Since the 1960s', by the symbol of "Fuzzy sets" of Zadek (1965), Fuzziness notion was swiftly formed in every branch of mathematics, and Fuzzy analysis method has been widely applied in many fields(William et al., 1995; Arasan et al., 1996; Zimmermann, 1996; Kivinen et al., 2002; Zhang et al., 2004). Stand diameter structure is the most important and basic structure of a forest stand. Generally, two kinds of methods were adopted to model stand structure, i.e. parameter methods and non-parameter methods(Maltamo et al., 1998). In the early studies of parameter methods, probability density function was widely applied, but the prediction precision of these methods was not satisfactory (Bailey et al., 1973; Davis et al., 1987; Zeide, 1993; Kangas et al., 2000; Li et al., 2002; Liu et al., 2002). After the mid 90' s in 20 centuries, Hui et al. (1995)based on the hypothesis that cumulative diameter frequency increment(dy/dx)was positively related to cumulative frequency and strained by the biggest cumulative frequency, applied "Logistic" population growth model to describe stand diameter cumulative distribution. Duan et al. (2003), on the basis of mathematical analysis, explored the application of theoretical growth equations to structure model, and resulting in good simulation effect. Studies on this aspect has received extensive attention(Gadow et al., 1998; Wu et al., 1998; Zhang et al., 2003). Non-parameter methods, such as k-nearest-neighbour estimation method, is a prediction method which is independent on any distribution function and based on weighted average of k-nearest-neighbour actual stand diameter distribution(Haara et al., 1997; Maltamo et al., 2000), this method has very strong flexibility, but is sophisticated. Some of the above-mentioned methods have showed great improvement in simulation precision, but still have some limits and shortages as follows: 1)While applying parameter methods to model stand diameter cumulative distribution, the theoretical foundation of the model does not directly build on distribution thought, but based on growth theory of population; 2)The relationship between stand characteristic factors and distribution parameters has not been sufficiently studied, which leads to having difficulty to confirm stand characteristic factors that can completely embody stand diameter structure; 3)The essential reason that modeling precision of different distribution functions produces discrepancy has not been made clear. This paper applied Fuzzy distribution functions to describe stand diameter structure. Based on studies of five common Fuzzy distributions, a generalized Fuzzy distribution was made and the change pattern of its inflection point with stand factors was analyzed. Comparative analyses of simulation precision of nine equations including wellknown logistic equation and two theoretical growth equations (Korf and Gompertz), were conducted to reveal the essential reason for the discrepancy of modeling precision of models, in order to further improve modeling technique of stand diameter structure, and to provide scientific and reliable foundation for direct silviculture of Cunninghamia lanceolata plantations.

1 Data and Methods 1.1 Data

Trial plots were located in Dagangshan Mountain Experiment Bureau, Fenyi County, Jiangxi Province, China, experiencing a subtropical climate. The altitude is 114°33′E, the latitude is 27°34′N. The elevation is 250 m. Soil is yellow brown, developed from sand-shale rock. Mean annual temperature, rainfall and evaporation are 16.8 ℃, 1 656 mm, 1 503 mm respectively.

Cunninghamia lanceolata stands were established in 1981, planting density was limited within an optimum range according to managerial purposes, respectively 1 667(A), 3 333(B), 5 000(C), 6 667(D), 10 000(E)tree·hm^-2, 15 plots, a random block design with 3 replications was used. The plot size was 0.06 hm². Two lines of trees were planted as buffer zone with the same spacing. All trees were marked for continuous measurement. Stem diameter at breast height (DBH)was measured after tree height reached 1.3 m. All stands were measured every year for 10 years, and every 2 years afterwards. Before 1999, measurements had been performed ten times. Self-thinning occurred in some stands over the experimental period, and all stands were unthinned. Information on stands was described in Tab. 1.

1.2 Selection of Fuzzy distribution function

After stands were measured, a series of stem DBH of stands were obtained. Size-class interval was 2 cm, Fuzzy statistics were made and many Fuzzy numbers were obtained, constituting a Fuzzy number group. Then tree stems, corresponding to the medium value of each size-class were counted. Relative frequency distribution was calculated and a cumulative distribution series was obtained, which falls within the range of [0, 1], and can be defined as a cumulative percentage distribution series that is smaller than or equal to some diameter classes.

