## Abstract

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.

## Keywords

Evolutionary Algorithm Mahalanobis Distance Positive Definite Matrix Cholesky Factor Fuzzy Measure## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Angeline, P. J., G.M. Saunders and J. B. Pollack, “An evolutionary algorithm that constructs recurrent neural networks,”
*IEEE Trans. on Neural Networks***5**:1, 54–65 (1994).CrossRefGoogle Scholar - [2]Bäck, T.,
*Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms*, Oxford University Press, New York (1996).zbMATHGoogle Scholar - [3]Bezdek, J. C.,
*Pattern Recognition with Fuzzy Objective Function Algorithms*, Plenum Press, New York (1981).zbMATHCrossRefGoogle Scholar - [4]Bezdek, J. C., “Some non-standard clustering algorithms,” in: Legendre, P. and L. Legendre, eds.,
*Developments in Numerical Ecology*, Springer-Verlag, 225–287 (1987).CrossRefGoogle Scholar - [5]Bezdek, J. C. and S. K. Pal, eds.,
*Fuzzy Models for Pattern Recognition: Methods That Search for Patterns in Data*, IEEE Press, New York (1992).Google Scholar - [6]Bobrowski, L. and J. C. Bezdek, “C-means clustering with the
*l*_{1}and*l*_{∞}norms,”*IEEE Trans. on Systems, Man, and Cybernetics***21**:3, 545–554 (1991).MathSciNetzbMATHCrossRefGoogle Scholar - [7]Duran, B. S. and P. L. Odell,
*Cluster Analysis: A Survey*, Springer-Verlag, New York (1974).zbMATHGoogle Scholar - [8]Fisher, R. A., “Multiple measurements in taxonomic problems,”
*Annals of Eugenics***7**:2, 179–188 (1936).Google Scholar - [9]Grabisch, M., H. T. Nguyen and E. A. Walker,
*Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference*, Kluwer, Boston (1995).Google Scholar - [10]Jain, A. K. and R. C. Dubes,
*Algorithms for Clustering Data*, Prentice Hall, New Jersey (1988).zbMATHGoogle Scholar - [11]Klir, G. J. and B. Yuan,
*Fuzzy Sets and Fuzzy Logic: Theory and Applications*, Prentice Hall, Upper Saddle River, NJ (1995).zbMATHGoogle Scholar - [12]Klir, G. J., B. Yuan and J. F. Swan-Stone, “Constructing fuzzy measures from given data.”
*Proc. of the Sixth IFSA Congress*1, Sao Paulo, Brazil, 61–64 (1995).Google Scholar - [13]Laarhoven, P. J. M. and E. H. Aarts,
*Simulated Annealing: Theory and Applications*, D. Reidel, Boston (1987).zbMATHGoogle Scholar - [14]Ruspini, E. H., “Numerical methods for fuzzy clustering,”
*Information Sciences***2**:3, 319–350 (1970).zbMATHCrossRefGoogle Scholar - [15]Tahani, H. and J. M. Keller, “Information fusion in computer vision using the fuzzy integral,”
*IEEE Trans. on Systems, Man, and Cybernetics***20**:3, 733–741 (1990).CrossRefGoogle Scholar - [16]Wang, Z. and G. J. Klir,
*Fuzzy Measure Theory*, Plenum Press, New York (1992).zbMATHGoogle Scholar - [17]Yuan, B. and G. J. Klir, “Data analysis based on evolutionary fuzzy c-means clustering,”
*Proc. of Joint Conference on Information Science*, 233–236 (1995).Google Scholar - [18]Yuan, B. and G. J. Klir, “Constructing fuzzy measures: a new method and its application in cluster analysis,”
*Proc. of NAFIPS’96*, University of California at Berkeley (1996).Google Scholar - [19]Yuan, B., G. J. Mir and J. F. Swan-Stone, “Evolutionary algorithm based fuzzy c-means algorithm,”
*Proc. FUZZY-IEEE/IFES’95*4, Yokohama, Japan, 2221–2226 (1995).Google Scholar