Membership function piecewise linear

Triangular (Figure 8.7), piecewise linear (Figure 8.8), or trapezoidal (Figure 8.9) functions are commonly used as membership functions because they are easily prepared and computationally fast. 

The output of a neurofuzzy (sub)model is simply a piecewise linear polynomial in other words, the continuous output is produced from a finite number of linear segments. How many pieces can be fitted (i.e., the complexity of the model) depends on the number of multivariate membership functions that can be used, and this relates directly to the amount of data that is... 

For any element x of universe X, membership function p,fx) equals the degree to which X is an element of set A. This degree represents the degree of membership, also known as the membership value of element x in set The most commonly used membership functions are triangular, trapezoidal, piecewise linear and Gaussian functions because they are easily prepared and computationally fast. The choice of membership functions is largely arbitrary because there is no theoretical justification for using one rather than another. The number of... 

It has been found in practice that the functional type of the membership function is less important than the fact that it is monotonic. Therefore, the substitution of piecewise linear functions, such as trapezoidal or triangular membership functions, by other more complicated functions, e.g., exponential ones, will often have little influence on the final conclusions to be drawn. Computational constraints may restrict the possible complexity of the membership functions or preferable mathematical properties (derivability) may prescribe their use. The concept can be generalized to specification of higherdimensional membership functions for multivariate problems. 

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