Information structures and entropy measurement for a fuzzy probabilistic information system

Abstract

An information system (IS), an important model in the field of artificial intelligence, takes information structure as the basic structure. A fuzzy probabilistic information system (FPIS) is the combination of some fuzzy relations in the same universe that satisfy probability distribution. A FPIS as an IS with fuzzy relations includes three types of uncertainties (i.e., roughness, fuzziness and probability). This paper studies information structures in a FPIS from the perspective of granular computing (GrC). Firstly, two types of information structures in a FPIS are defined by set vectors. Then, equality, dependence and independence between information structures in a FPIS are proposed, and they are depicted by means of the inclusion degree. Next, information distance between information structures in a FPIS is presented. Finally, entropy measurement for a FPIS is investigated based on the proposed information structures. These results may be helpful for understanding the nature of structures and uncertainty in a FPIS.

Keywords

Fuzzy relation FPIS GrC information structure dependence distance uncertainty measurement entropy

1 Introduction

1.1 Research background and related works

Information granule is a group of objects which are gathered together by similarity, indistinguishability and proximity of functionality [8, 20]. Information granulation, which can break a cosmic granule into a family of unrelated or overlapping information particles, refers to the process of constructing information granule [17, 18]. In mathematics, how to manage the construction, interpretation and expression of information granules is a significant problem.

Granular computing (GrC) is a term coined first proposed by Zadeh and Lin [41], which is an extremely important mathematical tool in artificial intelligence. This approach takes an information granule as the basic unit of calculation. Thus, GrC can establish an effective calculation model to cope with large-scale and complex data sets. A granular structure is a collection of information granules, in which the inner structure of apiece information granule is a visible substructure, and the interactions between granules are spotted by this visible substructure [9, 25]. This means that a granular structure can be characterized as a vector consisting of information granules. Therefore, some scholars have conducted research on it. For instance, Qian et al. [23] studied a framework for granular structures. Li et al. [12] investigated knowledge structures in a knowledge base.

Since Lin [10] and Yao [37] introduced the importance of GrC, more and more scholars pay attention to it. Thus, GrC has developed rapidly and has been applied to various fields, such as knowledge representation, data mining, machine learning [2 , 40]. Until now, there are four approaches to the research of GrC, i.e., rough set theory [21], fuzzy set theory [39], quotient space theory [43] and concept lattice theory [32].

Rough set theory is an important method for disposing uncertain knowledge. Until now, this theory has been successfully applied to various domains [6 , 29]. An information system (IS), which contains some objects and their attributes as knowledge representation, is the basis of rough set theory [22]. Many applications of rough set theory, such as reasoning with uncertainty [4, 36], classification [5, 28] and rule extraction [19, 31], are connected with an IS.

Information granularity and information structure play an important role in GrC of an IS. An equivalence relation represents a special kind of similarity between elements in a given set. Each attribute in an IS determines an equivalence relation. This equivalence relation divides the object set of an IS into disjoint classes, and these classes are called information granules.

If some objects in an IS belong to the same equivalence class, that is to say, they cannot be distinguished under this equivalence relationship. Therefore, each equivalence class is regarded as an information granule composed of indistinguishable objects [9, 24]. All these information granules form a vector and this vector is said to be an information structure in the IS induced by this attribute subset. Similarly, an information structure in an IS is also a granular structure in the sense of GrC. Zhang et al. [42] introduced information structures in a fully fuzzy IS. Qian et al. [26] investigated information granularity in fuzzy binary GrC model. Li et al. [11] looked into information structures in a covering IS.

Nowadays, information entropy [27] is an effective method to measure the uncertainty of an IS, which has been studied by many scholars. For example, Bianucci et al. [1] investigated entropy approaches for uncertainty measurements of coverings; Liang et al. [14, 15] inquired into information granules and entropy theory in ISs; Dai et al. [3] looked into entropy measures in a set-valued IS; Li et al. [13] gave entropy measurement based on Gaussian kernel for a fully fuzzy IS; Xie et al. [34] studied some new measures of uncertainty for interval-valued ISs. Li et al. [16] researched information entropy in a fuzzy relation information system (FRIS).

1.2 Motivation and inspiration

Fuzzy probability approximation space (FPA-space) [7] is an approximation space where probability distribution is put into a rough set model. And a fuzzy probabilistic information system (FPIS), combined some fuzzy relations in the same universe which satisfies the probability distribution, can be seen as an IS with fuzzy relations. A FPIS includes three types of uncertainties (i.e., roughness, fuzziness and probability). Its study is a good reference for dealing with various types of uncertainty. Thus, a FPIS is worth studying. FPA-spaces and FPISs have some similarities. Note that information structures and entropy measurement for a FPIS have not been reported. Then according to the research idea of a FPA-space, this paper studies information structures in a FPIS by means of set vectors, and investigates entropy measurement for a FPIS by using its information structures.

Fig.1

Flowchart of this paper.

1.3 Contribution

In this part, we discuss the potential values and contributions of this paper. At present, there are more and more researches on ISs. This paper studies a new IS, i.e. FPIS, which extends the category of IS. We may do some the related research, such as information structures, uncertainty measurement, decision-making, attribute reduction, etc. This makes our future research more rich. Because there are three types of uncertainties (i.e., roughness, fuzziness and probability) in a FPIS, we should combine rough set theory and fuzzy set theory to deal with it, and consider the impact of the probability distribution. These two theories are important and have application value.

This paper first considers the intersection of fuzzy relations by means of the ideal of rough set theory and obtains a new fuzzy relation. Then, let the probability distribution be combined with this new fuzzy relation. We use a fuzzy relation matrix to handle this operational process. Thus, the obtained fuzzy relation matrix expression contains not only the similarity between objects but also the probability distribution. This depicts the internal characteristics of a FPIS. This is the main innovation or contribution of this paper.

1.4 Organization

The remaining part of this paper is organized as follows. In Section 2, we briefly recall some concepts about fuzzy relations and FPISs. In Section 3, we propose information structures in a FPIS through set vectors. In Section 4, we introduce the concepts of equality, dependence, independence and inclusion degree between information structures in a FPIS. In Section 5, we investigate information distance between information structures. In Section 6, we study the entropy measurement for a FPIS. In Section 7, we get to make a summary of this paper.

The work flow of this paper is shown in Figure 1.

2 Preliminaries

In this section, we briefly introduce some concepts about fuzzy sets, fuzzy relations and FPISs.

Throughout this paper, U denotes a non-empty finite set and I expresses [0, 1] .

Put $U = {x_{1}, x_{2}, \dots, x_{n}} .$

2.1 Fuzzy relations

Recall that F is a fuzzy set whenever F is a function defined by F : U → I.

For a ∈ I, $\bar{a}$ indicates the constant fuzzy set on U, i.e., ∀ x ∈ U, $\bar{a} (x) = a$ .

In this article, I^U shows the collection of fuzzy sets on U.

Given F ∈ I^U. Then F is denoted as $F = \frac{F (x_{1})}{x_{1}} + \frac{F (x_{2})}{x_{2}} + \dots + \frac{F (x_{n})}{x_{n}}$ and $| F | = \sum_{i = 1}^{n} F (x_{i})$ expresses the cardinality of F .

If R is a fuzzy set in U × U, then R is called a fuzzy relation on U.

In this article, I^U×U denotes the collection of all fuzzy relations on U, and [I^U×U] ^<ω denotes the subfamily of all finite subsets of I^U×U.

Given R ∈ I^U×U. Then R can be denoted by the following matrix $M (R) = (\begin{matrix} r_{11} & r_{12} & \dots & r_{1 n} \\ r_{21} & r_{22} & \dots & r_{2 n} \\ \dots & \dots & \dots & \dots \\ r_{n 1} & r_{n 2} & \dots & r_{nn} \end{matrix}),$ where r_ij = R (x_i, x_j) means the degree of similarity between x_i and x_j.

If R = E (here, E is an identity matrix), then R is said to be a fuzzy identity relation on U, and we write as R =▵; if R = 0, then R is said to be a fuzzy zero relation on U, and we write as R = o; if R (x_i, x_j) =1 for any i, j, then R is said to be a fuzzy universal relation on U, and we write as R = ω.

Below, we give some relations on I^U×U [39].

Put R, R′ ∈ I^U×U with $M (R) = (r_{ij})_{nn}, M (R^{'}) = (r_{ij}^{'})_{nn}$ . Then

(1) R = R′ ⇔ $\forall i, j, r_{ij} = r_{ij}^{'}$ ;

(2) R ⊆ R′ ⇔ $\forall i, j, r_{ij} \leq r_{ij}^{'}$ ;

(3) R ⊂ R′ ⇔ R ⊆ R′ and R ≠ R′.

Suppose R ∈ I^U×U . Given x ∈ U . Then two fuzzy sets on U are defined as follows: $[x]^{R} (y) = R (x, y), \forall y \in U;$ $[x]_{R} (y) = R (y, x), \forall y \in U .$ Then [x] ^R and [x] _R are called the upper-fuzzy and lower-fuzzy sets of x with respect to R, respectively, and they can be viewed as the upper-fuzzy and lower-fuzzy neighbourhoods of point x with respect to R. Based on the perspective of GrC, they can also be regarded as the upper-fuzzy and lower-fuzzy information granules of x with respect to R.

Obviously, $[x_{i}]^{R} = \frac{r_{i 1}}{x_{1}} + \frac{r_{i 2}}{x_{2}} + \dots + \frac{r_{in}}{x_{n}},$ $[x_{i}]_{R} = \frac{r_{1 i}}{x_{1}} + \frac{r_{2 i}}{x_{2}} + \dots + \frac{r_{ni}}{x_{n}} .$ Then $| [x_{i}]^{R} | = \sum_{j = 1}^{n} r_{ij}, | [x_{i}]_{R} | = \sum_{j = 1}^{n} r_{ji} .$

For any $R \in [I^{U \times U}]^{< ω}$ , put $ind (R) = ⋂_{R \in R} R .$

Obviously, $ind (R) \in I^{U \times U}$ . Suppose $M (ind (R)) = (r_{ij})_{n \times n}$ . Then ∀ i, j, 0 ≤ r_ij ≤ 1 .

2.2 FPA-spaces

Definition 2.1. Let U be a finite set of objects called the universe. Then the ordered pair (U, R) is referred to as a fuzzy approximation space (FA-space), if R ∈ I^U×U.

Definition 2.2. Let U = {x₁, x₂, ⋯ , x_n}. Suppose that probability of occurrence on x_i is p_i (i = 1, 2, ⋯ , n) . If 0 ≤ p_i ≤ 1 (i = 1, 2, ⋯ , n) and $\sum_{i = 1}^{n} p_{i} = 1,$ then $P = {\frac{x_{1}, x_{2}, \dots, x_{n}}{p_{1}, p_{2}, \dots, p_{n}}}$ is called a probability distribution over U.

