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Fuzzy rough set model for set-valued data

Jianhua Dai∗, Haowei Tian

College of Computer Science, Zhejiang University, Hangzhou 310027, China

Received 3 April 2012; received in revised form 13 March 2013; accepted 14 March 2013

Abstract

In many practical situations, some of the attribute values for an object may be set-valued. The existing crisp rough set model for set-valued information systems is based on a tolerance relation that examines whether two set values have a non-empty intersection.

Some information in the data will be lost according to the tolerance relation. Here we define a fuzzy relation and construct a fuzzy rough set model for set-valued information systems. The concepts of reduct, core, discernibility matrix and discernibility function for set-valued information systems, and relative reduct, core, discernibility matrix and discernibility function for set-valued decision systems are defined and examined. Attribute reduction in set-valued information systems based on discernibility matrices and functions are investigated. © 2013 Elsevier B.V. All rights reserved.

Keywords: Set-valued data; Fuzzy rough set model; Discernibility matrix; Discernibility function; Attribute reduction 1. Introduction

Rough set theory [28,29] is a relatively new soft computing tool for the analysis of vague descriptions of objects.

Research into rough set theory has attracted great theoretical and practical interest.

Classical rough set philosophy is based on the assumption that every object in the universe of discourse is associated with some information. In many practical situations, some of the attribute values for an object may be set-valued; these are used to characterize uncertain and missing information in information systems [2]. Orlowska and Pawlak [23,24] investigated set-valued information systems considering non-deterministic information and introduced the concept of a non-deterministic information system. Lipski studied set-valued information systems under a framework of incomplete information systems [19,20]. Yao explicitly used the concept of set-based information systems [34,35]. Set-valued information systems can be viewed as generalized models of single-valued information systems [14,30]. Moreover, set-valued information systems can be used to handle incomplete information systems in which all missing values can be represented by the set of all possible values for each attribute [12,13,16–18].

Fuzzy rough sets were first proposed by Dubois and Prade to extend crisp rough set models [8,9]. Fuzzy rough sets encapsulate the related but distinct concepts of vagueness (for fuzzy sets [36]) and indiscernibility (for rough sets), both of which occur as a result of knowledge uncertainty. Fuzzy rough set models have been a popular topic in recent years and have been applied to many fields, including decision making [27], machine learning [1], business data processing ∗ Corresponding author.

E-mail addresses: jhdai@zju.edu.cn, david.joshua@foxmail.com (J. Dai). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.03.005

Please cite this article as: J. Dai, H. Tian, Fuzzy rough set model for set-valued data, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.005 2 J. Dai, H. Tian / Fuzzy Sets and Systems ( ) – [33], web categorization [15], complex systems monitoring [31], tumor classification [5], bioinformatics [21,25] and image analysis [22,26]. However, set-valued information systems have not been investigated under the framework of fuzzy rough set model.

The main objective of this study was to introduce a fuzzy rough set model for set-valued information systems by proposing a strategy to define fuzzy relations for set values. Discernibility matrices for set-valued information systems and set-valued decision tables are constructed under the framework of our fuzzy rough set model. The problem of attribute reduction for set-valued information systems is also investigated based on the indiscernibility matrices defined.

The remainder of the paper is organized as follows. A fuzzy rough set model for a set-valued information system is constructed in Section 2. The concepts of reduct, core, discernibility matrix and discernibility function for setvalued information systems without a decision attribute are defined and studied in Section 3. The concepts of relative reduct, core, discernibility matrix and discernibility function for set-valued decision tables are defined and studied in

Section 4. Section 5 provides two examples in which the proposed approach is applied. Section 6 concludes. 2. Fuzzy rough set model for set-valued information systems

An information system is a quadruple I S = 〈U, A, V, f 〉, where the universe U is a non-empty finite set of objects,

A is a non-empty finite set of attributes, V is the union of attribute domains (V =⋃a∈A Va), Va is the set of all possible values for attribute a ∈ A, and f : U × A → V is a function that assigns particular values from attribute domains to objects, for example, ∀a ∈ A, x ∈ U, f (a, x) ∈ Va , f (a, x) is the value of attribute a for object x. If, for all a and x, f (a, x) is a single value, then the information system is called a single-valued information system; if a system is not a single-valued information system, it is called a set-valued (multi-valued) information system.

Table 1 shows a set-valued information system. For brevity, f (a, x) is also written as a(x).

In some real applications, we need to consider a decision or classification problem. Hence, we need to handle setvalued decision systems. A set-valued decision system is a quadruple (U,C ∪{d}, V, f ), where U is a non-empty finite set of objects, C is a finite set of conditional attributes, d is a decision attribute with C ∩ {d} = ∅; V = VC ∪ Vd , where

VC is the set of conditional attribute values, Vd is the set of decision attribute values, f is a mapping from U × (C ∪{d}) to V such that f : U × C → 2VC is a set-valued mapping, and f : U × {d} → Vd is a single-valued mapping. The set-valued decision system can also be expressed as a table, called a set-valued decision table. Table 2 illustrates a set-valued decision system.