Decision-theoretic rough set approach for fuzzy decisions based on fuzzy probability measure and decision making

Abstract

Most existing studies on decision-theoretic rough set approach focus on crisp decisions. However, in many cases, we need to handle fuzzy concepts. This paper investigates decision-theoretic rough set approach for fuzzy decisions. By introducing fuzzy operators, we extend the traditional decision-theoretic rough set approach with crisp decisions to fuzzy concepts. Decision-theoretic rough set approach with fuzzy concepts is investigated based on fuzzy probabilistic approximation space. Bayesian decision procedure based on fuzzy probabilistic approximation space with fuzzy concepts and fuzzy probabilistic decision systems with fuzzy or continuous decision attributes are studied. The application of the proposed approach in decision making is also illustrated by an example. This study may expand the applications of decision-theoretic rough set model.

Keywords

Decision-theoretic rough set fuzzy concepts fuzzy decision decision making

1 Introduction

Rough set theory, originally proposed by Pawlak [72], has become a popular mathematical tool to deal with uncertainty and vagueness in many fields, such as data mining, data analysis and decision making [1 , 112]. To enhance the robustness of rough set models, probabilistic rough set model (PRS) was proposed by Pawlak et al. in which thresholds were used to determine the lower and upper approximations of a set. The 0.5-probabilistic rough set model was well adopted, i.e. a threshold of 0.5 on probability is used to define probabilistic lower and upper approximations of a set. Yao et al. [94, 100], proposed a generalized probabilistic rough set model called Decision-Theoretic Rough Set (DTRS) model. Compared with related probabilistic rough set models, DTRS provides a solution on computing the threshold parameters for inclusion degree by introducing Bayesian decision theory. DTRS has been widely disussed in the research area of rough set [38 , 113].

In DTRS approach, decision (also called state), conditional probability and loss function play an important role. Many existing studies focus on different types of loss function or different types of conditional probability. For example, Li gave a general loss function for supervised learning in which the cost and benefit of assigning an instance to a specific subcategory was studied [48]. Liang and Liu gave decision-theoretic rough set with triangular fuzzy loss function and interval valued loss function in [56] investigated DTRS approach in the frameworks of fuzzy and IVF probabilistic approximation spaces, respectively. In spite of all these studies on loss function and conditional probability, there are few studies on studying the decisions. In these studies, only crisp decision concepts are considered. As well known, the DTRS approach adopts two crisp decisions (also called states) and three actions to describe the decision process. The set of two decisions indicating an object is in a decision or not. The set of actions is deciding that an object belongs to a decision; an object does not belong to a decision and an object is not sure.

However, there are many fuzzy concepts in the real world, which we can’t make a definite division. We can describe them as fuzzy sets other than traditional sets. In this paper, we extend crisp decisions in DTRS to fuzzy decisions. Decision-theoretic rough set approach with fuzzy decisions based on fuzzy probability measure is investigated.

The remainder of this paper is organized as follows. In Section 2, some basic concepts of fuzzy sets and DTRS approach based on fuzzy probability are reviewed. In Section 3, DTRS approach with fuzzy decision based on fuzzy probability measure is investigated. Section 4 concludes this paper.

2 Preliminaries

2.1 Fuzzy logical operator

Throughout this paper, let T denote a triangular norm (t-norm for short). A triangular norm (t-norm for short) T is any increasing (T (a, b) ≤ T (c, d) if a ≤ c and b ≤ d), commutative (T (a, b) = T (b, a)) and associative (T (a, T (b, c)) = T (T (a, b) , c)) [0, 1] ² → [0, 1] mapping satisfying T (a, 1) = a, for all a in [0, 1].

Three well-known T operators are listed as follows: $\begin{matrix} T_{M} (a, b) = min (a, b) \\ T_{P} (a, b) = ab \\ T_{W} (a, b) = max (a + b - 1, 0) \end{matrix}$ where a, b ∈ [0, 1].

2.2 Fuzzy decision-theoretic rough set approach with traditional decision

For the purpose of comparison, we recall some useful concepts of fuzzy decision-theoretic rough set approach with traditional decisions. The details can be found in [107, 111].

Let $(U, \tilde{R}, P)$ be a fuzzy probabilistic approximation space in which U is a universe of discourse, $\tilde{R}$ is a fuzzy relation on U, and P is a fuzzy probability measure of U.

Definition 1. [111] The conditional probabilities P (X| [x] _R) and P (X^C| [x] _R) are fuzzy probabilities since R is a fuzzy relation. Let U = {u₁,... . . u_n} and P (u_i) = p_i, the conditional probabilities are defined by $\begin{matrix} P (X | [x]_{\tilde{R}}) & = & \frac{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{j} \in X} \tilde{R} (x, u_{j}) p_{j}} \\ P (X^{C} | [x]_{\tilde{R}}) & = & \frac{\sum_{u_{i} \in X^{C}} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{j} \in X^{C}} \tilde{R} (x, u_{j}) p_{j}} \end{matrix}$ where X is a classic set and $[x]_{\tilde{R}}$ is a fuzzy equivalence class induced by fuzzy equivalence relation $\tilde{R}$ .

It should be noticed that this computation of condition probability is different from that in [95 , 104] which are based on cardinalities.

