A fast approach to attribute reduction from perspective of attribute measures in incomplete decision systems

An incremental approach to attribute reduction from dynamic incomplete decision systems in rough set theory

“…How-55 ever, in many real-world tasks, it may occur that some of the attribute values for an object are incomplete (missing) due to the restriction 56 of access, the errors of measurement and so on. However, rules must be extracted from incomplete data, which motivates many re-57 searchers to study various approaches to address incomplete decision systems [8,11,16,17,[19][20][21][22][23][24]30,39,40]. Generally speaking, accord-58 ing to whether a given decision system has missing attribute values, it can be classified into two categories: complete decision systems 59 and incomplete decision systems.…”

“…In this con-64 text, such incomplete attribute values can be further divided into three categories by different interpretations: "do not care" conditions, 65 restricted "do not care" conditions and attribute-concept values. The way in which incomplete attribute values are considered as missing 66value semantics is relatively representative [8,11,17,[19][20][21][22][23][24]30], and it can be more easily modified to solve the absent value semantics. …”

“…Most existing attribute reduction algorithms rely on the information gathered from the positive region [17,22,25,53]. A pos-40 itive region contains all the objects that can be classified into the decision attribute set using the information in the condition attribute set.…”

“…Otherwise, it is called a In general, it is involved in using condition knowledge to approximate the decision and finding the mapping from the conditions to 164 decisions. Various attribute reduction approaches using rough set theory have been proposed to discover decision rules from data sets 165 taking the form of decision systems [3,14,19,22,24,47,53]. 166 A decision system (DS) is expressed as DS(U, C∪D,V, f), where, 167 (1) C is a set of condition attributes, D is decision attribute set, and C∩D = ∅; 168 (2) V is the union of attribute domain, V = V C ∪V D = ∪{Va|a∈C}∪{V d |d∈D}.…”

“…Some properties that hold for 184 singleton approximations do not hold, in general, for subset approximations and for concept approximations. Since the singleton ap- 185 proximations are representative and extensively studied in [8,17,[19][20][21][22][23][24], we use that way to define the approximations in this paper. 186 Given any subset of attributes P ⊆ A and subset X ⊆ U, the lower and upper approximations of X in terms of tolerance relation TR(P) 187 are respectively defined as apr P ðXÞ ¼ fx∈UjS P ðxÞ⊆Xg and apr P ðXÞ ¼ fx∈UjS P ðxÞ∩X≠∅g.…”