Matrix approach to decision-theoretic rough sets for evolving data

Decision-theoretic rough sets is a generalized probabilistic model for the expression of uncertainties and the representation of knowledge from data. It provides a semantic explanation and systematically computation of probabilistic thresholds to define probabilistic rough set approximations, which offers a ternary classification framework based on Bayesian decision theory. In practice, data for decision making process resides in a dynamic database whose data is typically evolving through the periodical or occasional updating, e.g., new data are appended and obsolete data are removed. It is impractical to have a maturity decision model, stalled until the preparation of all helpful training data. To address this issue, incremental learning appeared to be a feasible solution for continuous knowledge modeling from evolving data with the incorporation of unlearned knowledge embedded in the updating data. In this paper, we exploit matrix approaches to study incremental decision-theoretic rough set approach for evolving data. Starting from the representation of object subset and indiscernibility relation in matrix form, we obtain a matrix characterization of probabilistic rough set approximations in decision-theoretic rough sets by using matrix properties associated with the multiplication operator. We also develop incremental algorithms for updating probabilistic rough set approximations with respect to the addition/deletion of objects, which enables decision theoretic rough sets to deal gracefully with evolving data. A detailed experimental study is conducted to examine the performance of the proposed incremental algorithms on UCI data sets.