Feature selection for incomplete set-valued data

Abstract

Set-valued data is a significant kind of data, such as data obtained from different search engines, market data, patients’ symptoms and behaviours. An information system (IS) based on incomplete set-valued data is called an incomplete set-valued information system (ISVIS), which generalized model of a single-valued incomplete information system. This paper gives feature selection for an ISVIS by means of uncertainty measurement. Firstly, the similarity degree between two information values on a given feature of an ISVIS is proposed. Then, the tolerance relation on the object set with respect to a given feature subset in an ISVIS is obtained. Next, λ-reduction in an ISVIS is presented. What’s more, connections between the proposed feature selection and uncertainty measurement are exhibited. Lastly, feature selection algorithms based on λ-discernibility matrix, λ-information granulation, λ-information entropy and λ-significance in an ISVIS are provided. In order to better prove the practical significance of the provided algorithms, a numerical experiment is carried out, and experiment results show the number of features and average size of features by each feature selection algorithm.

Keywords

Rough set theory ISVIS feature selection similarity degree algorithm

1 Introduction

1.1 Research background and related works

Rough set theory (RST) put forward by Pawlak [15]. The rough set model has been extensively extended. One is to put equivalence relations into tolerance (reflexive and symmetric) relations, dominance (reflexive and transitive) relations, and even reflexive relations; the other is to extend the division of equivalence classes to cover, which can be used to deal with more complex knowledge acquisition and feature selection. RST is a significant method for dealing with uncertainty. Many applications of RST are connected with an information system (IS) [1 , 15].

An IS based on RST was also introduced by Pawlak. A set-valued information system (SVIS) means an IS where its information values are sets. As a common IS, a SVIS has been studied by many scholars.

As an important tool for estimating information, uncertainty measurement has been applied. We refer to the papers about uncertainty measurement of other scholars. For instance, Dai et al. [4] thought about entropy and granularity measures for a SVIS. Dai et al. [4] thought about entropy measure in a SVIS; Li et al. [14] used four kinds of entropy to measure the uncertainty of a fuzzy relation IS.

Feature selection is a significant issue in artificial intelligence. In the framework of RST [15], feature selection is also called attribute reduction, which is to find a minimal feature subset that provides the same discriminating information as the whole set of features. Specifically, some features in an IS are redundant. We want to find a reduct that has the fewest features. Feature selection in an IS mean deleting redundant features under keeping the classification ability. Feature selection has attracted a great deal of attention. For instance, Song et al. [19] discussed feature selection in a set-valued decision information system (SVDIS); Liu et al. [14] presented feature selection in a SVDIS based on dominance relation; Chen et al. [3] proposed feature selection in a SVIS based on a tolerance relation.

Sometimes we encounter incomplete data sets with missing values when analyzing data. Many scholars have carried out research on this situation. Tkachenko et al. [20] proposed a method to deal with the missing data in the Internet of Things (IoT) based on the GRNN-SGTM Ensemble. Izonin et al. [9] used high-performance extended-input neural-like structure to recover the missing values of IoT sensed Data. Izonin et al. [10] presented a prediction method for probable recovery of partially missing or completely lost data based on the improved GRNN-SGTM ensemble method. Li et al. [11, 13] studied fast assignment feature selection and quick feature selection in inconsistent incomplete decision information systems, respectively.

1.2 Motivation and inspiration

Handling incomplete data is becoming increasingly significant. However, the problem of feature selection for incomplete set-valued data has not been reported. So this paper focuses on it. Similarity measurement in an incomplete set-valued information system (ISVIS) is studied and the tolerance relations based on this measurement are put forward. Then, λ-discernibility matrix in an ISVIS is obtained by means of the given threshold λ. Next, λ-information granulation, λ-information entropy and λ-significance in an ISVIS are defined. Thus, some feature selection algorithms in an ISVIS based on them can given. And the work process is displayed in Fig. 1.

Fig. 1

The work process of the paper.

1.3 Comparison and discussion

In this part, we discuss several references for ISVISs so as to see the innovation and contribution of this paper.

1) Lang et al. [12] proposed three relations for feature selection of a SVDIS. Then, they designed an incremental algorithm to compress a dynamic SVDIS. Concretely, they mainly addressed the compression updating from three aspects: variations of attribute set, immigration and emigration of objects and alterations of attribute values.

2) Dai et al. [5] gave the relative bound difference similarity degree between two objects and presented a λ-tolerance relation by using this similarity degree. Moreover, they brought up some information theory concepts, including entropy, conditional entropy, and joint entropy. Based on these concepts, they provided an information theory view for feature selection in a SVDIS.

3) Wang et al. [21] dealt with feature selection of a SVDIS based on λ-level tolerance relation. They introduced the concepts of distribution reduct based on a SVDIS and examined the judgement theorems and discernibility matrices associated with the λ-level tolerance relation.

4) Liu et al. [14] defined a dominance relation in a SVDIS based on similarity degree of attribute value sets. They a discernibility matrix by this dominance relation and studied the problem of feature selection and the judgment in a SVDIS.

5) Singh et al. [18] introduced a new similarity degree between two set-valued attribute values and then proposed a fuzzy similarity-based rough set approach based on this fuzzy tolerance relation. In addition, feature selection of a SVDIS based on degree of dependency is postulated.

6) This paper aims to concentrate on feature selection for incomplete set-valued data. Given an ISVIS, we propose similarity measurement in an ISVIS and put forward the tolerance relations in an ISVIS based on the similarity measurement. Then, we can obtain θ-conditional entropy in an ISVIS by means of this tolerance relation. Lastly, we propose feature selection algorithms based on λ-discernibility matrix, λ-information granulation, λ-information entropy and λ-significance in an ISVIS. Generally, people study feature selection from two aspects of relation reduction and information entropy reduction. In this way, two methods of feature selection are formed. It has been proved that these two reducts are essentially the same. Thus, these two methods are actually the same. This is the main contribution of this paper.

The comparison and discussion between this paper and several representative literatures are shown in Table 1.

1.4 Organization and structure

The rest of this paper is designed as below. Section 2 retrospects an ISVIS. Section 3 gives the tolerance relation in an ISVIS. Section 4 investigates λ-reduction in an ISVIS. Section 5 gives feature selection algorithms in an ISVIS based on λ-discernibility matrix, λ-information granulation, λ-information entropy and λ-significance. Section 6 concludes this paper.

2 Preliminaries

In this paper, U and AT denote two finite universes, 2^U expresses a set of all subsets of U and |X| means the number of elements in X ∈ 2^U.

Put $U = {u_{1}, u_{2}, \dots, u_{n}}, AT = {a_{1}, a_{2}, \dots, a_{m}},$ $δ = U \times U, ▵ = {(u, u) : u \in U} .$

Let R be a binary relation on U. Then R is said to be a universal relation on U whenever R = δ; R is said to be an identity relation on U whenever R =▵.

Consider that U is a finite object set and AT is a finite feature set. If each feature a ∈ AT determines an information function a : U → V_a, where V_a = {a (u) : u ∈ U} is the set of information function values of the feature a, then (U, AT) is said to be an information system (IS).

Let (U, AT) be an IS. If there are u ∈ U and a ∈ AT such that a (u) is missing, then (U, AT) is regarded as an incomplete information system (IIS). For each a ∈ AT, denote

$V_{a}^{*} = V_{a} - {a (u) : a (u) = *} .$

Definition 2.1. Given that (U, AT) is an IIS. Then (U, AT) is said to be an incomplete set-valued information system (ISVIS), if ∀ a ∈ AT and u ∈ U, a (u) is a set.

If P ⊆ AT, then (U, P) is referred to as a subsystem of (U, AT).

Example 2.2. Table 1 depicts an ISVIS (U, AT), where U = {u₁, u₂, ⋯ , u₁₂} and AT = {a₁, a₂, ⋯ , a₈}.

Table 1
The comparison of this paper with other literatures

Literature Similarity degree Binary relation Attribute selection method Research tools

Difference Dai et al. [5], 2013 $S_{uv}^{a} = \frac{| a (u) \cap a (v) |}{| a (u) \cup a (v) |}$ Fuzzy tolerance relation Discernibility matrix Fuzzy rough set

Lang et al. [12], 2014 $c_{ij}^{l} = \frac{| f (a_{l}, u_{i}) \cap f (a_{l}, u_{j}) |}{| f (a_{l}, u_{i}) \cup f (a_{l}, u_{j}) |}$ Tolerance relation Discernibility matrix RST

Liu et al. [14], 2016 $c_{ij}^{l} = \frac{| f (a_{l}, u_{i} |}{| f (a_{l}, u_{j}) |}$ Dominance relation Discernibility matrix RST

Singh et al. [18], 2019 $S^{a} (u, v) = \frac{2 | a (u) \cap a (v) |}{| a (u) | + | a (v) |}$ Fuzzy tolerance relation Degree of dependency RST

Wang et al. [21], 2014 $S_{ij}^{l} = \min {\frac{| f (a_{l}, u_{i}) \cap f (a_{l}, u_{j}) |}{| f (a_{l}, u_{i}) |}, \frac{| f (a_{l}, u_{i}) \cap f (a_{l}, u_{j}) |}{| f (a_{l}, u_{j}) |}}$ Tolerance relation Discernibility matrix RST

This paper s (a (u) , a (v)) = Tolerance relation Relation reduction Information entropy reduction RST

${\begin{matrix} 1, & u = v; \\ \frac{1}{| V_{a}^{} |^{2}}, & u \neq v, a (u) = , a (v) = ; \\ \frac{1}{| V_{a}^{} |}, & u \neq v, a (u) \neq , a (v) = ; \\ \frac{1}{| V_{a}^{} |}, & u \neq v, a (u) = , a (v) \neq ; \\ 1, & u \neq v, a (u) \neq , a (v) \neq , a (u) = a (v); \\ \frac{| a (u) ⋂ a (v) |}{| a (u) ⋃ a (v) |}, & u \neq v, a (u) \neq , a (v) \neq *, a (u) \neq a (v) . \end{matrix}$

Similarity (1) All these papers have defined similarity degree of set-valued attributes.

