Three-Way Classification Models

Deng, Xiaofei
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Faculty of Graduate Studies and Research, University of Regina
In 1982, Pawlak proposed a theory of rough sets for data analysis. A fundamental concept of rough sets is the approximation of a set by a pair of lower and upper approximations or three pair-wise disjoint positive, negative and boundary regions. In 1990, Yao et al. introduced a probabilistic version called decision-theoretic rough sets (DTRS). Motivated by a three-way interpretation of the probabilistic positive, negative and boundary regions, in 2012 Yao proposed a theory of three-way decisions based on the actions of acceptance, rejection and non-commitment. The theory pro- vides a uni ed framework that integrates rough sets, interval sets, approximations of fuzzy sets, and several others. This thesis focuses particularly on classification models based on the theory of three-way decisions. An analysis of probabilistic two-way classifications shows that one cannot reduce incorrect acceptance and incorrect rejection errors simultaneously. In contrast, the probabilistic three-way classifications can obtain both low incorrect acceptance and incorrect rejection errors at the expense of non-commitment for some objects. A systematic study of three-way classification models may bring new insights into classification problems. Two classes of set-based three-way classification models are studied, namely, three- way classification with rough sets and three-way classification for approximations of fuzzy sets. The investigation of the first class consists of two parts. The first part is a review of existing three-way decision models using rough sets. To integrate existing models, the second part is a proposal of a framework of quantitative rough sets based on subsethood measures. The Pawlak rough sets do not allow any in- correct acceptance and incorrect rejection errors; they are a qualitative three-way decision model. Decision-theoretic rough sets and its various extensions are studied and categorized as different families of quantitative rough set models. In particular, an information-theoretic rough set model is proposed to give an alternative solution to a fundamental issue of probabilistic rough sets, that is, the interpretation and construction of an optimal pair of thresholds. The contribution from the investigation of the second class is a proposal of a decision-theoretic three-way decision model of approximations of fuzzy sets. A pair of thresholds ( ; ) is used to produce a three-way approximation of a fuzzy set. An object with membership grade greater than -level is accepted as a member of a fuzzy set, an object with membership grade smaller than -level is rejected as a member of the fuzzy set, and an object with membership grade between - and -level is neither accepted nor rejected. The thesis presents a semantically meaningful and computa- tionally effcient approach to compute an appropriate three-way approximation of a fuzzy set, that is, an optimal pair of thresholds ( ; ).
A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In Computer Science, University of Regina. xiii, 210 p.