An Overview of Function Based Three-Way Decisions - Springer

An Overview of Function Based Three-Way Decisions Dun Liu1 and Decui Liang2 1

2

School of Economics and Management, Southwest Jiaotong University Chengdu 610031, P.R. China [email protected] School of Management and Economics, University of Electronic Science and Technology of China Chengdu 610054, P.R. China [email protected]

Abstract. By considering the various of studies on loss functions with three-way decisions, a function based three-way decisions is proposed to generalize the existing models. A “four-level” approach with granular perspective is built, and the existing models can be categorized to a “four-level” framework through different decision criteria. Our work provides a novel “granularity” viewpoint on the current three-way decision researches. Keywords: Three-way decisions, loss functions, decision-theoretic rough sets, decision-making.

1

Introduction

Three-way decisions (TWD), a new perspective of probabilistic rough sets, which were proposed by Y.Y. Yao, have drawn more and more attentions in nearly five years [38, 39, 42, 43]. A theory of three-way decisions is constructed based on the notions of the acceptance, rejection or noncommitment, which can be directly generated by the three regions of probabilistic rough sets. The rules generated by the positive region are used to make a decision of acceptance, the rules generated by the negative region are used to make a decision of rejection, the rules generated by the boundary region are used for making a decision of noncommitment [38, 40]. In general, three-way decisions describe the human cognitive process during decision making, and establish an intimate connection between rough sets and decision theory. With carefully investigate current studies, there are three main research directions on three-way decisions. (1). The extended models of three-way decisions. By considering the key ingredient in three-way decisions is the loss functions, some researches adopted stochastic numbers [21], intervals [14, 23], fuzzy intervals [24, 27], triangular fuzzy numbers [12], intuitionistic fuzzy numbers [15], hesitant fuzzy numbers [13], shadow sets [5] to estimate the losses in three-way decisions, D. Miao et al. (Eds.): RSKT 2014, LNAI 8818, pp. 812–823, 2014. c Springer International Publishing Switzerland 2014 DOI: 10.1007/978-3-319-11740-9_74 

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which are the extension of losses under uncertain decision environments. Another extending of three-way decisions emphasized on the methodologies, e.g., multiple-classification three-way decisions [20, 46, 47], multi-agent three-way decisions [33], cost-sensitive three-way decisions [30], information-theoretic based three-way decisions [4], sequential three-way decisions [41], dynamic three-way decisions [28], game-theoretic based three-way decisions [1, 7], clusters based three-way decisions [16, 44], three-way decisions with two universes [29], etc. (2). The attributes reduction methods and rules acquisitions approaches of three-way decisions. The main attribute reduction methods in three-way decisions include probabilistic attribute reduct [37, 45], non-monotonicity of probabilistic positive region reduct [10], minimum cost attribute reduction [8], cost-sensitive attribute reduction [11] and some machine learning based attribute reduction [3]. (3). The applications of three-way decisions. The essential ideas of three-way decisions are commonly used in many domains, e.g., information sciences, engineering, management sciences, medical decision-making, etc [17, 26]. The aforementioned studies indicate that the three-way decisions has gradually been a hot research topic in granular computing and rough sets. In this paper, we focus on investigating the loss functions in three-way decisions. The remainder of this paper is organized as follows: Section 2 provides the basic concepts of three-way decisions and it’s extensions. A generalized threeway decision model with functional perspective is proposed in Section 3. Then, the similarity and difference of existing three-way decision models are carefully analyzed in Section 4, and a “four level” structure model for three-way decisions is built. Section 5 concludes the paper and outlines the future work.

2

Preliminaries

Basic concepts, notations and results of three-way decisions are briefly reviewed in this section [2, 18, 22, 31, 32, 34–36, 38, 48]. As Yao stated in [42, 43], many generalizations of sets have been proposed and studied with three-way decisions, including interval sets and three-valued logic, Pawlak rough sets, Decision-theoretic rough sets (DTRS), three-valued approximations in many-valued logic, fuzzy sets and shadowed sets. In our following discussions, we mainly discuss three-way decisions in DTRS. The DTRS model is inspired by the Bayesian decision theory, a well known theorem in decision analysis [6]. It considers 2 states Ω = {X, ¬X} and 3 actions A = {aP , aB , aN } during decision process. The set of states is given by Ω indicating that an object is in X and not in X, respectively. Meanwhile, aP , aB , and aN in A represent the three actions in classifying an object x, namely, deciding x ∈ POS(X), deciding x should be further investigated x ∈ BND(X), and deciding x ∈ NEG(X), respectively. The loss function λ regarding the risk or cost of actions in different states is given by the 3 × 2 matrix: X (P ) ¬X (N ) aP λPP λPN aB λBP λBN aN λNP λNN