Definition 2.3. ([7]) Let U be a finite set of objects called the universe. Suppose R ∈ I^U×U. Then the ordered pair (U, R, P) is referred to as a fuzzy probabilistic approximation space (FPA-space), if P is a probability distribution over U.

2.2 FPISs

Definition 2.4 ([21]). Let U be a set of objects and A a set of attributes. Suppose that U and A are finite sets. If each attribute a ∈ A determines a information function a : U → V_a, where V_a is the set of function values of attribute a, then the pair (U, A) is called an information system (IS).

Definition 2.5. ([30]). A pair $(U, R)$ is called a fuzzy relation information system (FRIS), if $R \in [I^{U \times U}]^{< ω}$ . Moreover, if $P \subseteq R$ , then $(U, P)$ is called a subsystem of $(U, R)$ .

Definition 2.6. Let U be a finite set of objects called the universe. Suppose P is a probability distribution over U. Then the ordered pair $(U, R, P)$ is called a fuzzy probabilistic information system (FPIS), if $R \in [I^{U \times U}]^{< ω}$ .

If $P \subseteq R$ , then $(U, P, P)$ is called a subsystem of $(U, R, P)$ .

In fact, a FRIS as the combination of some fuzzy relations on the same universe is not an IS. But we may view it as an IS with fuzzy relations. A FPIS a FPIS as combination of some fuzzy relations in the same universe that satisfy probability distribution is also not an IS. Similarly, a FPIS can be seen as an IS with fuzzy relations in the same universe that satisfy probability distribution. In this way, we can handle a FPIS like an IS.

Let $(U, R, P)$ be a FPIS. Put $R = {R_{1}, R_{2}, \dots, R_{m}}, P = {\frac{x_{1}, x_{2}, \dots, x_{n}}{p_{1}, p_{2}, \dots, p_{n}}} .$

If $p_{1} = p_{2} = \dots = p_{n} = \frac{1}{n}$ , then the FPIS $(U, R, P)$ is actually a FRIS $(U, R)$ .

Denote $ind (R) = ⋂_{R \in R} R = ⋂_{i = 1}^{m} R_{i} .$

Suppose $M (ind (R)) = (r_{ij})_{n \times n}$ .

Define $M ((ind (R))_{P}) = (\begin{matrix} p_{1} r_{11} & p_{2} r_{12} & \dots & p_{n} r_{1 n} \\ p_{1} r_{21} & p_{2} r_{22} & \dots & p_{n} r_{2 n} \\ \dots & \dots & \dots & \dots \\ p_{1} r_{n 1} & p_{2} r_{n 2} & \dots & p_{n} r_{nn} \end{matrix}) .$ Then $(ind (R))_{P}$ is called the fuzzy relation induced by $(U, R, P)$ .

The matrix expression $M ((ind (R))_{P})$ of the fuzzy relation $(ind (R))_{P}$ contains not only the similarity between objects but also the probability distribution. Thus, $(ind (R))_{P}$ reflects the internal characteristics of $(U, R, P)$ .

Obviously, $((ind (R)_{P}) (x_{i}, x_{j}) = p_{j} (ind (R)) (x_{i}, x_{j});$ $ind (Q) = ind (T) \to ind (Q)_{P} = ind (T)_{P},$ $ind (Q) \subseteq ind (T) \to ind (Q)_{P} \subseteq ind (T)_{P},$ $ind (Q) \subset ind (T) \to ind (Q)_{P} \subset ind (T)_{P} .$

Example 2.7. Let U = {x₁, x₂, ⋯ , x₇}, $P = {\frac{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}}{0.11, 0.12, 0.37, 0.12, 0.08, 0.15, 0.05}} .$ Put $R_{1} = (\begin{matrix} 0.46 & 0.16 & 0.53 & 0.13 & 0.02 & 0.03 & 0.89 \\ 0.57 & 0.17 & 0.57 & 0.11 & 0.22 & 0.15 & 0.55 \\ 0.46 & 0.06 & 0.49 & 0.11 & 0.28 & 0.09 & 0.43 \\ 0.46 & 0.04 & 0.43 & 0.32 & 0.03 & 0.31 & 0.41 \\ 0.56 & 0.02 & 0.42 & 0.17 & 0.04 & 0.33 & 0.43 \\ 0.65 & 0.26 & 0.47 & 0.19 & 0.03 & 0.04 & 0.48 \\ 0.63 & 0.01 & 0.62 & 0.23 & 0.09 & 0.18 & 0.66 \end{matrix}),$ $R_{2} = (\begin{matrix} 0.56 & 0.16 & 0.55 & 0.23 & 0.88 & 0.83 & 0.59 \\ 0.65 & 0.17 & 0.27 & 0.21 & 0.52 & 0.95 & 0.65 \\ 0.87 & 0.06 & 0.56 & 0.21 & 0.29 & 0.66 & 0.63 \\ 0.76 & 0.04 & 0.33 & 0.62 & 0.43 & 0.65 & 0.61 \\ 0.66 & 0.02 & 0.57 & 0.67 & 0.42 & 0.53 & 0.93 \\ 0.44 & 0.26 & 0.33 & 0.66 & 0.42 & 0.54 & 0.98 \\ 0.44 & 0.01 & 0.35 & 0.85 & 0.22 & 0.48 & 0.86 \end{matrix}),$ $R_{3} = (\begin{matrix} 0.63 & 0.36 & 0.55 & 0.74 & 0.55 & 0.83 & 0.68 \\ 0.56 & 0.56 & 0.56 & 0.66 & 0.52 & 0.95 & 0.65 \\ 0.66 & 0.33 & 0.56 & 0.35 & 0.56 & 0.75 & 0.55 \\ 0.56 & 0.33 & 0.55 & 0.57 & 0.66 & 0.65 & 0.61 \\ 0.75 & 0.42 & 0.46 & 0.68 & 0.66 & 0.84 & 0.55 \\ 0.74 & 0.44 & 0.74 & 0.77 & 0.73 & 0.54 & 0.98 \\ 0.44 & 0.41 & 0.74 & 0.75 & 0.54 & 0.48 & 0.55 \end{matrix}),$ $R_{4} = (\begin{matrix} 0.86 & 0.86 & 0.33 & 0.13 & 0.02 & 0.55 & 0.59 \\ 0.87 & 0.67 & 0.27 & 0.11 & 0.22 & 0.66 & 0.65 \\ 0.96 & 0.65 & 0.09 & 0.11 & 0.28 & 0.69 & 0.66 \\ 0.96 & 0.54 & 0.33 & 0.32 & 0.03 & 0.38 & 0.61 \\ 0.56 & 0.44 & 0.32 & 0.17 & 0.04 & 0.65 & 0.63 \\ 0.66 & 0.47 & 0.07 & 0.19 & 0.03 & 0.21 & 0.48 \\ 0.87 & 0.41 & 0.22 & 0.23 & 0.09 & 0.55 & 0.56 \end{matrix}),$ $R_{5} = (\begin{matrix} 0.66 & 0.86 & 0.53 & 0.23 & 0.45 & 0.03 & 0.44 \\ 0.95 & 0.67 & 0.57 & 0.33 & 0.54 & 0.15 & 0.44 \\ 0.46 & 0.86 & 0.59 & 0.33 & 0.45 & 0.09 & 0.63 \\ 0.56 & 0.66 & 0.43 & 0.54 & 0.45 & 0.31 & 0.41 \\ 0.55 & 0.35 & 0.42 & 0.35 & 0.33 & 0.33 & 0.54 \\ 0.62 & 0.36 & 0.67 & 0.65 & 0.15 & 0.04 & 0.44 \\ 0.73 & 0.21 & 0.62 & 0.55 & 0.29 & 0.18 & 0.47 \end{matrix}),$ $R_{6} = (\begin{matrix} 0.06 & 0.26 & 0.03 & 0.43 & 0.55 & 0.53 & 0.50 \\ 0.05 & 0.37 & 0.03 & 0.33 & 0.40 & 0.22 & 0.44 \\ 0.36 & 0.36 & 0.14 & 0.33 & 0.65 & 0.19 & 0.43 \\ 0.36 & 0.36 & 0.09 & 0.44 & 0.45 & 0.44 & 0.41 \\ 0.15 & 0.45 & 0.15 & 0.45 & 0.33 & 0.46 & 0.54 \\ 0.02 & 0.46 & 0.22 & 0.45 & 0.15 & 0.34 & 0.64 \\ 0.03 & 0.06 & 0.32 & 0.45 & 0.29 & 0.30 & 0.41 \end{matrix}),$ $R_{7} = (\begin{matrix} 0.88 & 0.68 & 0.75 & 0.66 & 0.75 & 0.68 & 0.50 \\ 0.89 & 0.88 & 0.66 & 0.75 & 0.68 & 0.58 & 0.56 \\ 0.65 & 0.65 & 0.86 & 0.66 & 0.55 & 0.55 & 0.75 \\ 0.67 & 0.68 & 0.55 & 0.66 & 0.75 & 0.55 & 0.84 \\ 0.75 & 0.77 & 0.55 & 0.75 & 0.92 & 0.64 & 0.66 \\ 0.55 & 0.79 & 0.55 & 0.88 & 0.33 & 0.47 & 0.85 \\ 0.75 & 0.96 & 0.56 & 0.76 & 0.48 & 0.54 & 0.66 \end{matrix}),$ $R_{8} = (\begin{matrix} 0.41 & 0.33 & 0.13 & 0.55 & 0.66 & 0.68 & 0.05 \\ 0.15 & 0.42 & 0.13 & 0.45 & 0.68 & 0.58 & 0.16 \\ 0.44 & 0.40 & 0.15 & 0.36 & 0.55 & 0.65 & 0.16 \\ 0.40 & 0.36 & 0.22 & 0.54 & 0.75 & 0.66 & 0.04 \\ 0.22 & 0.44 & 0.32 & 0.55 & 0.92 & 0.64 & 0.23 \\ 0.12 & 0.46 & 0.33 & 0.45 & 0.33 & 0.47 & 0.33 \\ 0.03 & 0.06 & 0.33 & 0.40 & 0.48 & 0.54 & 0.21 \end{matrix}),$ $R_{9} = (\begin{matrix} 0.58 & 0.88 & 0.75 & 0.56 & 0.55 & 0.58 & 0.60 \\ 0.89 & 0.68 & 0.55 & 0.45 & 0.58 & 0.44 & 0.74 \\ 0.45 & 0.65 & 0.76 & 0.66 & 0.55 & 0.55 & 0.75 \\ 0.67 & 0.68 & 0.55 & 0.86 & 0.85 & 0.55 & 0.84 \\ 0.53 & 0.67 & 0.88 & 0.75 & 0.56 & 0.77 & 0.66 \\ 0.55 & 0.79 & 0.55 & 0.88 & 0.33 & 0.47 & 0.85 \\ 0.71 & 0.96 & 0.51 & 0.74 & 0.48 & 0.57 & 0.76 \end{matrix}),$ $R_{10} = (\begin{matrix} 0.77 & 0.65 & 0.63 & 0.91 & 0.75 & 0.98 & 0.99 \\ 0.55 & 0.55 & 0.85 & 0.45 & 0.48 & 0.64 & 0.48 \\ 0.85 & 0.54 & 0.57 & 0.81 & 0.82 & 0.93 & 0.74 \\ 0.67 & 0.68 & 0.85 & 0.67 & 0.85 & 0.45 & 0.44 \\ 0.43 & 0.47 & 0.44 & 0.55 & 0.57 & 0.77 & 0.88 \\ 0.85 & 0.66 & 0.67 & 0.87 & 0.45 & 0.99 & 0.75 \\ 0.56 & 0.66 & 0.85 & 0.56 & 0.66 & 0.66 & 0.76 \end{matrix}) .$

Pick $R = {R_{1}, R_{2}, \dots, R_{10}}$ . Then $(U, R, P)$ is a FPIS.

