Exploring measure of uncertainty via a discernibility relation for partially labeled real-valued data

Abstract

In practical applications of machine learning, only part of data is labeled because the cost of assessing class label is relatively high. Measure of uncertainty is abbreviated as MU. This paper explores MU for partially labeled real-valued data via a discernibility relation. First, a decision information system with partially labeled real-valued data (p-RVDIS) is separated into two decision information systems: one is the decision information system with labeled real-valued data (l-RVDIS) and the other is the decision information system with unlabeled real-valued data (u-RVDIS). Then, based on a discernibility relation, dependence function, conditional information entropy and conditional information amount, four degrees of importance on an attribute subset in a p-RVDIS are defined. They are calculated by taking the weighted sum of l-RVDIS and u-RVDIS based on the missing rate, which can be considered as four MUs for a p-RVDIS. Combining l-RVDIS and u-RVDIS provides a more accurate assessment of the importance and classification ability of attribute subsets in a p-RVDIS. This is precisely the novelty of this paper. Finally, experimental analysis on several datasets verify the effectiveness of these MUs. These findings will contribute to the comprehension of the essence of the uncertainty in a p-RVDIS.

Keywords

Partially labeled real-valued data p-RVDIS Discernibility relation Uncertainty Measure

1 Introduction

1.1 Research background

There are a lot of uncertainties in data. The uncertainty of data is typically caused by the limited resolution and incomplete depiction of the data. This uncertainty is an inherent aspect of the real world and the capturing of it is becoming increasingly widespread. Addressing uncertainty is crucial in artificial intelligence. To a certain extent, the level of artificial intelligence depends on the extent to which uncertain problems are solved. Therefore, measure of uncertainty (MU) emerges as a pivotal subject of study in numerous disciplines, such as environmental conflict analysis [13], face identification [8] and medical decision making [22]. Furthermore, Zhan et al. [38 , 43] studied three-way behavioral decision making with hesitant fuzzy information systems, proposed a three-way decision methodology with regret theory via triangular fuzzy numbers in incomplete multi-scale decision information systems and gave a novel group decision-making approach in multi-scale environments. Wang et al. [35] discussed regret theory-based three-way decision model in hesitant fuzzy environments and its application to medical decision. Zhu et al. [39] proposed a probabilistic linguistic three-way decision method with regret theory via fuzzy c-means clustering algorithm. Atef et al. [2] researched fuzzy topological structures via fuzzy graphs. El-Bably et al. [16, 17] considered medical diagnosis for the problem of Chikungunya disease using soft rough sets and came up with new topological approaches to generalized soft rough approximations with medical applications. El-Gayar et al. [18] investigated economic decision-making using rough topological structures. Abu-Gdairi et al. [1] studied topological visualization and graph analysis of rough sets via neighborhoods.

Rough set theory is a tool to handle uncertainty, and it has been applied to pattern recognition, data mining, image processing, as well as medical diagnosis [30]. Pawlak presented the concept of an information system (IS) based on rough set theory. Many applications of rough set theory, such as uncertain reasoning, feature selection, rule extraction and classification, are implemented in an IS [5]. MU has always been an important issue of rough set theory. It plays a significant role in attribute reduction and rule acquisition. Classification accuracy, rough membership, attribute dependence and attribute importance are the basic MUs in rough set theory.

Information entropy, proposed by Shannon [31], is a significant tool for estimating uncertainty as well. Some researchers utilized information entropy to measure the uncertainty of an IS or rough sets. For instance, D $\ddot{u}$ ntsch et al. [10] utilized Shannon’s entropy to measure decision rules in rough set theory; Beaubouef et al. [4] proposed a method to measure uncertainty in rough set theory; Dai et al. [14] considered entropy measurement in a set-valued IS; Li et al. [28] measured uncertainty in a fuzzy relation IS; Hempelmann et al. [20] presented an information entropy-based technique for evaluating medical decision. Navarrete et al. [29] considered color smoothing for RGB-D data utilizing entropy information. Wan et al. [36] proposed a more efficient semi-supervised feature selection approach that utilizes information entropy. Wang et al. [33] suggested data mining algorithms by using conditional information entropy. Zhang et al. [41] presented MU for a fully fuzzy IS; Li et al. [27] used Gaussian kernel for MU in a fully fuzzy IS; Tan et al. [32] put forward entropy measurement for intuitionistic fuzzy information; Huang et al. [19] proposed discernibility measures for a fuzzy β-covering; Dai et al. [15] investigated MU for an incomplete decision IS. Jo et al. [25] explored the improvement measures of redundancy and correlation of mRMR feature selection. Kadkhodaei et al. [26] proposed a heterogeneous boosting-based ensemble classifier based on entropy measurement. Yang et al. [37] considered MU in a multi-source fuzzy IS in the view of multi-granulation. Delgado et al. [13] gave the entropy weight method-based environmental conflict analysis.

The comparative of this paper with the research results about some above literatures is shown in Table 1.

Table 1
The comparison of this paper with some recent research results

Literature Data type Research tool

[13] Real-valued data Information entropy

[14] Set-valued data Information entropy

[19] Real-valued data Significance degree, Fuzzy β covering

[20] Medical data Information entropy

[25] Real-valued data Information entropy

[26] Keterogeneous data Information entropy

[27] Real-valued data Information entropy, Granular computing

[28] Fuzzy relation Information entropy, Granular computing

[29] RGB-D data Information theory

[32] Intuitionistic fuzzy information Information entropy

[33] Real-valued data data Conditional information entropy

[36] Partially labeled data Information entropy

[37] Multi-source data Granular computing

Our paper Labeled real-valued data Information entropy, Rough set theory

Literature	Data type	Research tool
[13]	Real-valued data	Information entropy
[14]	Set-valued data	Information entropy
[19]	Real-valued data	Significance degree, Fuzzy β covering
[20]	Medical data	Information entropy
[25]	Real-valued data	Information entropy
[26]	Keterogeneous data	Information entropy
[27]	Real-valued data	Information entropy, Granular computing
[28]	Fuzzy relation	Information entropy, Granular computing
[29]	RGB-D data	Information theory
[32]	Intuitionistic fuzzy information	Information entropy
[33]	Real-valued data data	Conditional information entropy
[36]	Partially labeled data	Information entropy
[37]	Multi-source data	Granular computing
Our paper	Labeled real-valued data	Information entropy, Rough set theory

1.2 Motivation and contributions

There are many real-valued data in many real world applications. It requires a considerable amount of human resources to label these data. In practical scenarios, these data typically consist of only a limited number of labeled data. Considering the expense of determining class information, a small portion of real-valued data can be labeled with class information, while the majority remains unlabeled, referred to as unlabeled real-valued data. Due to the limited availability of labeled data, effectively utilizing unlabeled data for attribute reduction has become a prominent issue in the realm of big data. Bao et al. [3] studied partial label dimensionality reduction via confidence-based dependence maximization. Han et al. [24] proposed a semi-supervised attribute reduction algorithm. Campagner et al. [6, 7] introduced rough-set based genetic algorithms for weakly supervised feature selection and presented rough set-based feature selection for weakly labeled data. Dai et al. [12] introduced the concept of distinguish pair and studied partially labeled categorical data by means of distinguish pair. They also provided an importance for each attribute subset based on distinguished pairs and presented an attribute reduction method utilizing this importance. However, the provided importance of partially labeled categorical data did not consider the missing rate of labels, and they only considered a single importance.

This paper investigates MU for partially labeled real-valued data based on a discernibility relation. The major contributions are summarized as below.

(1) In view of labeled and unlabeled data of a decision information system for partially labeled real-valued data (p-RVDIS), the missing rate of labels in a p-RVDIS is defined. A p-RVDIS is induced into two decision information systems: one is the l-RVDIS, and the other one is u-RVDIS. The u-RVDIS is counted as an IS without decision attribute.

(2) Based on a discernibility relation, distinguishable relation, dependency function, conditional information entropy and conditional information amount, four importance of each attribute subset are proposed. They are the weighted sum of the importance of the corresponding subsystem of a l-RVDIS and the corresponding subsystem of a u-RVDIS determined by the missing rate of labels, which can be regarded as four MUs of the corresponding subsystem of a p-RVDIS.

(3) From the perspective of statistical analysis, numerical analysis, discrete analysis, correlation analysis, Friedman test and Nemenyi test are carried out to verify four MUs’ advantages and disadvantages.

The remaining portion of this paper is structured as follows. In Section 2, a p-RVDIS is defined. In Section 3, MU in a p-RVDIS is investigated. In Section 4, numerical analysis and statistical analysis are conducted. In Section 5, a summary of this paper is presented.