(2) All these papers have investigated feature selection methods for a SVDIS.

(3) All these papers have took advantage of RST for a SVDIS.

	Literature	Similarity degree	Binary relation	Attribute selection method	Research tools
Difference	Dai et al. [5], 2013	$S_{uv}^{a} = \frac{\| a (u) \cap a (v) \|}{\| a (u) \cup a (v) \|}$	Fuzzy tolerance relation	Discernibility matrix	Fuzzy rough set
	Lang et al. [12], 2014	$c_{ij}^{l} = \frac{\| f (a_{l}, u_{i}) \cap f (a_{l}, u_{j}) \|}{\| f (a_{l}, u_{i}) \cup f (a_{l}, u_{j}) \|}$	Tolerance relation	Discernibility matrix	RST
	Liu et al. [14], 2016	$c_{ij}^{l} = \frac{\| f (a_{l}, u_{i} \|}{\| f (a_{l}, u_{j}) \|}$	Dominance relation	Discernibility matrix	RST
	Singh et al. [18], 2019	$S^{a} (u, v) = \frac{2 \| a (u) \cap a (v) \|}{\| a (u) \| + \| a (v) \|}$	Fuzzy tolerance relation	Degree of dependency	RST
	Wang et al. [21], 2014	$S_{ij}^{l} = \min {\frac{\| f (a_{l}, u_{i}) \cap f (a_{l}, u_{j}) \|}{\| f (a_{l}, u_{i}) \|}, \frac{\| f (a_{l}, u_{i}) \cap f (a_{l}, u_{j}) \|}{\| f (a_{l}, u_{j}) \|}}$	Tolerance relation	Discernibility matrix	RST
	This paper	s (a (u) , a (v)) =	Tolerance relation	Relation reduction Information entropy reduction	RST
		${\begin{matrix} 1, & u = v; \\ \frac{1}{\| V_{a}^{} \|^{2}}, & u \neq v, a (u) = , a (v) = ; \\ \frac{1}{\| V_{a}^{} \|}, & u \neq v, a (u) \neq , a (v) = ; \\ \frac{1}{\| V_{a}^{} \|}, & u \neq v, a (u) = , a (v) \neq ; \\ 1, & u \neq v, a (u) \neq , a (v) \neq , a (u) = a (v); \\ \frac{\| a (u) ⋂ a (v) \|}{\| a (u) ⋃ a (v) \|}, & u \neq v, a (u) \neq , a (v) \neq *, a (u) \neq a (v) . \end{matrix}$
Similarity	(1) All these papers have defined similarity degree of set-valued attributes.
(2) All these papers have investigated feature selection methods for a SVDIS.
(3) All these papers have took advantage of RST for a SVDIS.

We have

$V_{a_{1}}^{*} = {{1}, {2}}, V_{a_{2}}^{*} = {{1, 2}, {1, 3}},$ $V_{a_{3}}^{*} = {{1, 2}, {2, 3}}, V_{a_{4}}^{*} = {{1, 2, 3}, {2}, {3}},$ $V_{a_{5}}^{*} = {\emptyset, {2, 3}}, V_{a_{6}}^{*} = {{1, 3}, {2}},$ $V_{a_{7}}^{*} = {{1, 3}, {2, 3}}, V_{a_{8}}^{*} = V_{a_{8}} = {{1}, {2, 3}, {1, 2, 3}} .$

3 Tolerance relations in an ISVIS

In this section, the tolerance relation induced by each subsystem in an ISVIS is given.

3.1 The similarity degree between information values on a feature

Definition 3.1. Let (U, AT) be an ISVIS. Then ∀ u, v ∈ U, a ∈ AT, the similarity degree of a (u) and a (v) is defined as follows:

s (a (u) , a (v)) = ${\begin{matrix} 1, & u = v; \\ \frac{1}{| V_{a}^{*} |^{2}}, & u \neq v, a (u) = *, a (v) = *; \\ \frac{1}{| V_{a}^{*} |}, & u \neq v, a (u) \neq *, a (v) = *; \\ \frac{1}{| V_{a}^{*} |}, & u \neq v, a (u) = *, a (v) \neq *; \\ 1, & u \neq v, a (u) \neq *, a (v) \neq *, a (u) = a (v); \\ \frac{| a (u) ⋂ a (v) |}{| a (u) ⋃ a (v) |}, & u \neq v, a (u) \neq *, a (v) \neq *, a (u) \neq a (v) . \end{matrix}$

For the convenience of expression, denote $s_{ij}^{k} = s (a_{k} (u_{i}), a_{k} (u_{j})) .$ $s_{ij}^{k}$ indicates the similarity degree between a_k (u_i) and a_k (u_j). This also expresses the similarity degree between two objects u_i and u_j with respect to the feature a_k.

Example 3.2. (Continued from Example 2.2) ∀ i, j, k, $s_{ij}^{k}$ is obtained as follows (see Tables 2-9).

Table 2
  An ISVIS

a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈

u ₁ * {1, 2} {1, 2} {1, 2, 3} ∅ {2} {2, 3} {1}

u ₂ {2} {1, 3} {2, 3} {3} {2, 3} {1, 3} * {1, 2, 3}

u ₃ {2} * {2, 3} * ∅ {1, 3} {1, 3} {2, 3}

u ₄ {2} {1, 2} {1, 2} {2} ∅ {2} {2, 3} {2, 3}

u ₅ {1} {1, 2} * {2} ∅ {2} {1, 3} {2, 3}

u ₆ {2} {1, 3} {2, 3} {1, 2, 3} {2, 3} {2} * {2, 3}

u ₇ {2} {1, 2} {2, 3} * * * {2, 3} {1}

u ₈ * {1, 2} * {2} ∅ {2} {2, 3} {1}

u ₉ {1} {1, 2} {1, 2} {1, 2, 3} ∅ {1, 3} * {2, 3}

u ₁₀ {2} {1, 3} {2, 3} {2} {2, 3} {1, 3} {2, 3} {1, 2, 3}

u ₁₁ * * * {3} {2, 3} * {2, 3} {1, 2, 3}

u ₁₂ {1} {1, 3} {1, 2} {1, 2, 3} ∅ {2} {1, 2} {2, 3}

	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈
u ₁	*	{1, 2}	{1, 2}	{1, 2, 3}	∅	{2}	{2, 3}	{1}
u ₂	{2}	{1, 3}	{2, 3}	{3}	{2, 3}	{1, 3}	*	{1, 2, 3}
u ₃	{2}	*	{2, 3}	*	∅	{1, 3}	{1, 3}	{2, 3}
u ₄	{2}	{1, 2}	{1, 2}	{2}	∅	{2}	{2, 3}	{2, 3}
u ₅	{1}	{1, 2}	*	{2}	∅	{2}	{1, 3}	{2, 3}
u ₆	{2}	{1, 3}	{2, 3}	{1, 2, 3}	{2, 3}	{2}	*	{2, 3}
u ₇	{2}	{1, 2}	{2, 3}	*	*	*	{2, 3}	{1}
u ₈	*	{1, 2}	*	{2}	∅	{2}	{2, 3}	{1}
u ₉	{1}	{1, 2}	{1, 2}	{1, 2, 3}	∅	{1, 3}	*	{2, 3}
u ₁₀	{2}	{1, 3}	{2, 3}	{2}	{2, 3}	{1, 3}	{2, 3}	{1, 2, 3}
u ₁₁	*	*	*	{3}	{2, 3}	*	{2, 3}	{1, 2, 3}
u ₁₂	{1}	{1, 3}	{1, 2}	{1, 2, 3}	∅	{2}	{1, 2}	{2, 3}

Table 3

$s_{ij}^{1}$

$s_{ij}^{1}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0.5	0.5	0.5	0.5	0.5	0.5	0.25	0.5	0.5	0.25	0.5
u ₂	0.5	1	1	1	0	1	1	0.5	0	1	0.5	0
u ₃	0.5	1	1	1	0	1	1	0.5	0	1	0.5	0
u ₄	0.5	1	1	1	0	1	1	0.5	0	1	0.5	0
u ₅	0.5	0	0	0	1	0	0	0.5	1	0	0.5	1
u ₆	0.5	1	1	1	0	1	1	0.5	0	1	0.5	0
u ₇	0.5	1	1	1	0	1	1	0.5	0	1	0.5	0
u ₈	0.25	0.5	0.5	0.5	0.5	0.5	0.5	1	0.5	0.5	0.25	0.5
u ₉	0.5	0	0	0	1	0	0	0.5	1	0	0.5	1
u ₁₀	0.5	1	1	1	0	1	1	0.5	0	1	0.5	0
u ₁₁	0.25	0.5	0.5	0.5	0.5	0.5	0.5	0.25	0.5	0.5	1	0.5
u ₁₂	0.5	0	0	0	1	0	0	0.5	1	0	0.5	1