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In the matrix, λPP , λBP and λNP denote the losses incurred for taking actions of aP , aB and aN , respectively, when an object belongs to X. Similarly, λPN , λBN and λNN denote the losses incurred for taking the same actions when the object belongs to ¬X. P r(X|[x]) is the conditional probability of an object x belonging to X given that the object is described by its equivalence class [x]. For an object x, the expected loss R(ai |[x]) associated with taking the individual actions can be expressed as: R(aP |[x]) = λPP P r(X|[x]) + λPN P r(¬X|[x]), R(aB |[x]) = λBP P r(X|[x]) + λBN P r(¬X|[x]), R(aN |[x]) = λNP P r(X|[x]) + λNN P r(¬X|[x]).

The Bayesian decision procedure suggests the following minimum-cost decision rules: (P) If R(aP |[x]) ≤ R(aB |[x]) and R(aP |[x]) ≤ R(aN |[x]), decide x ∈ POS(X); (B) If R(aB |[x]) ≤ R(aP |[x]) and R(aB |[x]) ≤ R(aN |[x]), decide x ∈ BND(X); (N) If R(aN |[x]) ≤ R(aP |[x]) and R(aN |[x]) ≤ R(aB |[x]), decide x ∈ NEG(X).

Since P r(X|[x]) + P r(¬X|[x]) = 1, we simplify the rules based only on the probability P r(X|[x]) and the loss function. By considering a reasonable kind of loss functions with λPP ≤ λBP < λNP and λNN ≤ λBN < λPN , the decision rules (P)-(N) can be expressed concisely as: (P) If P r(X|[x]) ≥ α and P r(X|[x]) ≥ γ, decide x ∈ POS(X); (B) If P r(X|[x]) ≤ α and P r(X|[x]) ≥ β, decide x ∈ BND(X); (N) If P r(X|[x]) ≤ β and P r(X|[x]) ≤ γ, decide x ∈ NEG(X).

The thresholds values α, β, γ generated by DTRS are given by: (λPN − λBN ) ; (λP N − λBN ) + (λBP − λP P ) (λBN − λNN ) ; = (λBN − λNN ) + (λNP − λBP ) (λPN − λNN ) . = (λPN − λNN ) + (λNP − λPP )

αDT RS = β DT RS γ DT RS

(1)

In addition, as a well-defined boundary region, the conditions of rule (B) suggest that α > β, which implies 0 ≤ β < γ < α ≤ 1. ∀X ⊆ U , the (α, β)-lower approximation, (α, β)-upper approximation of DTRS are defined as follows: (λPN − λBN ) }; (λP N − λBN ) + (λBP − λP P ) (λBN − λNN ) RS }. aprDT (α,β) (X) = {x ∈ U |P r(X|[x]) > (λBN − λNN ) + (λNP − λBP )

RS apr DT (X) = {x ∈ U |P r(X|[x]) ≥ (α,β)

(2)

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The three regions of (α, β)- positive, boundary and negative regions in DTRS can be written as: (λPN − λBN ) }, (λP N − λBN ) + (λBP − λP P ) (λBN − λNN ) RS BNDDT < P r(X|[x]) (α,β) (X) = {x ∈ U | (λBN − λNN ) + (λNP − λBP ) (λPN − λBN ) < }, (λP N − λBN ) + (λBP − λP P ) (λBN − λNN ) RS }. NEGDT (α,β) (X) = {x ∈ U | P r(X|[x]) ≤ (λBN − λNN ) + (λNP − λBP ) RS POSDT (α,β) (X) = {x ∈ U | P r(X|[x]) ≥

Specially, if the assumption “α > β” does not hold, the three-way decisions convert to two-way decisions, and it can be rewritten as: (λPN − λNN ) }, (λPN − λNN ) + (λNP − λPP ) (λPN − λNN ) RS NEGDT }. (γ,γ) (X) = {x ∈ U | P r(X|[x]) < (λPN − λNN ) + (λNP − λPP ) RS POSDT (γ,γ) (X) = {x ∈ U | P r(X|[x]) ≥

According to above discussions, DTRS gives a brief semantics explanation with minimum decision risks when comparing with other rough set models (e.g., probabilistic rough set model). Note that, the two thresholds α and β in DTRS are not setting in advance by human’s experiments, and they are associated to decision risk with different loss functions [19, 38, 39, 42, 43].