2 Preliminaries

W = {w₁, ⋯ , w_n}, 2^W and |Z| represent a finite set, the power set of W and the cardinality of Z ∈ 2^W, respectively. Put $δ = W \times W, ▵ = {(w, w) : w \in W} .$ Then, δ is know as a universal relation on W; ▵ is know as an identity relation on W.

Suppose that (W, A) is an information system (IS) [30]. For B ⊆ A, put $ind (B) = {(w, w^{'}) \in W \times W : \forall a \in B, a (w) = a (w^{'})},$ Then ind (B) is known as the indiscernibility relation on W with respect to B.

Definition 2.1. ([12]) Let (W, A) be an IS with B ⊆ A. Put $dis (B) = {(w, w^{'}) \in W \times W : \exists a \in B, a (w) \neq a (w^{'})} .$ Then dis (B) is known as the discernibility relation of B on W with respect to B.

Obviously, $dis (B) = W \times W - ind (B) .$

We refer to (W, A, d) as a decision information system, if (W, A) be an IS and d is a decision attribute.

Definition 2.2. ([12]) For a decision information system (W, A, d), let B ⊆ A, put

dis_d (B) = {(w, w′) ∈ W × W : ∃ a ∈ B, a (w) ≠ a (w′) and d (w) ≠ d (w′)} . Then dis_d (B) is known as the discernibility relation of B on W with respect to d.

Let (W, A, d) be a decision information system. If ∀ a ∈ A and w ∈ A, a (w) is a real number, then (W, A, d) is referred to as a real-valued decision information system (RVDIS).

Definition 2.3. For a p-RVDIS (W, A, d).

(1) (W, A, d) is known as a decision information system for labeled real-valued data (l-RVDIS), if ∀ w ∈ W, d (w)≠ *.

(2) (W, A, d) is known as a decision information system for partially labeled real-valued data (p-RVDIS), if $V_{d}^{*} \neq \emptyset$ and there exists w ∈ W, d (w) =*.

(3) (W, A, d) is known as a decision information system for unlabeled real-valued data (u-RVDIS), if ∀ w ∈ W, d (w) =*.

Because each object lacks label in a u-RVDIS (W, A, d), we think that (W, A, d) can be seen as (W, A).

Definition 2.4. For a p-RVDIS (W, A, d), put $W^{l} = {w \in W : d (w) \neq *}, W^{u} = {w \in W : d (w) = *} .$ Then W^l∪ W^u = W, W^l ∩ W^u = ∅. Here, (W^l, A, d) and (W^u, A, d) are called the l-RVDIS and u-RVDIS induced by (W, A, d), respectively.

(W, A, d) can be interpreted as the outcome of information fusion of (W^l, A, d) and (W^u, A, d).

Definition 2.5. The missing rate of labels in a p-RVDIS (W, A, d) is defined as $λ = \frac{| W^{u} |}{| W |} .$

Example 2.6.Table 1 depicts p-RVDIS (W, A, d), where W = {w₁, w₂, ⋯ , w₁₀} and A = {a₁, a₂, a₃, a₄, a₅}.

Table 2
A p-RVDIS (W, A, d)

W a ₁ a ₂ a ₃ a ₄ a ₅ d

w ₁ 23.6 4 14 143 8.5 1

w ₂ 15.5 8 40 190 4.7 2

w ₃ 35.7 4 97 800 9.9 *

w ₄ 18.4 6 23 100 3.1 3

w ₅ 23.6 4 14 303 2.2 2

w ₆ 44.3 4 90 480 2.2 1

w ₇ 23.6 5 14 770 3.9 *

w ₈ 34.5 4 23 100 1.8 1

w ₉ 44.1 9 97 521 2.2 *

w ₁₀ 40.9 4 97 800 1.8 3

W	a ₁	a ₂	a ₃	a ₄	a ₅	d
w ₁	23.6	4	14	143	8.5	1
w ₂	15.5	8	40	190	4.7	2
w ₃	35.7	4	97	800	9.9	*
w ₄	18.4	6	23	100	3.1	3
w ₅	23.6	4	14	303	2.2	2
w ₆	44.3	4	90	480	2.2	1
w ₇	23.6	5	14	770	3.9	*
w ₈	34.5	4	23	100	1.8	1
w ₉	44.1	9	97	521	2.2	*
w ₁₀	40.9	4	97	800	1.8	3

It is obvious that $W^{l} = {w_{1}, w_{2}, w_{4}, w_{5}, w_{6}, w_{8}, w_{10}}, W^{u} = {w_{3}, w_{7}, w_{9}} .$ Then $λ = \frac{3}{10} .$

Definition 2.7. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $| W^{l} | = n_{l};$ d (a (w) , a (w′)) $= \frac{| a (w) - a (w^{'}) |}{\max {a (x) : x \in W} - \min {a (x) : x \in W}},$ $B_{θ}^{l} = {(w, w^{'}) \in W^{l} \times W^{l} : \forall a \in B, d (a (w), a (w^{'})) \leq θ},$ $B_{θ}^{l} (w) = {w^{'} \in W^{l} : (w, w^{'}) \in B_{θ}^{l}};$ $R_{d}^{l} = {(w, w^{'}) \in W^{l} \times W^{l} : d (w) = d (w^{'})},$ $R_{d}^{l} (w) = {w^{'} \in W^{l} : (w, w^{'}) \in R_{d}^{l}};$ $\underline{B_{θ}^{l}} (X) = {w \in W^{l} : B_{θ}^{l} (w) \subseteq X}, X \subseteq W^{l};$ $W^{l} / R_{d}^{l} = {R_{d}^{l} (w) : w \in W^{l}} = {𝔻_{1}, \dots, 𝔻_{r}};$ ${POS}_{θ}^{l} (B) = ⋃_{i = 1}^{r} \underline{B_{θ}^{l}} (𝔻_{i}) .$

Remark 2.8. (1) d (a (w) , a (w′)) means the distance between a (w) and a (w′).

(2) $B_{θ}^{l}$ expresses a binary relation on W^l. Clearly, $B_{θ}^{l}$ is a tolerance (reflexive and symmetric) relation on W^l. $B_{θ}^{l} (w)$ is the tolerance class of w.

(3) $R_{d}^{l}$ refers to the decision function, and $R_{d}^{l} (w)$ is the decision class of w.

(4) $\underline{B_{θ}^{l}} (X)$ is lower approximation of X on W^l with respect to θ.

(5) $W^{l} / R_{d}^{l}$ is is the quotient set composed of all decision classes.

(6) ${POS}_{θ}^{l} (B)$ denotes the positive region of B.

Example 2.9. (Continued from Example 2.6) We have $W^{l} = {w_{1}, w_{2}, w_{4}, w_{5}, w_{6}, w_{8}, w_{10}} .$ Pick B = {a₁, a₂}, θ = 0.4. Then $B_{θ}^{l} (w_{1}) = {w_{1}, w_{5}, w_{8}}, B_{θ}^{l} (w_{2}) = {w_{2}},$ $B_{θ}^{l} (w_{4}) = {w_{4}}, B_{θ}^{l} (w_{5}) = {w_{1}, w_{5}, w_{8}},$ $B_{θ}^{l} (w_{6}) = {w_{6}, w_{8}, w_{10}}, B_{θ}^{l} (w_{8}) = {w_{1}, w_{5}, w_{6}, w_{8}, w_{10}},$ $B_{θ}^{l} (w_{10}) = {w_{6}, w_{8}, w_{10}};$ $W^{l} / R_{d}^{l} = {𝔻_{1}, 𝔻_{2}, 𝔻_{3}}$ , where $𝔻_{1} = {w_{1}, w_{6}, w_{8}}$ , $𝔻_{2} = {w_{2}, w_{5}}$ , $𝔻_{3} = {w_{4}, w_{10}} .$ Thus ${POS}_{θ}^{l} (B) = {w_{2}, w_{4}} .$

Definition 2.10. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put

${dis}_{θ}^{l, d} (B) = {(w, w^{'}) \in W^{l} \times W^{l} : \exists a \in B, d (a (w), a (w^{'})) \geq θ and d (w) \neq d (w^{'})} .$ Then ${dis}_{θ}^{l, d} (B)$ is known as the relative discernibility relation of B on W^l with respect to d.

Definition 2.11. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put

${dis}_{θ}^{u} (B) = {(w, w^{'}) \in W^{u} \times W^{u} : \exists a \in B, d (a (w), a (w^{'})) \geq θ} .$ Then ${dis}_{θ}^{u} (B)$ is known as the discernibility relation of B on W^u.

