An extension of rough fuzzy set

Abstract

In rough fuzzy set, equivalence relation is used for approximating the target concept, which is a fuzzy set. In this paper, we extend the equivalence relation in rough fuzzy set to tolerance relation, and propose an extended model named tolerance rough fuzzy set. Furthermore, the properties of the extended model are investigated, and the proofs of the properties are given in the paper. The advantage of the extended model is that it can directly deal with continuous-valued attributes, and it does not require the process of discretization.

Keywords

Rough set equivalence relation rough fuzzy set tolerance relationship tolerance rough fuzzy set

1 Introduction

Rough Set [1] was proposed by Pawlak in 1982, which uses a pair of operators $(\underline{R}, \bar{R})$ to approximate target concept X, the lower approximation operator $\underline{R}$ approximates X from its inside, while upper approximation operator $\bar{R}$ approximates X from its outside. The objects dealt with by RS is called information system IS = (U, A), where U is a set of objects named universe, A is a set of attributes. In the framework of classification, the objects dealt with by RS is called decision table DT = (U, A ∪ C), where C is a set of decision attributes. In this paper, C includes only a decision attribute, whose values is in the set d = {1, 2, …, k}. In other words, the objects in U are categorized into k classes: C₁, C₂, …, C_k. In decision table DT, A is called conditional attribute set. In classical RS, R is an equivalent relation, which is equivalent to a subset of A, X is a subset of U. Since the rough set was put forward in 1982, many extended models were presented, these extensions can be roughly classified into two categories: the extensions in certain circumstance and the extensions in uncertainty circumstance. The extensions in certainty circumstance can further be classified into two categories: the improvements of the definitions and the extensions of the equivalent relation, see fig. 1.

In the definition of the lower approximation operator $\underline{R}$ in classical rough set, it is required that the equivalence class belongs to the target concept X completely, in other words, it is required that all elements of the equivalence class belong to the target concept X. While in the definition of the upper approximation operator $\bar{R}$ in classical rough set, if even only one element of the equivalence class belong to the target concept X, then the equivalence class belong to the upper approximation. This limitation severely restricts the applicability of the rough set approach to many problems. In 1993, Ziarko proposed variable precision rough set [2] by introducing multiple inclusion relation into RS to deal with the problems mentioned above. Along with this technical route, in order to deal with the same problems, Yao proposed probabilistic rough set [3] in 2003 by introducing rough membership function into RS, and decision rough set [4] in 1992 by introducing Bayes risk into RS respectively. In this route, the hot research topics are focused on the study of three-way decision, which is based on decision rough set, many researchers pay their attention to this study[5 –9].

In the first category, along with the technical route of extension of equivalent relation, Yao and Lin proposed generalized rough set [10] in 1996 by replacing equivalent relation with generalization relation. Skowron proposed tolerance rough set [11] in 2000 by replacing equivalent relation with tolerance relation also named similarity relation. Grecoet al. proposed dominance rough set [12] by replacing equivalent relation with dominance relation. In this category, the hot research topics are focused on tolerance rough set and dominance rough set, especially the former[13 –15].

In the second category, Dubois and Prade proposed rough fuzzy set [16] in 1990 by replacing crisp target concept with a fuzzy target concept. They also proposed fuzzy rough set [16] in the same year by replacing equivalent relation with a fuzzy similarity relation and replacing crisp target concept with a fuzzy target concept. In addition, three other kinds of fuzzy rough sets were independently proposed by Kuncheva [17], Nanda [18] and Yao [19]. Rough fuzzy set is a special case of fuzzy rough set [16]. There are three hot topics: attributes reduction [20-24] of fuzzy rough set, the extensions of fuzzy rough set, and the applications of fuzzy rough set. Jensen and Parthaláin applied method of fuzzy-rough sets to feature selection for alleviating computational effort, and proposed two approaches [25]. The first approach calculates nearest neighbours prior to search and then uses only these neighbours for the subsequent fuzzy-rough dependency calculations. The second approach is an attempt to tackle the problem of larger data from the perspective of large dimensionality. Based on extended t-norms with respect to type-2 fuzzy relations, a model of type-2 fuzzy rough sets were proposed by Wang [26]. This model is defined on two finite universes of discourse and is characterized with both constructive and axiomatic approaches. By using the maximal and minimal membership degrees of an object with respect to a fuzzy set based on multi-fuzzy tolerance relations, Feng and Mi proposed two types of variable precision multi-granulation fuzzy rough sets [27]. Based on fuzzy rough sets, Vluymans et al. proposed a fuzzy rough classifier to deal with the problems of class imbalanced multi-instance data [28]. Wu et al. comprehensively investigated the axiomatic characterizations of (S, T)-fuzzy rough approximation operators [29].

