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that form a suitable measure quantifying the approximation effect. This function, in turn, can be used to construct a membership function of the term A describing an acceptable linearization error. The membership function of A at x characterizes an extent to which the linearization of the function is acceptable. Assuming that the expression is finite, the membership function of A is defined as In general, as a more flexible model of A we can admit any monotonically increasing function such that φ(0) =0 and φ(1) = 1, and 3.5.5. Membership estimation as a problem of parametric optimizationIn a nutshell, this method is not aimed at constructing a certain membership function from scratch. The intent here is to carry out a standard curve fitting to some experimental data consisting of ordered pairs (element, membership value), denoted by (xk, M(xk)). The form of the parameterized membership function is given in advance. It should reflect the nature of the concept to be described. Several examples of commonly used membership functions have been already discussed. For example, the notion of a low temperature can be described by a Γ membership function The parametric membership estimation procedure is as follows. Let us assume a parameterized membership function A(x;p) where x ∈ X, and p is a vector of its parameters in the appropriate parameter space P. For a given pair of data (xk,M(xk)), k = 1, , N, the vector of parameters p of the assumed membership function A(x;p) needs to be determined. A commonly used procedure exploits the Mean Squared Errors (MSE) as the estimation criterion. Therefore the problem reads which actually is a nonlinear optimization task. In the estimation of the membership functions, one can look at some additional constraints or intuitive hints. To maintain a smooth transition between membership values, we can assume that the rate of change of a membership value where k is a positive proportion factor. In what follows, we illustrate the problem of membership estimation alluding to some experimental data described by McCloskey and Glucksberg (1978). The experiment deals with a number of objects which are classified as belonging to a category of fruits. The membership grades are determined for each object individually. The results are normalized to the unit interval. The results listed below are also plotted in Fig. 3.3 - note a smooth transition between elements fully belonging to the class and those excluded from it.
3.6. Fuzzy relationsIn comparison to fuzzy sets that are defined in a single universe of discourse, fuzzy relations are defined in Cartesian product of some universes of discourse. For instance, a fuzzy relation R in X × Y is defined via a membership function The difference between fuzzy relations and relations is illustrated in Fig. 3.3.
While formally equivalent with multidimensional fuzzy sets (when X is replaced by the corresponding Cartesian product and one is willing to accept this notation), fuzzy relations stand on their own, forming a separate area of studies. Likewise, fuzzy sets, fuzzy relations are very common and we can easily enumerate a number of examples:
Each fuzzy relation in X ×Y can be expressed via its two projections
There are two interesting interpretations of the projection operation, Fig. 3.4.
Finally, we would like to emphasize that a reconstruction of R out of its projections in the form of the Cartesian product of these does not result in the original fuzzy relation but rather its upper estimate,
Copyright © CRC Press LLC
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