Example 2.12. (Continued from Example 2.6) We have

${dis}_{θ}^{l, d} (B) = {(w_{1}, w_{2}), (w_{1}, w_{4}), (w_{1}, w_{10}), (w_{2}, w_{1}), (w_{2}, w_{4}), (w_{2}, w_{6}), (w_{2}, w_{8}), (w_{2}, w_{10}), (w_{4}, w_{1}), (w_{4}, w_{2}), (w_{4}, w_{5}), (w_{4}, w_{6}), (w_{4}, w_{8}), (w_{5}, w_{4}), (w_{5}, w_{6}), (w_{5}, w_{10}), (w_{6}, w_{2}), (w_{6}, w_{4}), (w_{6}, w_{5}), (w_{8}, w_{2}), (w_{8}, w_{4}), (w_{10}, w_{1}), (w_{10}, w_{2}), (w_{10}, w_{5})}$ , ${dis}_{θ}^{w} (B) = {(w_{3}, w_{7}), (w_{3}, w_{9}), (w_{7}, w_{3}), (w_{7}, w_{9}), (w_{9}, w_{3}), (w_{9}, w_{7})} .$

3 Measure of uncertainty in a p-RVDIS

In this section, we explore measure of uncertainty in a p-RVDIS via a discernibility relation.

3.1 The type 1 importance of a subsystem in a p-RVDIS

Definition 3.1. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $Γ_{θ}^{l} (B) = \frac{1}{n_{l}} | {POS}_{θ}^{l} (B) |;$ Then $Γ_{θ}^{l} (B)$ is known as the dependence of B in W^l with respect to d.

Example 3.2. (Continued from Example 2.6) We have

$Γ_{θ}^{l} (B) = \frac{2}{7} \approx 0.2857 .$

Proposition 3.3.For a p-RVDIS (W, A, d), let θ ∈ [0, 1].

(1) $Γ_{θ}^{l} (B) = \frac{1}{n_{l}} \sum_{i = 1}^{r} | \underline{B_{θ}^{l}} (𝔻_{i}) | .$

(2) $0 \leq Γ_{θ}^{l} (B) \leq 1$ .

(3) If B ⊂ C ⊆ A, then ∀ θ $Γ_{θ}^{l} (B) \leq Γ_{θ}^{l} (C) .$

(4) If 0 ≤ θ₁ < θ₂ ≤ 1, then ∀ B $Γ_{θ_{2}}^{l} (B) \leq Γ_{θ_{1}}^{l} (B) .$

Proof. (1) Obviously, ∀ i, $\underline{B_{θ}^{l}} (𝔻_{i}) \subseteq 𝔻_{i}$ .

Since ${𝔻_{1}, \dots, 𝔻_{r}}$ is a partition of W^l, we have $| {POS}_{θ}^{l} (B) | = | ⋃_{i = 1}^{r} \underline{B_{θ}^{l}} (𝔻_{i}) | = \sum_{i = 1}^{r} | \underline{B_{θ}^{l}} (𝔻_{i}) | .$

Thus $Γ_{θ}^{l} (B) = \frac{1}{n_{l}} \sum_{i = 1}^{r} | \underline{B_{θ}^{l}} (𝔻_{i}) | .$

(2) This holds by (1).

(3) Suppose B ⊆ C ⊆ A. Then ∀ w ∈ W, $C_{θ}^{l} (w) \subseteq B_{θ}^{l} (w)$ . So $\forall i, \underline{B_{θ}^{l}} (𝔻_{i}) \subseteq \underline{C_{θ}^{l}} (𝔻_{i}) .$

This suggests that $\forall i, | \underline{B_{θ}^{l}} (𝔻_{i}) \leq | \underline{C_{θ}^{l}} (𝔻_{i}) | .$

By (1), $Γ_{θ}^{l} (B) \leq Γ_{θ}^{l} (C) .$

(4) Suppose 0 ≤ θ₁ < θ₂ ≤ 1. Then ∀ w ∈ W, $B_{θ_{1}}^{l} (w) \subseteq B_{θ_{2}}^{l} (w)$ . So $\forall i, \underline{B_{θ_{2}}^{l}} (𝔻_{i}) \subseteq \underline{B_{θ_{1}}^{l}} (𝔻_{i}) .$

This suggests that $\forall i, | \underline{B_{θ_{2}}^{l}} (𝔻_{i}) | \leq | \underline{B_{θ_{1}}^{l}} (𝔻_{i}) | .$

By (1), $Γ_{θ_{2}}^{l} (B) \leq Γ_{θ_{1}}^{l} (B) .$ □

Definition 3.4. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put ${imp}_{λ, θ}^{(1)} (B) = (1 - λ) \frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$ Then ${imp}_{λ, θ}^{(1)} (B)$ is known as the type 1 importance of (W, B, d).

Example 3.5. (Continued from Example 2.6) We have

${imp}_{λ, θ}^{(1)} (B) \approx 0.5800 .$

Proposition 3.6.For a p-RVDIS (W, A, d), let θ ∈ [0, 1].

(1) $0 \leq {imp}_{λ, θ}^{(1)} (B) \leq 1$ ;

(2) ${imp}_{λ, θ}^{(1)} (A) = 1$ ;

(3) B ⊆ C ⊆ A implies ${imp}_{λ, θ}^{(1)} (B) \leq {imp}_{λ, θ}^{(1)} (C)$ ;

(4) ${imp}_{λ, θ}^{(1)} (B) = 1$ ⇔ $Γ_{θ}^{l} (B) = Γ_{θ}^{l} (A)$ , $| {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) |$ .

Proof. It is evident that both “(1) and (2)" hold.

(3) B ⊆ C ⊆ A implies $Γ_{θ}^{l} (B) \leq Γ_{θ}^{l} (C), {dis}_{θ}^{w} (B) \subseteq {dis}_{θ}^{w} (C) .$

Then $\frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} \leq \frac{Γ_{θ}^{l} (C)}{Γ_{θ}^{l} (A)}, \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \leq \frac{| {dis}_{θ}^{w} (C) |}{| {dis}_{θ}^{w} (A) |} .$

Thus $(1 - λ) \frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} \leq (1 - λ) \frac{Γ_{θ}^{l} (C)}{Γ_{θ}^{l} (A)},$ $λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \leq λ \frac{| {dis}_{θ}^{w} (C) |}{| {dis}_{θ}^{w} (A) |} .$

Hence ${imp}_{λ, θ}^{(1)} (B) \leq {imp}_{λ, θ}^{(1)} (C)$ .

(4) “ ⇐ " is clear. Next, we provide a proof for the implication “ ⇒ ".

Suppose ${imp}_{λ, θ}^{(1)} (B) = 1$ . Then $(1 - λ) \frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} = 1 = (1 - λ) + λ .$

This suggests that $(1 - λ) (1 - \frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)}) + λ (1 - \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |}) = 0 .$

Note that $1 - \frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} = \frac{Γ_{θ}^{l} (A) - Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} \geq 0$ , $1 - \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} = \frac{| {dis}_{θ}^{w} (A) | - | {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \geq 0$ , Then $1 - \frac{Γ_{θ}^{l} (B)}{Γ_{θ}^{l} (A)} = 0$ , $1 - \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} = 0$ . Thus $Γ_{θ}^{l} (B) = Γ_{θ}^{l} (A), | {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) | .$ □

3.2 The type 2 importance of a subsystem in a p-RVDIS

Definition 3.7. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put ${imp}_{λ, θ}^{(2)} (B) = (1 - λ) \frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$ Then ${imp}_{λ, θ}^{(2)} (B)$ is known as the type 2 importance of (W, B, d).

Example 3.8. (Continued from Example 2.6) We have

${imp}_{λ, θ}^{(2)} (B) \approx 0.8600 .$

Proposition 3.9.For a p-RVDIS (W, A, d), let θ ∈ [0, 1].

(1) $0 \leq {imp}_{λ, θ}^{(2)} (B) \leq 1$ ;

(2) ${imp}_{λ, θ}^{(2)} (A) = 1$ ;

(3) B ⊆ C ⊆ A implies ${imp}_{λ, θ}^{(2)} (B) \leq {imp}_{λ, θ}^{(2)} (C)$ ;

(4) ${imp}_{λ, θ}^{(2)} (B) = 1$ ⇔ $| {dis}_{θ}^{l, d} (B) | = | {dis}_{θ}^{l, d} (A) |$ , $| {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) |$ .

Proof. It is evident that both “(1) and (2)" hold.