Table 4

$s_{ij}^{2}$

$s_{ij}^{2}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0.33	0.5	1	1	0.33	1	1	1	0.33	0.5	0.33
u ₂	0.33	1	0.5	0.33	0.33	1	0.33	0.33	0.33	1	0.5	1
u ₃	0.5	0.5	1	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.25	0.5
u ₄	1	0.33	0.5	1	1	0.33	1	1	1	0.33	0.5	0.33
u ₅	1	0.33	0.5	1	1	0.33	1	1	1	0.33	0.5	0.33
u ₆	0.33	1	0.5	0.33	0.33	1	0.33	0.33	0.33	1	0.5	1
u ₇	1	0.33	0.5	1	1	0.33	1	1	1	0.33	0.5	0.33
u ₈	1	0.33	0.5	1	1	0.33	1	1	1	0.33	0.5	0.33
u ₉	1	0.33	0.5	1	1	0.33	1	1	1	0.33	0.5	0.33
u ₁₀	0.33	1	0.5	0.33	0.33	1	0.33	0.33	0.33	1	0.5	1
u ₁₁	0.5	0.5	0.25	0.5	0.5	0.5	0.5	0.5	0.5	0.5	1	0.5
u ₁₂	0.33	1	0.5	0.33	0.33	1	0.33	0.33	0.33	1	0.5	1

Table 5

$s_{ij}^{3}$

$s_{ij}^{3}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0.33	0.33	1	0.5	0.33	0.33	0.5	1	0.33	0.5	1
u ₂	0.33	1	1	0.33	0.5	1	1	0.5	0.33	1	0.5	0.33
u ₃	0.33	1	1	0.33	0.5	1	1	0.5	0.33	1	0.5	0.33
u ₄	1	0.33	0.33	1	0.5	0.33	0.33	0.5	1	0.33	0.5	1
u ₅	0.5	0.5	0.5	0.5	1	0.5	0.5	0.25	0.5	0.5	0.25	0.5
u ₆	0.33	1	1	0.33	0.5	1	1	0.5	0.33	1	0.5	0.33
u ₇	0.33	1	1	0.33	0.5	1	1	0.5	0.33	1	0.5	0.33
u ₈	0.5	0.5	0.5	0.5	0.25	0.5	0.5	1	0.5	0.5	0.25	0.5
u ₉	1	0.33	0.33	1	0.5	0.33	0.33	0.5	1	0.33	0.5	1
u ₁₀	0.33	1	1	0.33	0.5	1	1	0.5	0.33	1	0.5	0.33
u ₁₁	0.5	0.5	0.5	0.5	0.25	0.5	0.5	0.25	0.5	0.5	1	0.5
u ₁₂	1	0.33	0.33	1	0.5	0.33	0.33	0.5	1	0.33	0.5	1

Table 6

$s_{ij}^{4}$

$s_{ij}^{4}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0.33	0.33	0.33	0.33	1	0.33	0.33	1	0.33	0.33	1
u ₂	0.33	1	0.33	0	0	0.33	0.33	0	0.33	0	1	0.33
u ₃	0.33	0.33	1	0.33	0.33	0.33	0.11	0.33	0.33	0.33	0.33	0.33
u ₄	0.33	0	0.33	1	1	0.33	0.33	1	0.33	1	0	0.33
u ₅	0.33	0	0.33	1	1	0.33	0.33	1	0.33	1	0	0.33
u ₆	1	0.33	0.33	0.33	0.33	1	0.33	0.33	1	0.33	0.33	1
u ₇	0.33	0.33	0.11	0.33	0.33	0.33	1	0.33	0.33	0.33	0.33	0.33
u ₈	0.33	0	0.33	1	1	0.33	0.33	1	0.33	1	0	0.33
u ₉	1	0.33	0.33	0.33	0.33	1	0.33	0.33	1	0.33	0.33	1
u ₁₀	0.33	0	0.33	1	1	0.33	0.33	1	0.33	1	0	0.33
u ₁₁	0.33	1	0.33	0	0	0.33	0.33	0	0.33	0	1	0.33
u ₁₂	1	0.33	0.33	0.33	0.33	1	0.33	0.33	1	0.33	0.33	1

Table 7

$s_{ij}^{5}$

$s_{ij}^{5}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0	1	1	1	0	0.5	1	1	0	0	1
u ₂	0	1	0	0	0	1	0.5	0	0	1	1	0
u ₃	1	0	1	1	1	0	0.5	1	1	0	0	1
u ₄	1	0	1	1	1	0	0.5	1	1	0	0	1
u ₅	1	0	1	1	1	0	0.5	1	1	0	0	1
u ₆	0	1	0	0	0	1	0.5	0	0	1	1	0
u ₇	0.5	0.5	0.5	0.5	0.5	0.5	1	0.5	0.5	0.5	0.5	0.5
u ₈	1	0	1	1	1	0	0.5	1	1	0	0	1
u ₉	1	0	1	1	1	0	0.5	1	1	0	0	1
u ₁₀	0	1	0	0	0	1	0.5	0	0	1	1	0
u ₁₁	0	1	0	0	0	1	0.5	0	0	1	1	0
u ₁₂	1	0	1	1	1	0	0.5	1	1	0	0	1

Table 8

$s_{ij}^{6}$

$s_{ij}^{6}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0	0	1	1	1	0.5	1	0	0	0.5	1
u ₂	0	1	1	0	0	0	0.5	0	1	1	0.5	0
u ₃	0	1	1	0	0	0	0.5	0	1	1	0.5	0
u ₄	1	0	0	1	1	1	0.5	1	0	0	0.5	1
u ₅	1	0	0	1	1	1	0.5	1	0	0	0.5	1
u ₆	1	0	0	1	1	1	0.5	1	0	0	0.5	1
u ₇	0.5	0.5	0.5	0.5	0.5	0.5	1	0.5	0.5	0.5	0.25	0.5
u ₈	1	0	0	1	1	1	0.5	1	0	0	0.5	1
u ₉	0	1	1	0	0	0	0.5	0	1	1	0.5	0
u ₁₀	0	1	1	0	0	0	0.5	0	1	1	0.5	0
u ₁₁	0.5	0.5	0.5	0.5	0.5	0.5	0.25	0.5	0.5	0.5	1	0.5
u ₁₂	1	0	0	1	1	1	0.5	1	0	0	0.5	1

Table 9

$s_{ij}^{7}$

$s_{ij}^{7}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0.33	0.33	1	0.33	0.33	1	1	0.33	1	1	0.33
u ₂	0.33	1	0.33	0.33	0.33	0.11	0.33	0.33	0.11	0.33	0.33	0.33
u ₃	0.33	0.33	1	0.33	1	0.33	0.33	0.33	0.33	0.33	0.33	0.33
u ₄	1	0.33	0.33	1	0.33	0.33	1	1	0.33	1	1	0.33
u ₅	0.33	0.33	1	0.33	1	0.33	0.33	0.33	0.33	0.33	0.33	0.33
u ₆	0.33	0.11	0.33	0.33	0.33	1	0.33	0.33	0.11	0.33	0.33	0.33
u ₇	1	0.33	0.33	1	0.33	0.33	1	1	0.33	1	1	0.33
u ₈	1	0.33	0.33	1	0.33	0.33	1	1	0.33	1	1	0.33
u ₉	0.33	0.11	0.33	0.33	0.33	0.11	0.33	0.33	1	0.33	0.33	0.33
u ₁₀	1	0.33	0.33	1	0.33	0.33	1	1	0.33	1	1	0.33
u ₁₁	1	0.33	0.33	1	0.33	0.33	1	1	0.33	1	1	0.33
u ₁₂	0.33	0.33	0.33	0.33	0.33	0.33	0.33	0.33	0.33	0.33	0.33	1

3.2 Tolerance relations in an ISVIS

Definition 3.3. Consider that (U, AT) is an ISVIS. Given λ ∈ (0, 1] and P ⊆ AT. Then $R_{P}^{λ} \subseteq U \times U$ can be defined as below. $R_{P}^{λ} = {(u, v) \in U \times U : \forall a \in P, s (a (u), a (v)) \geq λ} .$

Obviously, $R_{P}^{λ} \subseteq U \times U$ is a tolerance relation and $R_{P}^{λ} = ⋂_{a \in P} R_{a}^{λ}$ .

In addition, if P = {a}, then $R_{P}^{λ} = R_{{a}}^{λ}$ , which can be briefly expressed as $R_{P}^{λ} = R_{a}^{λ}$ .

Proposition 3.4. Let (U, AT) be an ISVIS. We can get the following properties:

(1) If P₁ ⊆ P₂ ⊆ AT, then ∀ λ ∈ (0, 1], $R_{P_{2}}^{λ} \subseteq R_{P_{1}}^{λ};$

(2) If 0 ≤ λ₁ ≤ λ₂ ≤ 1, then ∀ P ⊆ AT, $R_{P}^{λ_{2}} \subseteq R_{P}^{λ_{1}} .$

Proof. This is clear.□

For any λ ∈ (0, 1], P ⊆ AT and u ∈ U, denote $R_{P}^{λ} (u) = {v \in U : (u, v) \in R_{P}^{λ}} .$

Clearly, $R_{P}^{λ} (u) = ⋂_{a \in P} R_{a}^{λ} (u) .$

In addition, express $U / R_{P}^{λ} = {R_{P}^{λ} (u) : u \in U}$

Corollary 3.5. Suppose that (U, AT) is an ISVIS.

(1) If C ⊆ P ⊆ AT, then ∀ λ ∈ (0, 1] and u ∈ U, $R_{P}^{λ} (u) \subseteq R_{C}^{λ} (u)$ .

(2) If 0 ≤ λ₁ ≤ λ₂ ≤ 1, then ∀ P ⊆ AT and u ∈ U, $R_{P}^{λ_{2}} (u) \subseteq R_{P}^{λ_{1}} (u)$ .