All works mentioned above can only deal with the fuzzy conditional attributes, but many data sets in practice are with continuous-valued attributes. It is necessary for fuzzy rough set and rough set to fuzzify conditional attributes. In this paper, we extend the rough fuzzy set by replacing equivalence relation with tolerance relation, and propose an extended model named tolerance rough fuzzy set, denoted by TRFS in short. TRFS can overcome the drawback mentioned before. In addition, the properties of the extended model are investigated, and the proofs of the properties are given in the paper.

The paper is organized as follows. The preliminaries used in this paper are given in Section 2. The proposed extended model of rough fuzzy set are presented in Section 3. The properties of tolerance rough fuzzy set model and their proofs are presented in Section 4. Section 5 concludes the paper.

2 Preliminaries

In this section, we briefly review the preliminaries in the framework of classification, including rough set, rough fuzzy set, and tolerance rough set.

2.1 Rough set

Rough set uses a pair of operators to approximate the target concepts. Let DT = (U, A ∪ C) be a decision table, where U = {x₁, x₂, …, x_n}, A = {a₁, a₂, …, a_d}, the instances in U are categorized into k classes: C₁, C₂, …, C_k, i.e. U/C = C₁ ∪ C₂ ∪ ⋯ ∪ C_k. Let x ∈ U and R is an equivalence relation induced by a subset of A, the equivalence class containing x is given by: $[x]_{R} = {y | xRy}$ (1)

Given a decision table, for arbitrary target concept C_i (1 ≤ i ≤ k), the lower approximation and the upper approximation of C_i with respect to R are defined by $\underline{R} (C_{i}) = {[x]_{R} | [x]_{R} \subseteq C_{i}}$ (2) and $\bar{R} (C_{i}) = {[x]_{R} | [x]_{R} \cap C_{i} \neq φ}$ (3)

Given an equivalence relation R, U can be divided into three disjoint regions: POS_R (U), NEG_R (U) and BND_R (U), which are called positive region, negative region and boundary region respectively. The definitions of POS_R (U), NEG_R (U) and BND_R (U) are given as follows. ${POS}_{R} (U) = ⋃_{i = 1}^{k} \underline{R} (C_{i})$ (4) ${NEG}_{R} (U) = U - {POS}_{R} (U)$ (5) ${BND}_{R} (U) = ⋃_{i = 1}^{k} \bar{R} (C_{i}) - ⋃_{i = 1}^{k} \underline{R} (C_{i})$ (6)

2.2 Rough fuzzy set

Rough fuzzy set (RFS) [16] is an extended model of rough set. In RFS, the used knowledge is also equivalence relation, while the target concept is a fuzzy set. Without loss of generality, we also use C_i (1 ≤ i ≤ k) as the k fuzzy target concepts, which are k fuzzy sets. The corresponding decision table is called fuzzy decision table, an example of fuzzy decision table with 5 instances is given in Table 1. In this example, there are three fuzzy target concepts: Poor, Good and Excellent, which are three fuzzy sets.

Given a fuzzy decision table, for arbitrary fuzzy target concept C_i (1 ≤ i ≤ k), the lower approximation and the upper approximation of C_i with respect to equivalence relation R are defined by

$\begin{matrix} \underline{R} (C_{i}) & = & μ_{\underline{R} (C_{i})} ([x]_{R}) \\ = & \inf {μ_{C_{i}} (y) | y \in [x]_{R}} \end{matrix}$ (7) and $\begin{matrix} \bar{R} (C_{i}) & = μ_{\bar{R} (C_{i})} ([x]_{R}) \\ = \sup {μ_{C_{i}} (y) | y \in [x]_{R}} \end{matrix}$ (8)

According to the fuzzy extension principle, (7) and (8) can be equivalently written as follows. $\begin{matrix} \underline{R} (C_{i}) & = μ_{\underline{R} (C_{i})} (x) \\ = \inf {\max {μ_{C_{i}} (y), 1 - μ_{R} (x, y)} | y \in U} \end{matrix}$ (9) and $\begin{matrix} \bar{R} (C_{i}) & = μ_{\bar{R} (C_{i})} (x) \\ = \sup {\min {μ_{C_{i}} (y), μ_{R} (x, y)} | y \in U} \end{matrix}$ (10)

2.3 Tolerance rough set

Tolerance rough set (TRS) [11] is another extension of rough set. TRS extends rough set by replacing an equivalence relation with a similarity relation. The target concept is same as in rough set, which is also a crisp decision class.

Given a decision table DT = (U, A ∪ C), R is a similarity relation defined on U, if and only if R satisfies the following conditions:

(1) Reflexivity, i.e. for each x ∈ U, xRx;

(2) Symmetry, i.e. for each x, y ∈ U, xRy, and yRx.

