Statistical Inference in Rough Set Theory Based on Kolmogorov–Smirnov Goodness-of-Fit Test

Dependence degree (DD) and importance degree (ID) of patterns are crucial for pattern appraisement and model reconstruction. In rough set data analysis (RSDA), DD and ID lack robustness because the lower approximation set is terribly unstable under the indiscernibility relation perturbation. Statistical inference is a good way to deal with this instability. However, the fixed-value hypothesis testing and interval estimation of DD and ID were only discussed by the $\chi ^2$ test, which merely suits for two contingency tables (CT) with the same number of elements, and their nonzero elements must exist at the same positions. These requirements are too strict for practical applications. In this paper, the Kolmogorov-Smirnov (K-S) goodness-of-fit test is introduced to generalize the statistical inferences of DD and ID. As the bridge between data and corresponding measures, CT lies at the core of RSDA. By transforming CT to a random sample of a hypothesized random variable, the elementary algorithm for the goodness-of-fit test of contingency tables and the advanced algorithm for the goodness-of-fit test of contingency tables are proposed based on K-S statistic to implement goodness-of-fit tests of CTs. Better than $\chi ^2$ test, all CTs, even with different numbers and different positions of nonzero elements, are permitted. Subsequently, by generating a CT with expected DD value, the fixed-value hypothesis of DD is transformed to the goodness-of fit test between the original and expected CTs. Three algorithms, i.e., hypothesis test of dependence degree based on the K-S test, region estimation of the dependence degree, and significance test and region estimation of importance degree of attribute set, are proposed as fixed-value hypothesis tests and region estimations of DD and ID. These algorithms can be used to verify the importance of attributes and choose the attribute subset with the highest likelihood of maintaining the original discrimination ability. Experiments verify that the discrimination ability, disturbance tolerance ability, and stability under varying discretization strategies of DD and ID are significantly enhanced by the proposed algorithms.