## Fuzzy Algorithm

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]

### 3. defuzzification. Defuzzification

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.)

## 10 Ways To Fight Off Cancer

Learning About 10 Ways Fight Off Cancer Can Have Amazing Benefits For Your Life The Best Tips On How To Keep This Killer At Bay Discovering that you or a loved one has cancer can be utterly terrifying. All the same, once you comprehend the causes of cancer and learn how to reverse those causes, you or your loved one may have more than a fighting chance of beating out cancer.

Get My Free Ebook