We can define many similarity relations on U, such as, the definitions given in (11), (12) and (13). $R_{a} (x, y) = 1 - \frac{| a (x) - a (y) |}{| a_{\max} - a_{\min} |}$ (11) $R_{a} (x, y) = \exp (- \frac{(a (x) - a (y))^{2}}{2 {σ_{a}}^{2}})$ (12)

$R_{a} (x, y) = \max {\min {\frac{a (y) - (a (x) - σ_{a})}{a (x) - (a (x) - σ_{a})}, \frac{(a (x) + σ_{a}) - a (y)}{(a (x) + σ_{a}) - a (x)}, 0}}$ (13)

In (11)-(13), a ∈ A, x ∈ U, and a_max and a_min denote the maximum value and minimum value of a respectively, σ_a is variance of attribute a.

For ∀R ⊆ A, we can define the similarity relations induced by subset R as follows.

$R_{τ} (x, y) = \frac{\sum_{a \in R} R_{a} (x, y)}{| R |} \geq τ$ (14) or $R_{τ} (x, y) = \prod_{a \in R} R_{a} (x, y) \geq τ$ (15) where τ is a similarity threshold.

For each x ∈ U, the τ tolerance class generated by a given similarity relation R is defined as: $[x]_{R_{τ}} = {y | (y \in U) \land ({xR}_{τ} y)}$ (16)

Given a decision table and a similarity threshold τ, for arbitrary target concept C_i (1 ≤ i ≤ k), the tolerance lower approximation and the tolerance upper approximation of C_i with respect to R are defined by ${\underline{R}}_{τ} (C_{i}) = {x | (x \in U) \land ([x]_{R_{τ}} \subseteq C_{i})}$ (17) and ${\bar{R}}_{τ} (C_{i}) = {x | (x \in U) \land ([x]_{R_{τ}} \cap C_{i} \neq φ)}$ (18)

3 The proposed extended model of rough fuzzy set

In this section, we present the proposed extended model named tolerance rough fuzzy set(TRFS), which can deal with the fuzzy decision table with continuous-valued conditional attributes and fuzzy-valued decision attribute, Table 2 is an example of this kind of fuzzy decision table with 6 instances. In this fuzzy decision table, there are three continuous-valued conditional attributes a₁, a₂, a₃ and two fuzzy

decision classes, i.e. two fuzzy target concept C₁, C₂. In the following, we first present the definitions of tolerance rough fuzzy lower approximation and tolerance rough fuzzy upper approximation in Subsection 3.1, and then present an example in Subsection 3.2 to illustrate the related concepts.

3.1 The definitions of tolerance rough fuzzy lower approximation and tolerance rough fuzzy upper approximation

Given a fuzzy decision table DT = (U, A ∪ C), R is a similarity relation defined on U, τ is a similarity threshold, C_i is the ith fuzzy decision class which is a fuzzy set. The tolerance rough fuzzy lower approximation and tolerance rough fuzzy upper approximation of C_i with respect to R are defined by $\begin{matrix} {\underline{R}}_{τ} (C_{i}) & = μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \end{matrix}$ (19) and $\begin{matrix} {\bar{R}}_{τ} (C_{i}) & = μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \end{matrix}$ (20)

Similarly to (9) and (10), we have the following equivalent definitions: $\begin{matrix} {\underline{R}}_{τ} (C_{i}) & = μ_{{\underline{R}}_{τ} (C_{i})} (x) \\ = \inf {\max {(μ_{C_{i}} (y), 1 - μ_{R_{τ}} (x, y)} | y \in U} \end{matrix}$ (21) and

$\begin{matrix} {\bar{R}}_{τ} (C_{i}) = μ_{{\bar{R}}_{τ} (C_{i})} (x) \\ = \sup {\min {μ_{C_{i}} (y), μ_{R_{τ}} (x, y)} | y \in U} \end{matrix}$ (22)

Regarding the effect of threshold τ on ${\underline{R}}_{τ} (C_{i})$ and ${\bar{R}}_{τ} (C_{i})$ , we have the following proposition.

Proposition 3.1. If τ₁ ≤ τ₂, then we have $μ_{{\underline{R}}_{τ_{1}} (C_{i})} ([x]_{R_{τ_{1}}}) \leq μ_{{\underline{R}}_{τ_{2}} (C_{i})} ([x]_{R_{τ_{2}}}) .$ and $μ_{{\bar{R}}_{τ_{1}} (C_{i})} ([x]_{R_{τ_{1}}}) \geq μ_{{\bar{R}}_{τ_{2}} (C_{i})} ([x]_{R_{τ_{2}}})$

Proof 3.1. If τ₁ ≤ τ₂, then according to (16), we have [x] _{R
_{τ
₁}} ⊇ [x] _{R
_{τ
₂}}. It is obvious that the Proposition 3.1 is hold.

In other words, the bigger the τ is, the greater the ${\bar{R}}_{τ} (C_{i})$ is, the smaller ${\underline{R}}_{τ} (C_{i})$ is.