Proof. This can be obtained by Proposition 3.4.

□

Example 3.6. (Continued from Example 3.2)

Pick λ = 0.4. Then ∀ i and u ∈ U, the tolerance class $R_{a_{i}}^{λ} (u)$ of the object u is obtained (see Tables 10-13).

Table 10
$s_{ij}^{8}$

$s_{ij}^{8}$ u₁ u ₂ u ₃ u ₄ u ₅ u ₆ u ₇ u ₈ u ₉ u ₁₀ u ₁₁ u ₁₂

u ₁ 1 0.33 0 0 0 0 1 1 0 0.33 0.33 0

u ₂ 0.33 1 0.67 0.67 0.67 0.67 0.33 0.33 0.67 1 1 0.67

u ₃ 0 0.67 1 1 1 1 0 0 1 0.67 0.67 1

u ₄ 0 0.67 1 1 1 1 0 0 1 0.67 0.67 1

u ₅ 0 0.67 1 1 1 1 0 0 1 0.67 0.67 1

u ₆ 0 0.67 1 1 1 1 0 0 1 0.67 0.67 1

u ₇ 1 0.33 0 0 0 0 1 1 0 0.33 0.33 0

u ₈ 1 0.33 0 0 0 0 1 1 0 0.33 0.33 0

u ₉ 0 0.67 1 1 1 1 0 0 1 0.67 0.67 1

u ₁₀ 0.33 1 0.67 0.67 0.67 0.67 0.33 0.33 0.67 1 1 0.67

u ₁₁ 0.33 1 0.67 0.67 0.67 0.67 0.33 0.33 0.67 1 1 0.67

u ₁₂ 0 0.67 1 1 1 1 0 0 1 0.67 0.67 1

$s_{ij}^{8}$	u₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂
u ₁	1	0.33	0	0	0	0	1	1	0	0.33	0.33	0
u ₂	0.33	1	0.67	0.67	0.67	0.67	0.33	0.33	0.67	1	1	0.67
u ₃	0	0.67	1	1	1	1	0	0	1	0.67	0.67	1
u ₄	0	0.67	1	1	1	1	0	0	1	0.67	0.67	1
u ₅	0	0.67	1	1	1	1	0	0	1	0.67	0.67	1
u ₆	0	0.67	1	1	1	1	0	0	1	0.67	0.67	1
u ₇	1	0.33	0	0	0	0	1	1	0	0.33	0.33	0
u ₈	1	0.33	0	0	0	0	1	1	0	0.33	0.33	0
u ₉	0	0.67	1	1	1	1	0	0	1	0.67	0.67	1
u ₁₀	0.33	1	0.67	0.67	0.67	0.67	0.33	0.33	0.67	1	1	0.67
u ₁₁	0.33	1	0.67	0.67	0.67	0.67	0.33	0.33	0.67	1	1	0.67
u ₁₂	0	0.67	1	1	1	1	0	0	1	0.67	0.67	1

Table 11

The tolerance class of each object under $R_{a}^{λ}$ (a = a₁, a₂, λ = 0.4)

	$R_{a_{1}}^{λ} (u)$	$R_{a_{2}}^{λ} (u)$
u ₁	U - {a₈, a₁₁}	U - {a₂, a₆, a₁₀, a₁₁}
u ₂	U - {a₅, a₉, a₁₂}	{a₂, a₃, a₆, a₁₀, a₁₁, a₁₂}
u ₃	U - {a₅, a₉, a₁₂}	U - {a₁₁}
u ₄	U - {a₅, a₉, a₁₂}	U - {a₂, a₆, a₁₀, a₁₁}
u ₅	{a₁, a₅, a₈, a₉, a₁₁, a₁₂}	U - {a₂, a₆, a₁₀, a₁₁}
u ₆	U - {a₅, a₉, a₁₂}	{a₂, a₃, a₆, a₁₀, a₁₁, a₁₂}
u ₇	U - {a₅, a₉, a₁₂}	U - {a₂, a₆, a₁₀, a₁₁}
u ₈	U - {a₁, a₁₁}	U - {a₂, a₆, a₁₀, a₁₁}
u ₉	{a₁, a₅, a₈, a₉, a₁₁, a₁₂}	U - {a₂, a₆, a₁₀, a₁₁}
u ₁₀	U - {a₅, a₉, a₁₂}	{a₂, a₃, a₆, a₁₀, a₁₁, a₁₂}
u ₁₁	U - {a₁, a₈}	U - {a₃}
u ₁₂	{a₁, a₅, a₈, a₉, a₁₁, a₁₂}	{a₂, a₃, a₆, a₁₀, a₁₁, a₁₂}

Table 12

The tolerance class of each object under $R_{a}^{λ}$ (a = a₃, a₄, λ = 0.4)

	$R_{a_{3}}^{λ} (u)$	$R_{a_{4}}^{λ} (u)$
u ₁	U - {a₂, a₃, a₆, a₇, a₁₀}	{a₁, a₆, a₉, a₁₂}
u ₂	U - {a₁, a₄, a₉, a₁₂}	{a₂, a₁₁}
u ₃	U - {a₁, a₄, a₉, a₁₂}	{a₃}
u ₄	U - {a₂, a₃, a₆, a₇, a₁₀}	{a₄, a₅, a₈, a₁₀}
u ₅	U - {a₈, a₁₁}	{a₄, a₅, a₈, a₁₀}
u ₆	U - {a₁, a₄, a₉, a₁₂}	{a₁, a₆, a₉, a₁₂}
u ₇	U - {a₁, a₄, a₉, a₁₂}	{a₇}
u ₈	U - {a₅, a₁₁}	{a₄, a₅, a₈, a₁₀}
u ₉	U - {a₂, a₃, a₆, a₇, a₁₀}	{a₁, a₆, a₉, a₁₂}
u ₁₀	U - {a₁, a₄, a₉, a₁₂}	{a₄, a₅, a₈, a₁₀}
u ₁₁	U - {a₅, a₈}	{a₂, a₁₁}
u ₁₂	U - {a₂, a₃, a₆, a₇, a₁₀}	{a₁, a₆, a₉, a₁₂}

Table 13

The tolerance class of each object under $R_{a}^{λ}$ (a = a₅, a₆, λ = 0.4)

	$R_{a_{5}}^{λ} (u)$	$R_{a_{6}}^{λ} (u)$
u ₁	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₂	{a₂, a₆, a₇, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₃	U - {a₂, a₆, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₄	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₅	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₆	{a₂, a₆, a₇, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₇	U	U - {a₁₁}
u ₈	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₉	U - {a₂, a₆, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₁₀	{a₂, a₆, a₇, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₁₁	{a₂, a₆, a₇, a₁₀, a₁₁}	U - {a₇}
u ₁₂	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}

4 Feature selection in an ISVIS

4.1 λ-reduction, λ-core and λ-discernibility matrix in an ISVIS

Given an ISVIS (U, AT), each subset P of AT determines a tolerance relation (or indiscernibility relation) $R_{P}^{λ}$ .

Definition 4.1. Consider that (U, AT) is an ISVIS. Suppose P ⊆ AT and λ ∈ (0, 1].

(1) P is regarded as a λ-coordinate subset of AT, if $R_{AT}^{λ} = R_{P}^{λ}$ .

(2) a ∈ P is said to be λ-independent in P, if $R_{P - {a}}^{λ} \neq R_{P}^{λ}$ ; P is regarded as a λ-independent subset of AT, if for each a ∈ P, a is λ-independent in P.

(3) P is regarded as a λ-reduct of AT, if P is both λ-coordinate and λ-independent.

“ $R_{AT}^{λ} = R_{P}^{λ}$ " means that the subsystem (U, P) has the same classification ability as the system (U, AT) with respect to λ.

In this paper, the family of all λ-coordinate subsets (resp., λ-reducts) of AT is denoted by co^λ (AT) (resp., red^λ (AT)).

Obviously, $P \in {red}^{λ} (AT) \Leftrightarrow P \in {co}^{λ} (AT) and \forall a \in P,$ $P - {a} \notin {co}^{λ} (AT) .$

P ∈ red^λ (AT) means that P is a minimal feature subset where the subsystem (U, P) has the same classification ability as the system (U, AT) with respect to a given λ.

Proposition 4.2. Assume that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1]. Then there always exists a λ-reduct of AT.

Proof. If ∀ a ∈ AT, AT - {a} ∉ co^λ (AT), then AT ∈ red^λ (AT).

If ∃ a₁ ∈ AT, AT - {a₁} ∈ co^λ (AT), then we consider AT - {a₁}. Again if ∀ a ∈ AT - {a₁}, (AT - {a₁}) - {a} ∉ co^λ (AT), then AT - {a₁} ∈ red^λ (AT). Again if ∃ a₂ ∈ AT - {a₁}, (AT - {a₁}) - {a₂} ∈ co^λ (AT), then we consider AT - {a₁, a₂}. Repeat this process. Since AT is finite, a λ-reduct of AT can be found.□

Definition 4.3. Suppose that (U, AT) is an ISVIS. For any u, v ∈ U and λ ∈ (0, 1], put $D^{λ} (u, v) = {a \in AT : s (a (u), a (v)) < λ} .$

Then

(1) D^λ (u, v) is said to be the λ-discernibility set on u and v.

(2) $D^{λ}$ (AT) = (d_ij) _n×n is regarded as the λ-discernibility matrix where U = {u₁, u₂, ·· · , u_n} and d_ij = D^λ (u_i, u_j) (1 ≤ i, j ≤ n).