Because this cumulative percentage distribution series presents non-linear characteristics, five typical membership functions were selected such as Γ-typed distribution, normal type, Cauchy distribution and so on, these functions are all non-linear and Sshaped. In addition, because the distribution series presented increasing trend, inclined-bigness types of above-mentioned distribution forms were selected. When the distribution interval is real number set, the membership function is called Fuzzy distribution. In order to describe conveniently, five Fuzzy distribution functions were respectively marked by Fuzzy-Γ₁、Fuzzy-Γ₂、Fuzzy-Γ₃、Fuzzy-Γ₄及Fuzzy-C. Mathematical expression of each distribution was presented as follows.

1) Fuzzy-Γ₁(Inclined-bigness type of Γdistribution)

2) Fuzzy-Γ₂(Inclined-bigness type of normal distribution)

3) Fuzzy-Γ₃

4) Fuzzy-Γ₄

5) Fuzzy-C(Cauchy distribution)

Where a >0, b >0, k >0. Fig. 1 presents the basic figures of the first four distributions.

1.3 Mathematical analysis of three equations including Korf

In order to analyze the characteristics of Fuzzy distribution functions, three theoretical growth equations including Logistic, Korf and Gompertz were introduced to make comparisons, their mathematical expressions were described in Tab. 2. When parameter k of each equation is equal to 1, the value range of each equation is [0, 1], so, these equations can be applied to model cumulative diameter distribution of stand (Duan et al., 2003).

1.4 Parameter estimation methods and comparison of simulation properties

Due to distributions used being non-linear functions, the original values of distribution parameters were obtained by empirical value or evaluated through stand diameter cumulative distribution curve. Then all Fuzzy functions were fitted to data sets using the NLIN procedure of SAS (SAS Institute, 1991). To ensure the stability of parameter estimates, the models were refitted several times with the initial values estimated by the previous fittings. Validation statistics used in the sectional performance test included the coefficient of determination (R²) and the standard error of the estimate (S). Through the calculation of two-rank derivative, the distribution interval of inflection point of each function was obtained. The calculation formula of S was set out as follows, where n indicates diameter-class numbers of stand, i indicates each diameter class.

2 Results and discussion 2.1 Simulation performance of five Fuzzy distribution functions

The parameters and fit statistics of five Fuzzy distribution functions were presented in Tab. 3.

Each parameter of the five Fuzzy distribution functions had a distribution range, and from Fuzzy-Γ₁ to Fuzzy-Γ₄, the values of parameters k and a of functions gradually descended. Except Fuzzy-Γ₁, the other four distributions had relatively high precision, their coefficients of determination (R²)were all above 0.97 (Tab. 3), showing that the four distribution functions had better simulation properties for stand diameter cumulative distribution. The maximum R² of Fuzzy-Γ₁ was 0.971 3, less than the minimum R² (0.972 7) of the other four distributions. This showed that Fuzzy distribution functions without inflection point had worse simulation properties than those with inflection point. For S statistics, the order of five Fuzzy distributions was Fuzzy-Γ₃ >Fuzzy-Γ₄ > Fuzzy-C > Fuzzy-Γ₂ > Fuzzy-Γ₁. Considering the values of inflection points of five Fuzzy distributions, conclusion could be made that Fuzzy distributions with inflection point at about 0.5 had better simulation properties for stand diameter cumulative distribution (Tab. 3).

The maximum and minimum coefficients of multiple determination of Fuzzy-Γ₃ were relatively high, so this distribution was selected to study the relationship between distribution parameters and stand factors.

Parameters k and a had obviously descending trend with the change of stand age and density (Fig. 2). After comparative analysis, relationship between parameter k and stand age could be expressed by power function: k =0.158 4x^{-1.650 3}, the correlation coefficient (R) was 0.9. The relationship formula between parameter a and stand density adopted multinomial: a = 10.367 0-0.002 5x +1 ×10^-7 x², its correlation coefficient was 0.915 2. The ANOVA showed that two coefficients were significant at the 0.000 1 level. It was obvious that parameter k and a respectively had very close relationship with age and density. The ANOVA showed that parameter k, a respectively had weak relationship with density and age.

