Fuzzy probabilistic rough sets and their corresponding three-way decisions
Introduction
Probabilistic rough sets (short for PRSs), as a combination of rough set theory [35], [36] and probability theory [9], have been studied at length in literatures [5], [31], [32], [37], [41], [49], [67], [68]. The pair of probabilistic approximation operators are built in terms of conditional probabilities and parameter(s) (representing to what degree we can bare the uncertainty or misclassification). When applying PRSs to some concrete situation, parameters, playing a key role in establishing probabilistic approximation operators, are usually provided by experts who are familiar with that situation. It cannot make up for the lack of mathematical foundation even with the help of experts when deciding values of parameters; and it is not confirmed which way is most reasonable when choosing these values. However, from the mathematical viewpoint, we prefer such methods with solid mathematical foundations when there is no concordant criterion. The decision-theoretic rough set (DTRS, for short) approach, proposed by Yao [46], [47], [48], [49], [50], [51], [52], [55], [56], studies rough sets in terms of the Bayesian decision procedure. This approach, just like we said before that it is closely connected with PRSs, provides a mathematic and systematic way to explain and calculate the parameters on the basis of losses or costs of various decisions. The DTRS approach approximates a given concept or a set by three regions (positive, negative and boundary regions) which correspond to positive, negative and boundary rules in three-way decisions (3WDs), respectively [11], [50], [51]. It has been applied to text classification [20], [23], web-based support systems [54], cluster analysis [27], [58], investment decisions [30], multi-classification [6], [26], [27], email filtering [17], [63], government decisions [28], face recognition [22] etc. In the following, we first review developments of PRSs, DTRSs and 3WDs; then we present the main work of this paper.
Let U be a finite universe of discourse and R be an equivalence relation on U. Two subsets of U are denoted by X and Y.
- (1)
Probabilistic approaches to rough sets were first studied by Wong and Ziarko [44]. The PRS model was proposed in 1988 by Pawlak et al. [37]. Their model was built based on a fixed precision 0.5, called 0.5-PRS [45]. The variable precision rough sets (VPRSs) [18], [64] were formulated by a graded set-inclusion relation, i.e. . They can be regarded as special kinds of PRSs if the conditional probabilities are estimated by cardinalities of sets, i.e., [68]. In order to avoid the parameter appeared in VPRSs, Ślȩzak and Ziarko [39], [41] proposed the non-parametric Bayesian rough set (BRS) model, where the set approximations are defined by adopting the prior probability as references. (Further studies of BRS can be found in [34], [40], [61], [66].)
- (2)
The DTRS approach was first introduced by Yao in 1990 [45]. The lower and upper approximations of a concept are derived from the Bayesian minimum risk decision procedure, where the universe of objects are partitioned into three disjoint regions—positive, negative and boundary regions. It finds out that the Pawlak rough set can be obtained based on a special restriction on the loss function. Also, the 0.5-PRS can be obtained as a special case of DTRS by setting another restriction on the loss function. Yao further studied about DTRS approach in [46], [47], [48], [53] where he did detailed discussions on different restrictions of loss function and thus obtained the basic (α, β)-PRS model (0 ≤ β < α ≤ 1), the α-PRS model, etc.
- (3)
Li and Yang studied the axiomatic characterization of PRSs [21]. They derived two sets of axioms using the probabilistic approximation operators. The proposed approach helps to understand PRSs from an axiomatic way. Probabilistic rough set model on two universes was first discussed by Ma and Sun [31], [32]. They have studied the interrelationship between the Bayesian risk decision and probabilistic approximations on two universes. The rough entropy for this generalized PRS model was proposed based on Shannon entropy. Based on the local rough set and the dynamic granulation principle Sang et al. proposed a new DTRS model under dynamic granulation which satisfies the monotonicity of positive regions [38]. The two parameters α and β dynamically update for each granulation. Game-theoretic rough set was obtained by combining the DTRS approach with game theory [1], [10] and it has been applied into recommender system [3].
