Elsevier

Information Sciences

Volume 271, 1 July 2014, Pages 93-114
Information Sciences

Approximate distribution reducts in inconsistent interval-valued ordered decision tables

https://doi.org/10.1016/j.ins.2014.02.070Get rights and content

Abstract

Many methods based on the rough set theory to deal with information systems have been proposed in recent decades. In practice, some information systems are based on dominance relations and may be inconsistent because of various factors. Moreover, taking the imprecise evaluations and assignments in the description of objects into account, single-valued information systems have been generalized to interval-valued information systems. In this paper, by introducing a dominance relation to interval-valued ordered information systems, we establish a dominance-based rough set approach, which is mainly based on substitution of the indiscernibility relation by the dominance relation. To extract the minimal decision rules, approximate distribution reducts are proposed in inconsistent interval-valued ordered decision tables. This paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct. And two equivalent definitions of approximate distribution reducts are also introduced. In addition, numerical examples are employed to examine the validity of the approaches proposed in this paper.

Introduction

The rough set theory (RST), initiated by Pawlak in the early 1980s [35] (see also [37]), serves as an extension of the classical set theory for the study of intelligent systems characterized by insufficient and incomplete information [36]. The starting point of this theory is an observation that objects having the same description are indiscernible in the view of the available information about them. Within thirty years of development, it has been successfully applied in many fields such as machine learning, intelligent systems, inductive reasoning, pattern recognition, knowledge discovery, decision analysis, expert systems.

The original rough set theory is not able, however, to discover inconsistencies coming from consideration of criteria, that is, attributes with preference-ordered domains (scales), such as product quality, market share, and debt ratio. To address this issue, Greco, Matarazzo and Słowiński proposed an extension of the rough set theory, which is called the dominance-based rough set approach (DRSA) to take into account the ordering properties of criteria [14], [15]. This innovation is mainly based on the substitution of the indiscernibility relation by a dominance relation, which permits approximation of ordered sets in multiple criteria sorting problems. In DRSA, where condition attributes are criteria and decision classes are preference-ordered, the knowledge approximated is a collection of upward and downward unions of classes and the granules of knowledge are sets of objects defined using a dominance relation [25], [26], [27], [45], [46], [54].

Dubois and Prade presented the concept of rough fuzzy sets and fuzzy rough sets [12] by means of integrating the RST with the fuzzy set theory [11]. Likewise, the DRSA could be combined with fuzzy set theory as well. In [16], fuzzy set extensions of the DRSA are proposed to deal with fuzziness in preference representation. Dominance-based fuzzy rough set [13] based on t-norm, s-norm and implication operations serves as an extension of (I,T)-fuzzy rough set [44] established in RST and can be used to handle uncertain information. To handle with preference analysis in intelligent data analysis and machine learning, fuzzy preference based rough set approach [21] was presented by Hu et al. which is mainly based on the construction of the fuzzy preference relations from samples characterized by numerical criteria.

From the logical viewpoint, a decision table may also be seen as a series of decision rules. How to simplify these decision rules is one of the major topics discussed in DRSA. As pointed out by Greco et al. [4], [14], the set of decision rules induced from the approximations defined using dominance relations gives, in general, a more synthetic representation of knowledge contained in the decision table than the set of rules induced from classical approximations defined using indiscernibility relations. And they are more understandable and more applicable for the users because of the more general syntax of the rules.

According to Inuiguchi et al. [28], there exist at least four sources of inconstancy in decision tables, which are listed as follows: (1) hesitation in evaluation of decision attribute values, (2) errors in recording, measurement and observation, (3) missing condition attributes related to the evaluation of decision attribute values, (4) the unstable nature of the system represented by the decision table and the like. These inconsistencies cannot be considered as a simple error or noise. They may convey important information that should be taken into account. To acquire brief decision rules from inconsistent decision tables, relative attribute reducts are needed. Skowron and Rauszer introduced the discernibility matrix method which became a popular approach to enumerate all reducts in the RST [47]. Susmaga et al. introduced a discernibility matrix to ordered information systems and addressed the computation of reducts in DRSA [49]. Recently, a general definition of discernibility matrices was suggested by Miao et al. to summarize the common structures of the existed ones [33].

In the RST, this problem has already been faced in the literature. The concept of the approximate distribution reduct (μ-decision reduct) was proposed by Kryszkiewicz to deal with inconsistent systems [30]. However, the practical approach to compute all approximate distribution reducts was proposed by Zhang et al. [59]. Using the approximate distribution consistent set, the derived rules are compatible with the ones from the original system, that is to say, if two decision rules derived respectively from the reduced and the original system are supported by a same object, then their decision parts must be the same. The approximate distribution reduct has also been studied in other types of information systems, for instance, inconsistent incomplete decision tables [42]. On the other hand, β-reduct proposed by Ziarko was studied in the variable precision rough set model by reducing boundary area in decision tables [63]. But the derived decision rules from the β-reduct may be in conflict with the ones from the original system. To overcome this kind of drawback, Mi et al. introduced concepts of β lower distribution reduct and β upper distribution reduct [32]. Some algorithms have also been proposed for extracting decision rules from an inconsistent decision table, for example, algorithm REBCA [40].