Example 4.4. (Continued from Example 2.2)Pick λ = 0.4, $D^{λ}$ (AT) of AT is obtained as follows: $(\begin{matrix} \tilde{\emptyset} & AT - {a_{1}} & {a_{3}, a_{4}, a_{6}, a_{7}, a_{8}} & {a_{4}, a_{8}} \\ \emptyset & {a_{4}, a_{5}, a_{7}} & AT - {a_{1}, a_{8}} \\ \emptyset & {a_{3}, a_{4}, a_{6}, a_{7}} \\ \emptyset \end{matrix}$ $\begin{matrix} {a_{4}, a_{7}, a_{8}} & {a_{2}, a_{3}, a_{5}, a_{7}, a_{8}} & {a_{3}, a_{4}} \\ AT - {a_{3}, a_{8}} & {a_{4}, a_{6}, a_{7}} & {a_{2}, a_{4}, a_{7}, a_{8}} \\ {a_{1}, a_{4}, a_{6}} & {a_{4}, a_{5}, a_{6}, a_{7}} & {a_{4}, a_{7}, a_{8}} \\ {a_{1}, a_{7}} & {a_{2}, a_{3}, a_{4}, a_{5}, a_{7}} & {a_{3}, a_{4}, a_{8}} \\ \emptyset & {a_{1}, a_{2}, a_{4}, a_{5}, a_{7}} & {a_{1}, a_{4}, a_{7}, a_{8}} \\ \emptyset & {a_{2}, a_{4}, a_{7}, a_{8}} \\ \emptyset \end{matrix}$ $\begin{matrix} {a_{1}, a_{4}} & {a_{6}, a_{7}, a_{8}} & AT - {a_{1}, a_{7}} \\ AT - {a_{1}, a_{3}} & AT - {a_{6}, a_{8}} & {a_{4}, a_{7}} \\ {a_{4}, a_{6}, a_{7}, a_{8}} & {a_{1}, a_{3}, a_{4}, a_{7}} & {a_{4}, a_{5}, a_{7}} \\ {a_{8}} & {a_{1}, a_{4}, a_{6}, a_{7}} & {a_{2}, a_{3}, a_{5}, a_{6}} \\ {a_{3}, a_{7}, a_{8}} & {a_{4}, a_{6}, a_{7}} & {a_{1}, a_{2}, a_{5}, a_{6}, a_{7}} \\ {a_{2}, a_{4}, a_{5}, a_{7}, a_{8}} & AT - {a_{4}, a_{8}} & {a_{4}, a_{6}, a_{7}} \\ {a_{4}} & {a_{1}, a_{3}, a_{4}, a_{7}, a_{8}} & {a_{2}, a_{4}, a_{8}} \\ \emptyset & {a_{4}, a_{6}, a_{7}, a_{8}} & {a_{2}, a_{5}, a_{6}, a_{8}} \\ \emptyset & AT - {a_{6}, a_{8}} \\ \emptyset \end{matrix}$ $\begin{matrix} {a_{1}, a_{4}, a_{5}, a_{8}} & {a_{2}, a_{7}, a_{8}} \\ {a_{7}} & AT - {a_{2}, a_{8}} \\ {a_{2}, a_{4}, a_{5}, a_{7}} & {a_{1}, a_{3}, a_{4}, a_{6}, a_{7}} \\ {a_{4}, a_{5}} & {a_{1}, a_{2}, a_{4}, a_{7}} \\ {a_{3}, a_{4}, a_{5}, a_{7}} & {a_{2}, a_{4}, a_{7}} \\ {a_{4}, a_{7}} & {a_{1}, a_{3}, a_{5}, a_{7}} \\ {a_{4}, a_{6}, a_{8}} & AT - {a_{5}, a_{6}} \\ {a_{1}, a_{3}, a_{4}, a_{5}, a_{8}} & {a_{2}, a_{4}, a_{7}, a_{8}} \\ {a_{4}, a_{5}, a_{7}} & {a_{2}, a_{6}, a_{7}} \\ {a_{4}} & AT - {a_{2}, a_{8}} \\ \emptyset & {a_{4}, a_{5}, a_{7}} \\ \emptyset \end{matrix})$

4.2 Some properties

Proposition 4.5. Suppose that (U, AT) is an ISVIS. Then for any u, v, w ∈ U and λ ∈ (0, 1],

(1) D^λ (u, u) =∅.

(2) D^λ (u, v) = D^λ (v, u).

(3) D^λ (u, v) ⊆ D^λ (u, w) ∪ D^λ (w, v).

Proof. (1) and (2) are obvious.

(3) Suppose that D^λ (u, v) ⊊ D^λ (u, w) ∪ D^λ (w, v). Then D^λ (u, v) - D^λ (u, w)∪ D^λ (w, v) ≠ ∅. Pick $a \in D^{λ} (u, v) - D^{λ} (u, w) \cup D^{λ} (w, v) .$

a ∈ D^λ (u, v) implies $(u, v) \notin R_{{a}}^{λ}$ .

Since a ∉ D^λ (u, w) ∪ D^λ (w, v), we have a ∉ D^λ (u, w) and a ∉ D^λ (w, v). Then $(u, w) \in R_{{a}}^{λ}$ and $(w, v) \in R_{{a}}^{λ}$ .

Thus D^λ (u, v) ⊆ D^λ (u, w) ∪ D^λ (w, v).□

Corollary 4.6. Let (U, AT) be an ISVIS. Then for any u, v ∈ U and λ ∈ (0, 1]. Then d is a distance function on U where $d (u, v) = | D^{λ} (u, v) | .$

Proof. It can be obtained by Proposition 4.5.□

Proposition 4.7. Assume that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1]. Then

$P \in {co}^{λ} (AT) \Leftrightarrow if (u, v) \notin R_{AT}^{λ}, then P \cap D^{λ} (u, v) \neq \emptyset .$

Proof. “⇒". Let $(u, v) \notin R_{AT}^{λ}$ . Since P ∈ co^λ (AT), we have $R_{P}^{λ} = R_{AT}^{λ} .$ Then $(u, v) \notin R_{P}^{λ} .$ So there exists a ∈ P such that s (a (u) , a (v)) < λ. This implies a ∈ D^λ (u, v). Then a ∈ P ∩ D^λ (u, v).

Thus P ∩ D^λ (u, v) ≠ ∅ .

“⇐". Suppose P ∉ co^λ (AT). Then $R_{P}^{λ} \neq R_{AT}^{λ} .$ This implies $R_{P}^{λ} - R_{AT}^{λ} \neq \emptyset .$ Pick $(u, v) \in R_{P}^{λ} - R_{AT}^{λ} .$

Since $(u, v) \notin R_{AT}^{λ}$ , we have P ∩ D^λ (u, v) ≠ ∅ .

Note that $(u, v) \in R_{P}^{λ}$ . Then for each a ∈ P, s (a (u) , a (v)) ≥ λ. So a ∉ D^λ (u, v) . Thus P ∩ D^λ (u, v) = ∅ . This is a contradiction.

Hence P ∈ co^λ (AT).□ λ-discernibility sets can easily determine λ-reducts.

Theorem 4.8. Let (U, AT) be an ISVIS. Then ∀ u, v ∈ U and λ ∈ (0, 1],

P ∈ red^λ (AT) ⇔ (i) if $(u, v) \notin R_{AT}^{λ}$ , then P∩ D^λ (u, v) ≠ ∅ ;

(ii) ∀ a ∈ P, $\exists (u_{a}, v_{a}) \in R_{AT}^{λ}$ , (P - {b})∩ D^λ (u_a, v_a) = ∅.

Proof. It can be proved by Proposition 4.7.□

Definition 4.9. Suppose that (U, AT) is an ISVIS. Put λ ∈ (0, 1] ${core}^{λ} (AT) = ⋂_{P \in {red}^{λ} (AT)} P .$ Then core^λ (AT) is said to be the λ-core of AT. Moreover,

(1) a ∈ AT is regarded as a necessary λ-feature, if a ∈ core^λ (AT).

(2) a ∈ AT is said to be a relatively necessary λ-feature, if a ∈ ⋃ _{P∈red^λ(AT)}P - core^λ (AT).

(3) a ∈ AT is regarded as an unnecessary λ-feature, if a ∈ AT - ⋃ _{P∈red^λ(AT)}P.

Proposition 4.10. Assume that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1], $| {red}^{λ} (AT) | = 1 \Leftrightarrow {core}^{λ} (AT) \in {red}^{λ} (AT) .$

Proof. “⇒" is obvious.

“⇐". Denote red^λ (AT) = {P_k : 1 ≤ k ≤ n}. We only need to prove n = 1.

Suppose n ≥ 2. Since core^λ (AT) ∈ red^λ (AT), there exists i such that core^λ (AT) = P_i. Pick j ≠ i. Then $P_{i} = ⋂_{k = 1}^{n} P_{k} \subseteq P_{j}$ . Put P_i ≠ P_j. Thus P_i ⊂ P_j. Since P_j ∈ red^λ (AT), we have P_i ∉ co^λ (AT). Then P_i ∉ red^λ (AT). This is a contradiction.

Thus n = 1.□ λ-discernibility sets can easily determine the λ-core.

Proposition 4.11. Provided that (U, AT) is an ISVIS. Put λ ∈ (0, 1]. Then the following conditions are equivalent:

(1) a is a necessary λ-feature;

(2) a is λ-independent in AT;

(3) ∃ u, v ∈ U, D^λ (u, v) = {a}.

Proof.

