Twofold rough approximations under incomplete information

Abstract

A method using possible equivalence classes has been developed on information tables with missing values. The method essentially differs from the other methods in having the two features. One is to directly deal with missing values by using not actual but possible equivalence classes. The other is to consider both aspects of discernibility and indiscernibility of a missing value from another value. When information tables contain incomplete information, rough approximations are not unique. We have lower and upper bounds of the actual rough approximations. The lower and upper bounds correspond to certain and possible rough approximations, respectively. Therefore, rough approximations are twofold under incomplete information. The certain and possible rough approximations are linked with each other. The method creates the same rough approximations as the method of possible worlds. This justifies the method of possible equivalence classes. The method is free from the difficulty of computational complexity for the growth of the number of missing values. Furthermore, the method is free from the restriction that missing values may occur for only some specified attributes. Therefore, we can efficiently obtain certain and possible rough approximations between arbitrary sets of attributes having missing values.

Keywords:

• rough sets
• incomplete information
• missing values
• possible equivalence classes
• lower and upper approximations

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable comments and Professor Dominik Ślȩzak for his suggestions concerning the improvement of this paper. This work has been partially supported by the Grant-in-Aid for Scientific Research C, Japan Society for the Promotion of Science, No. 22500204.

Notes

is formally . is usually omitted for the sake of simplicity.

is obtained by replacing and by and in formulae (1)–(3).

There exists an equivalence class that supports multiple rules. In such a case, equivalence classes with different rules are described, although they are identical as mathematical sets.

Lipski used possible tables based on all attributes. However, possible tables on all attributes are not necessary. It is sufficient to have possible tables on a set of attributes used to obtain rough approximations. Therefore, we describe possible tables on attribute , because we deal with rough approximations on .

is formally and the subscript is omitted for simplicity.

When classes of objects said to be equivalent are obtained by using discretization techniques (Grzymala-Busse 2010) or interval pattern structures (Kaytoue et al. 2011) to numerical domains, we can use the method of possible equivalence classes.