3.2 An example

Given a fuzzy decision Table 2. Let R₁ = {a₁}, R₂ = {a₁, a₂}, R₃ = {a₁, a₂, a₃}, and τ = 0.8. Compute ${\underline{R_{i}}}_{τ} (C_{1})$ and ${\bar{R_{i}}}_{τ} (C_{1})$ , i = 1, 2, 3.

(1) Compute ${\underline{R_{1}}}_{τ} (C_{1})$ . Because $\begin{matrix} \max {μ_{C_{1}} (x_{1}), 1 - μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{1})} = 0.15, \\ \max {μ_{C_{1}} (x_{2}), 1 - μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{2})} = 0.91, \\ \max {μ_{C_{1}} (x_{3}), 1 - μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{3})} = 0.35, \\ \max {μ_{C_{1}} (x_{4}), 1 - μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{4})} = 1.00, \\ \max {μ_{C_{1}} (x_{5}), 1 - μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{5})} = 1.00, \\ \max {μ_{C_{1}} (x_{6}), 1 - μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{6})} = 1.00 . \end{matrix}$

Hence, according to (21), we have $\begin{matrix} μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{1}) = \\ \inf {0.15, 0.91, 0.35, 1.00, 1.00, 1.00} = 0.15 . \end{matrix}$

Similarly, we can obtain $\begin{matrix} μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{2}) & = μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{3}) = 0.15, \\ μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{4}) & = μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{5}) \\ = μ_{{\underline{R_{1}}}_{0.8} (C_{1})} (x_{6}) = 0.39 . \end{matrix}$

(2) Compute ${\bar{R_{1}}}_{τ} (C_{1})$ . Because $\begin{matrix} \min {μ_{C_{1}} (x_{1}), μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{1})} = 0.15, \\ \min {μ_{C_{1}} (x_{2}), μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{2})} = 0.91, \\ \min {μ_{C_{1}} (x_{3}), μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{3})} = 0.35, \\ \min {μ_{C_{1}} (x_{4}), μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{4})} = 0.00, \\ \min {μ_{C_{1}} (x_{5}), μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{5})} = 0.00, \\ \min {μ_{C_{1}} (x_{6}), μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}, x_{6})} = 0.00 . \end{matrix}$

Hence, according to (22), we have $\begin{matrix} μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{1}) = \\ \sup {0.15, 0.91, 0.35, 0.00, 0.00, 0.00} = 0.91 . \end{matrix}$

Similarly, we can obtain $\begin{matrix} μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{2}) & = 0.91, μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{3}) = 0.86, \\ μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{4}) & = μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{5}) \\ = μ_{{\bar{R_{1}}}_{0.8} (C_{1})} (x_{6}) = 0.75 . \end{matrix}$

(3) Compute ${\underline{R_{2}}}_{τ} (C_{1})$ . Because $\begin{matrix} \max {μ_{C_{1}} (x_{1}), 1 - μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{1})} = 0.15, \\ \max {μ_{C_{1}} (x_{2}), 1 - μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{2})} = 1.00, \\ \max {μ_{C_{1}} (x_{3}), 1 - μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{3})} = 0.35, \\ \max {μ_{C_{1}} (x_{4}), 1 - μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{4})} = 1.00, \\ \max {μ_{C_{1}} (x_{5}), 1 - μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{5})} = 1.00, \\ \max {μ_{C_{1}} (x_{6}), 1 - μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{6})} = 1.00 . \end{matrix}$

Hence, according to (21), we have $\begin{matrix} μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{1}) = \\ \inf {0.15, 1.00, 0.35, 1.00, 1.00, 1.00} = 0.15 . \end{matrix}$

Similarly, we can obtain $\begin{matrix} μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{2}) & = 0.91, μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{3}) = 0.15, \\ μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{4}) & = μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{5}) = 0.67, \\ μ_{{\underline{R_{2}}}_{0.8} (C_{1})} (x_{6}) & = 0.39 . \end{matrix}$

(4) Compute ${\bar{R_{2}}}_{τ} (C_{1})$ . Because $\begin{matrix} \min {μ_{C_{1}} (x_{1}), μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{1})} = 0.15, \\ \min {μ_{C_{1}} (x_{2}), μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{2})} = 0.00, \\ \min {μ_{C_{1}} (x_{3}), μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{3})} = 0.35, \\ \min {μ_{C_{1}} (x_{4}), μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{4})} = 0.00, \\ \min {μ_{C_{1}} (x_{5}), μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{5})} = 0.00, \\ \min {μ_{C_{1}} (x_{6}), μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}, x_{6})} = 0.00 . \end{matrix}$

