Estimating fuzzy sets

Abstract

Willing to use fuzzy set models described in the previous chapter, we face with the similar problem typically encountered with the probabilistic framework of adapting models to the operational problem at hand. This passes through the identification of the model free parameters from a set of experimental data, a task that we call estimation by analogy with statistical inference. Vice versa, the difference between fuzzy estimation and statistical inference stands in the weight we give the data: uniform with statistics, ad hoc with fuzzy methods. This reflects either in the shape of the membership function, or in the function mapping data into parameters, or both. As for the former, the methods for shaping the functions are not requested to mimic through these functions histograms of huge sample as for probability models. Thus no any constraint is given to the area subtended by a membership function, rather its identification is mainly demanded to the criteria discussed in the previous chapter. As for the data weight, we are in any case drawn to affect them with equal weights in absence of counter-indications. The fact is that these weights are shared by local subgroups in place of the whole dataset. The main scheme is the following: on the one hand you (either implicitly or explicitly) decide a set of points around which to pivot the membership functions of a given fuzzy set. We refer to them as pivotal points as being a counterpart of the sample points in statistical frameworks. On the other hand, on each point you collect one or more membership degrees to the set. Finally you extract parameters of the membership function by mixing with equal weights specific subsets of pairs (point, degree). We pass from subgroups of a single element whose membership to a given fuzzy set is expressly asked to an expert, to same subgroups on which opinions are gathered by many experts, to local subsets of a dataset on which to compute statistics. © 2008 Springer-Verlag Berlin Heidelberg.