Parallel work have also been made for the DRSA along this line of research. Nevertheless, only a limited number of methods using the DRSA to acquire knowledge in inconsistent ordered decision tables have been proposed. The distribution reduct and maximum distribution reduct were proposed by Xu et al. to meet different requirements in inconsistent ordered decision tables [51]. Later, Xu et al. further proposed another two types of reducts, possible and compatible distribution reducts [53]. As mentioned before, some mislabeled samples may lead to inconsistent ordered decision tables. It was reported that dominance-based rough sets are heavily sensitive to these noisy samples. Inuiguchi et al. [28] presented a variable-precision dominance-based rough set approach (VP-DRSA) and studied union-based reducts in VP-DRSA. To design a robust metric of feature quality, Hu et al. introduced a metric function, which is called rank mutual information (RMI) for monotonic classification [22]. On the other hand, by introducing of rank entropy, they designed a decision tree technique (REMT) based on RMI [23]. It has been proved that the proposed technique can produce monotonically consistent decision trees if the given training sets are monotonically consistent.

An interval-valued information system is an important type of information system, and a generalization of single-valued information system. Some problems of decision making in the context of interval-valued information systems have been studied in [6], [19], [20], [24], [41], [55], most of which are based on the concept of a possibility degree between any two interval numbers. Some similarity/distance measures for interval-valued fuzzy sets have been proposed to measure the similarity/difference of interval-valued fuzzy sets [50], [60]. The principle of the maximum (minimum) degree of similarity (difference) between interval-valued fuzzy sets can be used to solve problems of pattern recognition and decision making. Precisely, the more (less) the similarity (difference) between the sample and patterns already known, the more likely the sample should be clarified to the pattern.

The knowledge reduction of interval-valued fuzzy information systems is studied under the circumstances of the condition and decision attributes are nominal, not ordinal [17], [48]. As for interval-valued ordered information systems (IvOISs), Qian et al. proposed the DRSA to attribute reducts in IvOISs based on a dominance relation for interval numbers [41]. However, they did not mention the underlying concepts of relative attribute reducts in inconsistent interval-valued ordered decision tables (IvODTs) and only proposed an approach to relative attribute reducts in consistent IvODTs. Therefore, the purpose of this paper is to develop approaches to relative attribute reducts in inconsistent IvODTs to make a completion of their work.

The other parts of this paper are organized as follows. In Section 2, the dominance relations in IvOISs are reviewed and some important properties of the dominance classes are also investigated. In Section 3, dominance-based rough sets are defined in IvODTs and several types of decision rules are introduced on the basis of these concepts. In Section 4, the concepts of approximate distribution reducts are proposed in inconsistent IvODTs, the discernibility matrix method is introduced to enumerate of all these reducts and practical approach on the basis of significance to find one reduct are presented. While, in Section 5, two equivalent definitions of approximate distribution reducts are introduced. Numerical examples are presented in Section 6 to illustrate the feasibility of the approaches proposed in this paper. Finally, some concluding remarks and suggestions for future work are given in Section 7.

Section snippets

Preliminaries

In this section we mainly recall several basic concepts and introduce some notations.

An interval-valued information system (IvIS) is a quadruple S=(U,AT,V,f), where U is a finite non-empty set of objects, AT is a finite non-empty set of attributes, V=aATVa and Va is a domain of attribute a, and f:U×ATV is a total function such that f(x,a)Va for every aAT,xU, called an information function, where Va is a set of interval-valued numbers. Denoted byf(x,a)=aL(x),aU(x)=p|aL(x)paU(x),aL(x),aU(x

Dominance-based rough set approach to interval-valued ordered decision tables

In this section, we investigate the approximation problem of a set with respect to dominance relations DA+ and DA- in interval-valued ordered decision tables.

An interval-valued ordered decision table (IvODT) is a special case of interval-valued ordered information system S=(U,AT{d},V,f), where dAT,f(x,d)(xU) is single-valued, d is an overall preference called the decision and all the elements of AT are criterions. Furthermore, assume that the decision attribute d makes a partition of U into

Approximate distribution reducts and dominance decision rules in inconsistent IvODTs

Attribute reduct is one of the major topics in the rough set theory and is also discussed in DRSA. In the following, approximate distribution reducts and their computing approaches are studied in inconsistent IvODTs.

Definition 4.1

Let S=(U,AT{d},V,f) be an IvODT and AAT. Let us denote byLA=A̲Cl1,A̲Cl2,,A̲Cln;LA=A̲Cl1,A̲Cl2,,A̲Cln;HA=ACl1,ACl2,,ACln;HA=ACl1,ACl2,,ACln.