(1) ⇒ (2). Let a be a necessary λ-feature. Suppose that a is not λ-independent in AT. Then $R_{AT - {a}}^{λ} = R_{AT}^{λ} .$ This implies AT - {a} ∈ co^λ (AT). Consider AT - {a}, by Proposition 4.2, ∃ P ⊆ AT - {a}, P ∈ red^λ (AT).

P ⊆ AT - {a} implies a ∉ P. Then a is not a necessary λ-feature. This is a contradiction.

(2) ⇒ (1). Let a be λ-independent in AT. Suppose that a is not a necessary λ-feature. Then ∃ P ∈ red^λ (AT), a ∉ P. So P ⊆ AT - {a} ⊆ AT. This implies $R_{P}^{λ} \supseteq R_{AT - {a}}^{λ} \supseteq R_{AT}^{λ} .$

Since P ∈ red^λ (AT), we have $R_{P}^{λ} = R_{AT}^{λ}$ . Then $R_{AT - {a}}^{λ} = R_{AT}^{λ} .$ So a is not λ-independent in AT. This is a contradiction.

(2) ⇒ (3). Since a is λ-independent in AT, we have $R_{AT - {a}}^{λ} \neq R_{AT}^{λ}$ . Then $R_{AT - {a}}^{λ} - R_{AT}^{λ} \neq \emptyset$ . Pick $(u, v) \in R_{AT - {a}}^{λ} - R_{AT}^{λ} .$ Denote AT = {a₁, a₂, …, a_n}. Then ∃ j, a = a_j. So $(u, v) \in ⋂_{1 \leq i \leq n, i \neq j} R_{{a_{i}}}^{λ} - ⋂_{1 \leq i \leq n} R_{{a_{i}}}^{λ} .$ This implies that $(u, v) \notin R_{{a_{j}}}^{λ}$ and $(u, v) \in R_{{a_{i}}}^{λ} (i \neq j) .$

Then a_j ∈ D^λ (u, v), a_i ∉ D^λ (u, v) (i ≠ j) .

Thus D^λ (u, v) = {a_j} = {a}.

(3) ⇒ (2). Since ∃ u, v ∈ U, D^λ (u, v) = {a}, we have $(u, v) \notin R_{{a}}^{λ}, (u, v) \in R_{{a^{'}}}^{λ} (a^{'} \neq a) .$

Then $(u, v) \in R_{AT - {a}}^{λ}$ . But $(u, v) \notin R_{AT}^{λ}$ .

Thus $R_{AT - {a}}^{λ} \neq R_{AT}^{λ} .$ Hence a is λ-independent in AT.□

Proposition 4.12. Let (U, AT) be an ISVIS. Suppose λ ∈ (0, 1]. Denote $R (a) = ⋃_{P \in {co}^{λ} (AT)} R_{P - {a}}^{λ} .$ Then the following conditions are equivalent:

(1) a is an unnecessary λ-feature;

(2) ∀ P ∈ co^λ (AT), P - {a}∈ co^λ (AT) ;

(3) $R (a) = R_{AT}^{λ}$ ;

(4) $R (a) \subseteq R_{{a}}^{λ}$ .

Proof.

(1) ⇒ (2). Suppose P ∈ co^λ (AT). By Proposition 4.2, ∃ C ⊆ P, C ∈ red^λ (AT). Since a is an unnecessary λ-feature, we have a ∉ C, which implies C ⊆ AT - {a}. Then $C \subseteq P \cap (AT - {a}) = P - {a} \subseteq P .$ We have $R_{C}^{λ} \supseteq R_{P - {a}}^{λ} \supseteq R_{P}^{λ} .$

Note that P ∈ co^λ (AT) and C ∈ red^λ (AT). Then $R_{P}^{λ} = R_{AT}^{λ} = R_{C}^{λ}$ .

Thus $R_{P - {a}}^{λ} = R_{AT}^{λ} .$

Hence P - {a} ∈ co^λ (AT) .

(2) ⇒ (3) ⇒ (4) are obvious.

(4) ⇒ (1). Suppose that a is not an unnecessary λ-feature. Then ∃ P ∈ red^λ (AT), a ∈ P. This implies P - {a} ⊂ P. Since P ∈ red^λ (AT), we have P - {a} ∉ co^λ (AT). Then $R_{P - {a}}^{λ} - R_{AT}^{λ} \neq \emptyset .$ P ∈ red^λ (AT) implies $R_{P}^{λ} = R_{AT}^{λ}$ . Then $R_{P - {a}}^{λ} - R_{P}^{λ} \neq \emptyset .$

Pick $(u, v) \in R_{P - {a}}^{λ} - R_{P}^{λ}$ . Note that $R_{P}^{λ} = R_{P - {a}}^{λ} \cap R_{{a}}^{λ}$ . Then $(u, v) \notin R_{{a}}^{λ}$ .

Since P ∈ co^λ (AT) and $R (a) \subseteq R_{{a}}^{λ}$ , we have $R_{P - {a}}^{λ} \subseteq R_{{a}}^{λ}$ . Then $(u, v) \in R_{{a}}^{λ}$ . This is a contradiction.□

Theorem 4.13. Suppose that (U, AT) is an ISVIS. Put λ ∈ (0, 1]. Then the following properties hold:

(1) a is a necessary λ-feature ⇔ AT - {a} ∉ co^λ (AT);

(2) a is a relatively necessary λ-feature ⇔ AT - {a} ∈ co^λ (AT) and $R (a) ⊊ R_{{a}}^{λ}$ ;

(3) a is an unnecessary λ-feature ⇔ ∀ P ∈ co^λ (AT), P - {a} ∈ co^λ (AT) .

Proof. It can be proved by Propositions 4.11 and 4.12.□

Example 4.14. (Continued from Example 4.4) Pick λ = 0.4, we have

(1) a₄, a₇ and a₈ are three necessary λ-feature;

(2) a₂, a₃, a₅ and a₆ are four relatively necessary λ-features;

(3) a₁ is an unnecessary λ-features.

4.3 Uncertainty measurement for an ISVIS

Definition 4.15. Consider that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1] and P ⊆ AT. Then λ-information granulation of the subsystem (U, P) is defined as $G^{λ} (P) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | R_{P}^{λ} (u_{i}) |^{2} .$

Proposition 4.16. Assume that (U, AT) is an ISVIS. Suppose P ⊆ AT and λ ∈ (0, 1]. Then $\frac{1}{n} \leq G^{λ} (P) \leq n .$ Moreover, if $R_{P}^{λ}$ is an identity relation on U, then G^λ achieves the minimum value $\frac{1}{n}$ ; if $R_{P}^{λ}$ is a universal relation on U, then G^λ achieves the maximum value n.

Proof. Since for each i, $1 \leq | R_{P}^{λ} (u_{i}) | \leq n$ , we have $n \leq \sum_{i = 1}^{n} | R_{P}^{λ} (u_{i}) |^{2} \leq n^{3}$ .

By Definition 4.15, $\frac{1}{n} \leq G^{λ} (P) \leq n .$

If $R_{P}^{λ}$ is an identity relation on U, then ∀ i, $| R_{P}^{λ} (u_{i}) | = 1$ . So $G^{λ} (P) = \frac{1}{n}$ .

If $R_{P}^{λ}$ is a universal relation on U, then ∀ i, $| R_{P}^{λ} (u_{i}) | = n$ . So G^λ (P) = n.□

Proposition 4.17. Assume that (U, AT) is an ISVIS.

(1) If P₁ ⊆ P₂ ⊆ AT, then ∀ λ ∈ (0, 1], G^λ (P₂) ≤ G^λ (P₁).

(2) If 0 < λ₁ ≤ λ₂ ≤ 1, then for any P ⊆ AT, G^{λ
₂} (P) ≤ G^{λ
₁} (P).

Proof. This follows from Proposition 4.16.□

Definition 4.18. Suppose that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1] and P ⊆ AT. Then λ-information entropy of the subsystem (U, P) is defined as $H^{λ} (P) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| R_{P}^{λ} (u_{i}) |}{n} .$

Proposition 4.19. Assume that (U, AT) is an ISVIS. Suppose P ⊆ AT and λ ∈ (0, 1]. Then $0 \leq H^{λ} (P) \leq {log}_{2} n .$ Moreover, if $R_{P}^{λ}$ is a universal relation on U, then H^λ achieves the minimum value 0; if $R_{P}^{λ}$ is an identity relation on U, then H^λ achieves the maximum value log ₂n.

Proof. Since ∀ i, $1 \leq | R_{P}^{λ} (u_{i}) | \leq n$ , we have $0 \leq - {log}_{2} \frac{| R_{P}^{λ} (u_{i}) |}{n} \leq {log}_{2} n .$

By Definition 4.18, $0 \leq H^{λ} (P) \leq \log_{2} n .$

If $R_{P}^{λ}$ is an identity relation on U, then ∀ i, $| R_{P}^{λ} (u_{i}) | = 1$ . So H^λ (P) = log₂n.

If $R_{P}^{λ}$ is a universal relation on U, then ∀ i, $| R_{P}^{λ} (u_{i}) | = n$ . So H^λ (P) =0.□

Proposition 4.20. Assume that (U, AT) is an ISVIS.

(1) If P₁ ⊆ P₂ ⊆ AT, then ∀ λ ∈ (0, 1], H^λ (P₁) ≤ H^λ (P₂).

(2) If 0 < λ₁ ≤ λ₂ ≤ 1, then for any P ⊆ AT, H^{λ
₁} (P) ≤ H^{λ
₂} (P).

Proof. These follow from Corrollary 3.5.□

Theorem 4.21. Let (U, AT) be an ISVIS. Suppose λ ∈ (0, 1] and P ⊆ AT. Then the following conditions are equivalent:

(1) P ∈ co (AT);

(2) G^λ (P) = G^λ (AT) ;

(3) H^λ (P) = H^λ (AT) .