We have $ind (R) = (\begin{matrix} 0.06 & 0.16 & 0.03 & 0.13 & 0.02 & 0.03 & 0.05 \\ 0.05 & 0.17 & 0.03 & 0.11 & 0.22 & 0.15 & 0.16 \\ 0.36 & 0.06 & 0.09 & 0.11 & 0.28 & 0.09 & 0.16 \\ 0.36 & 0.04 & 0.09 & 0.32 & 0.03 & 0.31 & 0.04 \\ 0.15 & 0.02 & 0.15 & 0.17 & 0.04 & 0.33 & 0.23 \\ 0.02 & 0.26 & 0.07 & 0.19 & 0.03 & 0.04 & 0.33 \\ 0.03 & 0.01 & 0.22 & 0.23 & 0.09 & 0.18 & 0.21 \end{matrix}),$ $(ind (R))_{P} = (\begin{matrix} 0.0066 & 0.0192 & 0.0111 & 0.0156 & 0.0016 & 0.0045 & 0.0025 \\ 0.0055 & 0.0204 & 0.0111 & 0.0132 & 0.0176 & 0.0225 & 0.0080 \\ 0.0396 & 0.0072 & 0.0333 & 0.0132 & 0.0224 & 0.0135 & 0.0080 \\ 0.0396 & 0.0048 & 0.0333 & 0.0384 & 0.0024 & 0.0465 & 0.0020 \\ 0.0165 & 0.0024 & 0.0555 & 0.0204 & 0.0032 & 0.0495 & 0.0115 \\ 0.0022 & 0.0312 & 0.0259 & 0.0228 & 0.0024 & 0.0060 & 0.0165 \\ 0.0033 & 0.0012 & 0.0814 & 0.0276 & 0.0072 & 0.027 & 0.0105 \end{matrix}) .$

3 The concept of information structures in a FPIS

In this part, we propose the concept of information structures in a FPIS.

Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . For any x ∈ U, define $[x]^{(ind (P))_{P}} = {y \in U : (x, y) \in ind (P))_{P}},$ $[x]_{(ind (P))_{P}} = {y \in U : (y, x) \in ind (P))_{P}} .$ Then $[x]^{(ind (P))_{P}}$ and $[x]_{(ind (P))_{P}}$ can be regarded as the upper-fuzzy and lower-fuzzy information granules of the point x with respect to $ind (P)$ , respectively.

Qian et al. [26] considered a fuzzy information granule family generated by a fuzzy relation from the universe can be called fuzzy granular structures. In addition, the set vector can display information structures better than the set family. According to this view, the following definitions describe information structures in a FPIS by means of set vectors.

Definition 3.1. Let $(U, R, P)$ be an FPIS. Given $P \subseteq R$ . Define $S_{u} (P, P) = ([x_{1}]^{(ind (P))_{P}}, \dots, [x_{n}]^{(ind (P))_{P}});$ $S_{l} (P, P) = ([x_{1}]_{(ind (P))_{P}}, \dots, [x_{n}]_{(ind (P))_{P}}) .$ Then $S_{u} (P, P)$ and $S_{l} (P, P)$ are called upper information and lower information structures of the subsystem $(U, P, P)$ , respectively.

Example 3.2. (Continue to Example 2.7.) Given $P \subseteq R$ , $P = {R_{1}, R_{3}, R_{5}, R_{6}, R_{9}, R_{10}}$ . Then

$ind (P) = (\begin{matrix} 0.06 & 0.16 & 0.03 & 0.13 & 0.02 & 0.03 & 0.44 \\ 0.05 & 0.17 & 0.03 & 0.11 & 0.22 & 0.15 & 0.44 \\ 0.36 & 0.06 & 0.14 & 0.11 & 0.28 & 0.09 & 0.43 \\ 0.36 & 0.04 & 0.09 & 0.32 & 0.03 & 0.31 & 0.41 \\ 0.15 & 0.02 & 0.15 & 0.17 & 0.04 & 0.33 & 0.43 \\ 0.02 & 0.26 & 0.22 & 0.19 & 0.03 & 0.04 & 0.44 \\ 0.03 & 0.01 & 0.32 & 0.23 & 0.09 & 0.18 & 0.41 \end{matrix}),$ $(ind (P))_{P} = (\begin{matrix} 0.0066 & 0.0192 & 0.0111 & 0.0156 & 0.0016 & 0.0045 & 0.0220 \\ 0.0055 & 0.0204 & 0.0111 & 0.0132 & 0.0176 & 0.0225 & 0.0220 \\ 0.0396 & 0.0072 & 0.0518 & 0.0132 & 0.0224 & 0.0135 & 0.0215 \\ 0.0396 & 0.0048 & 0.0333 & 0.0384 & 0.0024 & 0.0465 & 0.0205 \\ 0.0165 & 0.0024 & 0.0555 & 0.0204 & 0.0032 & 0.0495 & 0.0215 \\ 0.0022 & 0.0312 & 0.0814 & 0.0228 & 0.0024 & 0.0060 & 0.0220 \\ 0.0033 & 0.0012 & 0.1184 & 0.0276 & 0.0072 & 0.0270 & 0.0205 \end{matrix}) .$

Thus

$[x_{1}]^{(ind (P))_{P}} = 0.0066 / x_{1} + 0.0192 / x_{2} + 0.0111 / x_{3} + 0.0156 / x_{4} + 0.0016 / x_{5} + 0.0045 / x_{6} + 0.0220 / x_{7},$ $[x_{2}]^{(ind (P))_{P}} = 0.0055 / x_{1} + 0.0204 / x_{2} + 0.0111 / x_{3} + 0.0132 / x_{4} + 0.0176 / x_{5} + 0.0225 / x_{6} + 0.0220 / x_{7},$ $[x_{3}]^{(ind (P))_{P}} = 0.0396 / x_{1} + 0.0072 / x_{2} + 0.0518 / x_{3} + 0.0132 / x_{4} + 0.0224 / x_{5} + 0.0135 / x_{6} + 0.0215 / x_{7},$ $[x_{4}]^{(ind (P))_{P}} = 0.0396 / x_{1} + 0.0048 / x_{2} + 0.0333 / x_{3} + 0.0384 / x_{4} + 0.0024 / x_{5} + 0.0465 / x_{6} + 0.0205 / x_{7},$ $[x_{5}]^{(ind (P))_{P}} = 0.0165 / x_{1} + 0.0024 / x_{2} + 0.0555 / x_{3} + 0.0204 / x_{4} + 0.0032 / x_{5} + 0.0495 / x_{6} + 0.0215 / x_{7},$ $[x_{6}]^{(ind (P))_{P}} = 0.0022 / x_{1} + 0.0312 / x_{2} + 0.0814 / x_{3} + 0.0228 / x_{4} + 0.0024 / x_{5} + 0.0060 / x_{6} + 0.0220 / x_{7},$ $[x_{7}]^{(ind (P))_{P}} = 0.0033 / x_{1} + 0.0012 / x_{2} + 0.1184 / x_{3} + 0.0276 / x_{4} + 0.0072 / x_{5} + 0.0270 / x_{6} + 0.0205 / x_{7},$ $[x_{1}]_{(ind (P))_{P}} = 0.0066 / x_{1} + 0.0055 / x_{2} + 0.0396 / x_{3} + 0.0396 / x_{4} + 0.0165 / x_{5} + 0.0022 / x_{6} + 0.0033 / x_{7},$ $[x_{2}]_{(ind (P))_{P}} = 0.0192 / x_{1} + 0.0204 / x_{2} + 0.0072 / x_{3} + 0.0048 / x_{4} + 0.0024 / x_{5} + 0.0312 / x_{6} + 0.0012 / x_{7},$ $[x_{3}]_{(ind (P))_{P}} = 0.0111 / x_{1} + 0.0111 / x_{2} + 0.0518 / x_{3} + 0.0333 / x_{4} + 0.0555 / x_{5} + 0.0814 / x_{6} + 0.1184 / x_{7},$ $[x_{4}]_{(ind (P))_{P}} = 0.0111 / x_{1} + 0.0111 / x_{2} + 0.0518 / x_{3} + 0.0333 / x_{4} + 0.0555 / x_{5} + 0.0814 / x_{6} + 0.1184 / x_{7},$ $[x_{5}]_{(ind (P))_{P}} = 0.0016 / x_{1} + 0.0176 / x_{2} + 0.0224 / x_{3} + 0.0024 / x_{4} + 0.0032 / x_{5} + 0.0024 / x_{6} + 0.0072 / x_{7},$ $[x_{6}]_{(ind (P))_{P}} = 0.0045 / x_{1} + 0.0225 / x_{2} + 0.0135 / x_{3} + 0.0465 / x_{4} + 0.0495 / x_{5} + 0.0060 / x_{6} + 0.0270 / x_{7},$ $[x_{7}]_{(ind (P))_{P}} = 0.0220 / x_{1} + 0.0220 / x_{2} + 0.0215 / x_{3} + 0.0205 / x_{4} + 0.0215 / x_{5} + 0.0220 / x_{6} + 0.0205 / x_{7} .$ So

$S_{u} (P, P) = ([x_{1}]^{(ind (P))_{P}}, \dots, [x_{7}]^{(ind (P))_{P}});$

$S_{l} (P, P) = ([x_{1}]_{(ind (P))_{P}}, \dots, [x_{7}]_{(ind (P))_{P}}) .$

Definition 3.3. Let $(U, R, P)$ be a FPIS. Put $S_{u} (U, R, P) = {S_{u} (P, P) : P \in 2^{| R |}};$ $S_{l} (U, R, P) = {S_{l} (P, P) : P \in 2^{| R |}};$ Then $S_{u} (U, R, P)$ and $S_{l} (U, R, P)$ are called the upper information and lower informationstructures of the system $(U, R, P)$ , respectively.