Hence, according to (22), we have $\begin{matrix} μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{1}) = \\ \sup {0.15, 0.00, 0.35, 0.00, 0.00, 0.00} = 0.35 . \end{matrix}$ Similarly, we can obtain $\begin{matrix} μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{2}) & = 0.91, μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{3}) = 0.35, \\ μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{4}) & = μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{5}) = 0.75, \\ μ_{{\bar{R_{2}}}_{0.8} (C_{1})} (x_{6}) & = 0.39 . \end{matrix}$

(5) Compute ${\underline{R_{3}}}_{τ} (C_{1})$ . Because $\begin{matrix} \max {μ_{C_{1}} (x_{1}), 1 - μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{1})} = 0.15, \\ \max {μ_{C_{1}} (x_{2}), 1 - μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{2})} = 1.00, \\ \max {μ_{C_{1}} (x_{3}), 1 - μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{3})} = 1.00, \\ \max {μ_{C_{1}} (x_{4}), 1 - μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{4})} = 1.00, \\ \max {μ_{C_{1}} (x_{5}), 1 - μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{5})} = 1.00, \\ \max {μ_{C_{1}} (x_{6}), 1 - μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{6})} = 1.00 . \end{matrix}$

Hence, according to (21), we have $\begin{matrix} μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{1}) = \\ \inf {0.15, 1.00, 1.00, 1.00, 1.00, 1.00} = 0.15 . \end{matrix}$

Similarly, we can obtain $\begin{matrix} μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{2}) & = 0.91, \\ μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{3}) & = 0.35, \\ μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{4}) & = 0.67, \\ μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{5}) & = μ_{{\underline{R_{3}}}_{0.8} (C_{1})} (x_{6}) = 0.39 . \end{matrix}$

(6) Compute ${\bar{R_{3}}}_{τ} (C_{1})$ . Because $\begin{matrix} \min {μ_{C_{1}} (x_{1}), μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{1})} = 0.15, \\ \min {μ_{C_{1}} (x_{2}), μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{2})} = 0.00, \\ \min {μ_{C_{1}} (x_{3}), μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{3})} = 0.00, \\ \min {μ_{C_{1}} (x_{4}), μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{4})} = 0.00, \\ \min {μ_{C_{1}} (x_{5}), μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{5})} = 0.00, \\ \min {μ_{C_{1}} (x_{6}), μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}, x_{6})} = 0.00 . \end{matrix}$ Hence, according to (22), we have $\begin{matrix} μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{1}) \\ = \sup {0.15, 0.00, 0.00, 0.00, 0.00, 0.00} = 0.15 . \end{matrix}$

Similarly, we can obtain $\begin{matrix} μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{2}) & = 0.91, \\ μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{3}) & = 0.35, \\ μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{4}) & = μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{5}) = 0.75, \\ μ_{{\bar{R_{3}}}_{0.8} (C_{1})} (x_{6}) & = 0.67 . \end{matrix}$

4 The Properties of tolerance rough fuzzy set model

Given a fuzzy decision table DT = (U, A ∪ C), R is a similarity relation defined on U, τ is a similarity threshold, C_i is the ith fuzzy decision class, ∀x, y ∈ U, we have the following conclusions.

Theorem 4.1.

$\begin{matrix} (1) μ_{\underline{R} (C_{i})} ([x]_{R, τ} \cup [y]_{R, τ}) \\ = \min {μ_{\underline{R} (C_{i})} ([x]_{R, τ}), μ_{\underline{R} (C_{i})} ([y]_{R, τ}}, \\ (2) μ_{\bar{R} (C_{i})} ([x]_{R, τ} \cup [y]_{R, τ}) \\ = \max {μ_{\bar{R} (C_{i})} ([x]_{R, τ}), μ_{\bar{R} (C_{i})} ([y]_{R, τ}}, \\ (3) μ_{\underline{R} (C_{i})} ([x]_{R, τ} \cap [y]_{R, τ}) \\ \geq \min {μ_{\underline{R} (C_{i})} ([x]_{R, τ}), μ_{\underline{R} (C_{i})} ([y]_{R, τ}}, \\ (4) μ_{\bar{R} (C_{i})} ([x]_{R, τ} \cap [y]_{R, τ}) \\ \leq \max {μ_{\bar{R} (C_{i})} ([x]_{R, τ}), μ_{\bar{R} (C_{i})} ([y]_{R, τ}} . \end{matrix}$

Proof 4.1. (1) Because $\begin{matrix} μ_{\underline{R} (C_{i})} ([x]_{R, τ} \cup [y]_{R, τ}) \\ = \inf {μ_{C_{i}} (z) | z \in [x]_{R, τ} \cup [y]_{R, τ}} \\ = \inf {μ_{C_{i}} | z \in [x]_{R, τ}} \cup {μ_{C_{i}} | z \in [y]_{R, τ}} \\ = \min {μ_{\underline{R} (C_{i})} ([x]_{R, τ}), μ_{\underline{R} (C_{i})} ([y]_{R, τ})} \end{matrix}$

Hence, the property (1) is hold.

