Three-way decisions with decision-theoretic rough sets in multiset-valued information tables
Introduction
The concept of three-way decisions (3WD) was originally introduced by Yao [38], [39], [42] to describe the three regions of rough sets [25]. As a matter of fact, the theory of 3WD goes beyond rough sets and is related to three-valued sets, three-valued logics, shadowed sets, and orthopairs [3], [4], [26], [41]. Yao [42] formalized a more general framework of 3WD called the trisecting-and-acting model, which divides a universal set into three pair-wise disjoint parts and performs effective strategies on some or all of the parts. The ideas of 3WD have inspired many three-way approaches and applications, for example, three-way classification [38], three-way analysis [27], three-way clustering [45], [46], three-way recommendation [47], three-way fuzzy matroids [11], and three-way approximations [43]. In three-way classification, the three regions are known as the positive, boundary, and negative regions, and can be interpreted in terms of three types of decision rules, namely rules for acceptance, rules for non-commitment, and rules for rejection, respectively. Objects satisfying acceptance rules are put into the positive region, objects satisfying non-commitment rules are put into the boundary region, and objects satisfying rejection rules are put into the negative region.
The decision-theoretic rough set (DTRS) model, proposed by Yao [44], is one possible way to construct the three regions of 3WD. The DTRS model systematically calculates thresholds from loss functions using the well-known Bayesian decision procedure. Researchers have studied 3WD and DTRS models in various types of situations [5], [6], [7], [9], [10], [15], [16], [17], [18], [30], [48], [49]. For example, Liang et al. [15], [16], [17], [18] generalized loss functions with triangular numbers, interval numbers, point operators, dual hesitant fuzzy elements, etc. As a result, they established the triangular fuzzy decision-theoretic rough set model [15], interval-valued decision-theoretic rough set model [16], intuitionistic fuzzy decision-theoretic rough set model [18], and dual hesitant fuzzy decision-theoretic rough set model [17], respectively. Within the framework of fuzzy probabilistic approximation space and interval-valued fuzzy probabilistic approximation space, Zhao and Hu [48], [49] generalized classical relations to fuzzy relations and interval-valued fuzzy relations, and proposed the fuzzy decision-theoretic rough set (FDTRS) model and interval-valued fuzzy decision-theoretic rough set model. Hu et al. [5], [6], [7], [8], [9], [28], [36] studied 3WD in the framework of three-way decision spaces in an attempt to provide a solid mathematical foundation. The results of 3WD and DTRS models have been successfully applied to many fields, such as software defect prediction [14], cluster analysis [45], [46], [50], face recognition [12], [13], government decisions [19], decision-making [2], [29], [30], [31], [32], [33], [34], attribute reduction [20], [21], [35], pattern discovery [23], credit scoring [22], etc.
A multiset (or multiple membership set, bag, heap, bunch, etc.) is a collection of elements in which elements may occur more than once [1], [24], [37]. The idea of multiple instances of the same element has existed throughout the development of mathematics. For example, the prime factorization of an integer n > 0 is a multiset whose elements are primes, e.g., the number 360 has the prime factorization which gives the multiset {2, 2, 2, 3, 3, 5}. Every monic polynomial f(x) over the complex numbers corresponds in a natural way to a multiset of its “roots”, e.g., the roots of the polynomial constitute the multiset {1, 1, 3, 3, 3}. Multisets are not only of interest in mathematics, computer science, physics, etc., but are also very common in our daily life. For example, decision results for a paper from a group of reviewers, grades for interviewees from a group of interviewers, and evaluation results from a group of customers are all multisets. Suppose that four doctors make their decisions of whether a patient has a disease. The probabilites of having the disease, as given by four doctors, are 0.9, 0.9, 0.9, and 0.1, respectively. We have two means of representing the results: {0.9, 0.1} and {0.9, 0.9, 0.9, 0.1}. The former is a classic set which deletes all duplicate elements, while the latter is a multiset. The former cannot provide full information about the result, while the later causes a person to believe that the patient has the disease.
In this paper, we generalize the DTRS model to deal with multiset-valued data. We develop two generalized models known as the multiset-decision-theoretic rough set model and multiset-fuzzy-decision-theoretic rough set model. The remainder of this paper is organised as follows. Section 2 reviews basic concepts of a multiset, 3WD, and the DTRS model. Several new operations of multisets are introduced and their corresponding properties are discussed. Section 3 investigates 3WD in multiset-valued information tables. The multiset-decision-theoretic rough set model is established, which generalizes loss functions using multiset values. By integrating the multiset-decision-theoretic rough set model with the FDTRS model, the multiset-fuzzy-decision-theoretic rough set model is created for multiset-valued information tables. The conclusion follows in the last section.
Section snippets
Multisets and three-way decisions
In this section, we briefly review concepts of a multiset, 3WD, and the DTRS model. Some new operations of multisets are introduced and the corresponding properties are discussed. A new concept known as the p-length of finite normal multisets is proposed, based on which a new order relation and a similarity degree measure for finite normal multisets are defined.
Three-way decisions in multiset-valued information tables
By applying DTRS models, 3WD in multiset-valued information tables are investigated in this section. First, the multiset-decision-theoretic rough set model is established by generalizing loss functions with multiset values. Based on the multiplication and p-length, two methods of computing expected costs are introduced. Then, by integrating the multiset-decision-theoretic rough set model with FDTRS model, the multiset-fuzzy-decision-theoretic rough set model is created in multiset-valued
Conclusions
The goal of 3WD is to divide a whole into three parts and to act on some or all of the three parts. The most commonly used method to obtain the three parts is one based on decision functions and thresholds, namely evaluation-based 3WD. The DTRS model is a probabilistic extension of rough set models that systematically calculates the required thresholds using loss functions and the well-known Bayesian decision procedure. Multiset-valued information tables are a generalization of information
Acknowledgments
This research was partially supported by the Fundamental Research Funds for the South-Central University for Nationalities (Grant no. CZQ16013), the China Scholarship Council (Grant no. 201707780022) and the National Natural Science Foundation of China (Grant nos. 11571010, 61179038, 61502419, and 61374085).
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