4 Equality, dependence, independence, and inclusion degree between information structures in a FPIS

In this section, we study the concepts of equality, dependence, independence and inclusion degree between information structures in a FPIS.

Definition 4.1. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively.

(1) If ∀ i, $[x_{i}]^{(ind (Q))_{P}} = [x_{i}]^{(ind (T))_{P}}$ , then $S (Q, P)$ and $S (T, P)$ are said to be upper-same. We write $S (Q, P) =_{u} S (T, P) .$

(2) If ∀ i, $[x_{i}]_{(ind (Q))_{P}} = [x_{i}]_{(ind (T))_{P}}$ , then $S (Q, P)$ and $S (T, P)$ are said to be lower-same. We write $S (Q, P) =_{l} S (T, P) .$

Proposition 4.2. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. Then $S (Q, P) =_{u} S (T, P) \Leftrightarrow S (Q, P) =_{l} S (T, P) .$

Proof. Obviously.□

By Proposition 4.2,

$S (Q, P) =_{u} S (T, P)$ or $S (Q, P) =_{l} S (T, P)$ is denoted by $S (Q, P) = S (T, P) .$

Definition 4.3. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively.

(1) $S (T, P)$ is said to be upper-dependent on $S (Q, P)$ if ∀ i, $[x_{i}]^{(ind (Q))_{P}} \subseteq [x_{i}]^{(ind (T))_{P}}$ , we write $S (Q, P) ⪯_{u} S (T, P);$ $S (T, P)$ is said to be strictly upper-dependent on $S (Q, P)$ if $S (Q, P) ⪯_{u} S (T, P)$ and $S (Q, P) \neq_{u} S (T, P)$ , we write $S (Q, P) ≺_{u} S (T, P) .$

(2) $S (T, P)$ is said to be lower-dependent on $S (Q, P)$ if ∀ i, $[x_{i}]_{(ind (Q))_{P}} \subseteq [x_{i}]_{(ind (T))_{P}}$ , we write $S (Q, P) ⪯_{l} S (T, P);$ $S (T, P)$ is said to be strictly lower-dependent on $S (Q, P)$ if $S (Q, P) ⪯_{l} S (T, P)$ and $S (Q, P) \neq_{l} S (T, P)$ , we write $S (Q, P) ≺_{l} S (T, P) .$

“ $S (Q, P) ⪯_{u} S (T, P)$ ” means that there is an upper dependence of $(ind (Q))_{P}$ with respect to $(ind (T))_{P} .$

Analogously, “ $S (Q, P) ⪯_{l} S (T, P)$ ” indicates that there is a lower dependence of $(ind (Q))_{P}$ with respect to $(ind (T))_{P} .$

Proposition 4.4. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. Then $S (Q, P) ⪯_{u} S (T, P) \Leftrightarrow S (Q, P) ⪯_{l} S (T, P) .$

Proof. The proof is trivial. □

By Proposition 4.4,

$S (Q, P) ⪯_{u} S (T, P)$ or $S (Q, P) ⪯_{l} S (T, P)$ is denoted by $S (Q, P) ⪯ S (T, P)$ .

Definition 4.5. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively.

(1) $S (T, P)$ is said to be partially upper-dependent on $S (Q, P)$ , if ∃ i, $[x_{i}]^{(ind (Q))_{P}} \subseteq [x_{i}]^{(ind (T))_{P}},$ we write $S (Q, P) ⊑_{u} S (T, P);$ $S (T, P)$ is said to be strictly partially upper-dependent on $S (Q, P)$ if $S (Q, P) ⊑_{u} S (T, P)$ and $S (Q, P) \neq_{u} S (T, P)$ , we write $S (Q, P) ⊏_{u} S (T, P) .$

(2) $S (T, P)$ is said to be partially lower-dependent on $S (Q, P)$ , if ∃ i, $[x_{i}]_{(ind (Q))_{P}}$ $\subseteq [x_{i}]_{(ind (T))_{P}}$ , we write $S (Q, P) ⊑_{l} S (T, P);$ $S (T, P)$ is said to be strictly partially lower-dependent on $S (Q, P)$ if $S (Q, P) ⊑_{l} S (T, P)$ and $S (Q, P) \neq_{l} S (T, P)$ , we write $S (Q, P) ⊏_{l} S (T, P) .$

(3) $S (T, P)$ is said to be partially dependent on $S (Q, P)$ if $S (Q, P) ⊑_{u} S (T, P)$ or $S (Q, P) ⊑_{l} S (T, P)$ , we write $S (Q, P) ⊑ S (T, P);$ $S (T, P)$ is said to be strictly partially dependent on $S (Q, P)$ if $S (Q, P) ⊑ S (T, P)$ and $S (Q, P) \neq S (T, P)$ , we write $S (Q, P) ⊏ S (T, P) .$

Definition 4.6. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively.

(1) $S (T, P)$ is said to be upper-independent of $S (Q, P)$ if ∀ i, $[x_{i}]^{(ind (Q))_{P}} ⊈ [x_{i}]^{(ind (T))_{P}} .$ We write $S (Q, P) ⋈_{u} S (T, P) .$

(2) $S (T, P)$ is said to be lower-independent of $S (Q, P)$ if ∀ i, $[x_{i}]_{(ind (Q))_{P}} ⊈ [x_{i}]_{(ind (T))_{P}} .$ We write $S (Q, P) ⋈_{l} S (T, P) .$

(3) $S (T, P)$ is said to be independent of $S (Q, P)$ if $S (Q, P) ⋈_{u} S (T, P)$ and $S (Q, P) ⋈_{l} S (T, P) .$ We write $S (Q, P) ⋈ S (T, P) .$

Clearly, $S (Q, P) = S (T, P) \Leftrightarrow S (Q, P) ⪯ S (T, P),$ $S (T, P) ⪯ S (Q, P);$

$S (Q, P) ⪯ S (T, P) \to S (Q, P) ⊑ S (T, P)$ $\Leftrightarrow S (Q, P) ⊑_{u} S (T, P) or S (Q, P) ⊑_{l} S (T, P);$ $S (Q, P) ≺ S (T, P) \to S (Q, P) ⊏ S (T, P)$ $\Leftrightarrow S (Q, P) ⊏_{u} S (T, P) or S (Q, P) ⊏_{l} S (T, P);$ $S (Q, P) ⋈_{u} S (T, P) \Leftrightarrow S (Q, P) ⊈_{u} S (T, P);$ $S (Q, P) ⋈_{l} S (T, P) \Leftrightarrow S (Q, P) ⊈_{l} S (T, P);$ $S (Q, P) ⋈ S (T, P) \Leftrightarrow S (Q, P) ⊈ S (T, P) .$

Proposition 4.7. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. The the following properties hold:

(1) $S (Q, P) ⊑_{u} S (T, P) \Leftrightarrow S (Q^{- 1}, P) ⊑_{l} S (T^{- 1}, P);$

(2) $S (Q, P) ⊏_{u} S (T, P) \Leftrightarrow S (Q^{- 1}, P) ⊏_{l} S (T^{- 1}, P);$

(3) $S (Q, P) ⋈_{u} S (T, P) \Leftrightarrow S (Q^{- 1}, P) ⋈_{l} S (T^{- 1}, P) .$

Proof. Obviously. □

Definition 4.8. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively.

Define

$\begin{matrix} D_{u} (S (T, P) / S (Q, P)) \\ = \sum_{l = 1}^{n} \frac{| [x_{l}]^{(ind (T))_{P}} |}{\sum_{i = 1}^{n} | [x_{i}]^{(ind (T))_{P}} |} χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}), \end{matrix}$ where $\begin{matrix} χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}) \\ = {\begin{matrix} 1, & if [x_{l}]^{(ind (Q))_{P}} \subseteq [x_{l}]^{(ind (T))_{P}}, \\ 0, & if [x_{l}]^{(ind (Q))_{P}} ⊈ [x_{l}]^{(ind (T))_{P}}; \end{matrix} \end{matrix}$

$\begin{matrix} D_{l} (S (T, P) / S (Q, P)) \\ = \sum_{l = 1}^{n} \frac{| [x_{l}]_{(ind (T))_{P}} |}{\sum_{i = 1}^{n} | [x_{i}]_{(ind (T))_{P}} |} χ ([x_{l}]_{(ind (Q))_{P}}, [x_{l}]_{(ind (T))_{P}}), \end{matrix}$ where $\begin{matrix} χ ([x_{l}]_{(ind (Q))_{P}}, [x_{l}]_{(ind (T))_{P}}) \\ = {\begin{matrix} 1, & if [x_{l}]_{(ind (Q))_{P}} \subseteq [x_{l}]_{(ind (T))_{P}}, \\ 0, & if [x_{l}]_{(ind (Q))_{P}} ⊈ [x_{l}]_{(ind (T))_{P}}; \end{matrix} \end{matrix}$

$D (S (T, P) / S (Q, P)) = \frac{1}{2} (D_{u} (S (T, P) / S (Q, P)) + D_{l} (S (T, P) / S (Q, P))) .$

Example 4.9. (Continue to Example 2.7) Let $Q, T \subseteq R$ , $Q = {R_{2}, R_{4}, R_{5}, R_{8}}$ and $T = {R_{3}, R_{4}, R_{6}, R_{7}, R_{8}, R_{9}}$ .

