Similarity measure for multi-granularity rough approximations of vague sets

Abstract

Vague sets are a further extension of fuzzy sets. In rough set theory, target concept can be characterized by different rough approximation spaces when it is a vague concept. The uncertainty measure of vague sets in rough approximation spaces is an important issue. If the uncertainty measure is not accurate enough, different rough approximation spaces of a vague concept may possess the same result, which makes it impossible to distinguish these approximation spaces for charactering a vague concept strictly. In this paper, this problem will be solved from the perspective of similarity. Firstly, based on the similarity between vague information granules(VIGs), we proposed an uncertainty measure with strong distinguishing ability called rough vague similarity (RVS). Furthermore, by studying the multi-granularity rough approximations of a vague concept, we reveal the change rules of RVS with the changing granularities and conclude that the RVS between any two rough approximation spaces can degenerate to granularity measure and information measure. Finally, a case study and related experiments are listed to verify that RVS possesses a better performance for reflecting differences among rough approximation spaces for describing a vague concept.

Keywords

Vague sets uncertainty measure vague information granule rough vague similarity multi-granularity

1 Introduction

There are couple of effective tools for dealing with uncertain knowledge, i.e., fuzzy sets [44], rough sets [26], probability theory [25], and cloud model [19], etc.. Vague sets [11], as well as intuitionistic fuzzy sets (further generalization of fuzzy sets) [3], are essentially the same. The similarities and differences between vague sets and intuitionistic fuzzy sets are further investigated in literature [13], which concluded that vague sets are more universally applicable than intuitionistic fuzzy sets to some degree. Based on fuzzy sets, vague sets generalize the membership degree from single value to interval value. As a classical soft computing tool, vague sets have more powerful ability to handle uncertain information than fuzzy sets. Currently, vague sets have been applied in various fields, including decision making [1, 14], pattern recognition [35, 46], control and reasoning [5, 49], et al.

Rough sets [26] are effective tool to handle uncertain information by using the given information granulation. In rough set theory, target concepts can be described by upper and lower approximation sets. Generally speaking, the target concepts in classical rough sets are usually accurate and crisp. However, the target concept may be vague or uncertain in many real applications. To solve this problem, there are many research works [28 , 50] on the combination between rough sets and vague sets. Especially, the research on uncertainty of vague concepts in rough approximation spaces becomes a basic issue. Current works [4 , 48] mainly focus on measuring the uncertainty of vague sets itself, but it is necessary to consider the uncertainty of vague concepts in rough approximation spaces. Zhang [50] constructed the approximation sets of a vague set in rough approximation spaces and proved that average-step-vague set is an optimal step-vague set. Moreover, by defining the concept of interval fuzziness of vague sets, Zhang [52] proposed the average fuzziness of vague sets. By constructing rough approximations of vague sets in fuzzy approximation spaces, Shen [36] presented two types of roughness measure of vague sets. These works indicate that rough sets are suitable mathematical tool for studying vagueness and uncertainty.

Granular computing (GrC) [20 , 42] provides a method to simulate the human mechanism to deal with complex problems hierarchically. Different knowledge spaces can be induced by diverse attribute sets, thus the multi-granularity rough approximations of uncertain knowledge can be achieved [9 , 40]. The uncertainty of multi-granularity approximation spaces for describing a vague concept is an important issue, and the rules of uncertainty of vague sets in multi-granularity approximation spaces should be in line with human cognitive mechanism. However, when a vague concept is described by different rough approximation spaces respectively, these approximation spaces may possess the same measure result, and it does not mean these approximation spaces are completely equivalent. Moreover, the current similarity measures [4 , 39] focus on reflecting the similarity between two vague sets, and the difference among approximation spaces for characterizing a vague concept cannot be reflected. Actually, in some cases, such as attribute evaluation, pattern recognition and knowledge matching, it is necessary to discriminate these rough approximation spaces when describe a vague concept. Take attribute evaluation as an example, if the rough approximation spaces corresponding to different attributes cannot be distinguished, ranking by attribute significance will be not accurate. Therefore, it is necessary to establish an uncertainty measure model with strong distinguishing ability in multi-granularity spaces. This is one of motivations of this paper. In the view of GrC, the information granules will be subdivided into finer granules with granularity being finer gradually. That means a vague concept can be described by rough approximation spaces with different granularities with the increase of attributes. Is there any rules in describing the same vague concept in multi-granularity approximation spaces? From the above discussion, the description for a vague concept in multi-granulation rough approximation spaces still need further research.

The main contributions of this paper are as follows: (1) to reflect the similarity between equivalence classes for characterizing a vague concept, the similarity between vague information granules (VIGs) is defined; (2) based on the similarity between VIGs, rough vague similarity (RVS) is proposed to measure the differences among approximation spaces for characterizing a vague concept; (3) the multi-granularity approximations of a vague set is analyzed from the perspective of similarity, then the change rules of the RVS with the changing granularities are revealed.

The rest of this paper is organized as follows. Many preliminary concepts are briefly recalled in Section 2. In Section 3, the rough vague similarity is proposed and its relative properties are investigated. In Section 4, the multi-granularity rough approximations of a vague concept are discussed in the view of similarity. In Section 5, a case study for relevant illustrations and related experiments are presented. In Section 6, the conclusions are drawn.

2 Preliminaries

To facilitate the description of this paper, many basic concepts are reviewed briefly in this section.

Let S = (U, C ∪ D) be a decision system, where U = {x₁, x₂, …, x_N} is the universe of discourse, C is the set of condition attributes, and D is the decision attribute. If R ⊆ C, U/R = {[x] _R} = {[x] ₁, [x] ₂, …, [x] _l} denotes a partition of U induced by U/R, where [x] _R denotes the equivalence class of R. U/R is also called a knowledge space or approximation space.

Definition 1. (Rough Sets) [26] Given a decision system S = (U, C ∪ D), R ⊆ C and X ⊆ U, the lower and upper approximation sets of X are defined as follows: $\begin{matrix} \underline{R} (X) = {x \in U | [x]_{R} \subseteq X}, \\ \bar{R} (X) = {x \in U | [x]_{R} \cap X \neq φ}, \end{matrix}$ where [x] _R denotes the equivalence class induced by U/R, namely, U/R = {[x] _R} = {[x] ₁, [x] ₂, ⋯, [x] _m}.

