## Nonparametric Density Estimation Method

Once a set of climatic variables is identified, multivariate nonpara-metric density estimation is applied to estimate probability density functions of each vegetation type. Nonparametric density estimation methods are well understood and have been increasingly used in practice for both univariate and multivariate analysis in such areas as chemical and electrical engineering and medical biostatis-tics. A comprehensive review of different density estimation methods can be found in Silverman (1986). The advantages of this approach are: (1) the class conditional distribution estimates and the decision surfaces are more flexible, (2) a more direct and significant role is given to the data, (3) no a priori assumptions about the parametric family of distribution are required, and (4) both categorical and continuous variables can be used as explanatory variables.

In a nonparametric density approach, every observation represents a center of the sampling interval. By placing a kernel density K(x) around each observation one can define a probability density function:

N is a number of observations, h is a smoothing parameter or bin width,

X is the i- th observation of a variable x.

The kernel density estimator can be thought as a sum of "bumps" centered at each available observation (see Fig. 6.1, adapted from Silverman 1986). The type of a kernel density function determines the shape of the bumps. The bin width determines the smoothness of the estimator. Unlike histograms, kernel density estimators are continuous. There are different types of kernel den-

Figure 6.1. Schematic representation of a kernal density estimation.

K(x) is a kernel density function such as = 1,

Figure 6.1. Schematic representation of a kernal density estimation.

sity functions: Gaussian, triangular, rectangular, and so forth. The Gaussian kernel was used in this model. The window width affects the smoothness of the estimator. With wider windows, the distributions obtained become less jagged.

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