(3) B ⊆ C ⊆ A implies $| {dis}_{θ}^{l, d} (B) | \leq | {dis}_{θ}^{l, d} (C) |, | {dis}_{θ}^{w} (C) | \leq | {dis}_{θ}^{w} (B) | .$

Then $\frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} \leq \frac{| {dis}_{θ}^{l, d} (C) |}{| {dis}_{θ}^{l, d} (A) |}, \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \leq \frac{| {dis}_{θ}^{w} (A) |}{| {dis}_{θ}^{w} (C) |} .$

Thus $(1 - λ) \frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} \leq (1 - λ) \frac{| {dis}_{θ}^{l, d} (C) |}{| {dis}_{θ}^{l, d} (A) |},$ $λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \leq λ \frac{| {dis}_{θ}^{w} (A) |}{| {dis}_{θ}^{w} (C) |} .$

Hence ${imp}_{λ, θ}^{(1)} (B) \leq {imp}_{λ, θ}^{(1)} (C)$ .

(4) “ ⇐ " is clear. Next, we provide a proof for the implication “ ⇒ ".

Suppose ${imp}_{λ, θ}^{(2)} (B) = 1$ . Then $(1 - λ) \frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} = 1 = (1 - λ) + λ .$

This suggests that $(1 - λ) (1 - \frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |}) + λ (1 - \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |}) = 0 .$

Note that $1 - \frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} = \frac{| {dis}_{θ}^{l, d} (A) | - | {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} \geq 0$ , $1 - \frac{| {dis}_{θ}^{w} (A) |}{| {nd}^{u} (B) |} = \frac{| {dis}_{θ}^{w} (B) | - | {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \geq 0$ , Then $1 - \frac{| {dis}_{θ}^{l, d} (B) |}{| {dis}_{θ}^{l, d} (A) |} = 0$ , $1 - \frac{| {dis}_{θ}^{w} (A) |}{| {nd}^{u} (B) |} = 0$ . Thus $| {dis}_{θ}^{l, d} (B) | = | {dis}_{θ}^{l, d} (A) |, | {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) | .$ □

3.3 The type 3 importance of a subsystem in a p-RVDIS

Definition 3.10. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $H_{θ}^{l} (B) = - \sum_{i = 1}^{n_{l}} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} .$ Then $H_{θ}^{l} (B)$ is known as information entropy of B.

Proposition 3.11. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, then $0 \leq H_{θ}^{l} (B) \leq n_{l} {log}_{2} n_{l} .$ Furthermore, if $B_{θ}^{l} = ▵$ , then $H_{θ}^{l} (B) = {log}_{2} n_{l}$ ; if $B_{θ}^{l} = δ$ , then $H_{θ}^{l} (B) = 0$ .

Proof. Considering that ∀ i, $1 \leq | B_{θ}^{l} (w_{i}) | \leq n_{l}$ , we have $\frac{1}{n_{l}} \leq \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \leq 1,$ $0 \leq - {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \leq {log}_{2} n_{l} .$

Then $0 \leq - \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \leq {log}_{2} n_{l} .$

By Definition 3.4, $0 \leq H_{θ}^{l} (B) \leq n_{l} {log}_{2} n_{l} .$

If $B_{θ}^{l} = ▵$ , then ∀ i, $| B_{θ}^{l} (w_{i}) | = 1$ . So $H_{θ}^{l} (B) = {log}_{2} n_{l}$ .

If $B_{θ}^{l} = δ$ , then ∀ i, $| B_{θ}^{l} (w_{i}) | = n_{l}$ . So $H_{θ}^{l} (B) = 0$ . □

Definition 3.12. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $H_{θ}^{l} (B | d) = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{| B_{θ}^{l} (w_{i}) |} .$ Then $H_{θ}^{l} (B | d)$ is known as conditional information entropy of B in W^l with respect to d.

Proposition 3.13. For a p-RVDIS (W, A, d), let θ ∈ [0, 1], if B ⊆ C ⊆ A, then $H_{θ}^{l} (C | d) \leq H_{θ}^{l} (B | d) .$

Proof. Denote $p_{ij}^{(1)} = | B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |, p_{ij}^{(2)} = | B_{θ}^{l} (w_{i}) \cap (W^{l} - 𝔻_{j}) |;$ $q_{ij}^{(1)} = | C_{θ}^{l} (w_{i}) \cap 𝔻_{j} |, q_{ij}^{(2)} = | C_{θ}^{l} (w_{i}) \cap (W^{l} - 𝔻_{j}) | .$

Then $| B_{θ}^{l} (w_{i}) | = p_{ij}^{(1)} + p_{ij}^{(2)}, | C_{θ}^{l} (w_{i}) | = q_{ij}^{(1)} + q_{ij}^{(2)} .$

Obviously, ∀ i, $C_{θ}^{l} (w_{i}) \subseteq B_{θ}^{l} (w_{i}) .$

Then $\forall i, j, 0 \leq q_{ij}^{(1)} \leq p_{ij}^{(1)}, 0 \leq q_{ij}^{(2)} \leq p_{ij}^{(2)} .$

$\begin{matrix} H_{θ}^{l} (B | d) & = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{p_{ij}^{(1)}}{n_{l}} {log}_{2} \frac{p_{ij}^{(1)}}{p_{ij}^{(1)} + p_{ij}^{(2)}} \\ ≜ - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} f (p_{ij}^{(1)}, p_{ij}^{(2)}) . \end{matrix}$ $\begin{matrix} H_{θ}^{l} (C | d) & = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{w_{ij}^{(1)}}{n_{l}} {log}_{2} \frac{w_{ij}^{(1)}}{w_{ij}^{(1)} + w_{ij}^{(2)}} \\ ≜ - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} f (w_{ij}^{(1)}, q_{ij}^{(2)}) . \end{matrix}$

Put $f (x, y) = - x {log}_{2} \frac{x}{x + y} (x > 0, y \geq 0)$ . Then f (x, y) increases with respect to x and increases with respect to y, respectively.

Since $q_{ij}^{(1)} \leq p_{ij}^{(1)}, q_{ij}^{(2)} \leq p_{ij}^{(2)},$ we have $f (q_{ij}^{(1)}, q_{ij}^{(2)}) \leq f (p_{ij}^{(1)}, q_{ij}^{(2)}) \leq f (p_{ij}^{(1)}, p_{ij}^{(2)}) .$

Thus $H_{θ}^{l} (C | d) \leq H_{θ}^{l} (B | d) .$ □

Definition 3.14. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $H_{θ}^{l} (B \cup d) = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} .$ Then $H_{θ}^{l} (B \cup d)$ is known as joint information entropy of B and d.

Proposition 3.15. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, then $H_{θ}^{l} (B | d) = H_{θ}^{l} (B \cup d) - H_{θ}^{l} (B) .$

Proof. It should be noted that ${𝔻_{1}, \dots, 𝔻_{r}}$ constitutes a partition of W. Then ∀ i, $\sum_{j = 1}^{r} | B_{θ}^{l} (w_{i}) \cap 𝔻_{j} | = | B_{θ}^{l} (w_{i}) | .$ $\begin{matrix} H_{θ}^{l} (B | d) & = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} ({log}_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} \\ - {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}}) \\ = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} \\ + \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \\ = H_{θ}^{l} (B \cup d) + \sum_{i = 1}^{n_{l}} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \\ = H_{θ}^{l} (B \cup d) - H_{θ}^{l} (B) . \end{matrix}$ □

Proposition 3.16. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, then $H_{θ}^{l} (B | d) \geq 0$ .

Proof. By Definition 3.4, $H_{θ}^{l} (B) = - \sum_{i = 1}^{n_{l}} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} .$

Denote ${𝔻_{1}, \dots, 𝔻_{r}}$ as a partition of W. Then ∀ i, $\sum_{j = 1}^{r} | B_{θ}^{l} (w_{i}) \cap 𝔻_{j} | = | B_{θ}^{l} (w_{i}) | .$

Then $H_{θ}^{l} (B) = - \sum_{i = 1}^{n_{l}} \frac{\sum_{j = 1}^{r} | B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} .$ $= - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} .$

By Definition 3.3, $H_{θ}^{l} (B \cup d) = - \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} .$

∀ i, j, $\log_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} \leq \log_{2} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} .$

Then $H_{θ}^{l} (B) \leq H_{θ}^{l} (B \cup d) .$

By Proposition 3.3, $H_{θ}^{l} (B | d) = H_{θ}^{l} (B \cup d) - H_{θ}^{l} (B) .$

Hence $H_{θ}^{l} (B | d) \geq 0 .$ □

Definition 3.17. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put ${imp}_{λ, θ}^{(3)} (B) = (1 - λ) \frac{H_{θ}^{l} (A | d)}{H_{θ}^{l} (B | d)} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$ Then ${imp}_{λ, θ}^{(3)} (B)$ is known as the type 3 importance of (W, B, d).