The ultimate aim of all simulation trials is to predict future stands. And prediction methods are often pursued to be both scientific, reliable and simple, workable. For Fuzzy-Γ₃, its simulation precision was relatively high in the above-mentioned Fuzzy distributions, and its parameters had close correlation to stand factors, so parameter prediction method could be adopted to predict stand diameter distribution. For Fuzzy-Γ₃ distribution, it is only needed to know future stand age and current stand density, the future diameter distribution of un-thinned stand can be obtained by an effective prediction after stand natural-thinning model has been built.

2.2 A generalized Fuzzy distribution function

For the first four distributions in Tab. 3, their mathematical expressions were very similar, only the power index of independent variables was different, and this index determined the inflection point of each equation. Simulation precision of the four equations had obvious discrepancy when their inflection point was different. By introducing a parameter, a generalized Fuzzy distribution function was obtained, its mathematical expression is as follows:

While parameter c >1, the generalized Fuzzy distribution has a variable inflection point, so that it is expected to have the better simulation properties, we call it a generalized Γ-typed distribution, denoted Fuzzy-Γ₅.

The fit statistics of Fuzzy-Γ₅ were presented in Tab. 4.

The determination coefficient (R²) and statistics S indicated that Fuzzy-Γ₅ had the better simulation properties than the above-mentioned five distributions such as Fuzzy-Γ₃ and so on. The result of Duncan's multiple range test showed that the modeling precision of the six kinds of Fuzzy distribution including Fuzzy-Γ₅ were obviously different each other. The value of parameter c was bigger than 1, so Fuzzy-Γ₅ distribution had an inflection point. To some extent, this showed that diameter cumulative distribution of Chinese Fir plantation was a S-shaped distribution, with down convex first, followed by up convex. 87.33% of the values of parameter c was in [2.5, 5.5], and about 71.33 percent in [2.5, 4.5], which showed that the value of parameter c was mostly nearby 3 and 4, and the forms of cumulative diameter distribution mainly showed on the nearby types of Fuzzy-Γ₃ and Fuzzy-Γ₄.

While applying Fuzzy-Γ₅ distribution function to predict stand diameter distribution, parameter recovery method could be adopted.

2.3 Relationships between parameters of the generalized Fuzzy distribution function and stand factors

Discussion of relationship between parameters of distribution functions and stand factorsmay benefit making clear factors that affect stand diameter distribution and their relative action extend, and be beneficial to explore the composition pattern of stand factors, when either parameter prediction method or parameter recovery method are adopted.

The result of stepwise regression of parameters of Fuzzy-Γ₅ and stand factors was showed in Tab. 5, the stand factors were site index (x₁), age (x₂), density (x₃), average diameter (x₄), basal area diameter (x₅), average height (x₆) and dominant height (x₇), the order that stand factors entered regression equations and the main factors that got significant at 0.05 test levelwere showed in Tab. 5. Parameters a, k, c were respectively related to minimum diameter, relative accumulation rate and distribution figure. Parameter b, used to describe k^-1/c, stands for scale parameter of distribution function. From Tab. 5, we could get following conclusions:stand age, density and dominant height had relatively obvious effects on parameters. Parameters a and b both increased with the increasing of stand age and dominant height, while decreased with the increasing of stand density, which corresponded to the biological sense of two parameters. Parameter k firstly decreased then increased with the increasing of stand age, and presented positive correlation to stand density. Parameter c is the parameter of inflection point of Fuzzy-Γ₅ distribution function, its value decreased with the increasing of stand age and density.

For man-made forests, stand density and site are two main factors that can be controlled by man. These two factors can affect distribution parameters, further effect on stand diameter distribution, so, mankind can directly culture and accomplish satisfactory stand diameter structure through selecting proper site and density.

