Decision-theoretic rough fuzzy set model and application
Introduction
Rough set theory is an extension of set theory for studying intelligent systems characterized by insufficient and incomplete information [34], [36], [37], [52]. As a new mathematical tool to deal with vagueness and uncertainty, one of the main advantages of rough set theory is that it does not need any preliminary or additional information about data, such as probability distribution in statistics, basic probability assignment in the Dempster–Shafer theory, or grade of membership or the value of possibility in fuzzy set theory [55]. This new mathematical approach to imprecision, vagueness and uncertainty is founded on the assumption that objects in the universe of discourse can be associated with data or knowledge [31], [42], [43]. Recently, rough set theory has become an important method and tool for granular computing [34], [35], [36], [37], [49], [50], [52] and also has been demonstrated to be useful in decision-making [26], [51], [53], feature selection [48], [72], [74], [75], clustering analysis [54], machine learning [3], [12] and so on.
The theory of rough set was proposed in 1982. It has captured much attention from artificial intelligence and intelligent systems researchers and has been applied to knowledge discovery, data mining, etc. Many generalized rough set models have been proposed by scholars in past years. Broadly speaking, there are several directions for the generalization of the rough set theory. In [65], Yao summarizes various formulations of the standard rough set theory and demonstrates how those formulations can be adopted to develop different generalized rough set theories. Within the set-theoretic framework, generalizations of the element-based definition can be obtained by using non-equivalence binary relations [33], [63], generalizations of the granule-based definition can be obtained by using coverings [47], [48], [64], [65], and generalizations of the subsystem-based definition can be obtained by using other subsystems [65]. By the fact that the system is based on Boolean algebra, one can generalize rough set theory using other algebraic systems such as Boolean algebras, lattices, and posets [10], [35]. Subsystem-based definitions and algebraic methods are useful for such generalizations. Also, there are many other important generalizations of the theory, such as probabilistic and decision-theoretic rough sets [45], [55], [56] and rough membership functions [31], [44], [50].
However, many generalized rough set models are often too strict when including objects into the approximation regions and may require additional information. A lack of consideration for the degree of overlap between an equivalence class and the set to be approximated unnecessarily limits the applications of rough set and has motivated a good deal of research to investigate probabilistic generalization of the theory [4], [7], [28], [29], [41], [45], [46], [55], [59], [71]. In 1987, Wong and Ziarko [45] introduced probabilistic approximation space to the studies of rough set and then presented the concept of probabilistic rough set. Subsequently, Yao et al. [66] proposed a more general probabilistic rough set called decision-theoretic rough set. Then another perspective to deal with the degree of overlap of an equivalence class with the set to be approximated was given, and an approach to select the needed parameters in lower and upper approximations was presented. As far as the probabilistic approach to rough set theory, Pawlak and Skowron [31], Pawlak et al. [32] and Wong and Ziarko [45] proposed a method to characterize a rough set by a single membership function. By the definition of a rough membership function, elements in the same equivalence class have the same degree of membership. The rough membership may be interpreted as the probability of any element belonging to a set, given that the element belongs to an equivalence class. This interpretation leads to probabilistic rough set [56]. Greco et al. [6] introduced a new generalization of the original definition of rough sets and variable precision rough sets, named the parameterized rough set model. They aim at modeling data relationships expressed in terms of frequency distribution rather than in terms of a full inclusion relation, which is used in the classical definition of rough sets.