Example 3.18 (Continued from Example 2.6) We have

${imp}_{λ, θ}^{(3)} (B) \approx 0.4566 .$

Proposition 3.19.For a p-RVDIS (W, A, d), let θ ∈ [0, 1].

(1) $0 \leq {imp}_{λ, θ}^{(3)} (B) \leq 1$ ;

(2) ${imp}_{λ, θ}^{(3)} (A) = 1$ ;

(3) B ⊆ C ⊆ A implies ${imp}_{λ, θ}^{(3)} (C) \leq {imp}_{λ, θ}^{(3)} (B)$ ;

(4) ${imp}_{λ, θ}^{(3)} (B) = 1$ ⇔ $H_{θ}^{l} (B | d) = H_{θ}^{l} (A | d)$ , $| {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) |$ .

Proof. It is evident that both “(1) and (2)" hold.

(3) B ⊆ C ⊆ A implies $H_{θ}^{l} (C | d) \leq H_{θ}^{l} (B | d), | {dis}_{θ}^{w} (C) | \leq | {dis}_{θ}^{w} (B) | .$

Then $\frac{H_{θ}^{l} (C | d)}{H_{θ}^{l} (A | d)} \leq \frac{H_{θ}^{l} (B | d)}{H_{θ}^{l} (A | d)}, \frac{| {dis}_{θ}^{w} (C) |}{| {dis}_{θ}^{w} (A) |} \leq \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$

Thus $(1 - λ) \frac{H_{θ}^{l} (C | d)}{H_{θ}^{l} (A | d)} \leq (1 - λ) \frac{H_{θ}^{l} (B | d)}{H_{θ}^{l} (A | d)}, λ \frac{| {dis}_{θ}^{w} (C) |}{| {dis}_{θ}^{w} (A) |} \leq λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$

Hence ${imp}_{λ, θ}^{(1)} (C) \leq {imp}_{λ, θ}^{(1)} (B)$ .

(4) “ ⇐ " is clear. Next, we provide a proof for the implication “ ⇒ ".

Suppose ${imp}_{λ, θ}^{(3)} (B) = 1$ . Then $(1 - λ) \frac{H_{θ}^{l} (A | d)}{H_{θ}^{l} (B | d)} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} = 1 = (1 - λ) + λ .$

This suggests that $(1 - λ) (1 - \frac{H_{θ}^{l} (A | d)}{H_{θ}^{l} (B | d)}) + λ (1 - \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |}) = 0 .$

Note that $1 - \frac{H_{θ}^{l} (A | d)}{H_{θ}^{l} (B | d)} = \frac{H_{θ}^{l} (B | d) - H_{θ}^{l} (A | d)}{H_{θ}^{l} (B | d)} \geq 0$ , $1 - \frac{| {dis}_{θ}^{w} (A) |}{| {nd}^{u} (B) |} = \frac{| {dis}_{θ}^{w} (B) | - | {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \geq 0$ . Then $1 - \frac{H_{θ}^{l} (A | d)}{H_{θ}^{l} (B | d)} = 0$ , $1 - \frac{| {dis}_{θ}^{w} (A) |}{| {nd}^{u} (B) |} = 0$ . Thus $H_{θ}^{l} (A | d) = H_{θ}^{l} (B | d), | {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) | .$ □

3.4 The type 4 importance of a subsystem in a p-RVDIS

Definition 3.20. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $E_{θ}^{l} (B) = \sum_{i = 1}^{n_{l}} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \frac{| W^{l} - B_{θ}^{l} (w_{i}) |}{n_{l}} .$ Then $E_{θ}^{l} (B)$ is known as information entropy of B.

Obviously, $E_{θ}^{l} (B) = \sum_{i = 1}^{n_{l}} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} (1 - \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}}) .$

Proposition 3.21. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, then $0 \leq E_{θ}^{l} (B) \leq 1 - \frac{1}{n_{l}} .$ Furthermore, when $B_{θ}^{l} = ▵$ , then $E_{θ}^{l} = 1 - \frac{1}{n_{l}}$ ; Furthermore, when $E_{θ}^{l} = 0$ .

Proof. Since ∀ i, $1 \leq | B_{θ}^{l} (w_{i}) | \leq n_{l}$ , we have $\frac{1}{n_{l}} \leq \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \leq 1,$ $0 \leq 1 - \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} \leq 1 - \frac{1}{n_{l}} .$

Thus $0 \leq E_{θ}^{l} (B) \leq 1 - \frac{1}{n_{l}} .$

If $B_{θ}^{l} = ▵$ , then ∀ i, $| B_{θ}^{l} (w_{i}) | = 1$ . So $E_{θ}^{l} (B) = 1 - \frac{1}{n_{l}}$ .

If $B_{θ}^{l} = δ$ , then ∀ i, $| B_{θ}^{l} (w_{i}) | = n_{l}$ . So $E_{θ}^{l} (B) = 0$ . □

Definition 3.22. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $E_{θ}^{l} (B | d) = \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} \frac{| B_{θ}^{l} (w_{i}) - 𝔻_{j} |}{n_{l}} .$ Then $E_{θ}^{l} (B | d)$ are called conditional information amount of B in W^l with respect to d.

Proposition 3.23. For a p-RVDIS (W, A, d), let θ ∈ [0, 1], if B ⊆ C ⊆ A, then $E_{θ}^{l} (C | d) \leq E_{θ}^{l} (B | d) .$

Proof. Suppose B ⊆ C ⊆ A. Then ∀ i, $C_{θ}^{l} (w_{i}) \subseteq B_{θ}^{l} (w_{i})$ . So ∀ i, j, $C_{θ}^{l} (w_{i}) \cap 𝔻_{j} \subseteq B_{θ}^{l} (w_{i}) \cap 𝔻_{j}, C_{θ}^{l} (w_{i}) - 𝔻_{j} \subseteq B_{θ}^{l} (w_{i}) - 𝔻_{j} .$

This suggests that ∀ i, j,

$| C_{θ}^{l} (w_{i}) \cap 𝔻_{j} | \leq | B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |,$

$| C_{θ}^{l} (w_{i}) - 𝔻_{j} | \leq | B_{θ}^{l} (w_{i}) - 𝔻_{j} | .$

Thus $E_{θ}^{l} (C | d) \leq E_{θ}^{l} (B | d) .$ □

Definition 3.24. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, put $E_{θ}^{l} (B \cup d) = \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} {log}_{2} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} .$ Then $E_{θ}^{l} (B \cup d)$ is known as joint information entropy of B and d.

Obviously, $E_{θ}^{l} (B | d) \geq 0 .$

Proposition 3.25. For a p-RVDIS (W, A, d), let θ ∈ [0, 1] and B ⊆ A, then $E_{θ}^{l} (B | d) = E_{θ}^{l} (B \cup d) - E_{θ}^{l} (B) .$

Proof. Denote ${𝔻_{1}, \dots, 𝔻_{r}}$ as a partition of W. Then ∀ i, $\sum_{j = 1}^{r} | B_{θ}^{l} (w_{i}) \cap 𝔻_{j} | = | B_{θ}^{l} (w_{i}) | .$ $E_{θ}^{l} (B | d)$ $\begin{matrix} = \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} \frac{| B_{θ}^{l} (w_{i}) - B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} \\ = \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} ((1 - \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}}) \\ - (1 - \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}})) \\ = \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} (1 - \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}}) - \\ \sum_{i = 1}^{n_{l}} \sum_{j = 1}^{r} \frac{| B_{θ}^{l} (w_{i}) \cap 𝔻_{j} |}{n_{l}} (1 - \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}}) \\ = E_{θ}^{l} (B \cup d) - \sum_{i = 1}^{n_{l}} \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}} (1 - \frac{| B_{θ}^{l} (w_{i}) |}{n_{l}}) \\ = E_{θ}^{l} (B \cup d) - E_{θ}^{l} (B) . \end{matrix}$ □

Definition 3.26. For a p-RVDIS (W, A, d), let $λ = \frac{| W^{u} |}{| W |}$ , θ ∈ [0, 1] and B ⊆ A. Put ${imp}_{λ, θ}^{(4)} (B) = (1 - λ) \frac{E_{θ}^{l} (A | d)}{E_{θ}^{l} (B | d)} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$ Then ${imp}_{λ, θ}^{(4)} (B)$ is known as the type 4 importance of (W, B, d).

Example 3.27. (Continued from Example 2.6) We have ${imp}_{λ, θ}^{(4)} (B) \approx 0.3933 .$

Proposition 3.28.For a p-RVDIS (W, A, d), let θ ∈ [0, 1].