If $\bar{R} (X) = \underline{R} (X)$ , X is a defined set, otherwise X is a rough set. The universe U is divided by positive region $PO S_{R} (X) = \underline{R} (X)$ , boundary region $BN D_{R} (X) = \bar{R} (X) - \underline{R} (X)$ and negative region $NE G_{R} (X) = U - \bar{R} (X)$ .

Definition 2. (Vague Sets) [11] Let U = {x₁, x₂, …, x_N} be the universe of discourse, a vague set V on U is characterized by a truth membership function t_V and a false membership function f_V given by $\begin{matrix} t_{V} : U \to [0, 1], \\ f_{V} : U \to [0, 1], \end{matrix}$ where, t_V (x) is a lower bound on the grade of membership of x derived from the evidence for x, and f_V (x) is a lower bound on the negation of x derived from the evidence against x, t_V (x) + f_V (x) ≤1.

Definition 3. (Operational Rules) [2] Let V₁ and V₂ be two vague sets on U, then

(1) Equality: V₁ = V₂ iff ∀x ∈ U, t_{V
₁} (x) = t_{V
₂} (x) and f_{V
₁} (x) = f_{V
₂} (x);

(2) Containment: V₁ ⊆ V₂ iff ∀x ∈ U, t_{V
₁} (x) ≤ t_{V
₂} (x) and f_{V
₁} (x) ≥ f_{V
₂} (x);

(3) Intersection: V₃ = V₁ ∩ V₂ iff ∀x ∈ U, t_{V
₃} (x) = min {t_{V
₁} (x) , t_{V
₂} (x)}, f_{V
₃} (x) = max {f_{V
₁} (x) , f_{V
₂} (x)};

(4) Union: V₃ = V₁ ∪ V₂ iff ∀x ∈ U, t_{V
₃} (x) = max {t_{V
₁} (x) , t_{V
₂} (x)}, f_{V
₃} (x) = min {f_{V
₁} (x) , f_{V
₂} (x)};

(5) Complement: $V_{1}^{C} = (V_{1})^{C}$ iff ∀x ∈ U, $t_{V_{1}^{C}} (x) = f_{V_{1}} (x)$ , $f_{V_{1}^{C}} (x) = t_{V_{1}} (x)$ .

Definition 4. (Average-step-vague Set) [50] Given a decision system S = (U, C ∪ D), R ⊆ C and V is a vague set on U, U/R = {[x] _R} ={[x] ₁, [x] ₂, …, [x] _l}, where ∀x ∈ [x] _i, i = 1, 2, …, l, we have $\begin{matrix} \bar{V_{R}} (x) = [t_{{\bar{V}}_{R}} (x), 1 - f_{{\bar{V}}_{R}} (x)], \end{matrix}$ where, $t_{{\bar{V}}_{R}} (x) = \frac{\sum_{x \subseteq {[x]}_{i}} t_{V} (x)}{| {[x]}_{i} |}$ and $f_{{\bar{V}}_{R}} (x) = \frac{\sum_{x \subseteq {[x]}_{i}} f_{V} (x)}{| {[x]}_{i} |}$ . $\bar{V_{R}} (x)$ is called average-step-vague set on U/R.

Definition 5. (Intuitionistic Fuzziness) [51] Suppose that [t_V (x) , 1 - f_V (x)] is a vague value of x (x ∈ U), the intuitionistic fuzziness of x is defined as follows, $H_{V} (x) = 2 \int_{t_{V} (x)}^{1 - f_{V} (x)} [μ ln μ + (1 - μ) ln (1 - μ)] d μ$ (1)

Where, μ denotes the membership degree of x belongs to the vague concept V.

When U is continuous, the intuitionistic fuzziness of V can be denoted by $UN C_{V} (x) = \int_{a}^{b} H_{V} (x)$ .

When U is discrete, the intuitionistic fuzziness of V can be denoted by UNC_V (x) = ∑_x∈UH_V (x).

To measure the uncertainty of a rough approximation spaces U/R for describing a vague concept V, literature [51] further proposed the average intuitionistic fuzziness as follows: $\begin{matrix} {\bar{H}}_{V_{R}} (x) = 2 \int_{t_{{\bar{V}}_{R}} (x)}^{1 - f_{{\bar{V}}_{R}} (x)} [μ ln μ + (1 - μ) ln (1 - μ)] d μ \end{matrix}$ (2)

Where, μ denotes the membership degree of x belongs to the vague concept V.

When U is continuous, the average intuitionistic fuzziness of V can be denoted by $\begin{matrix} {\bar{UNC}}_{V_{R}} (x) = \int_{a}^{b} {\bar{H}}_{V_{R}} (x), \end{matrix}$ (3)

When U is discrete, the average intuitionistic fuzziness of V can be denoted by $\begin{matrix} {\bar{UNC}}_{V_{R}} (x) = \sum_{i = 1}^{| U |} {\bar{H}}_{V_{R}} (x), \end{matrix}$ (4)

Example 1. Given a decision system S = (U, C ∪ D), R ⊆ C. $V = \frac{[0.1, 0.3]}{x_{1}} + \frac{[0.3, 0.4]}{x_{2}} + \frac{[0.5, 0.5]}{x_{3}} + \frac{[0.6, 0.7]}{x_{4}} + \frac{[0.8, 0.9]}{x_{5}}$ is a vague set on U, and U/R = {{x₁, x₂, x₃} , {x₄, x₅}} .

According to Definition 4 and formula (2), we have $\begin{matrix} \bar{V_{R}} (x_{1}) = \bar{V_{R}} (x_{2}) = \bar{V_{R}} (x_{3}) \\ = [\frac{0.1 + 0.3 + 0.5}{3}, \frac{0.3 + 0.4 + 0.5}{3}] = [0.3, 0.4], \\ \bar{V_{R}} (x_{4}) = \bar{V_{R}} (x_{5}) \\ = [\frac{0.6 + 0.8}{2}, \frac{0.7 + 0.9}{2}] = [0.7, 0.8] . \end{matrix}$ Then, $\begin{matrix} \bar{V_{R}} = \frac{[0.3, 0.4]}{x_{1}} + \frac{[0.3, 0.4]}{x_{2}} + \frac{[0.3, 0.4]}{x_{3}} \\ + \frac{[0.7, 0.8]}{x_{4}} + \frac{[0.7, 0.8]}{x_{5}}, \\ {\bar{H}}_{V_{R}} (x_{1}) = {\bar{H}}_{V_{R}} (x_{2}) = {\bar{H}}_{V_{R}} (x_{3}) \\ = 2 \int_{0.3}^{0.4} [μ_{\bar{V}} (x) ln μ_{\bar{V}} (x) + (1 - μ_{\bar{V}} (x)) \\ ln (1 - μ_{\bar{V}} (x))] d μ_{\bar{V}} (x) = 0.129 . \end{matrix}$ Similarly, $\begin{matrix} {\bar{H}}_{V_{R}} (x_{4}) = {\bar{H}}_{V_{R}} (x_{5}) = 0.112 . \end{matrix}$ Therefore, $\begin{matrix} {\bar{UNC}}_{V_{R}} (x) = 0.129 * 3 + 0.112 * 2 = 0.611 . \end{matrix}$