- (4)
The probabilistic approximations of fuzzy sets have appeared in [7], [42], [57]. In literature [7], Deng and Yao deal with fuzzy sets actually based on cut sets of fuzzy sets. An element whose membership grade is greater than or equal to α is put into the positive region; an element whose membership grade is less than or equal to β is put into the negative region; and an element whose membership is between α and β is put into the boundary region. They gave two ways to determine parameters α and β: one is to minimize the total error caused by aforementioned operations on all elements in U; the other is based on Bayesian risk decision procedure like that in [45], [46], [47], [48]. The probabilistic rough fuzzy set model for a fuzzy set was defined based on an equivalence relation on U [42]. It is also studied the decision-theoretic rough fuzzy set from the viewpoint of Bayesian decision theory. The limitation of the model was that it depends on equivalence relation. Yang et al. proposed fuzzy probabilistic rough set model based on fuzzy relations [57]. Even though the fuzzy relation was adopted in their model, it is the λ-cut sets of fuzzy relation that really work. That means it is still based on classical relations.
- (5)
Interval-valued decision-theoretic rough set model has been studied by Liang and Liu [24]. However, their model is built within probabilistic approximation spaces (i.e., still based on classical equivalence relations). It is only the loss function that is interval-valued. In another paper [25], Liang et al. proposed the triangular fuzzy decision-theoretic rough set model in the framework of a probabilistic approximation space. Likewise, only the loss function is made up of triangular numbers. A new model for incomplete information system was studied by Liu et al. in reference [29] where the conditional probabilities are computed based on a new similarity relation and the loss function is represented by interval-numbers.
- (6)
The concept of three-way decisions actually coexists with rough set theory in which it is interpreted as positive, negative and boundary regions. It was first clearly proposed by Yao in [51], [52]. Then, Hu studied 3WDs from a mathematical viewpoint and proposed three-way decision spaces based on fuzzy lattices and partially ordered sets, respectively [11], [12].
The notion of an event in probability theory [9] is a precisely specified collection of elements in the sample space. However, in everyday experience one frequently encounters situations in which an “event” is fuzzy rather than crisp [60]. For example, the temperature is around 21°C; a student is most probable to pass the exam, etc. These events are fuzzy because of the ill-defined description “around” and “most probable”. Besides, some properties (such as, the using temperature range of different bolts; the best quantum of water sprinkling for a certain kind of plants) cannot be described by exact values for which reason interval values are more desirable. On the other hand, since measurement errors are unavoidable in principle, the measuring result is often accompanied by an error range. Instead of using a single value to represent the measuring result, an interval number is, sometimes, more reasonable. Considering these, there is a need to generalize PRSs and DTRS approach for fuzzy events and interval-valued fuzzy events within the frameworks of fuzzy and interval-valued fuzzy probabilistic approximation spaces, respectively, which is the main work of this paper.
In the fuzzy probabilistic approximation space, we first propose four types of fuzzy probabilistic approximation operators (defined for fuzzy events); then, applying Bayesian decision theory, we study three-way decisions for fuzzy events and figure out the relationship between DTRS approach and fuzzy probabilistic rough sets; finally, we study fuzzy probabilistic rough sets with two different universes of discourse. Compared with existing results [7], [42], [57], our model has several advantages listed below:
- (1)
the fuzzy probabilistic rough set models presented in our paper can deal with fuzzy set directly instead of its λ-cut set;
- (2)
the models are established within the framework of fuzzy probabilistic approximation space which ensures fuzzy relations to directly take a part in computing conditional probabilities instead of using their λ-cut sets;
- (3)
the models are constructed in terms of fuzzy probability instead of the cardinality-based estimation in most literatures [32], [46], [48], [51], [52], [54].
