Inclusion measure-based multi-granulation intuitionistic fuzzy decision-theoretic rough sets and their application to ISSA
Introduction
Rough set theory, proposed by Pawlak [42], [43], is viewed as a valid tool for knowledge discovery. This theory has two main factors, namely, set approximation and attribute reduction, which affect the description capability of knowledge. However, Pawlak’s classical rough set model has two limitations [25]. First, this model is sensitive to noisy data. Second, this model is incapable of handling real values directly on the basis of equivalence relation.
The decision-theoretic rough set (DTRS) model [62] introduces a generalized framework to solve the first limitation by considering the tolerance of classification error. As a generalized rough set model, DTRS provides a unified and comprehensive framework for interpreting and determining the required thresholds. On the basis of minimum Bayesian decision cost procedure, DTRS can compute the required thresholds from the given cost functions. Different thresholds for different probabilistic rough set models can be deduced from appropriate cost functions. The DTRS model has become increasingly popular in various theoretical and practical fields, and has produced many thorough results [4], [12], [16], [22], [23], [25], [30], [40], [41], [61], [69], [70].
However, DTRS cannot handle numerical (e.g., continuous, interval, fuzzy, and intuitionistic fuzzy (IF)) data directly; this disadvantage is also another limitation of Pawlak’s rough set model. Thus, in various real applications, numerous studies have adopted a main method to overcome this defect, which define all types of relations rather than equivalence relations, e.g., tolerance relations [31], similarity measures [37], [55], dominance relations [9], [10], covering [5], [56], inclusion measures [63], fuzzy relations [24], [36], hesitant fuzzy [28], [30] and IF relations [20], [27], [66], and can be used to measure and represent various real values.
In all types of real data, the IF value, which are introduced by Atanassov [1], [2], [3], is regarded as an intuitively direct extension of the fuzzy set value. Rough sets and IF sets (IFSs) capture particular facets of the same notion imprecision. Studies on combination of IFS and rough set theory have been considered to be a positive approach to rough set theory [65]. For example, Zhou et al. [71], [73] examined the IF rough set approximation operators through constructive and axiomatic approaches. The IF rough set approximations were characterized in [72]. Huang et al. discussed the uncertainty measures for an IF approximation space [18] and an IF graded covering-based rough set [19].
Although these generalized rough set models can be utilized to handle real data with noise, their target concept is approximated by upper and lower approximations through a single granulation. That is, this concept is depicted by available knowledge derived from a knowledge base. However, the target concept in several cases is described through multiple relations on the universe according to user requirements or problem-solving targets [46]. Qian et al. [46] extended Pawlak’s single-granulation rough set model to a multi-granulation rough set (MGRS) model to adopt rough set theory in practical applications. In the MGRS model, two basic models, namely, optimistic [46] and pessimistic MGRS [45], are defined. Since then, MGRSs have been developed rapidly [20], [21], [32], [33], [35], [38], [50], [58], [59]. The combination of MGRSs and DTRSs is an important topic among these developments [13], [26], [29], [34], [47], [48], [52], [57].
The domains of DTRSs, IF rough sets, and MGRSs have been described comprehensively given the advancements in related studies. However, few studies focused on their combination (MG-IF-DTRSs), where three basic problems on MG-IF-DTRSs have been encountered. The three problems are presented as follows. (1) The method for addressing IF-DTRSs and MG-IF-DTRSs should be examined. (2) In information granule reduction, the use of fewer basic granules (IF relations) in a multi-granulation IF approximation space should be addressed form the perspective of MG-IF-DTRSs to achieve the same approximation results. (3) The presented MG-IF-DTRSs should be used to solve a real application problem.
On the basis of the above mentioned analysis, although utilizing MG-DTRSs for information analysis is significant, MGRSs and DTRSs can only handle crisp information systems and have limitations in terms of processing IF information. Hence, the motivation of this study is to explore MG-IF-DTRSs in the information system security (ISS) audit under the IF information environment. The main contribution of this paper is constructing MG-IF-DTRSs and further exploring their applications to expand the application domain of MG-IF-DTRSs by combining DTRSs and MGRSs under the IF environment. We also propose the inclusion measure-based MG-IF-DTRSs with application to the information system security audit. Table 1 describes the connections of some related models in detail and the existing rough set models presented in the corresponding references are displayed in Table 2.
This paper is organized as follows. Section 2 briefly introduces the preliminary notions considered in this study. Section 3 presents the concept of the inclusion measure-based IF rough sets in an IF approximation space. Section 4 presents the inclusion measure-based optimistic and pessimistic MG-IF-DTRSs, and examines their rough and precision degrees. Section 5 describes the reductions of the presented inclusion measure-based MG-IF-DTRSs and develops their discernibility-function-based reduction methods. Section 6 provides the application in information system security audit, which shows the feasibility of the presented method in this study. Section 7 shows further possible generalizations related to the inclusion measure-based MG-IF-DTRSs. Section 8 concludes this paper.