3 Rough vague similarity

Since Zadeh proposed fuzzy sets, many researchers pay attentions to similarity measures between fuzzy sets from different viewpoints. As a generalization of fuzzy sets, large of similarity measures [4 , 27] are proposed for vague sets, which has been applied in various areas, including data mining [39], approximate reasoning [4], pattern recognition [6], et al. However, when a vague concept is described by different rough approximation spaces respectively, the existed similarity measures failed to reflect the similarity among these approximation spaces. Moreover, different rough approximation spaces of a vague concept may possess the same measure result, and the differences among them for characterizing a vague concept cannot be reflected. To solve the above problems, in this section, we proposed a similarity measure for rough approximations of vague concepts based on the idea of Earth Mover’s Distance [32, 33].

Suppose that U is the universe of discourse, V is a vague set on U and A is a finite set on U. Then, VIG_A is a vague set formed by A to characterize V, and VIG_A is called a vague information granule (VIG).

(continued) Example 1. Suppose that A = {x₁, x₂, x₃}, B = {x₄, x₅}, then $\begin{matrix} VI G_{A} = \frac{[0.1, 0.3]}{x_{1}} + \frac{[0.3, 0.4]}{x_{2}} + \frac{[0.5, 0.5]}{x_{3}}, \\ VI G_{B} = \frac{[0.6, 0.7]}{x_{4}} + \frac{[0.8, 0.9]}{x_{5}} . \end{matrix}$

Definition 6. (Similarity Measure) Let U be the universe of discourse, A, B, C be three finite sets, ∀A, B, C ⊆ U. A function s (· , ·) is a similarity measure if it satisfies the following conditions:

(1) Positive: s (A, B) ≥0;

(2) Symmetric: s (A, B) = s (B, A);

(3) Triangle inequality: s (A, B) + s (B, C) ≤1 + s (A, C).

Definition 7. [18] Let U be a nonempty finite set, A, B, C be three finite sets, and ∀A, B, C ⊆ U . A function d (· , ·) is a distance measure if it satisfies the following conditions:

(1) Positive: d (A, B) ≥0;

(2) Symmetric: d (A, B) = d (B, A);

(3) Triangle inequality: d (A, B) + d (B, C) ≥ d (A, C).

Definition 8. Let U be the universe of discourse, V be a vague set on U, A and B be two finite sets on U. If VIG_A, VIG_B are two vague information granules on U, then the similarity between VIG_A and VIG_B is defined as $\begin{matrix} s_{VIG} (A, B) \\ = 1 - \frac{\sum_{x \in U} H_{VI G_{A} \cup VI G_{B}} (x) - \sum_{x \in U} H_{VI G_{A} \cap VI G_{B}} (x)}{UN C_{V} (x)} \end{matrix}$ (5) where, $UN C_{V} (x) = \frac{1}{| U |} \sum_{x \in U} H_{V} (x)$ .

Theorem 1. Let U be the universe of discourse, then s_VIG (· , ·) is a similarity measure on U.

Proof. Suppose that V is a vague set on U, A, B and C are finite sets on U. VIG_A, VIG_B and VIG_C are vague information granules formed by A, B and C. Obviously, s_VIG (· , ·) is positive and symmetric. We have the following assumption a = ∑_x∈UH_{VIG_A∪VIG_B} (x) - ∑_x∈UH_{VIG_A∩VIG_B} (x), b = ∑_x∈UH_{VIG_B∪VIG_C} (x) -∑_x∈UH_{VIG_B∩VIG_C} (x), and c = ∑_x∈UH_{VIG_A∪VIG_C} (x) - ∑_x∈UH_{VIG_A∩VIG_C} (x). Because (A ∪ B - A ∩ B) + (B ∪ C - B ∩ C) ≥ A ∪ C - A ∩ C, then a + b ≥ c.

Obviously, $\begin{matrix} \begin{matrix} \frac{a}{UN C_{V} (x)} + \frac{b}{UN C_{V} (x)} \geq \frac{c}{UN C_{V} (x)}, \end{matrix} \end{matrix}$ we have $\begin{matrix} (1 - \frac{a}{UN C_{V} (x)}) + (1 - \frac{b}{UN C_{V} (x)}) \\ \geq 1 + (1 - \frac{c}{UN C_{V} (x)}) . \end{matrix}$ Therefore, s_VIG (A, B) + s_VIG (B, C) ≤1 + s_VIG (A, C).□

According to Definition 6, s_VIG (· , ·) is a similarity measure between VIGs. We denote d_VIG (· , ·) =1 - s_VIG (· , ·). From Theorem 1, we have the following Lemma:

Lemma 1. Let U be the universe of discourse, then d (· , ·) is a distance measure on U.

Example 2. Given a decision system S = (U, C ∪ D) and $V = \frac{[0.2, 0.4]}{x_{1}} + \frac{[0.3, 0.5]}{x_{2}} + \frac{[0.4, 0.7]}{x_{3}} + \frac{[0.6, 0.8]}{x_{4}} + \frac{[0.7, 0.9]}{x_{5}} + \frac{[0.8, 0.9]}{x_{6}}$ is a vague set on U. A = {x₁, x₂, x₃, x₄}, B = {x₃, x₄, x₅, x₆} are two finite sets on U. We have $\begin{matrix} VI G_{A} = \frac{[0.2, 0.4]}{x_{1}} + \frac{[0.3, 0.5]}{x_{2}} + \frac{[0.4, 0.7]}{x_{3}} \\ + \frac{[0.6, 0.8]}{x_{4}}, \\ VI G_{B} = \frac{[0.4, 0.7]}{x_{3}} + \frac{[0.6, 0.8]}{x_{4}} + \frac{[0.7, 0.9]}{x_{5}} \\ + \frac{[0.9, 1]}{x_{6}} . \end{matrix}$ Then,

∑_x∈UH_{VIG_A∪VIG_B} (x) = 1.432,∑_x∈UH_{VIG_A∩VIG_B} (x) = 0.645,UNC_V (x) =1.432.