Then $ind (Q) = (\begin{matrix} 0.41 & 0.16 & 0.13 & 0.13 & 0.02 & 0.03 & 0.05 \\ 0.15 & 0.17 & 0.13 & 0.11 & 0.22 & 0.15 & 0.16 \\ 0.44 & 0.06 & 0.09 & 0.11 & 0.28 & 0.09 & 0.16 \\ 0.40 & 0.04 & 0.22 & 0.32 & 0.03 & 0.31 & 0.04 \\ 0.22 & 0.02 & 0.32 & 0.17 & 0.04 & 0.33 & 0.23 \\ 0.12 & 0.26 & 0.07 & 0.19 & 0.03 & 0.04 & 0.33 \\ 0.03 & 0.01 & 0.22 & 0.23 & 0.09 & 0.18 & 0.21 \end{matrix}),$ $ind (T) = (\begin{matrix} 0.06 & 0.26 & 0.03 & 0.13 & 0.02 & 0.53 & 0.05 \\ 0.05 & 0.37 & 0.03 & 0.11 & 0.22 & 0.22 & 0.16 \\ 0.36 & 0.33 & 0.09 & 0.11 & 0.28 & 0.19 & 0.16 \\ 0.36 & 0.33 & 0.09 & 0.32 & 0.03 & 0.38 & 0.04 \\ 0.15 & 0.42 & 0.15 & 0.17 & 0.04 & 0.46 & 0.23 \\ 0.02 & 0.44 & 0.07 & 0.19 & 0.03 & 0.21 & 0.33 \\ 0.03 & 0.06 & 0.22 & 0.23 & 0.09 & 0.3 & 0.21 \end{matrix}) .$ Thus $(ind (Q))_{P} = (\begin{matrix} 0.0451 & 0.0192 & 0.0481 & 0.0156 & 0.0016 & 0.0045 & 0.0025 \\ 0.0165 & 0.0204 & 0.0481 & 0.0132 & 0.0176 & 0.0225 & 0.0080 \\ 0.0484 & 0.0072 & 0.0333 & 0.0132 & 0.0224 & 0.0135 & 0.0080 \\ 0.0440 & 0.0048 & 0.0814 & 0.0384 & 0.0024 & 0.0465 & 0.0020 \\ 0.0242 & 0.0024 & 0.1184 & 0.0204 & 0.0032 & 0.0495 & 0.0115 \\ 0.0132 & 0.0312 & 0.0259 & 0.0228 & 0.0024 & 0.0060 & 0.0165 \\ 0.0033 & 0.0012 & 0.0814 & 0.0276 & 0.0072 & 0.0270 & 0.0105 \end{matrix}),$ $(ind (T))_{P} = (\begin{matrix} 0.0066 & 0.0312 & 0.0111 & 0.0156 & 0.0016 & 0.0795 & 0.0025 \\ 0.0055 & 0.0444 & 0.0111 & 0.0132 & 0.0176 & 0.0330 & 0.0080 \\ 0.0396 & 0.0396 & 0.0333 & 0.0132 & 0.0224 & 0.0285 & 0.0080 \\ 0.0396 & 0.0396 & 0.0333 & 0.0384 & 0.0024 & 0.0570 & 0.0020 \\ 0.0165 & 0.0504 & 0.0555 & 0.0204 & 0.0032 & 0.0690 & 0.0115 \\ 0.0022 & 0.0528 & 0.0259 & 0.0228 & 0.0024 & 0.0315 & 0.0165 \\ 0.0033 & 0.0072 & 0.0814 & 0.0276 & 0.0072 & 0.0450 & 0.0105 \end{matrix}) .$

Thus $\begin{matrix} D_{u} (S (T, P) / S (Q, P)) \\ = \sum_{l = 1}^{4} \frac{| [x_{l}]^{(ind (T))_{P}} |}{\sum_{i = 1}^{n} | [x_{i}]^{(ind (T))_{P}} |} χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}) \\ \approx 0.1469, \end{matrix}$ $\begin{matrix} D_{l} (S (T, P) / S (Q, P)) \\ = \sum_{l = 1}^{4} \frac{| [x_{l}]_{(ind (T))_{P}} |}{\sum_{i = 1}^{n} | [x_{i}]_{(ind (T))_{P}} |} χ ([x_{l}]_{(ind (Q))_{P}}, [x_{l}]_{(ind (T))_{P}}) \\ \approx 0.0476, \end{matrix}$

$D (S (T, P) / S (Q, P)) = \frac{1}{2} (D_{u} (S (T, P) / S (Q, P)) + D_{l} (S (T, P) / S (Q, P))) \approx 0.0972 .$

Relationships between information structures in a FPIS may be described by the inclusion degree as below.

Theorem 4.10. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. (1) $S (Q, P) ⪯_{u} S (T, P) \Leftrightarrow D_{u} (S (T, P) / S (Q, P)) = 1;$ (2) $S (Q, P) ⋈_{u} S (T, P) \Leftrightarrow D_{u} (S (T, P) / S (Q, P)) = 0;$ (3) $S (Q, P) ⊑_{u} S (T, P) \Leftrightarrow 0 < D_{u} (S (T, P) / S (Q, P)) \leq 1 .$

Proof. (1) “ $R ightarrow$ ” is obvious.

“⇐". Put $| [x_{l}]^{(ind (T))_{P}} | = q_{l}, \sum_{l = 1}^{n} q_{l} = q .$

Since $D_{u} (S (T, P) / S (Q, P)) = 1,$ we have $\sum_{l = 1}^{n} q_{l} χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}) = q = \sum_{l = 1}^{n} q_{l} .$

Then $\sum_{l = 1}^{n} q_{l} (1 - χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}})) = 0 .$

Thus $\forall l, 1 - χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}) = 0 .$

This implies that ∀ l, $[x_{l}]^{(ind (Q))_{P}} \subseteq [x_{l}]^{(ind (T))_{P}} .$

Consequently, $S (Q, P) ⪯_{u} S (T, P) .$

(2) “ $R ightarrow$ ". $S (Q, P) ⋈_{u} S (T, P)$ implies that ∀ l, $[x_{l}]^{(ind (Q))_{P}} ⊈ [x_{l}]^{(ind (T))_{P}} .$

Then ∀ l, $χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}) = 0 .$

Thus $D_{u} (S (T, P) / S (Q, P)) = 0 .$

“⇐". Note that $D_{u} (S (T, P) / S (Q, P)) = 0 .$ Then ∀ l, $χ ([x_{l}]^{(ind (Q))_{P}}, [x_{l}]^{(ind (T))_{P}}) = 0 .$

This means that ∀ l, $[x_{l}]^{(ind (Q))_{P}} ⊈ [x_{l}]^{(ind (T))_{P}} .$

Consequently, $S (Q, P) ⋈_{u} S (T, P) .$

(3) The proof is straightforward from (1) and (2). □

Theorem 4.11. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. Then one has the following properties: (1) $S (Q, P) ⪯_{l} S (T, P) \Leftrightarrow D_{l} (S (T, P) / S (Q, P)) = 1;$ (2) $S (Q, P) ⋈_{l} S (T, P) \Leftrightarrow D_{l} (S (T, P) / S (Q, P)) = 0;$ (3) $S (Q, P) ⊑_{l} S (T, P) \Leftrightarrow 0 < D_{l} (S (T, P) / S (Q, P)) \leq 1 .$

Proof. The proof is similar to Theorem 4.10. □

Theorem 4.12. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. Then one has the following properties: (1) $S (Q, P) ⪯ S (T, P) \Leftrightarrow D (S (T, P) / S (Q, P)) = 1;$ (2) $S (Q, P) ⋈ S (T, P) \Leftrightarrow D (S (T, P) / S (Q, P)) = 0;$ (3) $S (Q, P) ⊑ S (T, P) \Leftrightarrow 0 < D (S (T, P) / S (Q, P)) \leq 1 .$

Proof. It can be proved by Theorems 4.10 and 4.11. □

5 Information distance between information structures

In this section, we investigate the concept of information distance between information structures in a FPIS and obtain some of its properties.

For any F₁, F₂ ∈ I^U, denote $F_{1} \oplus F_{2} = (F_{1} - F_{2}) \cup (F_{2} - F_{1}) .$

Then F₁ ∪ F₂ is addressed as the symmetric difference between F₁ and F₂ .

If F₁ ⊆ F₂, then |F₁ ⊕ F₂| = |F₂ - F₁| = |F₂| - |F₁| .

Definition 5.1. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively.

(1) Upper-information distance between $S (Q, P)$ and $S (T, P)$ is defined as $d_{u} ((S (Q, P), S (T, P)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ .$

(2) Lower-information distance between $S (Q, P)$ and $S (T, P)$ is defined as $d_{l} (S (Q, P), S (T, P)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]_{(ind (Q))_{P}} \oplus [x_{i}]_{(ind (T))_{P}} ∣ .$

(3) Information distance between $S (Q, P)$ and $S (T, P)$ is defined as

$d (S (Q, P), S (T, P)) = \frac{1}{2} (d_{u} ((S (Q, P), S (T, P)) + d_{l} (S (Q, P), S (T, P))) .$

Proposition 5.2. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. Then

$d_{u} ((S (Q, P), S (T, P)) = d_{l} (S (Q, P), S (T, P)) = d (S (Q, P), S (T, P)) .$

Proof. By Lemma 5.4, we have

$d_{u} ((S (Q, P), S (T, P)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (\sum_{j = 1}^{n} \max (p_{j} r_{ij}, p_{j} x_{ij}) - \sum_{j = 1}^{n} \min (p_{j} r_{ij}, p_{j} x_{ij})) = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} \max (p_{j} r_{ij}, p_{j} x_{ij}) - \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} \min (p_{j} r_{ij}, p_{j} x_{ij})$

$d_{l} (S (Q, P), S (T, P)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]_{(ind (Q))_{P}} \oplus [x_{i}]_{(ind (T))_{P}} ∣ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (\sum_{j = 1}^{n} \max (p_{i} r_{ji}, p_{i} x_{ji}) - \sum_{j = 1}^{n} \min (p_{i} r_{ji}, p_{i} x_{ji})) = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} \max (p_{i} r_{ji}, p_{i} x_{ji}) - \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} \min (p_{i} r_{ji}, p_{i} x_{ji}) = \frac{1}{n^{2}} \sum_{j = 1}^{n} \sum_{i = 1}^{n} \max (p_{i} r_{ji}, p_{i} x_{ji}) - \frac{1}{n^{2}} \sum_{j = 1}^{n} \sum_{i = 1}^{n} \min (p_{i} r_{ji}, p_{i} x_{ji})$

Thus

$d_{u} ((S (Q, P), S (T, P)) - d_{l} (S (Q, P), S (T, P)) = 0 .$

$d_{u} ((S (Q, P), S (T, P)) = d_{l} (S (Q, P), S (T, P)) .$

Since

$d (S (Q, P), S (T, P)) = \frac{1}{2} (d_{u} ((S (Q, P), S (T, P)) + d_{l} (S (Q, P), S (T, P))) .$

Therefore

$d_{u} ((S (Q, P), S (T, P)) = d_{l} (S (Q, P), S (T, P)) = d (S (Q, P), S (T, P)) .$ □

Example 5.3. (Continue to Example 4.9) Let $Q, T \subseteq R$ , $Q = {R_{2}, R_{4}, R_{5}, R_{8}}$ and $T = {R_{3}, R_{4}, R_{6}, R_{7},$ R₈, R₉}.