For each object x ∈ U, $[x]_{\tilde{R}} (y) = \tilde{R} (x, y)$ for every y ∈ U. Assume that $P ([x]_{\tilde{R}}) \neq 0$ for all x ∈ U. The expected losses of taking different actions for x are as follows [111]: $\begin{matrix} L (a_{P} | [x]_{\tilde{R}}) & = & λ_{PP} P (X | [x]_{\tilde{R}}) + λ_{PN} P (X^{C} | [x]_{\tilde{R}}) \\ L (a_{N} | [x]_{\tilde{R}}) & = & λ_{NP} P (X | [x]_{\tilde{R}}) + λ_{NN} P (X^{C} | [x]_{\tilde{R}}) \end{matrix}$ (1) $L (a_{B} | [x]_{\tilde{R}}) = λ_{BP} P (X | [x]_{\tilde{R}}) + λ_{BN} P (X^{C} | [x]_{\tilde{R}})$ where λ_PP, λ_PN, λ_NP, λ_NN, λ_BP and λ_BN are defined by the loss function [λ] _X for X shown in Table 1.

Definition 2. Let $(U, \tilde{R}, P)$ be a fuzzy probabilistic approximation space and with loss function [λ], the [λ] - fuzzy probabilistic positive, negative and boundary regions are defined as follows: $\begin{matrix} PO S^{[λ]} (X) & = & {x \in U | L (a_{P} | [x]_{\tilde{R}}) \leq L (a_{B} | [x]_{\tilde{R}}), \\ L (a_{P} | [x]_{\tilde{R}}) \leq L (a_{N} | [x]_{\tilde{R}})} \\ NE G^{[λ]} (X) & = & {x \in U | L (a_{N} | [x]_{\tilde{R}}) < L (a_{P} | [x]_{R}), \\ L (a_{N} | [x]_{\tilde{R}}) \leq L (a_{B} | [x]_{\tilde{R}})} \\ BN D^{[λ]} (X) & = & (PO S^{[λ]} (X) \cup NE G^{[λ]} (X))^{C} \end{matrix}$ The [λ] - FPRS probabilistic lower and upper approximations of X are as follows: $\begin{matrix} {\underline{\tilde{R}}}^{[λ]} (X) & = & PO S^{[λ]} (X) \\ {\bar{\tilde{R}}}^{[λ]} (X) & = & (NE G^{[λ]} (X))^{C} \end{matrix}$ The pair of subsets, $({\underline{\tilde{R}}}^{[λ]} (X), {\bar{\tilde{R}}}^{[λ]} (X))$ is called the [λ] - FPRS probabilistic rough set of X([λ] - FPRS, for short).

3 Fuzzy decision-theoretic rough set approach with fuzzy decisions

3.1 Bayesian decision procedure based on fuzzy probabilistic approximation space with fuzzy concept $\tilde{X}$

Let $(U, \tilde{R}, P)$ be a fuzzy probabilistic approximation space in which U is a universe of discourse, $\tilde{R}$ is a fuzzy relation on U, and P is a fuzzy probability measure of U. As well known, decision-theoretic rough set approach adopts two decisions and three actions to describe the Bayesian decision process. The set of two decisions is {X, X^C}, (X ⊆ U is a classical set) indicating an object is in X or not, respectively. The set of actions is {a_P, a_N, a_B} deciding that x belongs to X; x does not belong to X and x is not sure (thus a deferment decision for x), respectively. In this paper, we extend the traditional decisions to fuzzy cases, i.e. the two decisions are $\tilde{X}$ and ${\tilde{X}}^{C}$ . It should be noted that $\tilde{X}$ is a fuzzy set representing a fuzzy concept. For example, the concept of “tall” for a person, is actually a fuzzy concept. It is not reasonable to treat this concept as a crisp concept X. Consequently, the loss function [λ] _X can given by a matrix in Table 2. The subscript $\tilde{X}$ represents this loss function is for $\tilde{X}$ , which is omitted in the following if no confusion arises.

For each object x ∈ U, $[x]_{\tilde{R}}$ is adopted as its description and $[x]_{\tilde{R}} (y) = \tilde{R} (x, y)$ for every y ∈ U. Assume that $P ([x]_{\tilde{R}}) \neq 0$ for all x ∈ U. Thus, the expected losses of taking different actions for x can be defined as follows: $\begin{matrix} L (a_{P} | [x]_{\tilde{R}}) & = & λ_{PP} P (\tilde{X} | [x]_{\tilde{R}}) + λ_{PN} P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) \\ L (a_{N} | [x]_{\tilde{R}}) & = & λ_{NP} P (\tilde{X} | [x]_{\tilde{R}}) + λ_{NN} P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) \end{matrix}$ (2) $L (a_{B} | [x]_{\tilde{R}}) = λ_{BP} P (\tilde{X} | [x]_{\tilde{R}}) + λ_{BN} P ({\tilde{X}}^{C} | [x]_{\tilde{R}})$

Definition 3. Suppose U = {u₁,... . . u_n} and P (u_i) = p_i. The conditional probabilities $P (\tilde{X} | [x]_{R})$ and $P ({\tilde{X}}^{C} |$ $[x]_{\tilde{R}})$ are defined as follows: $\begin{matrix} P (\tilde{X} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i}}{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i} + \sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}} \\ P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}}{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i} + \sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}} \end{matrix}$

where $\tilde{X}$ is a fuzzy set and [x] _R is a fuzzy equivalence class induced by fuzzy equivalence relation R, [x] _R (i) is the ith membership in fuzzy equivalence class [x] _R, and T is a t-norm.