According to Definition 8, we have

$\begin{matrix} s_{VIG} (A, B) \\ = 1 - \frac{\sum_{x \in U} H_{VI G_{A} \cup VI G_{B}} (x) - \sum_{x \in U} H_{VI G_{A} \cap VI G_{B}} (x)}{UN C_{V} (x)} \\ = 1 - \frac{1.432 - 0.645}{1.432} \\ = 0.45 \end{matrix}$

As well known, the Earth Mover’s Distance (EMD) [32, 33] is proposed by Rubner to calculate the differences between probability distributions in image retrieval. This prevents quantization errors by minimizing the cost of transformation. Moreover, the EMD achieves a many-to-many matching computation, which is in line with human cognition and applied in various fields, i.e., computer vision [24, 53], image processing [10, 34].

Definition 9. (Earth Mover’s Distance) [32] Suppose that P and Q are two partitions, where P = {(p₁, w_{p
₁}) , (p₂, w_{p
₂}) , …, (p_l, w_{p
_l})} and Q = {(q₁, w_{q
₁}) , (q₂, w_{q
₂}) , …, (q_m, w_{q
_m})}. p_i, q_j denote the granules of P and Q and w_{p
_i}, w_{q
_j} denote the weight of p_i and q_j, respectively. The EMD is defined as follows: $\begin{matrix} EMD (P, Q) = \frac{\sum_{i = 1}^{l} \sum_{j = 1}^{m} d_{ij} flo w_{ij}}{\sum_{i = 1}^{l} \sum_{j = 1}^{m} flo w_{ij}} \end{matrix}$ (6) $\begin{matrix} s . t . \\ flo w_{ij} \geq 0; 1 \leq i \leq l, 1 \leq j \leq m, \\ \sum_{j = 1}^{m} flo w_{ij} \leq w_{p_{i}}; 1 \leq i \leq l, \\ \sum_{i = 1}^{l} flo w_{ij} \leq w_{q_{j}}; 1 \leq j \leq m, \\ \sum_{i = 1}^{l} \sum_{j = 1}^{m} flo w_{ij} = min {\sum_{i = 1}^{l} w_{p_{i}}, \sum_{j = 1}^{m} w_{q_{j}}} . \end{matrix}$

In formula (6), d_ij is a distance measure between p_i and q_j. flow_ij is the flow from p_i to q_j. The purpose of the EMD is to search a matrix F = [flow_ij] to minimize the transformation cost.

Theorem 2. [32] The EMD is a distance measure, if it satisfies the following conditions:

(1) d (· , ·) is a distance measure;

(2) $\sum_{i = 1}^{l} w_{p_{i}} = \sum_{j = 1}^{m} w_{q_{j}}$ .

Based on the idea of EMD, a similarity measure for rough approximations of vague sets is presented, which is called rough vague similarity (RVS).

Definition 10. Given a decision system S = (U, C ∪ D),R₁, R₂ ⊆ C and V is a vague set on V. U/R₁ = {p₁, p₂, …, p_l} and U/R₂ = {q₁, q₂, …, q_m} are two rough approximation spaces on U. Then, the rough vague similarity between U/R₁ and U/R₂ for characterizing V is defined as follows: $\begin{matrix} RVS (U / R_{1}, U / R_{2}) = \frac{1}{| U |} \sum_{i = 1}^{l} \sum_{j = 1}^{m} s_{VI G_{ij}} f_{ij} \end{matrix}$ (7) where, s_{VIG_ij} = s_VIG(p_i, q_j) and f_ij = |p_i ∩ q_j| denotes the number of intersection between p_i and q_j. From the perspective of physical meaning, f_ij represents the flow from p_i to q_j or from q_j to p_i.

In formula (7), s_VIG (· , ·) can be any similarity measure, and it is chosen according to the special problem. Inheriting the merits of the EMD, RVS is more robust to achieves a many-to-many matching calculation, and is more in line with human cognition. Definition 10 is further explained by the following example.

Example 3. Given a decision system S = (U, C∪D), R₁, R₂ ⊆ C and $V = \frac{[0.1, 0.3]}{x_{1}} + \frac{[0.2, 0.4]}{x_{2}} + \frac{[0.5, 0.5]}{x_{3}} + \frac{[0.6, 0.7]}{x_{4}} + \frac{[0.8, 0.9]}{x_{5}}$ is a vague set on U. U/R₁ = {{x₁, x₂, x₃} , {x₄, x₅}},U/R₂ = {{x₁, x₄} , {x₂, x₅} , {x₃}}.

According to Definition 10, we have $\begin{matrix} RVS (U / R_{1}, U / R_{2}) = \frac{1}{| U |} \sum_{i = 1}^{2} \sum_{j = 1}^{3} s_{VI G_{ij}} f_{ij} = 0.466 \end{matrix}$

Moreover, Example 3 can be further described intuitively by Fig. 1. w_{p
_i} (i = 1, 2) denotes the weight of equivalent classes p_i, which is the number of elements in equivalent classes p_i. Similarly, w_{q
_j} (j = 1, 2, 3) denotes the weight of equivalent classes q_j, which is the number of elements in equivalent classes q_j. Then, f_ij represents the flow of elements from p_i to q_j or q_j to p_i.

Fig. 1

Schematic diagram of the RVS explained by Example 3.

Suppose that U/R₁ = {p₁, p₂, …, p_l} is a rough approximation spaces on U, where p_i = {x_i1, x_i2, …, x_{i|p_i|}}. Then p_i = g_{R
₁} (x_i1) = g_{R
₁} (x_i2) = ⋯ = g_{R
₁} (x_{i|p_i|}). The mechanism provides a uniform representation of granular spaces in a knowledge base. For example, $\begin{matrix} {U / R}_{1} = {{x_{1}, x_{2}, x_{3}}, {x_{4}, x_{5}}} \\ = {g_{R_{1}} (x_{1}), g_{R_{1}} (x_{2}), g_{R_{1}} (x_{3}), g_{R_{1}} (x_{4}), g_{R_{1}} (x_{5})} \\ = {{x_{1}, x_{2}, x_{3}}, {x_{1}, x_{2}, x_{3}}, {x_{1}, x_{2}, x_{3}}, {x_{4}, x_{5}}, \\ {x_{4}, x_{5}}} \end{matrix}$

Theorem 3. Let U be the universe of discourse, then RVS (· , ·) is a similarity measure on U.