Within the framework of interval-valued fuzzy probabilistic approximation space, we first provide two different ways to define probabilistic approximations for interval-valued fuzzy events. One is based on the interval-valued fuzzy probability; the other is based on fuzzy probability. Then, we study three-way decisions for interval-valued fuzzy events by employing Bayesian decision procedure. The case of two different universal sets is considered at last. Comparing with reference [24], our model is more flexible since it can directly deal with interval-valued fuzzy information systems.
It is worth reminding that investigations of DTRS approach for fuzzy events and interval-valued fuzzy events presented in this paper are actually generalizations of those introduced in reference [62] which studied DTRS approach for classical events in fuzzy and interval-valued fuzzy approximation spaces, respectively. Besides, the main purpose of this paper is to construct various (interval-valued) fuzzy probabilistic rough sets of (interval-valued) fuzzy sets for different application demands.
The remainder of this paper is organized as follows. Section 2 reviews basic notions of fuzzy set, fuzzy event, interval-valued fuzzy set, interval-valued fuzzy event, etc. Section 3 discusses fuzzy probabilistic rough sets and their three-way decisions. Section 4 studies interval-valued fuzzy probabilistic rough sets as well as associated three-way decisions in the framework of interval-valued fuzzy probabilistic approximation space. Brief examples are attached to Sections 3 and 4 to illustrate our models clearly. The last section concludes this paper.
Section snippets
Preliminaries
In order to make this paper self-contained, some useful concepts are recalled, such as fuzzy set, fuzzy event, interval-valued fuzzy set, and interval-valued fuzzy event, etc. More details can be found in literatures [2], [19], [59], [60], [62].
Fuzzy probabilistic rough sets and their corresponding three-way decisions
This section is mainly devoted to propose four types of fuzzy probabilistic rough sets (short for FPRSs) in terms of four groups of parameters. To provide a sensible way to determine the values of parameters or to determine which type of FPRSs is adopted, we study DTRS approach for fuzzy events within the framework of fuzzy probabilistic approximation space. To broaden the applicable ability of FPRSs, models based on two different universal sets (i.e., generalized fuzzy probabilistic rough
Interval-valued fuzzy probabilistic rough sets and their corresponding three-way decisions
Within the framework of IVF probabilistic approximation space, this section discusses establishments of interval-valued fuzzy probabilistic rough sets (short for IVF-PRSs) and interval-valued fuzzy decision-theoretic rough set (IVF-DTRS in short) approach. Note that for interval-valued data we have similar motivations for establishing four types of IVF-PRSs as for their counterparts of FPRSs.
Conclusion
This paper investigates FPRSs and IVF-PRSs within frameworks of fuzzy and IVF probabilistic approximation spaces, respectively. Instead of the usual computation of conditional probability, X⊆U, the fuzzy and IVF conditional probabilities are computed, respectively, as follows: and where and, A, R are a fuzzy set and a fuzzy
Acknowledgment
This research was supported by the National Natural Science Foundation of China (Grant Nos. 11571010, 61179038).
References (68)
- et al.
Interpretation of equilibria in game-theoretic rough sets
Inf. Sci.
(2015) - et al.
Recommender systems survey
Knowl.-Based Syst.
(2013) - et al.
Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets
Fuzzy Sets Syst.
(1996) - et al.
Credit scoring analysis using a fuzzy probabilistic rough set model
Comput. Stat. Data Anal.
(2012) - et al.
Decision-theoretic three-way approximations of fuzzy sets
Inf. Sci.
(2014) Three-way decisions space and three-way decisions
Inf. Sci.
(2014)- et al.
An axiomatic characterization of probabilistic rough sets
Int. J. Approx. Reason.
(2014) - et al.
An information filtering model on the Web and its application in JobAgent
Knowl.-Based Syst.
(2000) - et al.
Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets
Inf. Sci.
(2014) - et al.
Triangular fuzzy decision-theoretic rough sets
Int. J. Approx. Reason.
(2013)