On Probability of Matching in Probability Based Rough Set Definitions

Abstract

The original rough set theory deals with precise and complete data, while real applications frequently contain imperfect information. A typical imperfect data studied in rough set research is the missing values. Though there are many ideas proposed to solve the issue in the literature, the paper adopts a probabilistic approach, because it can incorporate other types of imperfect data including imprecise and uncertain values in a single approach. The paper first discusses probabilities of attribute values assuming different type of attributes in real applications, and proposes a generalized method of probability of matching. It also discusses the case of continuous data as well as discrete one. The proposed probability of matching could be used for defining valued tolerance/similarity relations in rough set approaches.

Rough Sets and Rule Induction in Imperfect Information Systems

“…The next step will summarize two possible solutions discussed in [20] with the adaptation of the representation of imperfect values in Definition 1.…”

“…P r a (f a (x) = v|f a (y) = v) denotes the conditional probability of f a (x) = v given f a (y) = v. Hereafter, the paper assumes that two events f a (x) = v and f a (y) = u, x, y ∈ U , a ∈ A are independent of each other for any u, v ∈ V a . The probability of matching for each type of missing, uncertain and imprecise values has been discussed in [20]. However, the concept of imperfectness introduced in this paper generalizes the way to calculate the probability of matching as follows:…”

“…The term imperfect data, hereafter, will represent the case of missing, uncertain and imprecise information. [20] and applying an probabilistic method [17,18,20,23,24]. This paper will introduce a representation of imperfectness including missing values, uncertainty as well as imprecision, then discuss the valued tolerance/similarity relation based rough set (VRRS) model in an imperfect information system, which is first introduced in [20].…”

An Incomplete Data Filling Approach Based on a New Valued Tolerance Relation

“…Therefore, in recent years, many scholars and experts focus on how to manage incomplete information. At present, According to documents [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] acknowledged there are two methods to deal with incomplete information systems.…”

“…Now a lot of generalized rough set models have been proposed. For example, Kryszkiewicz's tolerance relation [1], Stefanowski's non-symmetric similarity relation [2], and Guoyin Wang s limited tolerance relation [3], valued tolerance relation [4], On this basis, many other improved methods have proposed [5][6][7][8][9][10][11][12][13][14][15]. Compared with the other extension models, the valued tolerance relation has the following advantages.…”