If LA=LATHA=HAT, then A is referred to as the -lower (upper) approximate distribution consistent set of S. If A

Two equivalent definitions of approximate distribution reducts

In this section we introduce two equivalent definitions of approximate distribution reducts. First, denoted byγA=1|U|t=1n|A̲Clt|;ηA=1|U|t=1n|AClt|;γA=1|U|t=1n|A̲Clt|;ηA=1|U|t=1n|AClt|,where the notation |·| denotes the cardinality of set.

It is interesting to find that γA and γA are in the form of the quality of classification in the RST, and in DRSA, in essence, the quality of classification Cl by a set of attributes A is γA(Cl)=U-t=1nBnAClt|U|=U-t=1nBnAClt|U|. As to the

Case studies

In this section, a real-world problem about reviewers’ reports and some numerical experiments are conducted to test the proposed technique.

High quality papers help target journals for improving scientific impact and journals with high impact factors (IFs) make researchers’ publications more accessible for others. How to evaluate submitted papers is an important task for all journals. In general, originality, usefulness, technical soundness, presentation, linguistic quality and relevance to the

Conclusions

In this paper, inconsistent interval-valued ordered decision tables are considered. Throughout this paper, we adopt the most popular dominance relation LI of interval numbers in the literature. Based on this dominance relation, a rough set approach in this type of ordered decision tables has been established. In order to generate much simpler decision rules, based on the lower and upper approximation, approximate distribution reducts have been presented subsequently, which eliminate only

Acknowledgements

The authors are extremely grateful to the anonymous referees for their critical suggestions to improve the quality of this paper. This research was supported by the National Natural Science Foundation of China (Grant No. 61179038).

References (63)

  • Q.H. Hu et al.

    Fuzzy preference based rough sets

    Inform. Sci.

    (2010)
  • B. Huang

    Graded dominance interval-based fuzzy objective information systems

    Knowl.-Based Syst.

    (2011)
  • B. Huang et al.

    Dominance-based rough set model in intuitionistic fuzzy information systems

    Knowl.-Based Syst.

    (2012)
  • B. Huang et al.

    Using a rough set model to extract rules in dominance-based interval-valued intuitionistic fuzzy information systems

    Inform. Sci.

    (2013)
  • B. Huang et al.

    A dominance intuitionistic fuzzy-rough set approach and its applications

    Appl. Math. Modell.

    (2013)
  • M. Inuiguchi et al.

    Variable-precision dominance-based rough set approach and attribute reduction

    Int. J. Approx. Reason.

    (2009)
  • W. Kotłowski et al.

    Stochastic dominance-based rough set model for ordinal classification

    Inform. Sci.

    (2008)
  • J.S. Mi et al.

    Approaches to knowledge reduction based on variable precision rough set model

    Inform. Sci.

    (2004)
  • D.Q. Miao et al.

    Relative reducts in consistent and inconsistent decision tables of the Pawlak rough set model

    Inform. Sci.

    (2009)
  • Z. Pawlak et al.

    Rudiments of rough sets

    Inform. Sci.

    (2007)
  • Z. Pawlak et al.

    Rough sets and Boolean reasoning

    Inform. Sci.

    (2007)
  • Y.H. Qian et al.

    Set-valued ordered information systems

    Inform. Sci.

    (2009)
  • Y.H. Qian et al.

    Converse approximation and rule extraction from decision tables in rough set theory

    Comput. Math. Appl.

    (2008)
  • Y.H. Qian et al.

    Interval ordered information systems

    Comput. Math. Appl.

    (2008)
  • Y.H. Qian et al.

    Approximation reduction in inconsistent incomplete decision tables

    Knowl.-Based Syst.

    (2010)
  • A.M. Radzikowska et al.

    A comparative study of fuzzy rough sets

    Fuzzy Sets Syst.

    (2002)
  • B.Z. Sun et al.

    Fuzzy rough set theory for the interval-valued fuzzy information systems

    Inform. Sci.

    (2008)
  • C.P. Wei et al.

    Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications

    Inform. Sci.

    (2011)
  • W.H. Xu et al.

    Approaches to attribute reductions based on rough set and matrix computation in inconsistent ordered information systems

    Knowl. Inform. Syst.

    (2012)
  • X.B. Yang et al.

    Dominance-based rough set approach and knowledge reductions in incomplete ordered information system

    Inform. Sci.

    (2008)
  • X.B. Yang et al.

    Dominance-based rough set approach to incomplete interval-valued information system

    Data Knowl. Eng.

    (2009)
  • Cited by (47)

    • Three-way decisions method based on matrices approaches oriented dynamic interval-valued information system

      2022, International Journal of Approximate Reasoning
      Citation Excerpt :

      With a degree of possibility ranking method, decision rules under a certain risk attitude of decision maker was derived. Du and Hu [6] proposed approximate distribution reducts in inconsistent interval-valued ordered decision tables through dominance-based rough set, which was based on substitution of the indiscernibility relation by the dominance relation. With the discernibility matrix, the substitution of the indiscernibility relation by the dominance relation was extracted in IvIS.

    View all citing articles on Scopus
    View full text