process (LEiTE et al., 2006).
Making projections of diameter distribution is critical for the management of forests intended for multiple products, enabling a more accurate economic analysis (CLUTTEr et al., 1983).Projection could be done by using models that estimate diameter distribution based on a current distribution or based on the characteristics of the stand, enabling to simulate thinning procedures and to quantify multiple wood products (CLUTTEr et al., 1983;KNOBELL, 1986;NOGUEira et al., 2005;SiiPiLEHTO, 2007).

Binoti, D. H. B. et al.
a common characteristic in this type of modeling is the presence of a probability density function (pdf).
Despite the prevalence of the Weibull pdf, statistical studies have presented new functions with differing characteristics, flexibility and fitting capability.Depending on the type of stand and horizontal structure found, other functions might result in greater accuracy.Therefore, this study aimed to assess the effectiveness of Fatigue life, Frechet, Gamma, Generalized Gamma, Generalized Logistic, Log-logistic, Nakagami, Beta, Burr, Dagum, Weibull and Hyperbolic distributions in describing diameter distribution in teak stands subjected to thinning at different ages.

Data description
Data used in this study originated from 238 rectangular permanent plots 490 m 2 in area, installed in stands of Tectona grandis L. f. in Mato Grosso State, Brazil.These stands were located in a lowland region with geographical coordinates 15º 02' to 15º 11' south latitude and 56º 29' to 56º 35' west longitude, with initial spacing of 3.0 x 2.0 m.The average annual precipitation in the region is 1,300 to 1,600 mm, with six dry months, and the average annual temperature is 25.3 ºC.
The plots were measured at ages 34,43,55,68,81,82,92,104,105,120,134 and 145 months on average. in each plot and on each occasion, the diameter of all trees was measured 1.30 m above the ground (dbh).The stand was thinned on two occasions, the first thinning was systematic at 81months, with a basal area intensity of 36%, while the second was selective at 104 months and removed the poorer trees, reducing the basal area by 30% on average.
In order to test the goodness-of-fit of a function to data, the Kolmogorov-Smirnov test was used (GiBBONS;SUBHaBraTa, 1992;SOKaL;rOHLF, 1981).This test compares estimated cumulative frequency with observed frequency, the maximum difference being the test statistic (dn).All fittings were compared through graphical analysis of observed and estimated values and mean values of the K-S test.
Considering that these functions are used in diameter distribution models and considering, in this case, the need to obtain a significant correlation of their parameters with the characteristics of the stands (LEiTE, 1990) a correlation matrix was estimated between the parameters and the variables: maximum diameter, minimum diameter, quadratic diameter, age, number of trees, also correlating parameter values at age i with an earlier age i-1.

RESULTS AND DISCUSSION
Functions were fitted to each plot and measurement occasion, to a total of 2,856 fittings per function.Graphical analyses between estimated and observed values for three measurement occasions of a randomly selected plot are provided in Figure 1.Function fit assessment was based on K-S test values, the function providing the lowest mean values being considered the best function.results of the K-S test for function fitting are provided in Table 1.Table 2 illustrates correlations between stand characteristics and functions parameters.
The diameter distribution modeling in stands of Tectona grandis L. f. (teak) has been performed based on the prediction and/or projection of parameters of a statistical distribution, using regression models.The twoparameter Weibull function has been used in the majority of related studies on account of its flexibility and easy correlation of its parameters with stand characteristics (NOGUEira et al., 2005).The effectiveness of estimates generated by diameter distribution models is conditional not only on data quality and equation fitting quality but also on correlation capability between pdf parameters and easily measurable stand characteristics (LEiTE, 1990).
This study evaluated the application of functions with differing characteristics to describe the diameter structure of teak stands subjected to thinning.The Log-Logistic (3P), Burr (3P), Hyperbolic (3P), Burr (4P), Weibull (3P), Hyperbolic (2P), Fatigue Life (3P) and Nakagami functions provided more satisfactory values for the k-s test than the more commonly used Weibull function.
Nevertheless, all functions being analyzed in this study provided satisfactory results for use in growth and yield modeling of thinned teak stands.
The linear correlations between stand characteristics and function parameters revealed that inclusion of a location parameter may obstruct the construction of diameter distribution models.The Weibull (2P), Fatigue Life (2P), Frechet (2P), Generalized Logistic, Log-Logistic (2P), Logistic, Hyperbolic (2P) functions provided satisfactory linear correlations between parameters and stand characteristics, yet nonlinear correlations can be found and used in construction of diameter distribution models.

Table 1 -
Mean values of the Kolmogorov-Smirnov test for the tested functions.Numbers in brackets refer to the amount of parameters being used.