$(ind (Q))_{P} = (\begin{matrix} 0.0451 & 0.0192 & 0.0481 & 0.0156 & 0.0016 & 0.0045 & 0.0025 \\ 0.0165 & 0.0204 & 0.0481 & 0.0132 & 0.0176 & 0.0225 & 0.0080 \\ 0.0484 & 0.0072 & 0.0333 & 0.0132 & 0.0224 & 0.0135 & 0.0080 \\ 0.0440 & 0.0048 & 0.0814 & 0.0384 & 0.0024 & 0.0465 & 0.0020 \\ 0.0242 & 0.0024 & 0.1184 & 0.0204 & 0.0032 & 0.0495 & 0.0115 \\ 0.0132 & 0.0312 & 0.0259 & 0.0228 & 0.0024 & 0.0060 & 0.0165 \\ 0.0033 & 0.0012 & 0.0814 & 0.0276 & 0.0072 & 0.0270 & 0.0105 \end{matrix}),$

$(ind (T))_{P} = (\begin{matrix} 0.0066 & 0.0312 & 0.0111 & 0.0156 & 0.0016 & 0.0795 & 0.0025 \\ 0.0055 & 0.0444 & 0.0111 & 0.0132 & 0.0176 & 0.0330 & 0.0080 \\ 0.0396 & 0.0396 & 0.0333 & 0.0132 & 0.0224 & 0.0285 & 0.0080 \\ 0.0396 & 0.0396 & 0.0333 & 0.0384 & 0.0024 & 0.0570 & 0.0020 \\ 0.0165 & 0.0504 & 0.0555 & 0.0204 & 0.0032 & 0.0690 & 0.0115 \\ 0.0022 & 0.0528 & 0.0259 & 0.0228 & 0.0024 & 0.0315 & 0.0165 \\ 0.0033 & 0.0072 & 0.0814 & 0.0276 & 0.0072 & 0.0450 & 0.0105 \end{matrix}) .$

$d_{u} ((S (Q, P), S (T, P)) = \frac{1}{4^{2}} \sum_{i = 1}^{4} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ \approx 0.1264,$

$d_{l} (S (Q, P), S (T, P)) = \frac{1}{4^{2}} \sum_{i = 1}^{4} ∣ [x_{i}]_{(ind (Q))_{P}} \oplus [x_{i}]_{(ind (T))_{P}} ∣ \approx 0.1264 .$

Thus

$d (S (Q, P), S (T, P)) = \frac{1}{2} (d_{u} ((S (Q, P), S (T, P)) + d_{l} (S (Q, P), S (T, P)) \approx 0.1264 .$

Lemma 5.4. ([36]) Let F₁, F₂ ∈ I^U . Then |F₁ ⊕ F₂| = |F₁ ∪ F₂| - |F₁ ∩ F₂| .

Lemma 5.5. ([36]) Given F₁, F₂ ∈ I^U . Then F₁ = F₂ ⇔ |F₁ ⊕ F₂|=0 .

Lemma 5.6. ([36]) Suppose F₁, F₂, F₃ ∈ I^U . Then $| F_{1} \oplus F_{2} | + | F_{2} \oplus F_{3} | \geq | F_{1} \oplus F_{3} | .$ Moreover, if F₁ ⊆ F₂ ⊆ F₃ or F₃ ⊆ F₂ ⊆ F₁, then |F₁ ⊕ F₂| + |F₂ ⊕ F₃| = |F₁ ⊕ F₃| . s

Theorem 5.7. Suppose that $(U, R, P)$ is a FPIS. Then $(S (U, R, P), d_{u})$ , $(S (U, R, P), d_{l})$ and $(S (U, R, P), d)$ are three metric spaces.

Proof. Given that $S (Q, P)$ , $S (T, P)$ , $S (P, P)$ are information structures of $Q, T, P \subseteq R$ , respectively.

Obviously,

$d_{u} ((S (Q, P), S (T, P)) \geq 0,$ $d_{u} ((S (Q, P), S (T, P)) = d_{u} (S (T, P), (S (Q, P)) .$

By Lemma 5.5, $d_{u} ((S (Q, P), S (T, P)) = 0$ ⇔ ∀i, $| [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} | = 0$ ⇔ ∀i, $[x_{i}]^{(ind (Q))_{P}} = [x_{i}]^{(ind (T))_{P}}$ ⇔ $S (Q, P) = S (T, P) .$

By Lemma 5.6, $d_{u} ((S (Q, P), S (T, P)) + d_{u} (S (T, P), S (P, P))$

$= \frac{1}{n^{2}} \sum_{i = 1}^{n} (| [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} | +$

$| [x_{i}]^{(ind (T))_{P}} \oplus [x_{i}]^{(ind (P))_{P}} |)$

$\geq \frac{1}{n^{2}} \sum_{i = 1}^{n} | [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (P))_{P}} |$ $= d_{u} ((S (Q, P), S (P, P)) .$

Thus $(S (U, R, P), d_{u})$ is a metric space.

Similarly, one can prove that $(S (U, R, P), d_{l})$ is a metric space.

Since $(S (U, R, P), d_{u})$ and $(S (U, R, P), d_{l})$ are two metric spaces, we come to the conclusion that $(S (U, R, P), d)$ is also a metric space. □

Proposition 5.8. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ , $S (C, P)$ , $S (K, P)$ are information structures of $Q, T, C, K \subseteq R$ , respectively. Moreover $ind (C) = ▵, ind (K) = ω$ . Then one has the following properties: (1) $0 \leq d ((S (Q, P), S (T, P)) \leq 1;$ (2) If $ind (Q)$ and $ind (T)$ are reflexive, then $0 \leq d (S (Q, P), S (T, P)) \leq 1 - \frac{1}{n};$

(3) If $ind (Q)$ and $ind (T)$ are reflexive and $S (Q, P) ⪯ S (T, P)$ , then $d (S (Q, P), S (C, P)) \leq d (S (T, P), S (C, P));$

(4) If $S (Q, P) ⪯ S (T, P)$ , then $d (S (Q, P), S (K, P)) \geq d (S (T, P), S (K, P)) .$

Proof. (1) Since ∀ i, $1 \leq | [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} | \leq n$ , by Lemma 5.4, we have $\begin{matrix} d_{u} ((S (Q, P), S (T, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ - \\ ∣ [x_{i}]^{(ind (Q))_{P}} \cap [x_{i}]^{(ind (T))_{P}} ∣) \end{matrix}$ $\begin{matrix} \leq \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ \\ \leq \frac{1}{n^{2}} \sum_{i = 1}^{n} n \\ = 1 . \end{matrix}$

Similarly, we can prove that $d_{l} (S (Q, P), S (T, P)) \leq 1 .$

Thus $0 \leq d (S (Q, P), S (T, P)) \leq 1 .$

(2) Since $ind (Q)$ and $ind (T)$ are reflexive, ∀ i, we have $[x_{i}]^{(ind (Q))_{P}} (x_{i}) = [x_{i}]^{(ind (T))_{P}} (x_{i}) = 1 .$ Thus $\begin{matrix} ∣ [x_{i}]^{(ind (Q))_{P}} \cap [x_{i}]^{(ind (T))_{P}} ∣ \\ \geq ([x_{i}]^{(ind (Q))_{P}} \cap [x_{i}]^{(ind (T))_{P}}) (x_{i}) \\ = [x_{i}]^{(ind (Q))_{P}} (x_{i}) \land [x_{i}]^{(ind (T))_{P}} (x_{i}) \\ = 1 . \end{matrix}$ By Lemma 5.4, $\begin{matrix} d_{u} ((S (Q, P), S (T, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} \cup [x_{i}]^{(ind (T))_{P}} ∣ \end{matrix}$ $\begin{matrix} - ∣ [x_{i}]^{(ind (Q))_{P}} \cap [x_{i}]^{(ind (T))_{P}} ∣) \\ \leq \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (T))_{P}} ∣ - 1) \\ \leq \frac{1}{n^{2}} \sum_{i = 1}^{n} (n - 1) \\ = 1 - \frac{1}{n} . \end{matrix}$

Similarly, we can prove that $d_{l} (S (Q, P), S (T, P)) \leq 1 - \frac{1}{n} .$

Thus $0 \leq d (S (Q, P), S (T, P)) \leq 1 - \frac{1}{n} .$

(3) Note that $ind (C) = ▵$ and $ind (K) = ω$ , so ∀ i, j ≠ i,

$[x_{i}]^{(ind (C))_{P}} (x_{i}) = 1,$ and $[x_{i}]^{(ind (C))_{P}} (x_{j}) = 0,$

∀ i, j, $[x_{i}]^{(ind (K))_{P}} (x_{j}) = 1$ .

Since $S (Q, P) ⪯ S (T, P)$ , we have $S (Q, P) ⪯_{u} S (T, P) .$

Then ∀ i, $[x_{i}]^{(ind (Q))_{P}} \subseteq [x_{i}]^{(ind (T))_{P}} .$ So ∀ i, $∣ [x_{i}]^{(ind (Q))_{P}} ∣ \leq ∣ [x_{i}]^{(ind (T))_{P}} ∣ .$

Since $ind (Q)$ and $ind (T)$ are reflexive, ∀ i, we have $[x_{i}]^{(ind (Q))_{P}} (x_{i}) = [x_{i}]^{(ind (T))_{P}} (x_{i}) = 1 .$

By Lemma 5.4,

$\begin{matrix} d_{u} ((S (Q, P), S (C, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (C))_{P}} ∣ \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} \cup [x_{i}]^{(ind (C))_{P}} ∣ - \\ ∣ [x_{i}]^{(ind (Q))_{P}} \cap [x_{i}]^{(ind (C))_{P}} ∣) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ((∣ [x_{i}]^{(ind (Q))_{P}} ∣ + 1 - \\ [x_{i}]^{(ind (Q))_{P}} (x_{i})) - [x_{i}]^{(ind (Q))_{P}} (x_{i})) \end{matrix}$ $\begin{matrix} = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} ∣ - 1) \\ \leq \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (T))_{P}} ∣ - 1) \\ = d_{u} (S (T, P), S (C, P)) . \end{matrix}$

Similarly, we can prove that $d_{l} (S (Q, P), S (C, P)) \leq d_{l} (S (T, P), S (C, P)) .$

Thus $d (S (Q, P), S (C, P)) \leq d (S (T, P), S (C, P)) .$

(4) Note that $S (Q, P) ⪯ S (T, P) .$ Then ∀ i, $| [x_{i}]^{(ind (Q))_{P}} | \leq | [x_{i}]^{(ind (T))_{P}} | .$

