Continuous Analogical Proportions-Based Classifier

Abstract

Analogical proportions, often denoted, are statements of the form “A is to B as C is to D” that involve comparisons between items. They are at the basis of an inference mechanism that has been recognized as a suitable tool for classification and has led to a variety of analogical classifiers in the last decade. Given an object D to be classified, the basic idea of such classifiers is to look for triples of examples (A, B, C), in the learning set, that form an analogical proportion with D, on a maximum set of attributes. In the context of classification, objects A, B, C and D are assumed to be represented by vectors of feature values. Analogical inference relies on the fact that if a proportion is valid, one of the four components of the proportion can be computed from the three others. Based on this principle, analogical classifiers have a cubic complexity due to the search for all possible triples in a learning set to make a single prediction. A special case of analogical proportions involving only three items A, B and C are called continuous analogical proportions and are of the form “A is to B as B is to C” (hence denoted). In this paper, we develop a new classification algorithm based on continuous analogical proportions and applied to numerical features. Focusing on pairs rather than triples, the proposed classifier enables us to compute an unknown midpoint item B given a pair of items (A, C). Experimental results of such classifier show an efficiency close to the previous analogy-based classifier while maintaining a reduced quadratic complexity.