Probabilistic rough set extends the classical Pawlak rough set model. The major change is the consideration regarding the probability of an element being in a set to determine inclusion in approximation regions. Two probabilistic thresholds are used to determine the division between the boundary-positive region and boundary-negative region. Over the last two decades, probabilistic rough set theories, such as 0.5-probabilistic rough set [45], decision-theoretic rough set [57], rough membership function [31], parameterized rough set [38], Bayesian rough set [39], game-theoretic rough set [8] and naive Bayesian rough set [39], [60] have been proposed to solve probabilistic decision-making problems by allowing a certain acceptable level of error. All these models use two parameters and (a pair of thresholds) to define the lower and upper approximations. Actually, variable precision rough set [69] proposed by Ziarko also is one kind of probabilistic rough set; he firstly used a set inclusion function to define the lower and upper approximations. Moreover, only one parameter was used in the lower and upper approximations. Later on, he reformulated the theory of variable precision rough set by using probabilistic terms [69]. However, a fundamental difficulty with all generalizations of probabilistic rough set is the physical interpretation of the required threshold parameters [58], as well as the need for a systematic method for setting the parameters. In order to offer an effective approach for selecting the threshold parameters, Yao et al. [57] proposed the decision-theoretic rough set model by using the Bayesian decision theory, and then the above difficulty was resolved from the semantic point of view.
Decision-theoretic rough set, as a general probabilistic rough set model, has invoked the interest of many scholars and much valuable research has been done in recent years. We briefly review the studies of decision-theoretic rough set as follows: Herbert and Yao [8], [9] study the combination of the decision-theoretic rough set and the game rough set. Li and Zhou [14], [15] present a multi-perspective explanation of the decision-theoretic rough set and discuss attribute reduction and its application for the decision-theoretic rough set. Jia et al. [11] also discuss the attribute reduction problem for the decision-theoretic rough set theory. Liu et al. [16], [17], [18], [19] discuss multiple-category classification with decision-theoretic rough sets and its applications in the areas of management science. Li et al. [20], Lingras et al. [21], [22], [23] and Yu et al. [68] discuss the clustering analysis by using the decision-theoretic rough set theory. Yang [67] studies the multi-agent decision-theoretic rough set model. Li et al. and Liang et al. [24], [25], [27] discuss information retrieval and filtering by using the decision-theoretic rough set theory. Greco and Slowinski [5] combine the decision-theoretic rough set with the dominance-based rough set and then give a new generalized rough set model. Based on the basic idea of the decision-theoretic rough set model, Zhou [73] presents a new description of this model. Ma and Sun [28], [29] study the decision-theoretic rough set theory over two universes based on the idea of the classical decision-theoretic rough set.
In general, the objects approximated are crisp set or accurate concepts of the universe of discourse in both Yao’s decision-theoretic rough set theory and the existing probabilistic rough set models. For a decision-making problem, there are only two states, which are disjoint and opposite each other for a precise concept of the universe of discourse. For example, in the decision-making problems of diagnosis analysis and email spam filtering, there are only two states of Yes or No for a sufferer or an email. That is, a patient either has the disease or does not have the disease and an email either is junk mail or is not junk mail.
However, the objects of many decision-making problems, such as measuring student achievement in comprehensive testing or the credit evaluation of a credit card applicant, could have more than two states in practice. Moreover, the states of the decision object are not necessarily disjoint and opposite each other. For a given student or credit card applicant, the evaluation results may not be described by two completely opposite states with Yes or No. That would be the case if a student is either a good student or a bad student, or if a credit card applicant is either a good credit risk or a bad credit risk. As a matter of fact, the evaluation results could be a semantic state with preferences, such as Excellent or Good or Medium, High or Medium or Low and Large or Medium or Small. This means that the evaluation results of a student or a credit card applicant could be Excellent or Good or Medium. Obviously, these decision states are not completely opposite and disjoint, but they are fuzzy descriptions of the state of the object in the universe. So, for these decision-making problems, the states of the object approximated on the universe are a fuzzy set instead of a crisp set.