This computation of condition probability is different from that in [111]. The definition in this paper adopts a t-norm to perform the intersection of two fuzzy sets, and can handle the case when X becomes fuzzy. Moreover, it can degenerate to a classical case based on cardinalities.

Proposition 1. $P (\tilde{X} | [x]_{\tilde{R}}) + P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) = 1$ .

Proof. It is clear from the definition. $\begin{matrix} P (\tilde{X} | [x]_{\tilde{R}}) + P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i}}{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i} + \sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}} \\ + \frac{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}}{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i} + \sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}} = 1 \end{matrix}$

□

Proposition 2. Definition 3 is a generalization of Definition 1.

Proof. Recall the Definition 1 in [111], and we get $\begin{matrix} P (X | [x]_{\tilde{R}}) & = & \frac{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{j} \in U} \tilde{R} (x, u_{j}) p_{j}} \\ P (X^{C} | [x]_{\tilde{R}}) & = & \frac{\sum_{u_{i} \in X^{C}} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{j} \in U} \tilde{R} (x, u_{j}) p_{j}} \end{matrix}$ In our definition, when $\tilde{X}$ degenerates to a crisp set, it can still be denoted as a fuzzy set with memberships containing only 0s and 1s. By the characteristics of a t-norm, i.e. T (x, 1) = x and T (x, 0) =0, we can rewrite Definition 3 under the condition that X is a crisp set as follows: $\begin{matrix} P (\tilde{X} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i}}{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i} + \sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}} \end{matrix}$ (3)

It means that

$\begin{matrix} P (\tilde{X} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i} + \sum_{u_{i} \in X^{C}} \tilde{R} (x, u_{i}) p_{i}} \\ = \frac{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{i} \in U} \tilde{R} (x, u_{i}) p_{i}} \end{matrix}$ (4) At the same time, we know $\begin{matrix} P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}}{\sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), \tilde{X} (i)) p_{i} + \sum_{u_{i} \in U} T (\tilde{R} (x, u_{i}), {\tilde{X}}^{C} (i)) p_{i}} \end{matrix}$ (5) It follows that

$\begin{matrix} P ({\tilde{X}}^{C} | [x]_{\tilde{R}}) \\ = \frac{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{i} \in X} \tilde{R} (x, u_{i}) p_{i} + \sum_{u_{i} \in X^{C}} \tilde{R} (x, u_{i}) p_{i}} \\ = \frac{\sum_{u_{i} \in X^{C}} \tilde{R} (x, u_{i}) p_{i}}{\sum_{u_{i} \in U} \tilde{R} (x, u_{i}) p_{i}} \end{matrix}$ (6) Thus, we prove this proposition and know that the proposed definition is a generalization of that in literature [111]. □

Proposition 3. Definition 3 is a generalization of condition probability based on cardinalities.

Proof. Continued from the proof of Proposition 2, let the relation $\tilde{R}$ degenerate to an equivalence relation, then R (x, u_i) is either 0 or 1. Hence, the definition can be interpreted as

$\begin{matrix} P (\tilde{X} | [x]_{R}) & = & \frac{\sum_{u_{i} \in X} R (x, u_{i}) p_{i}}{\sum_{u_{i} \in U} R (x, u_{i}) p_{i}} \\ = & \frac{\sum_{u_{i} \in X, u_{i} \in [x]_{R}} p_{i}}{\sum_{u_{i} \in {[x]}_{R}} p_{i}} \end{matrix}$ (7) Suppose the universe is with the uniform probability distribution. The definition can be simplified as $P (\tilde{X} | [x]_{R}) = \frac{| X \cap {[x]}_{R} |}{| {[x]}_{R} |}$ In the same way, we can prove that $P ({\tilde{X}}^{C} | [x]_{R}) = \frac{| X^{C} \cap {[x]}_{R} |}{| {[x]}_{R} |}$ Hence, we know that the proposed definition is a generalization of the definition based on cardinalities. □

When any two or all actions have the same risk, we break the tie by taking an action according to the order a_P, a_N, a_B. Then the Bayesian decision procedure suggests the following three minimum-risk decision rules.

if $L (a_{P} | [x]_{\tilde{R}}) \leq L (a_{B} | [x]_{\tilde{R}})$ and $L (a_{P} | [x]_{\tilde{R}}) \leq L (a_{N} | [x]_{\tilde{R}})$ , then decide $x \in POS (\tilde{X})$ ;

if $L (a_{N} | [x]_{\tilde{R}}) < L (a_{P} | [x]_{\tilde{R}})$ and $L (a_{N} | [x]_{\tilde{R}}) \leq L (a_{B} | [x]_{\tilde{R}})$ , then decide $x \in NEG (\tilde{X})$ ;

if the element x satisfies neither P1 nor N1, then decide $x \in BND (\tilde{X})$ .