Proof.

(1) ⇒ (2). This is obvious.

(2) ⇒ (1). Suppose G^λ (P) = G^λ (AT). Then, we have $\frac{1}{n^{2}} \sum_{i = 1}^{n} | R_{P}^{λ} (u_{i}) | = \frac{1}{n^{2}} \sum_{i = 1}^{n} | R_{AT}^{λ} (u_{i}) | .$

So $\sum_{i = 1}^{n} (| R_{P}^{λ} (u_{i}) | - | R_{AT}^{λ} (u_{i}) |) = 0 .$

Note that $R_{AT}^{λ} \subseteq R_{P}^{λ}$ . Then ∀ i, $R_{AT}^{λ} (u_{i}) \subseteq R_{P}^{λ} (u_{i})$ . This implies that $\forall i, | R_{P}^{λ} (u_{i}) | - | R_{AT}^{λ} (u_{i}) | \geq 0 .$

So ∀ i, $| R_{AT}^{λ} (u_{i}) | = | R_{P}^{λ} (u_{i}) |$ .

Thus R_P = R_AT .

Hence $P \in co (AT) .$

(1) ⇒ (3). This is clear.

(3) ⇒ (1). Suppose H^λ (P) = H^λ (AT). Then, we have $- \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| R_{P}^{λ} (u_{i}) |}{n} = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| R_{AT}^{λ} (u_{i}) |}{n} .$

So $\sum_{i = 1}^{n} {log}_{2} \frac{| R_{P}^{λ} (u_{i}) |}{| R_{AT}^{λ} (u_{i}) |} = 0 .$

Note that $R_{AT}^{λ} \subseteq R_{P}^{λ}$ . Then ∀ i, $R_{AT}^{λ} (u_{i}) \subseteq R_{P}^{λ} (u_{i})$ . This implies that $\forall i, {log}_{2} \frac{| R_{P}^{λ} (u_{i}) |}{| R_{AT}^{λ} (u_{i}) |} \geq 0 .$

Thus ∀ i, ${log}_{2} \frac{| R_{P}^{λ} (u_{i}) |}{| R_{AT}^{λ} (u_{i}) |} = 0 .$ So ∀ i, $| R_{AT}^{λ} (u_{i}) | = | R_{P}^{λ} (u_{i}) |$ .

Consequently, R_P = R_AT .

Hence $P \in co (AT) .$ □

Definition 4.22. Assume that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1], P ⊆ AT and a ∈ AT - P. Then λ-significance of the feature a relative to P based on λ-information granulation is defined as ${Sig}_{G}^{λ} (a, P) = G^{λ} (P - {a}) - G^{λ} (P);$ λ-significance of the feature a relative to P based on λ-information entropy is defined as ${Sig}_{H}^{λ} (a, P) = H^{λ} (P) - H^{λ} (P - {a}) .$

We stipulate ${Sig}_{G}^{λ} (a, \emptyset) = 0;$ ${Sig}_{H}^{λ} (a, \emptyset) = 0 .$

Theorem 4.23. Let (U, AT) be an ISVIS. Suppose λ ∈ (0, 1] and P ⊆ AT. Then the following conditions are equivalent:

(1) P ∈ red (AT);

(2) G^λ (P) = G^λ (AT) and ∀ a ∈ P, G^λ (P - {a}) ≠ G^λ (AT);

(3) H^λ (P) = H^λ (AT) and ∀ a ∈ P, H^λ (P - {a}) ≠ H^λ (AT);

(4) G^λ (P) = G^λ (AT) and ∀ a ∈ P, ${Sig}_{G}^{λ} (a, P) > 0$ ;

(5) H^λ (P) = H^λ (AT) and ∀ a ∈ P, ${Sig}_{H}^{λ} (a, P) > 0$ .

Proof. It can be proved by Theorem 4.21.□

5 Algorithms on λ-reduction in an ISVIS

Below, we give an algorithm on λ-reduction in an ISVIS by using mathematical logic.

Let (U, AT) be an ISVIS. ∀ a ∈ AT, we specify a Boolean variable “a”. If D^λ (u, v) = {a₁, a₂, ·· · , a_k} with u, v ∈ U, then we specify a Boolean function a₁ ∨ a₂ ∨ ·· · ∨ a_k.

Denote $⋁ {a_{1}, a_{2}, \cdot \cdot \cdot, a_{k}} or ⋁_{i = 1}^{k} a_{i} = a_{1} \lor a_{2} \lor \cdot \cdot \cdot \lor a_{k},$ $⋀ {a_{1}, a_{2}, \cdot \cdot \cdot, a_{k}} or ⋀_{i = 1}^{k} a_{i} = a_{1} \land a_{2} \land \cdot \cdot \cdot \land a_{k} .$

We stipulate that ∨ ∅ =1 and ∧ ∅ =0 where 0 and 1 are two Boolean constants.

Definition 5.1. Provided that (U, AT) is an ISVIS. Suppose λ ∈ (0, 1]. Then the λ-discernibility matrix of AT, denoted by $D^{λ}$ (AT) = (d_ij) _n×n, is define as $Δ^{λ} (AT) = ⋀ (⋁ d_{ij}) .$

Example 5.2. (Continued from Example 4.4) Pick λ = 0.4, we have

▵^0.4 (AT) = a₈ ∧ a₄ ∧ a₇ ∧ (a₄ ∨ a₈) ∧ (a₃ ∨ a₄) ∧ (a₁ ∨ a₄) ∧ (a₄ ∨ a₇) ∧ (a₄ ∨ a₅) ∧ (a₁ ∨ a₇) ∧ (a₄ ∨ a₇ ∨ a₈) ∧ (a₄ ∨ a₅ ∨ a₇) ∧ (a₁ ∨ a₄ ∨ a₆) ∧ (a₄ ∨ a₆ ∨ a₇) ∧ (a₄ ∨ a₆ ∨ a₈) ∧ (a₃ ∨ a₄ ∨ a₈) ∧ (a₃ ∨ a₇ ∨ a₈) ∧ (a₂ ∨ a₄ ∨ a₈) ∧ (a₂ ∨ a₆ ∨ a₇) ∧ (a₂ ∨ a₄ ∨ a₇) ∧ (a₂ ∨ a₇ ∨ a₈) ∧ (a₆ ∨ a₇ ∨ a₈) ∧ (a₃ ∨ a₄ ∨ a₆ ∨ a₇) ∧ (a₄ ∨ a₅ ∨ a₆ ∨ a₇) ∧ (a₄ ∨ a₆ ∨ a₇ ∨ a₈) ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₇) ∧ (a₁ ∨ a₄ ∨ a₇ ∨ a₈) ∧ (a₂ ∨ a₄ ∨ a₇ ∨ a₈) ∧ (a₁ ∨ a₄ ∨ a₆ ∨ a₇) ∧ (a₂ ∨ a₅ ∨ a₆ ∨ a₈) ∧ (a₁ ∨ a₄ ∨ a₅ ∨ a₈) ∧ (a₂ ∨ a₄ ∨ a₅ ∨ a₇) ∧ (a₁ ∨ a₃ ∨ a₅ ∨ a₇) ∧ (a₁ ∨ a₂ ∨ a₄ ∨ a₇) ∧ (a₃ ∨ a₄ ∨ a₅ ∨ a₇) ∧ (a₂ ∨ a₃ ∨ a₅ ∨ a₆) ∧ (a₃ ∨ a₄ ∨ a₆ ∨ a₇ ∨ a₈) ∧ (a₂ ∨ a₃ ∨ a₅ ∨ a₇ ∨ a₈) ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₇) ∧ (a₁ ∨ a₂ ∨ a₄ ∨ a₅ ∨ a₇) ∧ (a₂ ∨ a₄ ∨ a₅ ∨ a₇ ∨ a₈) ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₇ ∨ a₈) ∧ (a₁ ∨ a₂ ∨ a₅ ∨ a₆ ∨ a₇) ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₈) ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₆ ∨ a₇) ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇) ∧ (a₁ ∨ a₂ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇) ∧ (a₂ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇ ∨ a₈) ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₇) ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₅ ∨ a₆ ∨ a₇) ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₈) ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇) ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄ ∨ a₇ ∨ a₈) ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇ ∨ a₈)

Denote $L (AT) = {⋁ d_{ij} : 1 \leq i, j \leq n} .$

Define

⋁d_ij ≤ ⋁ d_kl ⇔ d_ij ⊆ d_kl, ∀ ⋁ d_ij, ⋁ d_kl ∈ L (AT) .

Denote

(⋁ d_ij) ⋃ (⋁ d_kl) = ⋁ (d_ij ∪ d_kl) ,

(⋁ d_ij) ⋂ (⋁ d_kl) = ⋁ (d_ij ∩ d_kl) .

Proposition 5.3. (L (AT) , ≤) is a poset.

Proof. (1) ⋁d_ij ≤ ⋁ d_ij ∀ ⋁ d_ij ∈ L (AT). (2) Suppose ⋁d_ij, ⋁ d_kl ∈ L (AT). Suppose that ⋁d_ij ≤ ⋁ d_kl and ⋁d_kl ≤ ⋁ d_ij. Then d_ij ⊆ d_kl and d_kl ⊆ d_ij. This implies that d_ij = d_kl. So ⋁d_ij = ⋁ d_kl. (3) Suppose ⋁d_ij, ⋁ d_kl, ⋁ d_hv ∈ L (AT). Suppose that ⋁d_ij ≤ ⋁ d_kl and ⋁d_kl ≤ ⋁ d_hv. Then d_ij ⊆ d_kl and d_kl ⊆ d_hv. This implies that d_ij ⊆ d_hv. So ⋁d_ij ≤ ⋁ d_hv. Thus (L (AT) , ≤) is a poset.□

Proposition 5.4. Let (U, AT) be an ISVIS. Suppose U = {u₁, u₂, ·· · , u_n} and λ ∈ (0, 1]. If {d_ij : 1 ≤ i, j ≤ n} is a topology on AT, then (L (AT) , ≤ , ⊔ , ⊓) is a lattice with top element and bottom element.