(2) Because $\begin{matrix} μ_{\bar{R} (C_{i})} ([x]_{R, τ} \cup [y]_{R, τ}) \\ = \sup {μ_{C_{i}} (z) | z \in [x]_{R, τ} \cup [y]_{R, τ}} \\ = \sup {μ_{C_{i}} | z \in [x]_{R, τ}} \cup {μ_{C_{i}} | z \in [y]_{R, τ}} \\ = \max {μ_{\bar{R} (C_{i})} ([x]_{R, τ}), μ_{\bar{R} (C_{i})} ([y]_{R, τ})} \end{matrix}$ Hence, the property (2) is hold.

(3) Because $\begin{matrix} μ_{\underline{R} (C_{i})} ([x]_{R, τ} \cap [y]_{R, τ}) \\ = \inf {μ_{C_{i}} (z) | z \in [x]_{R, τ} \cap [y]_{R, τ}} \\ = \inf {μ_{C_{i}} | z \in [x]_{R, τ}} \cap {μ_{C_{i}} | z \in [y]_{R, τ}} \\ \geq \inf {μ_{C_{i}} | z \in [x]_{R, τ}} \cup {μ_{C_{i}} | z \in [y]_{R, τ}} \\ = \min {μ_{\underline{R} (C_{i})} ([x]_{R, τ}), μ_{\underline{R} (C_{i})} ([y]_{R, τ})} \end{matrix}$ Hence, the property (3) is hold.

(4) Because $\begin{matrix} μ_{\bar{R} (C_{i})} ([x]_{R, τ} \cap [y]_{R, τ}) \\ = \sup {μ_{C_{i}} (z) | z \in [x]_{R, τ} \cap [y]_{R, τ}} \\ = \sup {μ_{C_{i}} | z \in [x]_{R, τ}} \cap {μ_{C_{i}} | z \in [y]_{R, τ}} \\ \leq \sup {μ_{C_{i}} | z \in [x]_{R, τ}} \cup {μ_{C_{i}} | z \in [y]_{R, τ}} \\ = \max {μ_{\bar{R} (C_{i})} ([x]_{R, τ}), μ_{\bar{R} (C_{i})} ([y]_{R, τ})} \end{matrix}$ Hence, the property (4) is hold.

Given a fuzzy decision table DT = (U, A ∪ C), R is a similarity relation defined on U, τ is a similarity threshold, C_i and C_j are the ith and jth decision class respectively. The tolerance rough fuzzy lower approximation and tolerance rough fuzzy upper approximation satisfy the following properties.

Theorem 4.2.

$\begin{matrix} (1) {\underline{R}}_{τ} (C_{i}) \subseteq C_{i} \subseteq {\bar{R}}_{τ} (C_{i}), \\ (2) {\underline{R}}_{τ} (\emptyset) = {\bar{R}}_{τ} (\emptyset) = \emptyset, {\underline{R}}_{τ} (U) = {\bar{R}}_{τ} (U) = U, \\ (3) {\bar{R}}_{τ} (C_{i} \cup C_{j}) = {\bar{R}}_{τ} (C_{i}) \cup {\bar{R}}_{τ} (C_{j}), \\ (4) {\underline{R}}_{τ} (C_{i} \cap C_{j}) = {\underline{R}}_{τ} (C_{i}) \cap {\underline{R}}_{τ} (C_{j}), \\ (5) C_{i} \subseteq C_{j} \Rightarrow {\underline{R}}_{τ} (C_{i}) \subseteq {\underline{R}}_{τ} (C_{j}), \\ (6) C_{i} \subseteq C_{j} \Rightarrow {\bar{R}}_{τ} (C_{i}) \subseteq {\bar{R}}_{τ} (C_{j}), \\ (7) {\bar{R}}_{τ} (C_{i} \cap C_{j}) \subseteq {\bar{R}}_{τ} (C_{i}) \cap {\bar{R}}_{τ} (C_{j}), \\ (8) {\underline{R}}_{τ} (C_{i} \cup C_{j}) \supseteq {\underline{R}}_{τ} (C_{i}) \cup {\bar{R}}_{τ} (C_{j}), \\ (9) {\underline{R}}_{τ} (\sim C_{i}) = \sim {\bar{R}}_{τ} (C_{i}), \\ (10) {\bar{R}}_{τ} (\sim C_{i}) = \sim {\underline{R}}_{τ} (C_{i}), \\ (11) {\underline{R}}_{τ} ({\underline{R}}_{τ} (C_{i})) = {\bar{R}}_{τ} ({\underline{R}}_{τ} (C_{i})) = {\underline{R}}_{τ} (C_{i}), \\ (12) {\bar{R}}_{τ} ({\bar{R}}_{τ} (C_{i})) = {\underline{R}}_{τ} ({\bar{R}}_{τ} (C_{i})) = {\bar{R}}_{τ} (C_{i}), \\ (13) {\underline{R}}_{τ} (C_{i}) = \sim {\bar{R}}_{τ} (\sim C_{i}), \\ (14) {\bar{R}}_{τ} (C_{i}) = \sim {\underline{R}}_{τ} (\sim C_{i}) . \end{matrix}$

