Approximate Tests, Decision Trees and Rules

Abstract

When we use decision trees, rules and tests as ways for knowledge representation, we would like to have relatively simple trees, rules and tests. If exact decision rules, trees or tests have large complexity, we can consider approximate trees, rules and tests.

If we use tests, decision rules or trees in classifiers, then exact tests, rules and trees can be overfitted, i.e., dependent essentially on the noise or adjusted too much to the existing examples. In this case, it is more appropriate to work with approximate tests, rules and trees.

Therefore approximate reducts [83, 88], approximate decision rules [59, 67], and approximate decision trees [8, 52, 71] are studied intensively during many years.

This chapter is devoted to the consideration of α-tests, α-decision trees and α-decision rules which are special types of approximate tests, trees and rules. It contains nine sections. In Sect. 6.1, main notions are discussed. In Sect. 6.2, relationships among α-trees, α-rules and α-tests are studied. In Sects. 6.3 and 6.4, lower and upper bounds on complexity of α-rules, α-trees and α-tests are considered. Sections 6.5, 6.6 and 6.7 are devoted to the discussion of approximate algorithms for optimization of α-rules, α-trees and α-tests. In Sect. 6.8, exact algorithms for optimization of α-decision trees and rules are considered. Section 6.9 contains conclusions.