2.4 Precision comparison and analysis of Fuzzy distributions and theoretical growth equations

To further understand simulation properties of Fuzzy distribution functions and make clear substantial factors that make discrepancies in simulation precision of Fuzzy distribution functions, a population dynamic model (Logistic) and theoretical growth equations (Korf and Gompertz)were used to analyze the simulation precision. For Korf equation, it was the first time to be applied to stand diameter structure. The mathematical expressions of three equations were presented in Tab. 2.

When parameter k of each equation was defined as 1, the value intervals of equationswere [0, 1], so these equations can be used to model stand diameter cumulative distribution. The inflection points and simulation precision of the nine distributions were presented in Tab. 6.

For S statistics, the order of the nine equations was Fuzzy-Γ₅ >Logistic >Fuzzy-Γ₃ >Fuzzy-Γ₄ >Fuzzy-C >Fuzzy-Γ₂ >Gompertz > Korf > Fuzzy-Γ₁(Tab. 6). Except Fuzzy-Γ₁, all Fuzzy distribution functions, especially the generalized Fuzzy-Γ₅, displayed relatively high simulation precision. The precisions of the eight distributions were presented in Fig.3.

Inflection point indicates the place where the biggest variation takes place above the curve of diameter cumulative distribution. While modeling data sets, the size of inflection points of Fuzzy distribution functions may produce important affection on simulation precision.

Fuzzy-Γ₁ had no inflection point (Tab. 6), and the inflection points of Fuzzy-Γ₅, Korf and Gompertz had changeable intervals, while all the other five distributions had fixed inflection points. The distribution with the highest simulation precision had a flexible inflection point (Tab. 6, Fig.3), which showed that inflection point of diameter cumulative percentage distribution was not a fixed value, but a changeable range. And because the interval of inflection point of Fuzzy-Γ₅ distribution was 0.382 4~0.586 7, and 98 %of which was within the range of [0.4, 0.6], so it could be concluded that the main range of inflection points of stand diameter cumulative distribution was 0.4~0.6.

In addition, Tab. 6 showed that from left to right inflection points of distribution curve increased by subsequence. The simulation precision of Fuzzy-Γ₃, Logistic and Fuzzy-Γ₄ distribution functions whose inflection points were fixed and lay in the main interval of inflection point of stand diameter cumulative distribution were relatively high. Furthermore, for functions, with inflection points being fixed, their precision decreased from flection point about 0.5 or so to both sides. The results showed that inflection point of diameter cumulative distribution also had a central distribution point (0.5 or so)besides a main inflexible interval (0.4~0.6). For average precision of samples, when the inflection point of distribution function lied in the main interval, its precision was relatively high, and the closer to the central point, the higher the precision was.

3 Conclusion

In view of the correspondence of mathematical characteristics and distribution range of stand diameter cumulative percentage series, Fuzzy distribution functions were used to model diameter distribution of Chinese Fir plantation. Due to the difference of expression formula of Fuzzy distributions, their simulation properties produced obvious discrepancy. Among Fuzzy functions used, Fuzzy-Γ₁, with no inflection point, had the minimal simulation precision; Fuzzy-Γ₅, the generalized Fuzzy distribution function with variable figure parameter, proved to have the highest precision. The result of parameter analysis showed the parameters of Fuzzy-Γ₃ had deep correlation with stand age and density and the minimum diameter parameter, scale parameter, figure parameter of Fuzzy-Γ₅ had significant correlation with stand characteristic factors including age, density and dominant height. Based on comparison of simulation precision of Fuzzy distribution and theoretical equations such as Logistic, the results showed that inflection point of distribution function had the essential action, and Inflection points of diameter cumulative distribution had a main flexible interval (0.4~0.6) and a central distribution point(0.5 or so). For average simulation precision, when the inflection point of distribution function lay in the main interval, its precision was relatively high, and the closer to the central point, the higher the precision was.

Introduction of Fuzzy distribution functions into the simulation of stand diameter distribution is of benefit to the correct identification of diameter distribution and the development of simulation technique. The conjunction of stand factors and the key characteristics of distribution functions (inflection points)are beneficial to making clear the essential factor that causes discrepancy of the simulation properties of distribution functions, which has important implications in the selection of suitable stand diameter distribution for specific stands.