Proof 4.2. (1) According to the Definition (19) and (20), we have $\begin{matrix} {\underline{R}}_{τ} (C_{i}) & = μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \end{matrix}$

And $\begin{matrix} {\bar{R}}_{τ} (C_{i}) & = μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \end{matrix}$

Because $\begin{matrix} \inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} & \leq {μ_{C_{i}} (y) | \forall y \in [x]_{R_{τ}}} \\ \leq \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \end{matrix}$

Hence ${\underline{R}}_{τ} (C_{i}) \subseteq C_{i} \subseteq {\bar{R}}_{τ} (C_{i})$

(2) Because $μ_{R_{τ} (φ)} ([x]_{R_{τ}}) = 0$

Therefore $\begin{matrix} μ_{{\underline{R}}_{τ} (φ)} ([x]_{R_{τ}}) & = \inf {μ_{φ} (y) | y \in [x]_{R_{τ}}} \\ = μ_{{\bar{R}}_{τ} (φ)} ([x]_{R_{τ}}) \\ = \sup {μ_{φ} (y) | y \in [x]_{R_{τ}}} \\ = 0 \end{matrix}$

Hence ${\underline{R}}_{τ} (φ) = {\bar{R}}_{τ} (φ) = φ$

Because $μ_{R_{τ} (U)} ([x]_{R_{τ}}) = 1$

Therefore $\begin{matrix} μ_{{\underline{R}}_{τ} (U)} ([x]_{R_{τ}}) & = \inf {μ_{U} (y) | y \in [x]_{R_{τ}}} \\ = μ_{{\bar{R}}_{τ} (U)} ([x]_{R_{τ}}) \\ = \sup {μ_{U} (y) | y \in [x]_{R_{τ}}} \\ = 1 \end{matrix}$

Hence ${\underline{R}}_{τ} (U) = {\bar{R}}_{τ} (U) = U$

(3) Because $\begin{matrix} {\bar{R}}_{τ} (C_{i} \cup C_{j}) = μ_{{\bar{R}}_{τ} (C_{i} \cup C_{j})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{i} \cup C_{j}} (y) | y \in [x]_{R_{τ}}} \\ = \max (μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}), μ_{{\bar{R}}_{τ} (C_{j})} ([x]_{R_{τ}})) \\ = {\bar{R}}_{τ} (C_{i}) \cup {\bar{R}}_{τ} (C_{j}) \end{matrix}$ Hence, the property (3) is hold.

(4) Because $\begin{matrix} {\underline{R}}_{τ} (C_{i} \cap C_{j}) & = μ_{{\underline{R}}_{τ} (C_{i} \cap C_{j})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{i} \cap C_{j}} (y) | y \in [x]_{R_{τ}}} \\ = \min (μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}), μ_{{\underline{R}}_{τ} (C_{j})} ([x]_{R_{τ}})) \\ = {\underline{R}}_{τ} (C_{i}) \cap {\underline{R}}_{τ} (C_{j}) \end{matrix}$

Hence, the property (4) is hold.

(5) Because $\begin{matrix} C_{i} \subseteq C_{j} & \Rightarrow μ_{R_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \leq μ_{R_{τ} (C_{j})} ([x]_{R_{τ}}) \\ = {μ_{C_{j}} (y) | y \in [x]_{R_{τ}}} \\ \Rightarrow μ_{\underline{R} τ (C_{i})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \leq μ_{\underline{R} τ (C_{j})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{j}} (y) | y \in [x]_{R_{τ}}} \\ \Rightarrow {\underline{R}}_{τ} (C_{i}) \subseteq {\underline{R}}_{τ} (C_{j}) \end{matrix}$

Hence, the property (5) is hold.

(6) Because $\begin{matrix} C_{i} \subseteq C_{j} & \Rightarrow μ_{R_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \leq μ_{R_{τ} (C_{j})} ([x]_{R_{τ}}) \\ = {μ_{C_{j}} (y) | y \in [x]_{R_{τ}}} \\ \Rightarrow μ_{\bar{R} τ (C_{i})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \leq μ_{\bar{R} τ (C_{j})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{j}} (y) | y \in [x]_{R_{τ}}} \\ \Rightarrow {\bar{R}}_{τ} (C_{i}) \subseteq {\bar{R}}_{τ} (C_{j}) \end{matrix}$

Hence, the property (6) is hold.