3

A Function Based Perspective of Three-Way Decisions

In this section, we propose a more generalized model of three-way decisions. As we stated in Section 2, the decision rules generated by the three regions are closely related with loss functions. In DTRS, the values of losses are precise real numbers. In some cases, one can directly utilize the precise value (i.e., money, energy and time, etc) to estimate the costs [19]. However, in most of cases, it is rather difficult to utilize one value to illustrate the costs [12–14, 21, 23, 24, 27]. Inspired by the deficiencies, we propose a function based three-way decisions. In this model, we use two groups of loss functions f (λPP ), f (λBP ), f (λNP ) and f (λPN ), f (λBN ), f (λNN ) to instead of six single values, and the matrix refers to the cost of actions in different states can be rewritten as: X (P ) aP f (λPP ) aB f (λBP ) aN f (λNP )

¬X (N ) f (λPN ) f (λBN ) f (λNN )

In the matrix, λ•• (• = P, B, N ) is not a fixed number, but an independent variable. Given a fixed λ•• , by considering the losses of classifying an object x

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belonging to X into the positive region P OS(X) is less than or equal to the loss of classifying x into the boundary region BN D(X), and both of these losses are strictly less than the loss of classifying x into the negative region N EG(X). The reverse order of losses is used for classifying an object not in X. Therefore, the following two conditions should be considered: f (λPP ) ≤ f (λBP ) < f (λNP ), f (λNN ) ≤ f (λBN ) < f (λPN ).

(3)

Under (3) and P r(X|[x])+ P r(¬X|[x]) = 1, the thresholds values α, β, γ with a function based three-way decisions can be calculated as: f (λPN ) − f (λBN ) ; (f (λP N ) − f (λBN )) + (f (λBP ) − f (λP P )) f (λBN ) − f (λNN ) ; = (f (λBN ) − f (λNN )) + (f (λNP ) − f (λBP )) f (λPN ) − f (λNN ) . = (f (λPN ) − f (λNN )) + (f (λNP ) − f (λPP ))

αDT RS = β DT RS γ DT RS

(4)

Similarly, suppose 0 ≤ β < γ < α ≤ 1, the three regions of (α, β)- positive, boundary and negative regions with a function based three-way decisions can be written as: f (λPN ) − f (λBN )) }, (f (λP N ) − f (λBN )) + (f (λBP ) − f (λP P )) f (λBN ) − f (λNN ) RS BNDDT < P r(X|[x]) (α,β) (X) = {x ∈ U | (f (λBN ) − f (λNN ) + f (λNP ) − f (λBP )) f (λPN ) − f (λBN ) }, < (f (λP N ) − f (λBN )) + (f (λBP ) − f (λP P )) f (λBN ) − f (λNN ) RS NEGDT }. (α,β) (X) = {x ∈ U | P r(X|[x]) ≤ (f (λBN ) − f (λNN )) + (f (λNP ) − f (λBP )) RS POSDT (α,β) (X) = {x ∈ U | P r(X|[x]) ≥

Specially, if the assumption “α > β” does not hold, the three-way decisions converts to two-way decisions, and it can be rewritten as: f (λPN ) − f (λNN ) }, (f (λPN ) − f (λNN )) + (f (λNP ) − f (λPP )) f (λPN ) − f (λNN ) RS NEGDT }. (γ,γ) (X) = {x ∈ U | P r(X|[x]) < (f (λPN ) − f (λNN )) + (f (λNP ) − f (λPP )) RS POSDT (γ,γ) (X) = {x ∈ U | P r(X|[x]) ≥

4

The Framework and Comparison of Existing Three-Way Decision Models

In the following, we construct a “four-level” framework of existing three-way decision models, which can be regarded as the “granular structure” of three-way decisions with loss functions.