(1) $0 \leq {imp}_{λ, θ}^{(4)} (B) \leq 1$ ;

(2) ${imp}_{λ, θ}^{(4)} (A) = 1$ ;

(3) B ⊆ C ⊆ A implies ${imp}_{λ, θ}^{(4)} (C) \leq {imp}_{λ, θ}^{(4)} (B)$ ;

(4) ${imp}_{λ, θ}^{(4)} (B) = 1$ ⇔ $E_{θ}^{l} (B | d) = E_{θ}^{l} (A | d)$ , $| {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) |$ .

Proof. It is evident that both “(1) and (2)" hold.

(3) B ⊆ C ⊆ A implies $E_{θ}^{l} (C | d) \leq E_{θ}^{l} (B | d), | {dis}_{θ}^{w} (C) | \leq | {dis}_{θ}^{w} (B) | .$

Then $\frac{E_{θ}^{l} (C | d)}{E_{θ}^{l} (A | d)} \leq \frac{E_{θ}^{l} (B | d)}{E_{θ}^{l} (A | d)}, \frac{| {dis}_{θ}^{w} (C) |}{| {dis}_{θ}^{w} (A) |} \leq \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$

Thus

$(1 - λ) \frac{E_{θ}^{l} (C | d)}{E_{θ}^{l} (A | d)} \leq (1 - λ) \frac{E_{θ}^{l} (B | d)}{E_{θ}^{l} (A | d)},$

$λ \frac{| {dis}_{θ}^{w} (C) |}{| {dis}_{θ}^{w} (A) |} \leq λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} .$

Hence ${imp}_{λ, θ}^{(1)} (C) \leq {imp}_{λ, θ}^{(1)} (B)$ .

(4) “ ⇐ " is clear. Next, we provide a proof for the implication “ ⇒ ".

Suppose ${imp}_{λ, θ}^{(4)} (B) = 1$ . Then $(1 - λ) \frac{E_{θ}^{l} (A | d)}{E_{θ}^{l} (B | d)} + λ \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} = 1 = (1 - λ) + λ .$

This suggests that $(1 - λ) (1 - \frac{E_{θ}^{l} (A | d)}{E_{θ}^{l} (B | d)}) + λ (1 - \frac{| {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |}) = 0 .$

Note that $1 - \frac{E_{θ}^{l} (A | d)}{E_{θ}^{l} (B | d)} = \frac{E_{θ}^{l} (B | d) - E_{θ}^{l} (A | d)}{E_{θ}^{l} (B | d)} \geq 0$ , $1 - \frac{| {dis}_{θ}^{w} (A) |}{| {nd}^{u} (B) |} = \frac{| {dis}_{θ}^{w} (B) | - | {dis}_{θ}^{w} (B) |}{| {dis}_{θ}^{w} (A) |} \geq 0$ . Then $1 - \frac{E_{θ}^{l} (A | d)}{E_{θ}^{l} (B | d)} = 0$ , $1 - \frac{| {dis}_{θ}^{w} (A) |}{| {nd}^{u} (B) |} = 0$ . Thus $E_{θ}^{l} (A | d) = E_{θ}^{l} (B | d), | {dis}_{θ}^{w} (B) | = | {dis}_{θ}^{w} (A) | .$ □

4 Experimental analysis

This section designs experiments and performs effectiveness analysis on the proposed measures.

4.1 Datasets and experimental components

Eight datasets from UCI are selected (see Table 3). They are all real number type. Actually, these datasets consist of labeled real-valued data. However, our study focuses on partially labeled real-valued data. In our experiments, we randomly select and remove certain labeled values from the original dataset to create partially labeled datasets. The missing values are randomly distributed among the decision attributes with λ=20%. Here we take θ=0.4.

Table 3
Eight datasets from UCI

No. Datesets Abbr. Objects Attributes

1 Iris Ir 150 4

2 Ecoli Ec 336 7

3 Parkinsons Pa 197 23

4 Seeds Se 210 7

5 Sona So 208 30

6 Wdbc Wd 369 30

7 Wine Wi 178 13

8 Breast Br 106 9

No.	Datesets	Abbr.	Objects	Attributes
1	Iris	Ir	150	4
2	Ecoli	Ec	336	7
3	Parkinsons	Pa	197	23
4	Seeds	Se	210	7
5	Sona	So	208	30
6	Wdbc	Wd	369	30
7	Wine	Wi	178	13
8	Breast	Br	106	9

4.2 Numerical experiments

Regarding the dataset Ir, it is recorded as $B 1_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 4) .$ Put ${imp}_{λ, θ}^{(1)} (Ir) = {{imp}_{λ, θ}^{(1)} (B 1_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 1_{4}},$

${imp}_{λ, θ}^{(2)} (Ir) = {{imp}_{λ, θ}^{(2)} (B 1_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 1_{4})},$

${imp}_{λ, θ}^{(3)} (Ir) = {{imp}_{λ, θ}^{(3)} (B 1_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 1_{4})},$

${imp}_{λ, θ}^{(4)} (Ir) = {{imp}_{λ, θ}^{(4)} (B 1_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 1_{4})} .$ Regarding the dataset Ec, it is recorded as $B 2_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 7) .$ Put ${imp}_{λ, θ}^{(1)} (Ec) = {{imp}_{λ, θ}^{(1)} (B 1_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 2_{7})},$

${imp}_{λ, θ}^{(2)} (Ec) = {{imp}_{λ, θ}^{(2)} (B 2_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 2_{7})},$

${imp}_{λ, θ}^{(3)} (Ec) = {{imp}_{λ, θ}^{(3)} (B 2_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 2_{7})},$

${imp}_{λ, θ}^{(4)} (Ec) = {{imp}_{λ, θ}^{(4)} (B 2_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 2_{7})} .$ Regarding the dataset Pa, it is recorded as $B 3_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 23) .$ Put ${imp}_{λ, θ}^{(1)} (Pa) = {{imp}_{λ, θ}^{(1)} (B 3_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 3_{23})},$

${imp}_{λ, θ}^{(2)} (Pa) = {{imp}_{λ, θ}^{(2)} (B 3_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 3_{23})},$

${imp}_{λ, θ}^{(3)} (Pa) = {{imp}_{λ, θ}^{(3)} (B 3_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 3_{23})},$

${imp}_{λ, θ}^{(4)} (Pa) = {{imp}_{λ, θ}^{(4)} (B 3_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 3_{23})} .$ Regarding the dataset Se, it is recorded as $B 4_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 7) .$ Put ${imp}_{λ, θ}^{(1)} (Se) = {{imp}_{λ, θ}^{(1)} (B 4_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 4_{7})},$

${imp}_{λ, θ}^{(2)} (Se) = {{imp}_{λ, θ}^{(2)} (B 4_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 4_{7})},$

${imp}_{λ, θ}^{(3)} (Se) = {{imp}_{λ, θ}^{(3)} (B 4_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 4_{7})},$

${imp}_{λ, θ}^{(4)} (Se) = {{imp}_{λ, θ}^{(4)} (B 4_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 4_{7})} .$ Regarding the dataset So, it is recorded as $B 5_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 30) .$ Put ${imp}_{λ, θ}^{(1)} (So) = {{imp}_{λ, θ}^{(1)} (B 5_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 5_{30})},$

${imp}_{λ, θ}^{(2)} (So) = {{imp}_{λ, θ}^{(2)} (B 5_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 5_{30})},$

${imp}_{λ, θ}^{(3)} (So) = {{imp}_{λ, θ}^{(3)} (B 5_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 5_{30})},$

${imp}_{λ, θ}^{(4)} (So) = {{imp}_{λ, θ}^{(4)} (B 5_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 5_{30})} .$ Regarding the dataset Wd, it is recorded as $B 6_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 30) .$ Put ${imp}_{λ, θ}^{(1)} (Wd) = {{imp}_{λ, θ}^{(1)} (B 6_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 6_{30})},$

${imp}_{λ, θ}^{(2)} (Wd) = {{imp}_{λ, θ}^{(2)} (B 6_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 6_{30})},$

${imp}_{λ, θ}^{(3)} (Wd) = {{imp}_{λ, θ}^{(3)} (B 6_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 6_{30})},$

${imp}_{λ, θ}^{(4)} (Wd) = {{imp}_{λ, θ}^{(4)} (B 6_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 6_{30})} .$ Regarding the dataset Wi, it is recorded as $B 7_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 13) .$ Put ${imp}_{λ, θ}^{(1)} (Wi) = {{imp}_{λ, θ}^{(1)} (B 7_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 7_{13})},$

