## Training of Fuzzy Models

A fuzzy system is just a way to formulate heuristically basis functions for regression and controller functions with a rule table instead of by optimization. The fuzzy modeling approach is a transformation of expert knowledge into a mathematical model. The expert can control this process by defining the membership functions, the outputs and the rules. For modeling tasks like the one our example, this is the only possible way. Usually the expert has lots of experience but little observation data. (Otherwise the modeler should think about a artificial neural network.) When the bird is not as rare as the lesser spotted eagle used in the example below, more data are frequently available. In other investigations the data collection process is part of a project. So the modeler can construct a fuzzy model and wants to enhance it with the collected data set. Often used as an optimization strategy is the minimization of the mean square error (mse) between the modeling output and the observed data:

Figure 3 Fuzzy AND Prod, x-axis: coverage degree, mean distance to structure = 130m.

where y i is the measured value and the fuzzy (xn, x2¡, x3i) the output of the fuzzy model. There are three possible strategies to train such a fuzzy model:

• to change the membership function for the inputs;

• to change the outputs; and

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