Definition 4. Let $(U, \tilde{R}, P)$ be a fuzzy probabilistic approximation space and with loss function [λ], the [λ] - fuzzy probabilistic positive, negative and boundary regions are defined as follows: $\begin{matrix} PO S^{[λ]} (\tilde{X}) & = & {x \in U | L (a_{P} | [x]_{\tilde{R}}) \leq L (a_{B} | [x]_{\tilde{R}}), \\ L (a_{P} | [x]_{\tilde{R}}) \leq L (a_{N} | [x]_{\tilde{R}})} \\ NE G^{[λ]} (\tilde{X}) & = & {x \in U | L (a_{N} | [x]_{\tilde{R}}) <_{S} L (a_{P} | [x]_{\tilde{R}}), \\ L (a_{N} | [x]_{\tilde{R}}) \leq L (a_{B} | [x]_{\tilde{R}})} \\ BN D^{[λ]} (\tilde{X}) & = & (PO S^{[λ]} (\tilde{X}) \cup NE G^{[λ]} (\tilde{X}))^{C} \end{matrix}$ The [λ] - fuzzy probabilistic lower and upper approximations of $\tilde{X}$ are as follows: $\begin{matrix} {\underline{\tilde{R}}}^{[λ]} (\tilde{X}) = PO S^{[λ]} (\tilde{X}) \\ {\bar{\tilde{R}}}^{[λ]} (\tilde{X}) = (NE G^{[λ]} (\tilde{X}))^{C} \end{matrix}$

The pair of subsets, $({\underline{\tilde{R}}}^{[λ]} (\tilde{X}), {\bar{\tilde{R}}}^{[λ]} (\tilde{X}))$ is called the [λ] - fuzzy probabilistic rough set of X([λ] - fuzzy, for short).

Consider a special and reasonable kind of loss function [94, 100] with $λ_{PP} \leq λ_{BP} < λ_{NP} and λ_{NN} \leq λ_{BN} < λ_{PN}$

That is, the loss of classifying an object x belonging to $\tilde{X}$ into the positive region of $\tilde{X}$ is less than or equal to the loss of classifying x into the boundary region of $\tilde{X}$ and both of these losses are less than the loss of classifying an object x into the negative region of $\tilde{X}$ . The other order of losses is used for classifying an object that does not belong to $\tilde{X}$ . For this type of loss function the minimum-risk decision rules (P1)-(B1) can be rewritten as follows:

if $P (\tilde{X} | [x]_{\tilde{R}}) \geq α$ and $P (\tilde{X} | [x]_{\tilde{R}}) \geq γ$ , then decide $x \in POS (\tilde{X})$ ;

if $P (\tilde{X} | [x]_{\tilde{R}}) < γ$ and $P (\tilde{X} | [x]_{\tilde{R}}) \leq β$ , then decide $x \in NEG (\tilde{X})$ ;

if x satisfies neither P2 nor N2, then decide $x \in BND (\tilde{X})$ .

where

\begin{matrix} α & = & \frac{λ_{PN} - λ_{BN}}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})} \\ γ & = & \frac{λ_{PN} - λ_{NN}}{(λ_{PN} - λ_{NN}) + (λ_{NP} - λ_{PP})} \\ β & = & \frac{λ_{BN} - λ_{NN}}{(λ_{BN} - λ_{NN}) + (λ_{NP} - λ_{BP})} \end{matrix}

Remark 1. In this circumstance we have the following simplified fuzzy probabilistic regions: $\begin{matrix} PO S^{(α, γ)} (\tilde{X}) \\ = {x \in U : P (\tilde{X} | [x]_{\tilde{R}}) \geq α, P (\tilde{X} | [x]_{\tilde{R}}) \geq γ} \\ NE G^{(γ, β)} (\tilde{X}) \\ = {x \in U : P (\tilde{X} | [x]_{\tilde{R}}) < γ, P (\tilde{X} | [x]_{\tilde{R}}) \leq β} \\ BN D^{(α, β)} (\tilde{X}) \\ = {x \in U : P (\tilde{X} | [x]_{\tilde{R}}) < α, P (\tilde{X} | [x]_{\tilde{R}}) > β} \end{matrix}$

The (α, γ) - fuzzy probabilistic lower and (γ, β) - fuzzy upper approximations are as follows:

$\begin{matrix} {\tilde{\underline{R}}}^{(α, γ)} (\tilde{X}) = PO S^{(α, γ)} (\tilde{X}) \\ {\bar{\tilde{R}}}^{(γ, β)} (\tilde{X}) = (NE G^{(γ, β)} (\tilde{X}))^{C} \end{matrix}$ (8) Consider an additional condition on the loss function [94, 100] with

$\begin{matrix} (λ_{PN} - λ_{BN}) (λ_{NP} - λ_{BP}) \\ < (λ_{BN} - λ_{NN}) (λ_{BP} - λ_{PP}) \end{matrix}$ (9)

It follows that 1 ≥ α > γ > β ≥ 0. Thus, the following simplified rules are obtained:

if $P (\tilde{X} | [x]_{\tilde{R}}) \geq α$ , then decide $x \in POS (\tilde{X})$ ;

if $P (\tilde{X} | [x]_{\tilde{R}}) \leq β$ , then decide $x \in NEG (\tilde{X})$ ;

if x satisfies neither P3 nor N3, then decide $x \in BND (\tilde{X})$ .

Remark 2. The parameter γ is no longer needed. The α-fuzzy probabilistic positive region, β-fuzzy probabilistic negative region and (α, β)-fuzzy probabilistic boundary region are given by rules (P3)-(B3), respectively, as follows: $\begin{matrix} {POS}_{α} (\tilde{X}) = {x \in U : P (\tilde{X} | [x]_{\tilde{R}}) \geq α} \\ {NEG}_{β} (\tilde{X}) = {x \in U : P (\tilde{X} | [x]_{\tilde{R}}) \leq β} \\ {BND}_{(α, β)} (\tilde{X}) = {x \in U : β < P (\tilde{X} | [x]_{\tilde{R}} < α} \end{matrix}$