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Level 1: The basic expression of three-way decisions In this basic level, the loss function f (λ•• ) (• = P, B, N ) is set as a precise value number: f (λ•• ) = λ•• (• = P, B, N ), and the function based three-way decisions degenerate into precise-value based three-way decisions. Obviously, the basic model of three-way decisions is DTRS [34, 35]. Furthermore, due to the single valued losses in DTRS, the precise-value based three-way decisions can be regarded as static and certain decision model. Level 2: Three-way decisions with uncertainty As we known, “uncertainty” in decision making means the lack of certainty, it used to describe the state of having limited knowledge where it is impossible to exactly describe the existing state, a future outcome, or more than one possible outcome. The main mathematical theories to describe uncertainty are probability theory, fuzzy set theory, rough set theory, interval set theory, inclusion degree theory, evidence theory, etc. The former three theorems are most famous uncertainty methodologies and have been widely used in human’s decision process. (1). Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena, it utilizes “stochastic number” to express uncertainty and indicates that a particular subject is seen from point of view of randomness. The uncertainty in probability theory means one thing cannot definite happened, but may happen with a probability. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. (2). The uncertainty in fuzzy set theory comes from the fuzzification, which is used to describe the unclear classification for a concept. (e.g., the concept of “young”). The fuzzification comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. (3). The uncertainty in rough set theory comes from the inaccuracy for boundary sets. A rough set is a formal approximation of a crisp set in terms of a pair of sets which give the lower and the upper approximation of the original set, and can deal with inaccuracy, inconsistent, incomplete information in a decision system. With above discussions, we investigate three types of uncertain three-way decision models as follows. • Interval three-way decisions In this scenario, the loss function f (λ•• ) (• = P, B, N ) can be rewritten as: + − + f (λ•• ) = [λ− •• , λ•• ] (• = P, B, N ), λ•• and λ•• are the lower bound and the upper bound of λ•• . Liu et al. firstly introduced interval-valued loss functions to DTRS and carefully discussed the corresponding propositions and criteria of interval-valued three-way decisions [23]. Furthermore, Liu et al. used a fuzzy + ] and proposed fuzzy interval-valued three-way decisions − , λ interval number [λ •• •• [27]. Liang and Liu did systematic studies on three-way decisions with intervalvalued decision-theoretic rough sets, they derived three-way decisions with the aid of two conventional methods and proposed a new optimization method in the viewpoint of the flexibility of information granularity [14]. • Fuzzy three-way decisions

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In this scenario, the loss function f (λ•• ) (• = P, B, N ) can be rewritten as: •• (• = P, B, N ). Liu et al. firstly introduced fuzzy loss functions f (λ•• ) = λ to DTRS and investigated the corresponding propositions and criteria of fuzzy three-way decisions [24]. Liang et al. considered the triangular fuzzy loss func•• = (l•• , m•• , u•• ) (• = P, B, N ), and discussed 5 ranking functions for tions λ measure triangular fuzzy number [12]. In addition, Liang and Liu further took •• = hE (λ•• )) into account the losses of DTRS with hesitant fuzzy elements (λ and proposed a new model of hesitant fuzzy three-way decisions, a novel risk decision-making method with the aid of hesitant fuzzy DTRS was developed in their work [13]. As to intuitionistic fuzzy sets, Liang and Liu used the degree of membership μ and degree of non-membership ν to depict the fuzzification [15]. The loss function for the intuitionistic fuzzy three-way decisions can be •• = (λ•• , μ(λ•• ), ν(λ•• )). The decision rules generated by intuitionistic set as: λ fuzzy three-way decisions both considered the membership and non-membership functions. In addition, Deng and Yao investigated mean-value-based three-way shadowed sets, they introduced a generalized decision-theoretic shadowed set model using the mean value [5]. In a word, all the aforementioned work have solid contributions to develop fuzzy three-way decision theorems. • Stochastic three-way decisions In this scenario, the loss function f (λ•• ) (• = P, B, N ) can be rewritten as: f (λ•• ) = λε•• (ε denotes the loss function λε•• is a stochastic number). Liu et al. introduced stochastic loss functions to DTRS and proposed stochastic decisiontheoretic rough set theory [21]. In their studies, a model of stochastic three-way decisions was built with respect to the minimum bayesian expected risk, and they further investigated two special stochastic three-way models under uniform distribution and normal distribution, respectively. To sum up, Level 2 focuses on the decision of uncertainty, but these models are still within a static and close decision environment. Level 3: Three-way decisions with multi-stage decision making Level 3 begins to discuss the dynamic decision environment, and the rules acquisition by three-way decisions are not a single step decision, but a multi-stage process. Liu et al. considered the dynamic change of loss functions f (λ•• ) = g(λt•• ) (• = P, B, N ) in DTRS with the time t, and proposed dynamic three-way decisions [28]. Yang and Yao proposed a multi-agent three-way decision model, the new model is utilized to seek for synthesized or consensus decisions when there are multiple decision preferences and criteria adopted by different agents [33]. Yao and Deng discussed sequential three-way decisions when adding attributes [41]. It suggested that if a decision of acceptance or rejection with certain tolerable levels of errors can be made at a higher level, it is not necessary to move to a lower level. This work enabled the decision maker to consider both the cost of various mis-classifications and the cost of obtaining the necessary evidence for making a classification decision. Liu et al. investigated multiple-category classification problems with three-way decisions [20], and further proposed a two-stage method to choose the best candidate classification. Zhou also provided a new formulation of multi-class three-way decision model, which can be well suited