Thus $\begin{matrix} d_{u} ((S (Q, P), S (K, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (K))_{P}} ∣ \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (K))_{P}} ∣ - ∣ [x_{i}]^{(ind (Q))_{P}} ∣) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (n - ∣ [x_{i}]^{(ind (Q))_{P}} ∣) \\ \geq \frac{1}{n^{2}} \sum_{i = 1}^{n} (n - ∣ [x_{i}]^{(ind (T))_{P}} ∣) \\ = d_{u} (S (T, P), S (K, P)) . \end{matrix}$

Similarly, we can prove that $d_{l} (S (Q, P), S (K, P)) \geq d_{l} (S (T, P), S (K, P)) .$

Hence, $d (S (Q, P), S (K, P)) \geq d (S (T, P), S (K, P)) .$ □

Proposition 5.9. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (C, P)$ and $S (K, P)$ are information structures of $Q, C, K \subseteq R$ , respectively. If $ind (C) = ▵, ind (K) = ω$ , then $d (S (Q, P), S (C, P)) + d (S (Q, P), S (K, P)) = 1 - \frac{1}{n} .$

Proof. Since $ind (Q)$ is reflexive, ∀ i, we have $[x_{i}]^{(ind (Q))_{P}} (x_{i}) = 1 .$

Note that $ind (C) = ▵$ and $ind (K) = ω$ , then ∀ i, j ≠ i,

$[x_{i}]^{(ind (C))_{P}} (x_{i}) = 1, [x_{i}]^{(ind (C))_{P}} (x_{j}) = 0,$

∀ i, j, $[x_{i}]^{(ind (K))_{P}} (x_{j}) = 1$ .

By Lemma 5.4, we have $\begin{matrix} d_{u} ((S (Q, P), S (C, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} \cup [x_{i}]^{(ind (C))_{P}} ∣ - \\ ∣ [x_{i}]^{(ind (Q))_{P}} \cap [x_{i}]^{(ind (C))_{P}} ∣) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ((∣ [x_{i}]^{(ind (Q))_{P}} ∣ + 1 - \\ [x_{i}]^{(ind (Q))_{P}} (x_{i})) - [x_{i}]^{(ind (Q))_{P}} (x_{i})) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} ∣ - 1) . \end{matrix}$

$\begin{matrix} d_{u} ((S (Q, P), S (K, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]^{(ind (Q))_{P}} \oplus [x_{i}]^{(ind (K))_{P}} ∣ \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (K))_{P}} ∣ - ∣ [x_{i}]^{(ind (Q))_{P}} ∣) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (n - ∣ [x_{i}]^{(ind (Q))_{P}} ∣) . \end{matrix}$

Thus

$\begin{matrix} d_{u} ((S (Q, P), S (C, P)) + d_{u} ((S (Q, P), S (K, P)) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (∣ [x_{i}]^{(ind (Q))_{P}} ∣ - 1) \\ + \frac{1}{n^{2}} \sum_{i = 1}^{n} (n - ∣ [x_{i}]^{(ind (Q))_{P}} ∣) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} (n - 1) \\ = 1 - \frac{1}{n} . \end{matrix}$

Similarly, we can prove that

$d_{l} (S (Q, P), S (C, P)) + d_{l} (S (Q, P), S (K, P)) = 1 - \frac{1}{n} .$

Hence, $d (S (Q, P), S (C, P)) + d (S (Q, P), S (K, P)) = 1 - \frac{1}{n} .$ □

Proposition 5.10. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ , $S (P, P)$ are information structures of $Q, T, P \subseteq R$ , respectively. If $S (Q, P) ⪯ S (T, P) ⪯ S (P, P)$ or $S (P, P) ⪯ S (T, P) ⪯ S (Q, P)$ , then

$d (S (Q, P), S (T, P)) + d (S (T, P), S (P, P)) = d (S (Q, P), S (P, P)) .$

Proof. This holds by Lemma 5.6. □

6 Entropy measurement for a FPIS

In physics, entropy is often used to estimate the disorder degree of a system. Shannon [27] used the concept of entropy for measuring their uncertainty.

Let $P = {\frac{x_{1}, x_{2}, \dots, x_{n}}{p_{1}, p_{2}, \dots, p_{n}}} .$ Then Shannon entropy of P is defined as follows: $H (P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} p_{i} .$

The uncertainty of a given FPIS is derived from the uncertainty of fuzzy relations imposed on its universe. In this section, as an application for information structures in a FPIS, we study entropy measurement for a FPIS based on the proposed information structures.

Definition 6.1. Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . Then self-information quantity of $[x]^{ind (P))_{P}}$ and $[x]_{ind (P))_{P}}$ are denoted by $I ([x]^{ind (P))_{P}})$ and $I ([x]_{ind (P))_{P}})$ , respectively. They are defined as follows: $I ([x]^{ind (P))_{P}}) = {log}_{2} | [x]^{ind (P))_{P}} |,$ $I ([x]_{ind (P))_{P}}) = {log}_{2} | [x]_{ind (P))_{P}} | .$

Definition 6.2. Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . Then upper information, lower information and information entropy of the subsystem $(U, P, P)$ are respectively defined as follows: $H_{u} (P, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]^{(ind (P))_{P}}),$ $H_{l} (P, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]_{(ind (P))_{P}}),$ $H (P, P) = \frac{1}{2} (H_{u} (P, P) + H_{l} (P, P)) .$

Clearly, $H_{u} (P, P) = H_{l} (P^{- 1}, P) .$

Given $U = {x_{1}, x_{2}, \dots, x_{n}}, P = {\frac{x_{1}, x_{2}, \dots, x_{n}}{p_{1}, p_{2}, \dots, p_{n}}},$ $ind (P) = (r_{ij})_{n \times n} .$

Then $H_{u} (P, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}),$ $H_{l} (P, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji})$ $= - \sum_{i = 1}^{n} p_{i} {log}_{2} (p_{i} | [x_{i}]_{ind (P)} |) .$

If $p_{1} = p_{2} = \dots = p_{n} = \frac{1}{n}$ , then $H_{u} (P, P) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]^{ind (P)} |}{n},$ $H_{l} (P, P) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{ind (P)} |}{n} .$

$- \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]^{ind (P)} |}{n}$ and $- \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{ind (P)} |}{n}$ both are the generalizations of Shannon entropy. Thus, each of $H_{u} (P, P)$ and $H_{l} (P, P)$ can be regarded as generalized Shannon entropy.

Example 6.3. (Continue to Example 2.7) Let $P, Q, T \subseteq R$ , $P = {R_{1}, R_{3}, R_{5}, R_{6}, R_{9}, R_{10}}$ , $T = {R_{2}, R_{4}, R_{5}}$ and $T = {R_{3}, R_{4}, R_{6}, R_{7}, R_{8}, R_{9}}$ . Then $(ind (P))_{P} = (\begin{matrix} 0.0066 & 0.0192 & 0.0111 & 0.0156 & 0.0016 & 0.0045 & 0.0220 \\ 0.0055 & 0.0204 & 0.0111 & 0.0132 & 0.0176 & 0.0225 & 0.0220 \\ 0.0396 & 0.0072 & 0.0518 & 0.0132 & 0.0224 & 0.0135 & 0.0215 \\ 0.0396 & 0.0048 & 0.0333 & 0.0384 & 0.0024 & 0.0465 & 0.0205 \\ 0.0165 & 0.0024 & 0.0555 & 0.0204 & 0.0032 & 0.0495 & 0.0215 \\ 0.0022 & 0.0312 & 0.0814 & 0.0228 & 0.0024 & 0.0060 & 0.0220 \\ 0.0033 & 0.0012 & 0.1184 & 0.0276 & 0.0072 & 0.0270 & 0.0205 \end{matrix}),$

Thus

$H_{u} (P, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]^{(ind (P))_{P}}) \approx 2.7237,$ $H_{l} (P, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]_{(ind (P))_{P}}) \approx 2.4901,$ $H (P, P) = \frac{1}{2} (H_{u} (P, P) + H_{l} (P, P)) \approx 2.6069;$ $H_{u} (Q, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]^{(ind (Q))_{P}}) \approx 2.7036,$ $H_{l} (Q, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]_{(ind (Q))_{P}}) \approx 2.3723,$ $H (Q, P) = \frac{1}{2} (H_{u} (Q, P) + H_{l} (Q, P)) \approx 2.5380;$ $H_{u} (T, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]^{(ind (T))_{P}}) \approx 2.5217,$ $H_{l} (T, P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]_{(ind (T))_{P}}) \approx 2.4055,$ $H (T, P) = \frac{1}{2} (H_{u} (T, P) + H_{l} (T, P)) \approx 2.4636 .$ This example illustrates that H_u ≠ H_l . Thus, Definition 6 is reasonable.

Denote $p^{*} = - \sum_{i = 1}^{n} p_{i} {log}_{2} p_{i}, p_{*} = - \sum_{i = 1}^{n} p_{i} {log}_{2} ({np}_{i}) .$

Obviously, p_* < p, 0 ≤ p .

Proposition 6.4. Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . Then $0 \leq H_{u} (P, P) < + \infty .$ Moreover, if $ind (P)$ is reflexive, then $0 \leq H_{u} (P, P) \leq p^{*};$ if $ind (P) = ω$ or $ind (P)$ is reflexive, p_k ≠ 0 for some k and p_i = 0 for any i ≠ k, then H_u achieves the minimum value 0 ;

Proof. (1) Given $U = {x_{1}, x_{2}, \dots, x_{n}}, P = {\frac{x_{1}, x_{2}, \dots, x_{n}}{p_{1}, p_{2}, \dots, p_{n}}},$ $ind (P) = (r_{ij})_{n \times n} .$

Then $H_{u} (P, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}) .$

It should be noted that ∀ i, j, 0 ≤ r_ij ≤ 1 . Then $\forall i, 0 \leq \sum_{j = 1}^{n} p_{j} r_{ij} \leq \sum_{j = 1}^{n} p_{j} = 1 .$

Thus ∀ i, $- \infty < {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}) \leq 0 .$

This implies that $\forall i, 0 \leq - p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}) < + \infty .$

Hence $0 \leq H_{u} (P, P) < + \infty .$

(2) Suppose that $ind (P)$ is reflexive. Then ∀ i, r_ii = 1 . So $\forall i, p_{i} = p_{i} r_{ii} \leq \sum_{j = 1}^{n} p_{j} r_{ij} .$

Thus $\forall i, p_{i} {log}_{2} p_{i} \leq p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}) .$

Hence $H_{u} (P, P) \leq p^{*} .$

By (1), $0 \leq H_{u} (P, P) \leq p^{*} .$

(3) a) Suppose $ind (P) = ω .$ Then ∀ i, j, r_ij = 1 . Thus $\forall i, {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}) = {log}_{2} (\sum_{j = 1}^{n} p_{j}) = {log}_{2} 1 = 0 .$

This implies that $H_{u} (P, P) = 0$ .Hence, H_u achieves the minimum value 0 when $ind (P) = ω$ . (3) b) Suppose that $ind (P)$ is reflexive, p_k ≠ 0 for some k and p_i = 0 for any i ≠ k. Obviously, p_k = 1 .

