Abstract

Patterns are the key building blocks in the logical analysis of data (LAD). It has been observed in empirical studies and practical applications that some patterns are more “suitable” than others for use in LAD. In this paper, we model various such suitability criteria as partial preorders defined on the set of patterns. We introduce three such preferences, and describe patterns which are Pareto-optimal with respect to any one of them, or to certain combinations of them. We develop polynomial time algorithms for recognizing Pareto-optimal patterns, as well as for transforming an arbitrary pattern to a better Pareto-optimal one with respect to any one of the considered criteria, or their combinations. We obtain analytical representations characterizing some of the sets of Pareto-optimal patterns, and investigate the computational complexity of generating all Pareto-optimal patterns. The empirical evaluation of the relative merits of various types of Pareto-optimality is carried out by comparing the classification accuracy of Pareto-optimal theories on several real life data sets. This evaluation indicates the advantages of “strong patterns”, i.e. those patterns which are Pareto-optimal with respect to the “evidential preference” introduced in this paper.

Keywords: Extremal patterns, Data mining, Machine learning, Classification accuracy, Boolean functions

Article Outline

1. Introduction

2. Notation and terminology

3. Preferences and Pareto-optimal patterns

4. Pareto-optimization of patterns and theories

4.1. Prime patterns

4.2. Spanned patterns

4.3. Strong patterns

4.4. Pareto-optimization of theories

5. Complexity of generation of strong spanned patterns

6. Analytical description of Pareto-optimal patterns

6.1. Description of patterns and prime patterns

6.2. Description of spanned patterns

6.3. Description of strong spanned patterns

7. Empirical evaluation

8. Conclusions

References