${imp}_{λ, θ}^{(2)} (Wi) = {{imp}_{λ, θ}^{(2)} (B 7_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 7_{13})},$

${imp}_{λ, θ}^{(3)} (Wi) = {{imp}_{λ, θ}^{(3)} (B 7_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 7_{13})},$

${imp}_{λ, θ}^{(4)} (Wi) = {{imp}_{λ, θ}^{(4)} (B 7_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 7_{13})} .$ Regarding the dataset Br, it is recorded as $B 8_{i} = {a_{1}, \dots, a_{i}} (i = 1, \dots, 7) .$ Put ${imp}_{λ, θ}^{(1)} (Br) = {{imp}_{λ, θ}^{(1)} (B 8_{1}), \dots, {imp}_{λ, θ}^{(1)} (B 8_{7})},$

${imp}_{λ, θ}^{(2)} (Br) = {{imp}_{λ, θ}^{(2)} (B 8_{1}), \dots, {imp}_{λ, θ}^{(2)} (B 8_{7})},$

${imp}_{λ, θ}^{(3)} (Br) = {{imp}_{λ, θ}^{(3)} (B 8_{1}), \dots, {imp}_{λ, θ}^{(3)} (B 8_{7})},$

${imp}_{λ, θ}^{(4)} (Br) = {{imp}_{λ, θ}^{(4)} (B 8_{1}), \dots, {imp}_{λ, θ}^{(4)} (B 8_{7})} .$

4.3 Experimental results

The experimental results are shown in Figure 1.

Fig.1

Values of MU on eight datasets.

From Figure 1, the following conclusions are obtained:

${imp}_{λ, θ}^{(1)}, {imp}_{λ, θ}^{(2)}, {imp}_{λ, θ}^{(3)}$ and ${imp}_{λ, θ}^{(4)}$ exhibit a consistent upward trend as the cardinality of attribute subset increases. Additionally, ${imp}_{λ, θ}^{(1)}$ has a bigger range. As the attribute subset expands, these measurements suggest a reduction in the uncertainty of a p-RVDIS. Thus, ${imp}_{λ, θ}^{(1)}, {imp}_{λ, θ}^{(2)}, {imp}_{λ, θ}^{(3)}$ and ${imp}_{λ, θ}^{(4)}$ are be able to measure uncertainty of a p-RVDIS.

4.4 Dispersion analysis

Standard deviation is primarily employed for gauging the extent of dispersion in numerical data. A larger standard deviation signifies higher data dispersion, whereas a smaller value indicates lower data dispersion.

Suppose U = {u₁, ⋯ , u_n} is a dataset. The arithmetic average value, standard deviation and standard deviation coefficient of U are denoted as σ (U), $\bar{U}$ and CV (U), respectively. Their definitions are as follows:

$\bar{u} = \frac{1}{n} \sum_{i = 1}^{n} u_{i}, σ (U) = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} (u_{i} - \bar{u})^{2}},$

$CV (U) = \frac{σ (U)}{\bar{u}} .$

Continuing the aforementioned experiment, the coefficient of variation CV-values of four measurement sets were compared. The outcomes are illustrated in Figure 2.

Fig.2

Values of MU on eight datasets.

From the Figure 2, we are able to observe that the CV-values of ${imp}_{λ, θ}^{(1)}$ and ${imp}_{λ, θ}^{(4)}$ are significantly higher compared to ${imp}_{λ, θ}^{(2)}$ and ${imp}_{λ, θ}^{(3)}$ . Among these eight datasets, ${imp}_{λ, θ}^{(2)}$ has the smallest CV-value. This indicates that ${imp}_{λ, θ}^{(2)}$ has the lowest degree of dispersion, thus providing the best measurement effect for the uncertainty of these eight datasets. In conclusion, it is able to be seen that fuzzy rough entropy ${imp}_{λ, θ}^{(2)}$ performs better in measuring the uncertainty of a p-RVDIS.

4.5 Correlation analysis

Correlation analysis is a statistical analysis and Pearson correlation coefficient is a linear correlation measure that quantifies the degree and direction of the linear relationship between two datasets. Assuming U = {u₁, ⋯ , u_n} and V = {v₁, ⋯ , v_n} are two datasets. Pearson correlation coefficient between U and V, denoted as r (U, V), is defined as $r (U, V) = \frac{\sum_{i = 1}^{n} (u_{i} - \bar{u}) (v_{i} - \bar{v})}{\sqrt{\sum_{i = 1}^{n} (u_{i} - \bar{u})^{2}} \sqrt{\sum_{i = 1}^{n} (v_{i} - \bar{v})^{2}}},$ where $\bar{u} = \frac{1}{n} \sum_{i = 1}^{n} u_{i}$ , $\bar{v} = \frac{1}{n} \sum_{i = 1}^{n} v_{i}$ .

Obviously, $- 1 \leq r (U, V) \leq 1 .$

The correlation between U and V can be derived using Table 4.

Table 4
The corresponding correlation between U and V

r (U, V) Correlation between U and V Abbreviation

r (U, V) =1 Totally positive correlation TPC

0.7 ≤ r (U, V) <1 Strong positive correlation SPC

0.4 ≤ r (U, V) <0.7 Moderate positive correlation MPC

0 < r (U, V) <0.4 Weak positive correlation WPC

r (U, V) =0 No correlation NC

-0.4 < r (U, V) <0 Weak negative correlation WNC

-0.7 ≤ r (U, V) < -0.4 Moderate negative correlation MNC

-1 ≤ r (U, V) < -0.7 Strong negative correlation SNC

r (U, V) = -1 Totally negative correlation TNC

r (U, V)	Correlation between U and V	Abbreviation
r (U, V) =1	Totally positive correlation	TPC
0.7 ≤ r (U, V) <1	Strong positive correlation	SPC
0.4 ≤ r (U, V) <0.7	Moderate positive correlation	MPC
0 < r (U, V) <0.4	Weak positive correlation	WPC
r (U, V) =0	No correlation	NC
-0.4 < r (U, V) <0	Weak negative correlation	WNC
-0.7 ≤ r (U, V) < -0.4	Moderate negative correlation	MNC
-1 ≤ r (U, V) < -0.7	Strong negative correlation	SNC
r (U, V) = -1	Totally negative correlation	TNC

In continuation of the previous experiment, r-values between any two of four measurement sets are compared for each of the eight datasets. The outcomes are presented in Tables 5-12.

By referring to Tables 5-12, one have determined the correlation levels between four measurement metrics across the eight datasets.The evidence from Tables 13-20 indicates that the correlation levels across the 8 datasets are consistent. This confirms the stability of four newly proposed measurements.

4.6 Friedman test and Nemenyi test

To obtain the more comprehensive evaluation of the performance of the proposed measures, we conduct Friedman and Nemenyi test in this part.

The Friedman test is a statistical test based on ranking algorithms. The Friedman statistic is defined by the equation: $χ_{F}^{2} = \frac{12 N}{k (k + 1)} \sum_{i = 1}^{k} r_{i}^{2} - 3 N (k + 1) .$ Here, k represents the number of algorithms, N represents the number of datasets, r_i represents the average ranking of the i-th algorithm. Nonetheless, due to its excessively conservative nature, the Friedman test is often substituted with the subsequent statistic $F_{F} = \frac{(N - 1) χ_{F}^{2}}{N (k - 1) - χ_{F}^{2}} .$

If F_F surpasses the critical value of F_α (k - 1, (k - 1) (N - 1)), it implies rejecting the null hypothesis in the Friedman test. Afterwards, the Nemenyi test with critical distance CD_α is able to be employed to further investigate which algorithm exhibits superior statistical performance, it is defined as ${CD}_{α} = q_{α} \sqrt{\frac{k (k + 1)}{6 N}},$ where q_α represents the critical tabulated value for the test and α denotes the significance level. If the average distance exceeds the CD_α, it signifies that the performance between two algorithms is significantly different.

In the context provided, we consider these four MUs as separate algorithms, and proceed to assess their statistical significance using both the Friedman and Nemenyi test.