The α - fuzzy probabilistic lower and β - fuzzy upper approximations are as follows: $\begin{matrix} {\tilde{\underline{R}}}^{α} (\tilde{X}) & = & PO S^{α} (\tilde{X}) \\ {\bar{\tilde{R}}}^{β} (\tilde{X}) & = & (NE G^{β} (\tilde{X}))^{C} \end{matrix}$

Actually, it holds that $\begin{matrix} {\tilde{\underline{R}}}^{α} (\tilde{X}) & = & {x \in U : P (\tilde{X} | [x]_{\tilde{R}} \geq α)} \\ {\bar{\tilde{R}}}^{β} (\tilde{X}) & = & {x \in U : P (\tilde{X} | [x]_{\tilde{R}} > β)} \end{matrix}$

Example 3.1. Continued from the example in [111], Table 3 exhibits a reflexive and symmetric fuzzy relation among ten objects (only the lower triangle is presented). Assume that P (u_i) =0.1 for each i = 1, 2,... . . , 10. For $\tilde{X} = {0.1, 0.1, 0.5, 0.3, 0.7, 0.2, 0.8, 0.9, 0.3, 0.5}$ the loss function is as follows: $\begin{matrix} λ_{PP} = 0.25, λ_{PN} = 0.72, λ_{NP} = 0.72 \\ λ_{NN} = 0.14, λ_{BP} = 0.43, λ_{BN} = 0.40 \end{matrix}$ Then we can get α, β as follows: $\begin{matrix} α & = & \frac{0.72 - 0.4}{(0.72 - 0.4) + (0.43 - 0.25)} = 0.64 \\ β & = & \frac{0.4 - 0.14}{(0.4 - 0.14) + (0.72 - 0.43)} = 0.47 \end{matrix}$ The fuzzy conditional probabilities for each u_i ∈ U are as follows $\begin{matrix} P (\tilde{X} | [u_{1}]_{\tilde{R}}) \approx 0.421, P (\tilde{X} | [u_{2}]_{\tilde{R}}) \approx 0.440 \\ P (\tilde{X} | [u_{3}]_{\tilde{R}}) \approx 0.398, P (\tilde{X} | [u_{4}]_{\tilde{R}}) \approx 0.420 \\ P (\tilde{X} | [u_{5}]_{\tilde{R}}) \approx 0.510, P (\tilde{X} | [u_{6}]_{\tilde{R}}) \approx 0.493 \\ P (\tilde{X} | [u_{7}]_{\tilde{R}}) \approx 0.490, P (\tilde{X} | [u_{8}]_{\tilde{R}}) \approx 0.502 \\ P (\tilde{X} | [u_{9}]_{\tilde{R}}) \approx 0.414, P (\tilde{X} | [u_{10}]_{\tilde{R}}) \approx 0.475 \end{matrix}$ According to Remark 1, we have $\begin{matrix} PO S^{0.64} (\tilde{X}) & = & \emptyset \\ NE G^{0.47} (\tilde{X}) & = & {u_{1}, u_{2}, u_{3}, u_{4}, u_{9}} \\ BN D^{(0.64, 0.47)} & = & {u_{5}, u_{6}, u_{7}, u_{8}, u_{10}} \end{matrix}$ and $\begin{matrix} {\underline{\tilde{R}}}^{0.64} (\tilde{X}) & = \emptyset \\ {\bar{\tilde{R}}}^{0.47} (\tilde{X}) & = {u_{5}, u_{6}, u_{7}, u_{8}, u_{10}} \end{matrix}$

Let $\tilde{X}$ degenerate to a crisp set, the result obtained is the same as the result in [111]. Let X = {u₅, u₆, u₇, u₈, u₉, u₁₀}, if we denote it as a fuzzy set, it will be $\tilde{X} = {0, 0, 0, 0, 1, 1, 1, 1, 1, 1}$ .

The fuzzy conditional probabilities for each u_i ∈ U are as follows: $\begin{matrix} P (\tilde{X} | [u_{1}]_{\tilde{R}}) \approx 0.47, P (\tilde{X} | [u_{2}]_{\tilde{R}}) \approx 0.53 \\ P (\tilde{X} | [u_{3}]_{\tilde{R}}) \approx 0.45, P (\tilde{X} | [u_{4}]_{\tilde{R}}) \approx 0.44 \\ P (\tilde{X} | [u_{5}]_{\tilde{R}}) \approx 0.68, P (\tilde{X} | [u_{6}]_{\tilde{R}}) \approx 0.74 \\ P (\tilde{X} | [u_{7}]_{\tilde{R}}) \approx 0.71, P (\tilde{X} | [u_{8}]_{\tilde{R}}) \approx 0.66 \\ P (\tilde{X} | [u_{9}]_{\tilde{R}}) \approx 0.56, P (\tilde{X} | [u_{10}]_{\tilde{R}}) \approx 0.67 \end{matrix}$ which is the same as that in [111], and we have the same parameters α, β, γ. We can also get the same positive region, boundary region and negative region, as well as the lower approximation and upper approximation. From the example, we can see it clearly that our proposed definition is really a generalization of that in [111].