Proof. Suppose that decision system S = (U, C ∪ D), R₁, R₂ ⊆ C and V is a vague set on U. Let U = {x₁, x₂, . . . , x_N} be a non-empty finite universe, U/R₁ = {p₁, p₂, . . . , p_l}, U/R₂ = {q₁, q₂, . . . , q_m} and U/R₃ = {r₁, r₂, . . . , r_n}. Obviously, it is positive and symmetric. We have

$\begin{matrix} RVS (U / R_{1}, U / R_{2}) + RVS (U / R_{2}, U / R_{3}) \\ = \frac{1}{| U |} \sum_{i = 1}^{l} \sum_{j = 1}^{m} s_{ij} f_{ij} + \frac{1}{| U |} \sum_{j = 1}^{m} \sum_{k = 1}^{n} s_{jk} f_{jk} = \frac{1}{| U |} \sum_{x_{i} \in U} \\ (s (g_{R_{1}} (x_{i}), g_{R_{2}} (x_{i})) + s (g_{R_{2}} (x_{i}), g_{R_{3}} (x_{i}))) \end{matrix}$

Similarly,

$RVS (U / R_{1}, U / R_{3}) = \frac{1}{| U |} \sum_{i = 1}^{l} \sum_{k = 1}^{n} s_{ik} f_{ik} = \frac{1}{| U |} \sum_{x_{i} \in U} s (g_{R_{1}} (x_{i}), g_{R_{3}} (x_{i}))$ .

According to Theorem 1, we have

s (g_{R
₁} (x_i) , g_{R
₂} (x_i)) + s (g_{R
₂} (x_i) , g_{R
₃} (x_i))

≤1 + s (g_{R
₁} (x_i) , g_{R
₃} (x_i)) then,

$\frac{1}{| U |} \sum_{x_{i} \in U} (s (g_{R_{1}} (x_{i}), g_{R_{2}} (x_{i})) + s (g_{R_{2}} (x_{i}), g_{R_{3}} (x_{i})))$ $\leq 1 + \frac{1}{| U |} \sum_{x_{i} \in U} s (g_{R_{1}} (x_{i}), g_{R_{3}} (x_{i}))$ Therefore,

RVS (U/R₁, U/R₂)+ RVS (U/R₂, U/R₃) ≤1 + RVS (U/R₁, U/R₃). According to Definition 6, RVS (· , ·) is a similarity between rough approximation spaces for charactering a vague concept.□

4 Rough approximations of vague sets in multi-granularity spaces in the view of RVS

From the perspective of granular computing, with the granularity being finer gradually, the information granules in approximation space will be subdivided into finer granules. As a result, the hierarchical multi-granularity approximation spaces are formed, which means that a vague concept can be described by multilevel rough approximation spaces in a decision system. In this section, the changing rules will be discussed in multi-granularity rough approximation spaces for charactering a vague concept.

Lemma 2. Let A,B and C be finite sets on U, and A ⊆ B ⊆ C. VIG_A,VIG_B and VIG_C are three vague information granules formed by A,B and C, then s_VIG (A, B) > s_VIG (A, C).

Proof. According to formula (5), we have $\begin{matrix} s_{VIG} (A, B) \\ = 1 - \frac{\sum_{x \in U} H_{VI G_{A} \cup VI G_{B}} (x) - \sum_{x \in U} H_{VI G_{A} \cap VI G_{B}} (x)}{UN C_{V} (x)} \end{matrix}$ $\begin{matrix} s_{VIG} (A, C) \\ = 1 - \frac{\sum_{x \in U} H_{VI G_{A} \cup VI G_{C}} (x) - \sum_{x \in U} H_{VI G_{A} \cap VI G_{C}} (x)}{UN C_{V} (x)} \end{matrix}$ $\begin{matrix} s_{VIG} (A, B) - s_{VIG} (A, C) \\ = \frac{\sum_{x \in U} (H_{VI G_{B}} (x) - H_{VI G_{C}} (x))}{UN C_{V} (x)} \geq 0 . \end{matrix}$ Therefore, s_VIG (A, B) ≥ s_VIG (A, C).□

Theorem 4. Given a decision system S = (U, C ∪ D), R₁, R₂, R₃ ⊆ C and V be a vague set on U. If R₁ ⊆ R₂ ⊆ R₃, then RVS (U/R₁, U/R₂) ≥ RVS (U/R₁, U/R₃) holds.

Proof. Let U = {x₁, x₂, . . . , x_N} be a non-empty finite universe, U/R₁ = {p₁, p₂, . . . , p_l}, U/R₂ = {q₁, q₂, . . . , q_m} and U/R₃ = {r₁, r₂, . . . , r_n}. Because R₁ ⊆ R₂ ⊆ R₃, so $U / R_{3} \underline{≺} U / R_{2} \underline{≺} U / R_{1}$ . According to the conditions, for simplicity, supposing only one granule p₁ can be subdivided into two finer sub-granules q₁ and q₂ by ΔR = R₂ - R₁ and only one granule q₁ can be subdivided into two finer sub-granules r₁ and r₂ by ΔR = R₃ - R₂ (the more complicated cases can be transformed into this case, so we will not repeat them here).

Without loss of generality, let p₁ = q₁ ∪ q₂, p₂ = q₃, p₃ = q₄,...,p_l = q_m (m = l + 1), q₁ = r₁∪r₂, q₂ = r₃, q₃ = r₄,⋯,q_m = r_n (n = m + 1), namely, U/R₂ = {q₁, q₂, p₂, p₃, . . . , p_l}, U/R₃ ={r₁, r₂, q₂, q₃, . . . , q_m}. According to formula (7), we have $\begin{matrix} RVS (U / R_{1}, U / R_{2}) = \frac{1}{| U |} \sum_{i = 1}^{l} \sum_{j = 1}^{m} s_{ij} f_{ij} = \frac{1}{| U |} \\ (s (p_{1}, q_{1}) | q_{1} | + s (p_{1}, q_{2}) | q_{2} | + | U | - | q_{1} | - | q_{2} |) \end{matrix}$ Similarly, $\begin{matrix} RVS (U / R_{1}, U / R_{3}) \\ = \frac{1}{| U |} (s (p_{1}, r_{1}) | r_{1} | + s (p_{1}, r_{2}) | r_{2} | + s (p_{1}, r_{3}) | r_{3} | \\ + | U | - | r_{1} | - | r_{2} | - | r_{3} |) . \end{matrix}$ Because q₁ = r₁ ∪ r₂, q₂ = r₃, we have $\begin{matrix} RVS (U / R_{1}, U / R_{3}) - RVS (U / R_{1}, U / R_{2}) \\ = \frac{1}{| U |} [(s_{VIG} (p_{1}, q_{1}) - s_{VIG} (p_{1}, r_{1})) | r_{1} | \\ + (s_{VIG} (p_{1}, q_{1}) - s_{VIG} (p_{1}, r_{2}) | r_{2} |)] . \end{matrix}$