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for cost-sensitive classification tasks where different types of classification errors have different costs [46, 47]. In summary, all the above mentioned studies make effective interpretations on dynamic three-way decisions. Level 4: Function based three-way decisions As we stated in Levels 1-3, Level 4 is the generation level for three-way decisions because the loss function in former three levels can be treated as a special case of f (λ•• ) . In Level 4, f (λ•• ) can be treated as power function, exponeneial function, nonlinear function, etc. It is the generalization level for three-way decisions. For simplicity, we choose three dimensions (characters): certain or uncertain decisions, single-stage or multi-stage decisions, static or dynamic decisions, to illustrate the similarity and difference of different three-way decision models. Figure 1 outlines the framework of the four-level model in three-way decisions. Table 1 summarizes the characters for the models in Figure 1. Table 1. The characters for different three-way decision models Level Level 1

Level 2

Level 3 Level 4

Model Precise-value (DTRS) Interval-valued Fuzzy interval Interval Interval sets Triangular Intuitionistic Fuzzy Hesitant Shadow Uniform Normal Stochastic Other Dynamic Multi-agent Sequential Multiple-category Function

Character 1 Certain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Uncertain Both

Character 2 Single-stage Single-stage Single-stage Single-stage Single-stage Single-stage Single-stage Single-stage Single-stage Single-stage Single-stage Multi-stage Multi-stage Multi-stage Multi-stage Both

Character 3 Static Static Static Static Static Static Static Static Static Static Static Dynamic Dynamic Dynamic Dynamic Dynamic

In Figure 1 and Table 1, Level 1 and Level 2 are based on the static three-way decisions, Level 3 and Level 4 are based on the dynamic three-way decisions. If we consider the variation of loss functions, the static models can change to dynamic models. In addition, Level 1 is the specialization level, and Level 4 is the generalization level. Between Level 1 and Level 4, the middle two levels respectively consider the decision risk and decision stages. With the insightful gain from the top-down or bottom-up perspective in granular computing, DTRS model in three-way decisions corresponds finest granularity, and the function based three-way decision model corresponds the coarsest granularity. In general, one can choose a special model by using different decision criterion or viewpoint in a real decision problem.

Level 4

decisions

Shadow three-way decisions

Hesitant fuzzy three-way decisions

Fig. 1. The framework of the four-level model in three-way decisions (Specialization/Certain Level) (Single-step Decision Level)

Precise-value based three-way decisions (DTRS)

Level 1

Level 2

Other distribution

Normal distribution

Stochastic three-way decisions

Multiple-category three-way decisions

Uniform distribution

(Uncertain Level/ Single-step Decision Level)

Intuitionistic fuzzy three-way decisions

Triangular fuzzy three-way decisions

Fuzzy three-way decisions

Level 3

Sequential three-way

(Multi-stage Decision Level)

Multi-agent three-way decisions

Fuzzy interval three-way decisions

Interval sets three-way decisions

Interval-valued three-way decisions

Interval three-way decisions

Dynamic three-way decisions

Function-based three-way decisions (Generalization Level)

Static Level

Variation of loss functions

Dynamic Level

820 D. Liu and D. Liang

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Conclusions

In this paper, we systematically study the loss functions with three-way decisions in nearly two decades and propose a function based three-way decision model. The current models of three-way decisions are carefully investigated, then classified into a generalization research framework via three different dimensions. A “four-level” model in three-way decisions is built, which draw an intuitive and clear impression for different three-way decision models. Our future researches will focus on properties for different types of function based three-way decisions, the group decision method in three-way decisions will be our another future research topic. Acknowledgements. This work is partially supported by the National Science Foundation of China (No. 71201133), the Youth Social Science Foundation of the Chinese Education Commission (No. 11YJC630127), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120184120028) and the Fundamental Research Funds for the Central Universities of China (No. SWJTU12CX117).

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