$\begin{matrix} H_{u} (P, P) & = & - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij}) \\ = & - \sum_{i = 1}^{n} p_{i} {log}_{2} (p_{k} r_{ik}) \\ = & - p_{k} {log}_{2} (p_{k} r_{kk}) - \sum_{i \neq k} p_{i} {log}_{2} (p_{k} r_{ik}) \\ = & - p_{k} {log}_{2} (p_{k} r_{kk}) \\ = & - {log}_{2} r_{kk} . \end{matrix}$

Since $ind (P)$ is reflexive, we have $H_{u} (P, P) = - {log}_{2} r_{kk} = - {log}_{2} 1 = 0 .$

Hence, H_u achieves the minimum value 0 when $ind (P)$ is reflexive, p_k ≠ 0 for some k and p_i = 0 for any i ≠ k. □

Proposition 6.5. Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . Then $p_{*} \leq H_{l} (P, P) < + \infty .$ Moreover, if $ind (P)$ is reflexive, then $p_{*} \leq H_{l} (P, P) \leq p^{*};$ if $ind (P)$ is reflexive, p_k ≠ 0 for some k and p_i = 0 for any i ≠ k, then $p_{*} \leq H_{l} (P, P) \leq 0;$ if $ind (P) = ω$ , then H_l achieves the minimum value p_* .

Proof. (1) Given $U = {x_{1}, x_{2}, \dots, x_{n}}, P = {\frac{x_{1}, x_{2}, \dots, x_{n}}{p_{1}, p_{2}, \dots, p_{n}}},$ $ind (P) = (r_{ij})_{n \times n} .$

Then $H_{l} (P, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji}) .$

It should be noted that ∀ i, j, 0 ≤ r_ji ≤ 1 . Then $\forall i, 0 \leq \sum_{j = 1}^{n} p_{i} r_{ji} \leq \sum_{j = 1}^{n} p_{i} = {np}_{i} .$

Thus ∀ i, $- \infty < {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji}) \leq {log}_{2} ({np}_{i}) .$

This implies that ∀ i, $- p_{i} {log}_{2} ({np}_{i}) \leq - p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji}) < + \infty .$

Hence $p_{*} \leq H_{l} (P, P) < + \infty .$

(2) Suppose that $ind (P)$ is reflexive. Then ∀ i, r_ii = 1 . This implies that $\forall i, p_{i} = p_{i} r_{ii} \leq \sum_{j = 1}^{n} p_{i} r_{ji} .$

Thus $\forall i, p_{i} {log}_{2} p_{i} \leq p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji}) .$

Hence $H_{l} (P, P) \leq p^{*} .$

By (1), $p_{*} \leq H_{l} (P, P) \leq p^{*} .$

(3) Obviously, p_k = 1 .

$\begin{matrix} H_{l} (P, P) & = & - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji}) \\ = & - \sum_{i = 1}^{n} p_{i} {log}_{2} (p_{i} \sum_{j = 1}^{n} r_{ji}) \\ = & - p_{k} {log}_{2} (p_{k} \sum_{j = 1}^{n} r_{jk}) \\ - & \sum_{i \neq k} p_{i} {log}_{2} (p_{i} \sum_{j = 1}^{n} r_{ji}) \\ = & - p_{k} {log}_{2} (p_{k} \sum_{j = 1}^{n} r_{jk}) \\ = & - {log}_{2} (\sum_{j = 1}^{n} r_{jk}) . \end{matrix}$

Since $ind (P)$ is reflexive, we have $\sum_{j = 1}^{n} r_{jk} = r_{kk} + \sum_{j \neq k} r_{jk} \geq 1 .$

So $H_{l} (P, P) \leq 0 .$

By (2), we can obtain that $p_{*} \leq H_{l} (P, P) \leq 0 .$

(4) Suppose $ind (P) = ω .$ Then ∀ i, j, r_ij = 1 . Thus ∀ i, ${log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji}) = {log}_{2} (\sum_{j = 1}^{n} p_{i}) = {log}_{2} ({np}_{i}) .$

This implies that $H_{l} (P, P) = p_{*}$ .

Hence, H_l achieves the minimum value p_* when $ind (P) = ω$ . □

Theorem 6.6. Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . Then $\frac{p_{*}}{2} \leq H (P, P) < + \infty .$ Moreover, if $ind (P)$ is reflexive, then $\frac{p_{*}}{2} \leq H (P, P) \leq p^{*};$ if $ind (P)$ is reflexive, p_k ≠ 0 for some k and p_i = 0 for any i ≠ k, then $\frac{p_{*}}{2} \leq H (P, P) \leq 0;$ if $ind (P) = ω$ , then H achieves the minimum value $\frac{p_{*}}{2} .$

Proof. It can be proved by Propositions 6 and 6. □

Proposition 6.7. Let $(U, R, P)$ be a FPIS. Given $P \subseteq R$ . If $ind (P)$ is reflexive, then the following properties hold:

(1) $H_{u} (P, P) \leq H (P);$

(2) $H_{l} (P, P) \leq H (P);$

(3) $H (P, P) \leq H (P) .$

Proof.

(1) We have

$H_{u} (P, P) - H (P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]^{(ind (P))_{P}})$

$- (- \sum_{i = 1}^{n} p_{i} {log}_{2} p_{i}),$

$= - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{j} r_{ij})$

$- (- \sum_{i = 1}^{n} p_{i} {log}_{2} p_{i})$

$= - \sum_{i = 1}^{n} p_{i} {log}_{2} \frac{\sum_{j = 1}^{n} p_{j} r_{ij}}{p_{i}} .$

Note that $ind (P)$ is reflexive. Then $\sum_{j = 1}^{n} p_{j} r_{ij} = p_{i} r_{ii} + \sum_{j \neq i} p_{j} r_{ij} = p_{i} + \sum_{j \neq i} p_{j} r_{ij} .$

So $\sum_{j = 1}^{n} p_{j} r_{ij} \geq p_{i} .$

Thus $- \sum_{i = 1}^{n} p_{i} {log}_{2} (\frac{\sum_{j = 1}^{n} p_{j} r_{ij}}{p_{i}}) \leq 0 .$

Therefore, $H_{u} (P, P) \leq H (P) .$

(2) We have

$H_{l} (P, P) - H (P) = - \sum_{i = 1}^{n} p_{i} I ([x_{i}]_{(ind (P))_{P}})$

$- (- \sum_{i = 1}^{n} p_{i} {log}_{2} p_{i}) .$

$= - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} p_{i} r_{ji})$

$- (- \sum_{i = 1}^{n} p_{i} {log}_{2} p_{i})$

$= - \sum_{i = 1}^{n} p_{i} {log}_{2} (\frac{\sum_{j = 1}^{n} p_{i} r_{ji}}{p_{i}}) .$

Note that $\sum_{j = 1}^{n} p_{j} r_{ij} = p_{i} r_{ii} + \sum_{j \neq i} p_{i} r_{ji} = p_{i} + \sum_{j \neq i} p_{i} r_{ji} .$ Then $\sum_{j = 1}^{n} p_{i} r_{ji} \geq p_{i} .$

Thus $- \sum_{i = 1}^{n} p_{i} {log}_{2} (\frac{\sum_{j = 1}^{n} p_{i} r_{ji}}{p_{i}}) \leq 0 .$

Therefore, $H_{l} (P, P) \leq H (P) .$

(3) It can be proved by (1) and (2). □

Theorem 6.8. Suppose that $(U, R, P)$ is a FPIS. Given that $S (Q, P)$ , $S (T, P)$ are information structures of $Q, T \subseteq R$ , respectively. Then the following properties hold: (1) If $S (Q, P) ≺ S (T, P)$ , then $H (T, P) < H (Q, P);$ (2) If $S (Q, P) ⪯ S (T, P)$ , then $H (T, P) \leq H (Q, P) .$

Proof. (1) Since $S (Q, P) ≺ S (T, P)$ , we have $S (Q, P) ≺_{u} S (T, P) .$ Similar to that of Proposition 3.20, it obtained that∀ i, j,

$ind (Q) (x_{i}, x_{j}) \leq ind (T) (x_{i}, x_{j}),$ and ∃ i′, j′, $ind (Q) (x_{i^{'}}, x_{j^{'}}) < ind (T) (x_{i^{'}}, x_{j^{'}}) .$

By Definition 6, $H_{u} (Q, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} \sum_{j = 1}^{n} [x_{i}]^{(ind (Q))_{P}} (x_{j}) .$

Then $H_{u} (Q, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} (ind (Q))_{P} (x_{i}, x_{j})) .$

Homoplastically, we have $H_{u} (T, P) = - \sum_{i = 1}^{n} p_{i} {log}_{2} (\sum_{j = 1}^{n} (ind (T))_{P} (x_{i}, x_{j})) .$

Thus $H_{u} (T, P) < H_{u} (Q, P) .$

In the same manner, we can prove that $H_{l} (T, P) < H_{l} (Q, P) .$

Hence, $H (T, P) < H (Q, P) .$

(2) The proof is similar to (1) . □

The above proposition shows that information entropy increases while the fuzzy information structures in a FPIS become finer. Inversely, it decreases while the fuzzy information structures become coarser.

7 Conclusions

In this paper, a FPIS has been treated as an IS with fuzzy relations, where probability distribution is integrated into fuzzy relations. New fuzzy relation matrix expression has been proposed, which contains not only the similarity between objects but also the probability distribution. This depicts the internal characteristics of a FPIS. Information structures in a FPIS have been defined by set vectors from two aspects of upper fuzzy information structure and lower fuzzy information structure. Relationships between these structures have been investigated. Information distance between two information structures has been given. As an application for information structures in a FPIS, uncertainty measurement for a FPIS has been studied by using information structures. Because fuzzy relations are inconvenient to be generated by UCI (Machine Learning Repository), we do not make big data analysis on entropy measurement. This is the limitation of this paper. In the future, we will consider applications of the obtained results.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions, which have helped immensely in improving the quality of the paper.

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