According to Lemma 2, because s_VIG (p₁, q₁)>s_VIG (p₁, r₁) and s_VIG (p₁, q₂) > s_VIG (p₁, r₂), we have RVS (U/R₁, U/R₂) - RVS (U/R₁, U/R₃) ≥0, namely, RVS (U/R₁, U/R₂) ≥ RVS (U/R₁, U/R₃). Similarly, RVS (U/R₂, U/R₃) ≥ RVS(U/R₁, U/R₃) holds.□

Example 4. Given a decision system S = (U, C ∪ D), R₁, R₂, R₃ ⊆ C and $V = \frac{[0.1, 0.3]}{x_{1}} + \frac{[0.2, 0.4]}{x_{2}} + \frac{[0.5, 0.5]}{x_{3}} + \frac{[0.6, 0.7]}{x_{4}} + \frac{[0.8, 0.9]}{x_{5}} + \frac{[0.6, 0.8]}{x_{6}} + \frac{[0.3, 0.4]}{x_{7}}$ is a vague set on U. Suppose that U/R₁ = {{x₁, x₂, x₃} , {x₄, x₅, x₆, x₇}}, U/R₂ ={{x₁, x₂, x₃} , {x₄, x₅} , {x₆, x₇}} , U/R₃ = {{x₁, x₂} , {x₃} , {x₄} , {x₅} , {x₆, x₇}} .

According to conditions, we have $\begin{matrix} RVS (U / R_{1}, U / R_{2}) = \frac{1}{| U |} \sum_{i = 1}^{2} \sum_{j = 1}^{3} {s_{VIG}}_{ij} f_{ij} = 0.837 . \end{matrix}$ Similarly, $\begin{matrix} RVS (U / R_{2}, U / R_{3}) = \frac{1}{| U |} \sum_{j = 1}^{3} \sum_{k = 1}^{5} {s_{VIG}}_{jk} f_{jk} = 0.909, \\ RVS (U / R_{1}, U / R_{3}) = \frac{1}{| U |} \sum_{i = 1}^{2} \sum_{k = 1}^{5} {s_{VIG}}_{ik} f_{ik} = 0.746 . \end{matrix}$ Therefore, $\begin{matrix} RVS (U / R_{1}, U / R_{3}) \leq RVS (U / R_{1}, U / R_{2}), \\ RVS (U / R_{1}, U / R_{3}) \leq RVS (U / R_{2}, U / R_{3}) . \end{matrix}$

In a decision system, there are two special cases: the finest approximation space and the coarsest approximation space re denoted by ω and δ respectively. According to Theorem 5, the following corollaries hold:

Corollary 1. Given a decision system S = (U, C ∪ D), R₁, R₂ ⊆ C and V be a vague set on U. If R₁ ⊆ R₂, then RVS (U/R₁, ω) ≤ RVS (U/R₂, ω) holds.

Corollary 2. Given a decision system S = (U, C ∪ D), R₁, R₂ ⊆ C and V be a vague set on U. If R₁ ⊆ R₂, then RVS (U/R₁, δ) ≥ RVS (U/R₂, δ) holds.

Lemma 3. Let A, B and C be finite sets on U, VIG_A,VIG_A and VIG_A are vague information granules formed by A, B and C. If C ⊆ B ⊆ A then 1 + s (A, C) = s (A, B) + s (B, C).

Theorem 5. Given a decision system S = (U, C ∪ D), R₁, R₂, R₃ ⊆ C and V be a vague set on U. If R₁ ⊆ R₂ ⊆ R₃, then 1 + RVS (U/R₁, U/R₃) = RVS (U/R₁, U/R₂) + RVS (U/R₂, U/R₃) holds.

Proof. Based on the proof of Theorem 4, we have $\begin{matrix} RVS (U / R_{1}, U / R_{2}) \\ = \frac{1}{| U |} (s (p_{1}, q_{1}) | q_{1} | + s (p_{1}, q_{2}) | q_{2} | + | U | - | p_{1} |) \end{matrix}$ $\begin{matrix} RVS (U / R_{1}, U / R_{3}) \\ = \frac{1}{| U |} (s (p_{1}, r_{1}) | r_{1} | + s (p_{1}, r_{2}) | r_{2} | + s (p_{1}, r_{3}) | r_{3} | \\ + | U | - | p_{1} |) \end{matrix}$ Similarly, $\begin{matrix} RVS (U / R_{2}, U / R_{3}) \\ = \frac{1}{| U |} (s (q_{1}, r_{1}) | r_{1} | + s (q_{1}, r_{2}) | r_{2} | + | U | - | q_{1} |) \end{matrix}$ Because p₁ = q₁ ∪ q₂ and q₁ = r₁ ∪ r₂, according to Lemma 3, we have $\begin{matrix} RVS (U / R_{1}, U / R_{2}) + RVS (U / R_{2}, U / R_{3}) \\ - RVS (U / R_{1}, U / R_{3}) \\ = \frac{1}{| U |} (| q_{1} | + | U | - | q_{1} |) \\ = 1 \end{matrix}$ Therefore, $\begin{matrix} 1 + RVS (U / R_{1}, U / R_{3}) \\ = RVS (U / R_{1}, U / R_{2}) + RVS (U / R_{2}, U / R_{3}) . \end{matrix}$ □

(continued) Example 4. We have $\begin{matrix} 1 + RVS (U / R_{1}, U / R_{3}) \\ = RVS (U / R_{1}, U / R_{2}) + RVS (U / R_{2}, U / R_{3}) \\ = \frac{1}{| U |} \sum_{i = 1}^{2} \sum_{j = 1}^{3} {s_{VIG}}_{ij} f_{ij} + \frac{1}{| U |} \sum_{j = 1}^{3} \sum_{k = 1}^{5} {s_{VIG}}_{jk} f_{jk} \\ = 0.837 + 0.909 = 1.746 \end{matrix}$ According to Theorem 5, the following corollaries hold:

Corollary 3. Given a decision system S = (U, C ∪ D), R₁, R₂ ⊆ C and V be a vague set on U. If R₁ ⊆ R₂, then RVS (U/R₁, U/R₂) =1 + RVS (U/R₁, ω) - RVS (U/R₂, ω) holds.