3.2 Fuzzy probabilistic decision system with fuzzy or continuous decision attribute

Let (U, A) be a fuzzy information system and P be a fuzzy probability measure on U. Then (U, A, P) is called a fuzzy probabilistic information system. If A = C ∪ D and C∩ D = ∅ with C being the conditional attributes set and D the decision attribute, then (U, C ∪ D, P) is called a fuzzy probabilistic decision system and the decision classes generated by D are denoted by $π_{D} = {{\tilde{D}}_{1}, {\tilde{D}}_{2}, . . ., {\tilde{D}}_{t}}$ . Sometimes we have different loss functions for different decision classes. To avoid triviality, we only consider the special loss functions in the following discussion that satisfy $\begin{matrix} λ_{PP}^{i} \leq λ_{BP}^{i} < λ_{NP}^{i}, λ_{NN}^{i} \leq λ_{BN}^{i} < λ_{PN}^{i} and \\ (λ_{PN}^{i} - λ_{BN}^{i}) (λ_{NP}^{i} - λ_{BP}^{i}) \\ > (λ_{BN}^{i} - λ_{NN}^{i}) (λ_{BP}^{i} - λ_{PP}^{i}) \\ for all i = 1, 2, . . ., t \end{matrix}$ Then, each fuzzy decision class ${\tilde{D}}_{i}$ is associated with a group of parameters α_i, β_i and γ_i, denoted by three parameter vectors α = (α₁, α₁,... . . , α_t), β = (β₁, β₁,... . . , β_t) and γ = (γ₁, γ₁,... . . , γ_t).

The α_i - fuzzy probabilistic positive region, β_i - fuzzy probabilistic negative region and (α_i, β_i) probabilistic boundary region of ${\tilde{D}}_{i}$ related to C are given by rules (P3)-(B3), respectively, as follows: ${POS}_{C}^{α_{i}} (\tilde{D_{i}}) = {x \in U : P (\tilde{D_{i}} | [x]_{\tilde{R}}) \geq α_{i}}$ (10) ${NEG}_{C}^{β_{i}} (\tilde{D_{i}}) = {x \in U : P (\tilde{D_{i}} | [x]_{\tilde{R}}) \leq β_{i}}$ (11) $\begin{matrix} {BND}_{C}^{(α_{i}, β_{i})} (\tilde{D_{i}}) & = & {x \in U : β \\ < P (\tilde{D_{i}} | [x]_{\tilde{R}} < α_{i}} \end{matrix}$ (12) The α_i - fuzzy probabilistic lower and β_i - fuzzy probabilistic upper approximations of ${\tilde{D}}_{i}$ are as follows:

$\begin{matrix} {\underline{\tilde{R}}}_{C}^{α_{i}} ({\tilde{D}}_{i}) & = & {x \in U : P ({\tilde{D}}_{i} | [x]_{\tilde{R}}) \geq α_{i}} \\ {\bar{\tilde{R}}}_{C}^{α_{i}} ({\tilde{D}}_{i}) & = & {x \in U : P ({\tilde{D}}_{i} | [x]_{\tilde{R}}) > β_{i}} \end{matrix}$ (13)

Definition 5. Given a fuzzy probabilistic decision system (U, C ∪ D, P), then

$\begin{matrix} {\underline{Apr}}_{C}^{α} (π_{D}) & = & {{\underline{\tilde{R}}}_{C}^{α_{1}} ({\tilde{D}}_{1}), {\underline{\tilde{R}}}_{C}^{α_{2}} ({\tilde{D}}_{2}), . . ., {\underline{\tilde{R}}}_{C}^{α_{t}} ({\tilde{D}}_{t})} \\ {\bar{Apr}}_{C}^{α} (π_{D}) & = & {{\bar{\tilde{R}}}_{C}^{α_{1}} ({\tilde{D}}_{1}), {\bar{\tilde{R}}}_{C}^{α_{2}} ({\tilde{D}}_{2}), . . ., {\bar{\tilde{R}}}_{C}^{α_{t}} ({\tilde{D}}_{t})} \end{matrix}$ (14) are called, respectively, the α - fuzzy probabilistic lower approximation distribution and β - fuzzy probabilistic upper approximation distribution of π_D w.r.t. C (α - FP - UAD and β - FP - UAD for short, respectively). The α - fuzzy probabilistic positive region, (α, β) - fuzzy probabilistic negative region and (α, β) - fuzzy probabilistic boundary region of π_D w.r.t. C are defined as follows: $\begin{matrix} {POS}_{C}^{α} (π_{D}) & = & ⋃_{1 \leq i \leq t} {POS}_{C}^{α_{i}} (\tilde{D_{i}}), \\ {NEG}_{C}^{(α_{i}, β_{i})} (π_{D}) & = & ⋃_{1 \leq i \leq t} {NEG}_{C}^{β_{i}} (\tilde{D_{i}}) - \\ ⋃_{1 \leq i \leq t} ({BND}_{C}^{β_{i}} (\tilde{D_{i}}) \cup {POS}_{C}^{α_{i}} (\tilde{D_{i}})) \\ {BND}_{C}^{(α_{i}, β_{i})} (π_{D}) & = & ({POS}_{C}^{α_{i}} (π_{D}) \cup {NEG}_{C}^{β_{i}} (π_{D}))^{C} \end{matrix}$ The α - fuzzy probabilistic lower and (α, β) - fuzzy probabilistic upper approximations of π_D w.r.t. C are as follows:

$\begin{matrix} {\underline{\tilde{R}}}_{C}^{α} (π_{D}) & = & {POS}_{C}^{α} (π_{D}) \\ {\bar{\tilde{R}}}_{C}^{α} (π_{D}) & = & ({NEG}_{C}^{(α, β)} (π_{D}))^{C} \end{matrix}$ (15)

Remark 3. It should be noted that the decision classes generated by D is an extension of that in [111]. The decision classes are fuzzy sets denoted as $π_{D} = {{\tilde{D}}_{1}, {\tilde{D}}_{2}, . . ., {\tilde{D}}_{t}}$ .

