Data-Driven Identification of Key Variables

In this chapter, we investigate the following problem: given a data set involving n variables, determine key variables that contribute most to a specific partition of this data set. This problem has a broad applicability, even though it emerged in the context of a particular engineering application—the process of manufacturing electric circuit boards. Two distinct approaches are used for dealing with the problem, each resulting in a particular algorithm. Both algorithms employ evolutionary computation. The first approach is based on the well-known fuzzy c-means algorithm. The principal idea is that we use the full class of Mahalanobis distances, each of which weights the variables involved in a particular way. Using this class of distances, we search by an evolutionary algorithm for the optimal distance—one under which the fuzzy c-means algorithm produces a fuzzy partition of the given data set that is as close as possible to the given crisp partition. The contribution of each variable to this partition is then inferred from parameter values of the optimal Mahalanobis distance. The second approach is based on fuzzy measures. The principal idea is that we consider each data vector as an evaluation function of an object with respect to several features, represented by the variables involved. This allows us to aggregate values of the variables at each data vector by the fuzzy integral with respect to a particular fuzzy measure that specifies the significance of the various subsets of variables. The fuzzy c-means algorithm is then applied to the aggregated values under different fuzzy measures. An evolutionary algorithm is used to search for the optimal fuzzy measure—one under which the fuzzy c-means algorithm produces a fuzzy partition that is as close as possible to the given crisp partition. The contribution of each subset of variables to this partition is then inferred from the optimal fuzzy measure.