Corollary 4. Given a decision system S = (U, C ∪ D), R₁, R₂ ⊆ C and V be a vague set on U. If R₁ ⊆ R₂, then RVS (U/R₁, U/R₂) =1 + RVS (U/R₂, δ) - RVS (U/R₁, δ) holds.

Definition 11. (Information measure) [22, 43] Given a decision system S = (U, C ∪ D),R₁, R₂ ⊆ C and I be a mapping from the power set of C to the set of real numbers. If I satisfies the following conditions:

(1) I (U/R₁) ≥0, for any R₁ ⊆ C;

(2) U/R₁ ≺ U/R₂ ⇒ I (U/R₁) > I (U/R₂), for any R₁, R₂ ⊆ C;

(3) U/R₁ = U/R₂ ⇒ I (U/R₁) = I (U/R₂), for any R₁, R₂ ⊆ C.

Then I is an information measure.

Theorem 6. Given a decision system S = (U, C ∪ D), R₁ ⊆ C and V be a vague set on U. RVS (U/R₁, ω) is an information measure.

Proof.

(1) From Theorem 3, RVS (U/R₁, ω) ≥0.

(2) Assume that R₂ ⊆ C, when U/R₁ = U/R₂, obviously, RVS (U/R₁, ω) = RVS (U/R₂, ω).

(3) According to Corollary 1, if R₁ ⊆ R₂, RVS (U/R₁, ω) ≤ RVS (U/R₂, ω).□

According to Corollary 3 and Theorem 6, for a vague concept in hierarchical multi-granularity approximation spaces, the larger the information difference between rough approximation spaces is, the larger the rough vague similarity between them is.

Definition 12. (Granularity measure) [40, 41] Given a decision system S = (U, C ∪ D),R₁, R₂ ⊆ C and G be a mapping from the power set of C to the set of real numbers. If G satisfies the following conditions:

(1) G (U/R₁) ≥0, for any R₁ ⊆ C;

(2) U/R₁ ≺ U/R₂ ⇒ G (U/R₁) > G (U/R₂), for any R₁, R₂ ⊆ C;

(3) U/R₁ = U/R₂ ⇒ G (U/R₁) = G (U/R₂), for any R₁, R₂ ⊆ C.

Then G is a granularity measure.

Theorem 7. Given a decision system S = (U, C ∪ D), R₁ ⊆ C and V be a vague set on U. RVS (U/R₁, δ) is a granularity measure.

Proof.

(1) From Theorem 3, RVS (U/R₁, δ) ≥0.

(2) Assume that R₂ ⊆ C, when U/R₁ = U/R₂, obviously, RVS (U/R₁, δ) = RVS (U/R₂, δ).

(3) According to Corollary 2, if R₁ ⊆ R₂, RVS (U/R₁, δ) ≥ RVS (U/R₂, δ).

Therefore, RVS (U/R₁, δ) is a granularity measure.□

According to Corollary 4 and Theorem 7, for a vague concept in hierarchical multi-granularity approximation spaces, the larger the granularity difference between rough approximation spaces is, the larger the rough vague similarity between them is.

5 Applicable analysis

As mentioned above in Section 1, the rough vague similarity has the following potential applications: (1) In rough set theory, the rough vague similarity is helpful for effectively choosing the suitable rough approximation spaces for describing a vague concept more accurately.(2) The rough vague similarity can be used to construct a heuristic function in attribute reduction. In this section, we only discuss the former, that is, the effectiveness of the proposed rough vague similarity in the view of granular computing.

5.1 Case study

As well known, various knowledge spaces are formed by different attribute sets. This means that a vague concept can be described by multi-granularity rough approximation spaces in a decision system. There always exists an optimal knowledge space by attribute reduction [15–17 , 37] from the perspective of granular computing. For charactering a vague concept, although it is difficult to acquire the optimal attribute sets in some case, we can search the attribute sets that possess the similar approximation ability of optimal attribute sets by measuring the matching degree between optimal knowledge space and other knowledge spaces. The following example is presented to further illustrate this idea.

Example 5. Table 1 is a decision system for liver cancer diagnosis.Suppose that C = {A₁, A₂, A₃, A₄, A₅} represents all the detection items for liver cancer and there are two hospitals Ω₁ and Ω₂ that can check for liver cancer. For economic reasons, the hospital Ω₁ has only the detection items {A₁, A₂, A₃} and the hospital Ω₂ has only the detection items {A₂, A₄, A₅}. Unfortunately, owing to factor constraints, such as time, economic and environment, Ω₁ and Ω₂ may only possess a certain amount of detection items. Then, which hospital should patients choose for liver cancer diagnosis? Assume that C = {A₁, A₂, A₃, A₄, A₅}, R₁ = {A₁, A₂, A₃} and R₂ = {A₂, A₄, A₅}, according to Table 1, we have $V = \frac{[0.2, 0.4]}{x_{1}} + \frac{[0.3, 0.5]}{x_{2}} + \frac{[0.4, 0.6]}{x_{3}} + \frac{[0.5, 0.7]}{x_{4}} + \frac{[0.6, 0.8]}{x_{5}} + \frac{[0.4, 0.6]}{x_{6}} + \frac{[0.3, 0.5]}{x_{7}} + \frac{[0.2, 0.4]}{x_{8}}$ , U/R₁ = {{x₁, x₂, x₃} , {x₄, x₅, x₆} , {x₇, x₈}}, U/R₂ = {{x₁, x₂} , {x₃, x₄, x₅} , {x₆, x₇, x₈}}, U/C = {{x₁, x₂} , {x₃} , {x₄} , {x₅, x₆} , {x₇, x₈}}, then, ${\bar{UNC}}_{V_{R_{1}}} (x) = {\bar{UNC}}_{V_{R_{2}}} (x) = 2.072$ .