Example 3.2. A fuzzy decision table is shown in Table 4. There are five conditional attributes and one decision attribute. All the conditional attributes and the decision attribute are fuzzy attributes. The fuzzy relation induced by conditional attributes is shown in Table 5. ${sim}_{a} (u_{i}, u_{j}) = 1 - \frac{| a_{u_{j}} - a_{u_{i}} |}{max (a) - min (a)}$ , where a ∈ {a₁, a₂, a₃, a₄, a₅}.

Since the decision attribute is fuzzy, we also need to construct fuzzy relation from decision attribute. We define the similarity between different decision values by ${sim}_{d} (u_{i}, u_{j}) = 1 - \frac{| d_{u_{j}} - d_{u_{i}} |}{max (d) - min (d)}$ . For example, $sim (d_{1}, d_{2}) = 1 - \frac{0.2 - 0.1}{0.8 - 0.1} \approx 0.8571$ . Hence, we can get a fuzzy similarity relation by fuzzy decision attribute, which is a fuzzy equivalence class. For d₁ = d (u₁), we can get its fuzzy equivalence class as follows: $\begin{matrix} {\tilde{D}}_{1} & = & {1, 0.8571, 0.8571, 0.5714, 0.8571, 0.4286, \\ 0.1429, 0, 0.2857, 0.1429} \end{matrix}$ For simplicity, we suppose that different decision classes have the same loss function, shown as follows: $\begin{matrix} λ_{PP} & = & 0.25, λ_{PN} = 0.72, λ_{NP} = 0.72 \\ λ_{NN} & = & 0.14, λ_{BP} = 0.43, λ_{BN} = 0.40 \end{matrix}$ Consequently, we can get α, β as follows: $\begin{matrix} α & = & \frac{0.72 - 0.4}{(0.72 - 0.4) + (0.43 - 0.25)} = 0.64 \\ β & = & \frac{0.4 - 0.14}{(0.4 - 0.14) + (0.72 - 0.43)} = 0.47 \end{matrix}$

Then, we have the conditional probabilities of ${\tilde{D}}_{1}$ shown in Table 6. Consequently, we have the positive region, negative region and bound region as follows: $\begin{matrix} {POS}_{C}^{0.64} ({\tilde{D}}_{1}) & = & {x_{1}} \\ {NEG}_{C}^{0.47} ({\tilde{D}}_{1}) & = & {x_{6}, x_{7}, x_{8}, x_{9}, x_{10}} \\ {BND}_{C}^{(0.64, 0.47)} ({\tilde{D}}_{1}) & = & {x_{2}, x_{3}, x_{4}, x_{5}} \end{matrix}$ Similarly, we can also get the three regions of ${\tilde{D}}_{2}$ , ${\tilde{D}}_{3}$ ... ${\tilde{D}}_{10}$ . Finally, we get $\begin{matrix} {POS}_{C}^{0.64} (π_{D}) = {x_{1}, x_{2}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}} \\ {NEG}_{C}^{(0.64, 0.47)} (π_{D}) = \emptyset \\ {BND}_{C}^{(0.64, 0.47)} (π_{D}) = {x_{3}} \end{matrix}$ At the same time, we have $\begin{matrix} {\underline{\tilde{R}}}_{C}^{0.64} (π_{D}) = {x_{1}, x_{2}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}} \\ {\bar{\tilde{R}}}_{C}^{(0.64, 0.47)} (π_{D}) = U \end{matrix}$

Definition 6. Let (U, C ∪ D, P) be a fuzzy probabilistic decision system and B ⊆ C.

If ${\underline{Apr}}_{B}^{α} (π_{D}) = {\underline{Apr}}_{C}^{α} (π_{D}) ({\bar{Apr}}_{B}^{α} (π_{D}) = {\bar{Apr}}_{C}^{α} (π_{D}))$ , then B is an α-lower (β-upper) consistent set for α-FP-LAD (β-FP-UAD) of π_D. w.r.t. C. If B is α-lower (β-upper) consistent and no other proper subset of B is α-lower (β-upper) consistent, then B is an α-lower (β-upper) distribution reduct for α-FP-LAD (β-FP-UAD) of π_D w.r.t. C.

If ${POS}_{B}^{α} (π_{D}) = {POS}_{C}^{α} (π_{D})$ , then B is called an α-fuzzy probabilistic consistent set of π_D w.r.t. C. If B is a-fuzzy probabilistic consistent and no other proper subset of B is α-fuzzy probabilistic consistent, then B is an α-fuzzy probabilistic reduct of π_D w.r.t. C.

Remark 4. It should be noted that the decision classes π_D are fuzzy sets which are induced by fuzzy concept.

Proposition 4. Given a fuzzy probabilistic decision system (U, C ∪ D, P) and B ⊂ C, if B is an α-lower distribution reduct then B is an α-fuzzy probabilistic reduct.

Proof. Since ${\underline{\tilde{R}}}_{C}^{α} (π_{D}) = {POS}_{C}^{α} (π_{D})$ , i.e., the positive region is the same as the lower approximation, then an α-lower distribution reduct is the same as an α-fuzzy probabilistic reduct. □

3.3 Application in decision making

In this section, the application of the proposed approach in decision making is illustrated by an example. Table 7 shows a data set about Breast Cancer diagnosis. The data set records ten patients {x₁, x₂,... . . , x₁₀}. There are two conditional attributes representing two symptoms, i.e. the clump thickness and the uniformity of cell size. The decision attribute denotes the diagnosis results given by a doctor.

