The fuzzy algorithm uses the membership functions and the rules in three steps:

1. fuzzification: for every input x = x2, ...) all membership functions ^i(x) will be calculated;

2. inference:

(a) assigns a value akm using the minimum or product operator to each rule (k points to the outputs; m points to different rules with the same output k):

(b) selects the best rule from the m rules with same output k using maximum operator ak = max{ a], a],...} [8]

The defuzzification algorithm assigns the fuzzy output of the fuzzy inference to one floating point value. This value can be used for further calculation, for example, in a spatial model. The realization of the defuzzification algorithm depends on the character of the output values. For crisp outputs the algorithm is simple and fast:

J2kak x

J2kak

In eqn [9] the activity of a rule ak and its output value ok are used to calculate the output o of the fuzzy system. The realization is fast, which is a precondition for the usage in a spatial simulation. In a spatial simulation a huge number (up to some millions) of grid cells must often be calculated.

A further advantage of crisp output values is not so obvious. This simple algorithm can produce linear functions as well as nonlinear functions. Using a fuzzy set as output leads to a nonlinear behavior. Most ecological processes are nonlinear, but sometimes a piecewise linear part is essential. The modeler must be able to decide the functional behavior of the model. A more demanding modeling approach using outputs as fuzzy sets can be used at the expense of a bit of nonlinearity in the model.

When using fuzzy sets as outputs there are some possibilities for defuzzification. One of the most commonly used is the so-called 'center of gravity' method. To

Figure 1 Center of gravity defuzzification understand this method, a closer look at the fuzzy inference is necessary. The inference procedure assigns a value akm to each rule. After the selection of the best value for a specified output ak (eqn [8]), this value is multiplied to the output. In case of fuzzy set output the multiplication 'cuts' the fuzzy set. A result of the inference procedure may look like Figure 1.

To determine one floating point value from this area the following equation is used:

Equation [10] is a generalization of eqn [9]. The center-of-gravity algorithm is numerically more complex than the simple eqn [9]. Additionally, the integral in eqn [10] introduces the aforementioned nonlinearity in the system. (This can be checked using a simple fuzzy model with one fuzzy input and one fuzzy output.)

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