Based on these considerations, in this paper we focus on the rough approximation of a fuzzy concept on probabilistic approximation space. We propose the probabilistic rough fuzzy set model by defining the conditional probability of a fuzzy event. Meanwhile, the basic theory of the probabilistic rough fuzzy set is investigated in detail and several generalized models of the probabilistic rough fuzzy set are also given. At the same time, we define a decision-theoretic rough fuzzy set model in order to provide an approach to select the parameters needed in the lower and upper approximations of probabilistic rough fuzzy set by using the Bayesian decision procedure. This also can be regarded as a generalization of Yao’s [57] decision-theoretic rough set model in a fuzzy environment. In this paper, we firstly define the lower and upper approximations of a fuzzy set with respect to probabilistic approximation space according to the traditional idea which treats the required parameter values of and as a primitive notion. Then, we present the Bayesian risk decision with fuzzy objects based on the well-established Bayesian decision procedure. Subsequently, we establish the relationship between the probabilistic rough fuzzy set model and Bayesian risk decision-making. Moreover, the required parameter values of and can be calculated and interpreted systematically by the process of Bayesian risk decision with fuzzy objects. Finally, the basis of the formula of calculating the values of and is given by the process of Bayesian risk decision with fuzzy objects.
So, the motivation of this paper is to approximate a fuzzy object of the universe of discourse in probabilistic approximation space, as well as to present an approach to determine the parameters needed in the lower and upper approximations. Therefore, there are two aspects for the proposed model. One is to present decision-making for the fuzzy decision classes by approximating the fuzzy concepts in probabilistic approximation space. Another is to present a method of how to calculate the needed threshold values.
The rest of this paper is organized as follows. Section 2 provides the basic concept of rough set theory and briefly reviews probabilistic rough set theory. In Section 3, we propose the probabilistic rough fuzzy set model and discuss several properties. Moreover, we give three generalizations of the probabilistic rough fuzzy set model. In Section 4, we firstly review the Bayesian risk decision-making procedure and then propose a decision-theoretic rough fuzzy set model. Furthermore, a systematic method for setting the parameters is given. In Section 5, an example is presented to illustrate the concept and show approaches in handling the credit card applicant problem. At last, we conclude our research and suggest further research directions in Section 6.
Section snippets
Review of rough set models
In this section, we briefly review the concept of rough set theory as well as its extensions.
Probabilistic rough fuzzy set model and its extensions
As for the above analysis, we know that there is a solid necessity to approximate a fuzzy concept in probabilistic approximation space or to discuss the theory of probabilistic rough set in a fuzzy environment for management decision-making in practice. With the objective of bringing together existing studies on probabilistic rough set approximations in a fuzzy environment, we discuss the approximation of a fuzzy concept of the universe of discourse on a probabilistic approximation space in
Decision-theoretic rough fuzzy set
Like the traditional probabilistic rough set, all the rough set models defined in Section 3 are parametric definitions of positive and negative regions that depend on the setting of a permissible level of uncertainty associated with each of the approximation regions. In some applications, however, it is not clear how to define the parameters. Also, using parameters is sometimes not required, as the general objective can be to increase the certainty of a prediction that an event of interest will
A test example
To demonstrate the decision-theoretic rough fuzzy set model established in Section 4, we present a simplified example of an application in practice. The example does not necessarily reflect realistic practices in management decision-making but is used here as an illustration of the basic principal and steps given in Section 4. Below, Table 2 gives a decision information system for responding to a credit card applicant. Here, the salary of applicants can be describe by the linguistic value Low,
Conclusion and future work
In this paper, we systematically discuss the basic theory of the probabilistic rough fuzzy set. We propose several probabilistic rough fuzzy set models such as the 0.5-probabilistic rough fuzzy set model, variable precision probabilistic rough fuzzy set model and Bayesian rough fuzzy set model. That is, we establish the method of approximating a fuzzy concept in a probabilistic approximation space. Like the traditional probabilistic rough set, these models depend on the choice of parameters by
Acknowledgements
The authors are very grateful to the Editor in Chief Professor W. Pedrycz, and the anonymous referees for their thoughtful comments and valuable suggestions. Some remarks directly benefit from the referees’ comments. This research was partly supported by the National Science Foundation of China (Grant Nos. 71161016 and 71071113), the National Science Foundation of Gansu Province of China (1308RJYA020), and the Fundamental Research Funds for the Central Universities.
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