Algorithms for preprocessing databases with incomplete and imprecise data are seldom studied. For the most part, we lack numerical tools to quantify the mutual information between fuzzy random variables. Therefore, these algorithms (discretization, instance selection, feature selection, etc.) have to use crisp estimations of the interdependency between continuous variables, whose application to vague datasets is arguable. In particular, when we select features for being used in fuzzy rule-based classifiers, we often use a mutual information-based ranking of the relevance of inputs. But, either with crisp or fuzzy data, fuzzy rule-based systems route the input through a fuzzification interface. The fuzzification process may alter this ranking, as the partition of the input data does not need to be optimal. In our opinion, to discover the most important variables for a fuzzy rule-based system, we want to compute the mutual information between the fuzzified variables, and we should not assume that the ranking between the crisp variables is the best one. In this paper we address these problems, and propose an extended definition of the mutual information between two fuzzified continuous variables. We also introduce a numerical algorithm for estimating the mutual information from a sample of vague data. We will show that this estimation can be included in a feature selection algorithm, and also that, in combination with a genetic optimization, the same definition can be used to obtain the most informative fuzzy partition for the data. Both applications will be exemplified with the help of some benchmark problems.

Abstract-Algorithms for preprocessing databases with incomplete and imprecise data are seldom studied, partly because we lack numerical tools to quantify the interdependency between fuzzy random variables. In particular, many filtertype feature selection algorithms rely on crisp discretizations for estimating the mutual information between continuous variables, effectively preventing the use of vague data.Fuzzy rule based systems pass continuous input variables, in turn, through their own fuzzification interface. In the context of feature selection, should we rank the relevance of the inputs by means of their mutual information, it might happen that an apparently informative variable is useless after having been codified as a fuzzy subset of our catalog of linguistic terms.In this paper we propose to address both problems by estimating the mutual information with the same set of fuzzy partitions that will be used to codify the antecedents of the fuzzy rules. That is to say, we introduce a numerical algorithm for estimating the mutual information between two fuzzified continuous variables. This algorithm can be included in certain feature selection algorithms, and can also be used to obtain the most informative fuzzy partition for the data. The use of our definition will be exemplified with the help of some benchmark problems.

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