Section snippets
Preliminaries
This section reviews some notions, such as IFS, IF relation, IF inclusion measure, DTRS, MGRSs, and MG-DTRSs.
Definition 2.1 Given the universe of discourse U, an IFS A in U is an object with the form where μA: U → [0, 1] and γA: U → [0, 1] satisfy for all x ∈ U; μA(x) and γA(x) are called the degree of membership and non-membership of element x ∈ U to A, respectively. Furthermore, is the degree of the hesitancy of x ∈ U to A. The family of all IFSs [1], [2]
Inclusion measure-based IF-DTRSs
We define the concept of the inclusion measure-based IF-DTRSs in an IF approximation space as follows.
Definition 3.1 Let be an IF approximation space with a nonempty and finite universe of discourse U and an IF relation on U. For any A ∈ IF(U) and 0 ≤ β < α ≤ 1, the inclusion measure-based α-lower and β-upper approximations of A w.r.t. denoted by and respectively, are defined as follows:
where
Inclusion measure-based MG-IF-DTRSs
The investigation for all kinds of rough sets from the perspective of multi-granulation represents a promising direction in rough set theory, in which we can approximate all concepts using multiple relations. In this section, the inclusion measure-based MG-IF-DTRSs are discussed based on IF inclusion measure.
Definition 4.1 Let be a multi-granulation IF approximation space with nonempty and finite universe of discourse U and m IF relations on U. For any A ∈ IF(U) and 0 ≤ β < α ≤ 1, the
Reducts of the inclusion measure-based MG-IF-DTRSs
Attribute reduction is an important issue in classical and various generalized rough sets [14], [17], [39], [54], [60], [67], [68]. In this section, the reducts are defined based on the inclusion measure-based MG-IF-DTRSts presented in Section 4, and their discernibility-function-based reduction methods are examined.
Definition 5.1 Let be a multi-granulation IF approximation space. For an IF event A ∈ IF(U) and 0 ≤ β < α ≤ 1, if there exists such that
Application to ISSA
Information system security (ISS) denotes implementing specific IT environment protective measures (e.g., computers, networks, information systems, and data bases) against accidental damaging actions and intended attacks, such as espionage, sabotage, and murder [44]. Although different security guidelines and software tools have been developed for security evaluation and risk management, which cover different approaches and solve different security issues, a number of ISS issues persist through
Other generalizations related to the inclusion measure-based MG-IF-DTRSs
Further generalizations of the inclusion measure-based MG-IF-DTRSs will be examined in this section.
Conclusion
Among generalized rough set models, MGRSs and DTRSs are two essential generalizations of Pawlak’s classical rough set. The evaluation of MGRSs and DTRSs shows that their combination has been examined rarely in IF environments, although MGRSs and DTRSs have been studied separately by numerous researchers. This study aims to develop a novel inclusion measure-based IF-DTRS and two corresponding MG-IF-DTRSs. The results of this study can be considered as generalizations of the concepts described in
Acknowledgments
We would like to thank the EssayStar Company (http://essaystar.com/) for their assistance in improving the English language of this paper. We are grateful for the support provided by the Natural Science Foundation of China (Grant nos. 61473157, 71671086, 61170105, and 71201076) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
References (73)
Intuitionistic fuzzy sets
Fuzzy Sets Syst.
(1986)- et al.
Extensions and intensions in rough set theory
Inf. Sci. (NY)
(1998) - et al.
Structures on intuitionistic fuzzy relations
Fuzzy Sets Syst.
(1996) - et al.
On measurements of covering rough sets based on granules and evidence theory
Inf. Sci. (NY)
(2015) - et al.
Monotonic similarity measures between intuitionistic fuzzy sets and their relationship with entropy and inclusion measure
Inf. Sci. (NY)
(2015) - et al.
Dominance-based rough set approach to incomplete ordered information systems
Inf. Sci. (NY)
(2016) - et al.
Dominance-based rough fuzzy set approach and its application to rule induction
Eur. J. Oper. Res.
(2017) - et al.
An information systems security risk assessment model under uncertain environment
Appl. Soft Comput.
(2011) - et al.
Uncertainty and reduction of variable precision multigranulation fuzzy rough sets based on three-way decisions
Int. J. Approx. Reason.
(2017) - et al.
Variable precision multigranulation decision-theoretic fuzzy rough sets
Knowl. Based Syst.
(2016)