Table 5
r-values of eight pairs of four measurement metrics on Ir

r ${imp}_{λ, θ}^{(1)}$ ${imp}_{λ, θ}^{(2)}$ ${imp}_{λ, θ}^{(3)}$ ${imp}_{λ, θ}^{(1)}$

${imp}_{λ, θ}^{(1)}$ 1

${imp}_{λ, θ}^{(2)}$ 0.9497 1

${imp}_{λ, θ}^{(3)}$ 0.9740 0.9856 1

${imp}_{λ, θ}^{(4)}$ 0.9843 0.9822 0.9987 1

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.9497	1
${imp}_{λ, θ}^{(3)}$	0.9740	0.9856	1
${imp}_{λ, θ}^{(4)}$	0.9843	0.9822	0.9987	1

Table 6

r-values of sixteen pairs of four measurement metrics on Ec

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.9490	1
${imp}_{λ, θ}^{(3)}$	0.9766	0.9883	1
${imp}_{λ, θ}^{(4)}$	0.9696	0.9738	0.9948	1

Table 7

r-values of sixteen pairs of four measurement metrics on Pa

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.9240	1
${imp}_{λ, θ}^{(3)}$	0.8613	0.9452	1
${imp}_{λ, θ}^{(4)}$	0.8350	0.9195	0.9969	1

Table 8

r-values of sixteen pairs of four measurement metrics on Se

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.8342	1
${imp}_{λ, θ}^{(3)}$	0.7948	0.9866	1
${imp}_{λ, θ}^{(4)}$	0.7784	0.9798	0.9992	1

Table 9

r-values of sixteen pairs of four measurement metrics on So

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.8029	1
${imp}_{λ, θ}^{(3)}$	0.9415	0.8900	1
${imp}_{λ, θ}^{(4)}$	0.9235	0.8226	0.9897	1

Table 10

r-values of sixteen pairs of four measurement metrics on Wd

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.9297	1
${imp}_{λ, θ}^{(3)}$	0.9717	0.9822	1
${imp}_{λ, θ}^{(4)}$	0.9742	0.9714	0.9986	1

Table 11

r-values of sixteen pairs of four measurement metrics on Wi

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.8190	1
${imp}_{λ, θ}^{(3)}$	0.9940	0.8365	1
${imp}_{λ, θ}^{(4)}$	0.9865	0.7868	0.9943	1

Table 12

r-values of sixteen pairs of four measurement metrics on Br

r	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(1)}$
${imp}_{λ, θ}^{(1)}$	1
${imp}_{λ, θ}^{(2)}$	0.9073	1
${imp}_{λ, θ}^{(3)}$	0.9376	0.9963	1
${imp}_{λ, θ}^{(4)}$	0.9405	0.9952	0.9998	1

Table 13

The correlation between two measurement metrics on Ir

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 14

The correlation between two measurement metrics on Ec

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 15

The correlation between two measurement metrics on Pa

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 16

The correlation between two measurement metrics on Se

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 17

The correlation between two measurement metrics on So

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 18

The correlation between two measurement metrics on Wd

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 19

The correlation between two measurement metrics on Wi

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

Table 20

The correlation between two measurement metrics on Br

	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
${imp}_{λ, θ}^{(1)}$	TPC
${imp}_{λ, θ}^{(2)}$	SPC	TPC
${imp}_{λ, θ}^{(3)}$	SPC	SPC	TPC
${imp}_{λ, θ}^{(4)}$	SPC	SPC	SPC	TPC

(1)The rankings of CV-values is presented for four measurement metrics across eight datasets in Table 21.

Table 21

The type i importance (i = 1,2,3,4)

Date sets	${imp}_{λ, θ}^{(1)}$	${imp}_{λ, θ}^{(2)}$	${imp}_{λ, θ}^{(3)}$	${imp}_{λ, θ}^{(4)}$
Ir	4	1	2	3
Ec	4	1	2	3
Pa	3	1	2	4
Se	4	1	2	3
So	4	1	2	3
Wd	4	2	1	3
Wi	4	1	2	3
Br	4	1	2	3
Average	3.875	1.125	1.875	3.125

(2) The Friedman test is take to examine whether there are significant differences in the performances of four measures. Considering four measures and eight datasets, the F_F follows a distribution with 3 and 21 degrees of freedom. Notably, the critical value of the F_0.05 (3, 21) is 3.07. With F_F being equal to 73.00, it is evident that this value is considerably larger than 3.07. Consequently, at a significance level of α = 0.05, there is substantial evidence to reject the null hypothesis. This implies that the performances of four measures exhibit statistical significance.

(3) To further illustrate the significant difference among four measures, Nemenyi test is employed. Considering a significance level of α = 0.05, we can calculate the critical values as q_α = 2.5690 and ${CD}_{α} = 2.5690 \times \sqrt{\frac{4 \times (4 + 1)}{6 \times 8}} = 1.6583$ . Figure 3 displays the results and the dots represent the average rankings of four measures. When there is partial overlap of the confidence intervals of two measurements on the y-axis, it suggests that there is not a statistically significant difference between those MUs.

Based on the observation results in Figure 3, the following outcomes are derived:

a) The performance of ${imp}_{λ, θ}^{(2)}$ is statistically superior to that of ${imp}_{λ, θ}^{(1)}$ ; the performance of ${imp}_{λ, θ}^{(2)}$ is statistically superior to that of ${imp}_{λ, θ}^{(4)}$ ; the performance of ${imp}_{λ, θ}^{(3)}$ is statistically superior to that of ${imp}_{λ, θ}^{(1)}$ .

Fig. 3

Nemenyi test

b) No significant difference is found between the performance of ${imp}_{λ, θ}^{(2)}$ and ${imp}_{λ, θ}^{(3)}$ ; no significant difference is found between the performance of ${imp}_{λ, θ}^{(3)}$ and ${imp}_{λ, θ}^{(4)}$ ; no significant difference is found between the performance of ${imp}_{λ, θ}^{(1)}$ and ${imp}_{λ, θ}^{(4)}$ .

5 Conclusions

In this paper, a p-RVDIS has been defined. It has been divided into two DISs: l-RVDIS and u-RVDIS. Based on these two DISs, four degrees of importance on an attribute subset in a p-RVDIS have been presented. They are the weighted sum of l-RVDIS and u-RVDIS determined by the missing rate and may be regarded as MUs for a p-RVDIS. To evaluate the performance of the presented MUs, numerical experiments and statistical tests on eight datasets have been carried out. These findings will be significant in comprehending the core nature of uncertainty in a p-RVDIS. The limitation of the study is that the experimental sample is small and parametric experiment is not conducted. In the future work, we will apply the proposed measures of uncertainty to attribute reduction in a p-RVDIS and study partially labeled gene data.

Footnotes

Acknowledgment

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. This work is supported by Natural Science Research Project of Colleges and Universities in Anhui Province (2023AH040386).

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Exploring measure of uncertainty via a discernibility relation for partially labeled real-valued data

Abstract

Keywords

1 Introduction

1.1 Research background

2 Preliminaries

Table 2 A p-RVDIS (W, A, d) W a 1 a 2 a 3 a 4 a 5 d w 1 23.6 4 14 143 8.5 1 w 2 15.5 8 40 190 4.7 2 w 3 35.7 4 97 800 9.9 * w 4 18.4 6 23 100 3.1 3 w 5 23.6 4 14 303 2.2 2 w 6 44.3 4 90 480 2.2 1 w 7 23.6 5 14 770 3.9 * w 8 34.5 4 23 100 1.8 1 w 9 44.1 9 97 521 2.2 * w 10 40.9 4 97 800 1.8 3

3.1 The type 1 importance of a subsystem in a p-RVDIS

3.2 The type 2 importance of a subsystem in a p-RVDIS

3.3 The type 3 importance of a subsystem in a p-RVDIS

3.4 The type 4 importance of a subsystem in a p-RVDIS

4 Experimental analysis

4.1 Datasets and experimental components

Table 3 Eight datasets from UCI No. Datesets Abbr. Objects Attributes 1 Iris Ir 150 4 2 Ecoli Ec 336 7 3 Parkinsons Pa 197 23 4 Seeds Se 210 7 5 Sona So 208 30 6 Wdbc Wd 369 30 7 Wine Wi 178 13 8 Breast Br 106 9

4.3 Experimental results

Table 5 r-values of eight pairs of four measurement metrics on Ir r imp λ , θ ( 1 ) imp λ , θ ( 2 ) imp λ , θ ( 3 ) imp λ , θ ( 1 ) imp λ , θ ( 1 ) 1 imp λ , θ ( 2 ) 0.9497 1 imp λ , θ ( 3 ) 0.9740 0.9856 1 imp λ , θ ( 4 ) 0.9843 0.9822 0.9987 1

Footnotes

Acknowledgment

References

Table 3
Eight datasets from UCI

No. Datesets Abbr. Objects Attributes

1 Iris Ir 150 4

2 Ecoli Ec 336 7

3 Parkinsons Pa 197 23

4 Seeds Se 210 7

5 Sona So 208 30

6 Wdbc Wd 369 30

7 Wine Wi 178 13

8 Breast Br 106 9