(7) Because $\begin{matrix} {\bar{R}}_{τ} (C_{i} \cap C_{j}) & = μ_{{\bar{R}}_{τ} (C_{i} \cap C_{j})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{i} \cap C_{j}} (y) | y \in [x]_{R_{τ}}} \\ \leq \min (μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}), μ_{{\bar{R}}_{τ} (C_{j})} ([x]_{R_{τ}})) \\ = {\bar{R}}_{τ} (C_{i}) \cap {\bar{R}}_{τ} (C_{j}) \end{matrix}$

Hence, we have ${\bar{R}}_{τ} (C_{i} \cap C_{j}) \subseteq {\bar{R}}_{τ} (C_{i}) \cap {\bar{R}}_{τ} (C_{j})$ (8) Because $\begin{matrix} {\underline{R}}_{τ} (C_{i} \cup C_{j}) & = μ_{{\underline{R}}_{τ} (C_{i} \cup C_{j})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{i} \cup C_{j}} (y) | y \in [x]_{R_{τ}}} \\ \geq \max (μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}), μ_{{\underline{R}}_{τ} (C_{j})} ([x]_{R_{τ}})) \\ = {\underline{R}}_{τ} (C_{i}) \cup {\underline{R}}_{τ} (C_{j}) \end{matrix}$

Hence, we have ${\underline{R}}_{τ} (C_{i} \cup C_{j}) \supseteq {\underline{R}}_{τ} (C_{i}) \cup {\bar{R}}_{τ} (C_{j})$

(9) Because $\begin{matrix} {\underline{R}}_{τ} (\sim C_{i}) & = μ_{{\underline{R}}_{τ} (\sim C_{i})} ([x]_{R_{τ}}) \\ = \inf {1 - μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = {\bar{R}}_{τ} (C_{i}) \end{matrix}$

Hence, the property (9) is hold.

(10) Because $\begin{matrix} {\bar{R}}_{τ} (\sim C_{i}) & = μ_{{\bar{R}}_{τ} (\sim C_{i})} ([x]_{R_{τ}}) \\ = \sup {1 - μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - \inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \sim {\underline{R}}_{τ} (C_{i}) \end{matrix}$

Hence, the property (10) is hold.

(11) Because $\begin{matrix} {\underline{R}}_{τ} (C_{i}) & = μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \inf {1 - μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = \sup {\inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}}} \\ = {\bar{R}}_{τ} ({\underline{R}}_{τ} (C_{i})) \\ = \inf {\inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}}} \\ = {\underline{R}}_{τ} ({\underline{R}}_{τ} (C_{i})) \end{matrix}$

Hence, the property (11) is hold.

(12) Because $\begin{matrix} {\bar{R}}_{τ} (C_{i}) & = & μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = & \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = & \inf {\sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}}} \\ = & {\underline{R}}_{τ} ({\bar{R}}_{τ} (C_{i})) \\ = & \sup {\sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}}} \\ = & {\bar{R}}_{τ} ({\bar{R}}_{τ} (C_{i})) \end{matrix}$ Hence, the property (12) is hold.

(13) Because $\begin{matrix} {\underline{R}}_{τ} (C_{i}) & = μ_{{\underline{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \inf {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - \sup {1 - μ_{\sim C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - μ_{{\bar{R}}_{τ} (\sim C_{i})} ([x]_{R_{τ}}) \\ = \sim {\bar{R}}_{τ} (\sim C_{i}) \end{matrix}$

Hence, the property (13) is hold.

(14) Because $\begin{matrix} {\bar{R}}_{τ} (C_{i}) & = μ_{{\bar{R}}_{τ} (C_{i})} ([x]_{R_{τ}}) \\ = \sup {μ_{C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - \inf {1 - μ_{\sim C_{i}} (y) | y \in [x]_{R_{τ}}} \\ = 1 - μ_{{\underline{R}}_{τ} (\sim C_{i})} ([x]_{R_{τ}}) \\ = \sim {\underline{R}}_{τ} (\sim C_{i}) \end{matrix}$

Hence, the property (14) is hold.

5 Conclusions

This paper proposes an extended model of rough fuzzy set. The advantage of the proposed model is that it can directly deal with continuous-valued data sets without discretizing the conditional attributes. Accordingly, there is no information loss in the process of dealing with the fuzzy decision tables. In addition, the properties of the proposed model are investigated in details. The promising future works of this study include (1) the monotonicity of the lower approximation and the upper approximation with respect to similar parameter τ; (2) the numerical characteristics of the proposed TRFS model, including significance of knowledge, dependency of knowledge, rough degree, etc. (3) the attribute reductions and core, and the corresponding computing methods.

Footnotes

Acknowledgments

This research is supported by the national natural science foundation of China (61170040, 71371063), by the natural science foundation of Hebei Province (F2013201110, F2013201220), by the Key Scientific Research Foundation of Education Department of Hebei Province (ZD20131028) and by the Opening Fund of Zhejiang Provincial Top Key Discipline of Computer Science and Technology at Zhejiang Normal University, China.

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