For the decision value of sample x₁, (0 :0.9, 1 : 0.1) means that patient x₁ has a probability of 0.9 belonging to class ’0’ and a probability of 0.1 belonging to class ‘1’. It is significant to consider such a decision system where decision attribute is fuzzy since sometimes the doctor can’t make a definite decision.

Let’s denote the two fuzzy decision classes by $\tilde{D}$ and ${\tilde{D}}^{C}$ . From Table 7, we can get the two fuzzy sets as follows: $\begin{matrix} \tilde{D} & = & {0.9, 0.8, 0.7, 0.6, 0.6, 0.4, 0.3, 0.3, 0.1, 0.2} \\ {\tilde{D}}^{C} & = & {0.1, 0.2, 0.3, 0.4, 0.4, 0.6, 0.7, 0.7, 0.9, 0.8} \end{matrix}$ where class D is the class of benign and class D^C is the class of malignant.

The similarity relation generated by conditional attributes is shown in Table 8.

The loss function is shown in Table 9 which is given by experts previously.

From Table 9, we know the loss function for class D is as follows $\begin{matrix} λ_{PP} & = & 0.10, λ_{PN} = 0.80, λ_{NP} = 0.70 \\ λ_{NN} & = & 0.30, λ_{BP} = 0.40, λ_{BN} = 0.50 \end{matrix}$ It should be noticed that λ_PN > λ_NP. It is reasonable since the risk or danger of judging a malignant patient as a benign patient is greater than the versus.

Then we can get α, β as follows $\begin{matrix} α & = & \frac{0.80 - 0.50}{(0.80 - 0.50) + (0.40 - 0.10)} = 0.50 \\ β & = & \frac{0.50 - 0.30}{(0.50 - 0.30) + (0.70 - 0.40)} = 0.40 \end{matrix}$ The conditional probabilities of decision class $\tilde{D}$ are as follows $\begin{matrix} P (\tilde{D} | [x_{1}]_{\tilde{R}}) & = & 0.6668, P (\tilde{D} | [x_{2}]_{\tilde{R}}) = 0.6289, \\ P (\tilde{D} | [x_{3}]_{\tilde{R}}) & = & 0.5636, P (\tilde{D} | [x_{4}]_{\tilde{R}}) = 0.5948, \\ P (\tilde{D} | [x_{5}]_{\tilde{R}}) & = & 0.4431, P (\tilde{D} | [x_{6}]_{\tilde{R}}) = 0.5039, \\ P (\tilde{D} | [x_{7}]_{\tilde{R}}) & = & 0.4223, P (\tilde{D} | [x_{8}]_{\tilde{R}}) = 0.3768, \\ P (\tilde{D} | [x_{9}]_{\tilde{R}}) & = & 0.3515, P (\tilde{D} | [x_{10}]_{\tilde{R}}) = 0.3462 . \end{matrix}$

Hence, we can get the positive region, negative region and the boundary region of fuzzy decision class $\tilde{D}$ as follows $\begin{matrix} {POS}_{C}^{0.64} (\tilde{D}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}} \\ {NEG}_{C}^{0.47} (\tilde{D}) = {x_{8}, x_{9}, x_{10}} \\ {BND}_{C}^{(0.64, 0.47)} (\tilde{D}) = {x_{5}, x_{7}} \end{matrix}$

From the three regions and the meaning of rough sets, we can get that patients {x₁, x₂, x₃, x₄, x₆} are definitely belong to class D. In other words, under the loss function given, patients {x₁, x₂, x₃, x₄, x₆} must be benign. At the same time, patients {x₈, x₉, x₁₀} must be malignant. We are not quite sure about patients {x₅, x₇}, and these patients need to be diagnosed further.

4 Conclusion

Existing studies of decision-theoretic rough set approach mainly focus on crisp decisions. This paper investigates decision-theoretic rough set approach with fuzzy decisions in the framework of fuzzy approximation space. By introducing fuzzy operators, we construct new definition of conditional probability for decision-theoretic rough set approach with fuzzy decisions. Thus, the decision-theoretic rough set procedure of fuzzy concepts can be handled. Furthermore, some characteristics of the proposed method are studied. Theoretical analysis shows that the proposed approach is an extension of traditional decision-theoretic rough set model with crisp decisions.

In future studies, we plan to extend our study to two-universe case. Moreover, attribute reduction in fuzzy probabilistic decision system with fuzzy or continuous decision attribute is an interesting problem worthy of further research.

Footnotes

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (No. 61473259, No. 61070074, No. 60703038), the Zhejiang Provincial Natural Science Foundation (No. Y14F020118), the National Science & Technology Support Program of China (2015BAK26B00, 2015BAK26B02), the Major Projects of National Social Science Foundation of China (11&ZD189) and the PEIYANG Young Scholars Program of Tianjin University (2016XRX-0001).

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Decision-theoretic rough set approach for fuzzy decisions based on fuzzy probability measure and decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy logical operator

2.2 Fuzzy decision-theoretic rough set approach with traditional decision

3.1 Bayesian decision procedure based on fuzzy probabilistic approximation space with fuzzy concept X ˜

4 Conclusion

Footnotes

Acknowledgements

References

3.1 Bayesian decision procedure based on fuzzy probabilistic approximation space with fuzzy concept $\tilde{X}$