Dual Mathematical Models Based on Rough Approximations in Data Analysis

In rough set approach, the rough approximations called lower and upper ones have been discussed. This concept can be extended into a new research field of data analysis. The proposed approach to data modeling is to obtain dual mathematical models by using a similar concept to rough sets. The dual models called lower and upper models have an inclusion relation. In the other words, the proposed method can be described as two approximations to a phenomenon under consideration such that $$ Lower Model \subseteq Phenomenon \subseteq Upper Model. $$Thus, the lower and upper models are obtained by the greatest lower bound and the least upper bound, respectively. This property is illustrated by interval regression models which are not crisp, but have an interval relationship between inputs and outputs. Generally, the lower and upper models are formulated by greatest lower and least upper bounds, respectively. The given phenomenon can be expressed by the pair (lower model, upper model) corresponding to rough approximations.