Table 1
Table of student score

A ₁ A ₂ A ₃ A ₄ A ₅ V

x ₁ 2 1 2 1 3 [0.2,0.4]

x ₂ 2 1 2 1 3 [0.3,0.5]

x ₃ 2 1 2 3 2 [0.4,0.6]

x ₄ 3 1 1 3 2 [0.5,0.7]

x ₅ 3 1 1 3 2 [0.6,0.8]

x ₆ 3 1 1 2 1 [0.4,0.6]

x ₇ 1 2 3 2 1 [0.3,0.5]

x ₈ 1 2 3 2 1 [0.2,0.4]

	A ₁	A ₂	A ₃	A ₄	A ₅	V
x ₁	2	1	2	1	3	[0.2,0.4]
x ₂	2	1	2	1	3	[0.3,0.5]
x ₃	2	1	2	3	2	[0.4,0.6]
x ₄	3	1	1	3	2	[0.5,0.7]
x ₅	3	1	1	3	2	[0.6,0.8]
x ₆	3	1	1	2	1	[0.4,0.6]
x ₇	1	2	3	2	1	[0.3,0.5]
x ₈	1	2	3	2	1	[0.2,0.4]

Table 2

The descriptions of datasets

ID	Dataset	Attribute characteristics	Instances	Condition attributes
1	Air Quality	Real	9358	10
2	Bank	Integer	39	10
3	Concrete	Real	1030	8
4	ENB2012	Real	768	8

From the perspective of uncertainty, the significance of U/R₁ and U/R₂ for checking liver cancer can not be distinguished. However, according to rough vague similarity, we have $\begin{matrix} RVS (U / R_{1}, U / C) = \frac{1}{| U |} \sum_{i = 1}^{3} \sum_{k = 1}^{5} {s_{VIG}}_{ij} f_{ij} = 0.873, \\ RVS (U / R_{2}, U / C) = \frac{1}{| U |} \sum_{j = 1}^{3} \sum_{k = 1}^{5} {s_{VIG}}_{ij} f_{ij} = 0.811 . \end{matrix}$

Therefore, for describing the target concept V, attribute sets R₁ is more similar than attribute sets R₂ to the optimal attribute sets C. To sum up, although the local hospital does not possess the all detection items for liver cancer for some reasons, we can choose the hospital whose approximate ability is closer to the optimal attribute set by rough vague similarity.

5.2 Experiments and analysis

In this subsection, related experiments are conducted to validate the effectiveness of RVS. The experimental environments are Windows 7, Intel Core (TM) I5-4590 CPU (3.30 GHz) and 8GB RAM. The experimental platform is Matlab 2015b. The experimental datasets include one dataset from Greek industrial development bank [12] and three datasets from UCI Repository of machine learning databases [31] are listed in Table 1.

Supposing that GS = {GL₁, GL₂, GL₃, GL₄, GL₅, GL₆, GL₇}, and it denotes the seven granularity layers with the granularity being finer. In Fig. 2, the different colors represent the RVS of different rough approximation spaces. Figure 2 shows that each dataset behaves similarly, that is, the larger the granularity difference between two rough approximation spaces, the smaller the RVS between them. Conversely, the smaller the granularity difference between two rough approximation spaces, the larger the RVS between them. Figure 3 further shows the details of RVS in Fig. 2.

Fig. 2

The change of RVS among rough approximation spaces with partial order relation.

Fig. 3

The matrix value of RVS in Fig. 2.

In Fig. 4, rough vague similarity and average intuitionistic fuzziness are compared for distingushing approximation spaces coming from the same dataset. The pairs of approximation spaces distingushed by rough vague similarity and average intuitionistic fuzziness, respectively. With loss of generality, average intuitionistic fuzziness is only one representative of the fuzziness measure in this experiment. Based on the idea of literature [29], we have the following formula of the identifiable ratio: $\begin{matrix} IR = \frac{distinguished pairs from each other}{all pairs of rough approximation spaces} \end{matrix}$

Fig. 4

Comparison on rough vague similarity and average intuitionistic fuzziness.

In Fig. 4, the blue bar and the yellow bar represent the identifiable ratio of rough vague similarity and average intuitionistic fuzziness, respectively. Obviously, the identifiable ratio of rough vague similarity is greater than that of average intuitionistic fuzziness. Compared with average intuitionistic fuzziness, the rough vague similarity possesses better performance for reflecting differences among rough approximation spaces when they describe a vague concept at the same. That is, as long as the descriptive ability of two rough approximation spaces for describing a vague concept are not the same as each other, they can be distinguished by rough vague similarity.

6 Conclusions

Multi-granularity thinking of human intelligence is an effective method to establish a new model for data processing. Knowledge similarity can reflect the differences of different granularities in problem observation, which is in line with the information processing mechanism of human brain. In this paper, to reflect the similarity between rough approximation spaces for charactering a vague concept, the rough vague similarity (VGS) with strong distinguishing ability is proposed. Furthermore, a comprehensive analysis for multi-granularity rough approximations of vague sets from the perspective of RVS is presented and the relationship between RVS and uncertainty measure (granularity measure and information measure) is established. Finally, the effectiveness of RVS is verified by a case study and related experiments. These results are useful for knowledge discovery in multi-granularity spaces for describing an uncertain concept and important for establishing a GrC framework in knowledge bases.

Footnotes

Acknowledgment

This work is supported by the National Science Foundation of China (No. 61762089, 61663047, 61863036), Guizhou Provincial Natural Science Foundation (KY[2018]No.318), Innovation and exploration project of Guizhou Province (QKHPTRC [2017]5727-06), High level innovative talents of Guizhou Province (ZKG 2018 [15]).

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Similarity measure for multi-granularity rough approximations of vague sets

Abstract

Keywords

1 Introduction

2 Preliminaries

5 Applicable analysis

5.1 Case study

Table 1 Table of student score A 1 A 2 A 3 A 4 A 5 V x 1 2 1 2 1 3 [0.2,0.4] x 2 2 1 2 1 3 [0.3,0.5] x 3 2 1 2 3 2 [0.4,0.6] x 4 3 1 1 3 2 [0.5,0.7] x 5 3 1 1 3 2 [0.6,0.8] x 6 3 1 1 2 1 [0.4,0.6] x 7 1 2 3 2 1 [0.3,0.5] x 8 1 2 3 2 1 [0.2,0.4]

Footnotes

Acknowledgment

References

Table 1
Table of student score

A ₁ A ₂ A ₃ A ₄ A ₅ V

x ₁ 2 1 2 1 3 [0.2,0.4]

x ₂ 2 1 2 1 3 [0.3,0.5]

x ₃ 2 1 2 3 2 [0.4,0.6]

x ₄ 3 1 1 3 2 [0.5,0.7]

x ₅ 3 1 1 3 2 [0.6,0.8]

x ₆ 3 1 1 2 1 [0.4,0.6]

x ₇ 1 2 3 2 1 [0.3,0.5]

x ₈ 1 2 3 2 1 [0.2,0.4]