Proof. Denote τ = {d_ij : 1 ≤ i, j ≤ n}. By Proposition 5.3, (L (AT) , ≤) is a poset. For ⋁d_ij, ⋁ d_kl ∈ L (AT), since τ is a topology on AT, we have d_ij ∪ d_kl ∈ τ, d_ij ∩ d_kl ∈ τ. This implies $(⋁ d_{ij}) ⋃ (⋁ d_{kl}) = ⋁ (d_{ij} \cup d_{kl}) \in L (AT),$ $(⋁ d_{ij}) ⋂ (⋁ d_{kl}) = ⋁ (d_{ij} \cap d_{kl}) \in L (AT) .$ Obviously, 1_L(AT) = ∨ AT, 0_L(AT) =∨ ∅.

Thus (L (AT) , ≤ , ⋃ , ⋂) is a lattice with top element and bottom element.□

Definition 5.5. Given an ISVIS (U, AT). If λ ∈ (0, 1] and $Δ^{λ} (AT) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl})$ , where every P_k = {a_kl : l ≤ p_k} ⊆ AT has not repetitive elements, then $⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl})$ is called the standard minimum formula of Δ^λ (AT). We denote it by $Δ_{*}^{λ} (AT)$ . That is, $Δ_{*}^{λ} (AT) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl}) .$

Example 5.6. (Continued from Example 5.2)

Pick λ = 0.4, we have a₄ ∧ (a₄ ∨ a₈) = a₄, a₄ ∧ (a₃ ∨ a₄) = a₄, a₄ ∧ (a₁ ∨ a₄) = a₄, a₄ ∧ (a₄ ∨ a₇) = a₄, a₄ ∧ (a₄ ∨ a₅) = a₄, a₄ ∧ (a₄ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₄ ∨ a₅ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₄ ∨ a₆) = a₄, a₄ ∧ (a₄ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₄ ∨ a₆ ∨ a₈) = a₄, a₄ ∧ (a₃ ∨ a₄ ∨ a₈) = a₄, a₄ ∧ (a₂ ∨ a₄ ∨ a₈) = a₄, a₄ ∧ (a₂ ∨ a₄ ∨ a₇) = a₄, a₄ ∧ (a₃ ∨ a₄ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₄ ∨ a₅ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₄ ∨ a₆ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₄ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₂ ∨ a₄ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₄ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₄ ∨ a₅ ∨ a₈) = a₄, a₄ ∧ (a₂ ∨ a₄ ∨ a₅ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₂ ∨ a₄ ∨ a₇) = a₄, a₄ ∧ (a₃ ∨ a₄ ∨ a₅ ∨ a₇) = a₄, a₄ ∧ (a₃ ∨ a₄ ∨ a₆ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₂ ∨ a₄ ∨ a₅ ∨ a₇) = a₄, a₄ ∧ (a₂ ∨ a₄ ∨ a₅ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₂ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₂ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₇) = a₄, a₄ ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₈) = a₄, a₄ ∧ (a₁ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇) = a₄, a₄ ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄ ∨ a₇ ∨ a₈) = a₄, a₄ ∧ (a₂ ∨ a₃ ∨ a₄ ∨ a₅ ∨ a₆ ∨ a₇ ∨ a₈) = a₄, a₇ ∧ (a₁ ∨ a₇) = a₇, a₇ ∧ (a₃ ∨ a₇ ∨ a₈) = a₇, a₇ ∧ (a₂ ∨ a₆ ∨ a₇) = a₇, a₇ ∧ (a₂ ∨ a₇ ∨ a₈) = a₇, a₇ ∧ (a₁ ∨ a₃ ∨ a₅ ∨ a₇) = a₇, a₇ ∧ (a₂ ∨ a₃ ∨ a₅ ∨ a₇ ∨ a₈) = a₇, a₇ ∧ (a₁ ∨ a₂ ∨ a₅ ∨ a₆ ∨ a₇) = a₇, a₇ ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₅ ∨ a₆ ∨ a₇) = a₇, a₈ ∧ (a₂ ∨ a₅ ∨ a₆ ∨ a₈) = a₈ .

Then $Δ_{*}^{0.4} (AT) = a_{4} \land a_{7} \land a_{8} \land (a_{2} \lor a_{3} \lor a_{5} \lor a_{6}) = (a_{4} \land a_{7} \land a_{8} \land a_{2}) \lor (a_{4} \land a_{7} \land a_{8} \land a_{3}) \lor (a_{4} \land a_{7} \land a_{8} \land a_{5}) \lor (a_{4} \land a_{7} \land a_{8} \land a_{6})$

Thus $Δ_{*}^{0.4} (AT) = (a_{4} \land a_{7} \land a_{8} \land a_{2}) \lor (a_{4} \land a_{7} \land a_{8} \land a_{3}) \lor (a_{4} \land a_{7} \land a_{8} \land a_{5}) \lor (a_{4} \land a_{7} \land a_{8} \land a_{6})$ .

Theorem 5.7. Suppose that (U, AT) is an ISVIS. If λ ∈ (0, 1], $Δ_{*}^{λ} (AT) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl})$ is the standard minimum formula of Δ^λ (AT), then red^λ (AT) = {P_k : k ≤ q} where P_k = {a_kl : l ≤ p_k}.

Proof. (1) Suppose P_{k
₀} ∈ {P_k : k ≤ q}.

(i) Clearly, $Δ_{*}^{λ} (AT) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl}) = ⋁_{k = 1}^{q} (⋀ P_{k})$ . Then $⋀ P_{k_{0}} ⟶ Δ_{*}^{λ} (AT)$ .

Since $Δ_{*}^{λ} (AT) = Δ^{λ} (AT) = ⋀ (⋁ d_{ij})$ , we have $Δ_{*}^{λ} (AT) \Leftrightarrow ⋁ d_{ij} for any 1 \leq i, j \leq n .$

Then ∀ u, v ∈ U, ⋀P_{k
₀} ⟶ ⋁ D^λ (u, v).

So $\forall (u, v) \notin R_{AT}^{λ}$ , ⋀P_{k
₀} ⟶ ⋁ D^λ (u, v).

Now ⋀P_{k
₀} ⇔ a_{k
₀
l} ∀ l ≤ p_{k
₀} and ⋁D^λ (u, v) ⟷ a for some a ∈ D^λ (u, v). Then $\forall (u, v) \notin R_{AT}^{λ}$ , a_{k
₀
l} ∀ l ≤ p_{k
₀} ⟶ a for some a ∈ D^λ (u, v).

So $\forall (u, v) \notin R_{AT}^{λ}$ , there exists l₀ ≤ p_{k
₀} such that a = a_{k
₀
l
₀}, i.e., a ∈ P_{k
₀} ∩ D^λ (u, v).

Thus $\forall (u, v) \notin R_{AT}^{λ}$ , P_{k
₀}∩ D^λ (u, v) ≠ ∅.

By Proposition 4.7, P_{k
₀} ∈ co^λ (AT).

(ii) To prove P_{k
₀} ∈ red^λ (AT), by Theorem 3.8, we only need to show that

$\forall a \in P_{k_{0}}, \exists (u_{a}, v_{a}) \in R_{AT}^{λ}, (P_{k_{0}} - {a}) \cap D^{λ} (u_{a}, v_{a}) = \emptyset .$

Suppose that ∃ a₀ ∈ P_{k
₀} such that (P_{k
₀} - {a₀})∩ D^λ (u, v) ≠ ∅ for any $(u, v) \notin R_{AT}^{λ}$ . Pick a_uv ∈ (P_{k
₀} - {a₀}) ∩ D^λ (u, v). Then ⋀ (P_{k
₀} - {a₀}) ⟶ a_uv and a_uv ⟶ ⋁ D^λ (u, v).

Thus $\forall (u, v) \notin R_{AT}^{λ}$ , ⋀ (P_{k
₀} - {a₀}) ⟶ ⋁ D^λ (u, v) .

$\forall (u, v) \in R_{AT}^{λ}$ , we have D^λ (u, v) =∅. Then ⋀ (P_{k
₀} - {a₀}) ⟶ ⋁ D^λ (u, v) .

It follows that ∀ u, v ∈ U, $⋀ (P_{k_{0}} - {a_{0}}) ⟶ ⋁ D^{λ} (u, v) .$

Since $Δ_{*}^{λ} (AT)$ contains all true explanations of Δ^λ (AT), we have P_{k
₀} - {a₀} ∈ {P_k : k ≤ q}. Then

(⋀ P_{k
₀}) ⋁ (⋀ (P_{k
₀} - {a₀}))

= ((⋀ (P_{k
₀} - {a₀})) ⋀ {a₀}) ⋁ ((⋀ (P_{k
₀} - {a₀})) ⋀1)

= (⋀ (P_{k
₀} - {a₀})) ⋀ ({a₀} ⋁1)

= (⋀ (P_{k
₀} - {a₀})) ⋀1

= ⋀ (P_{k
₀} - {a₀}).

So P_{k
₀} ∉ {P_k : k ≤ q}. This is a contradiction.

Thus P_{k
₀} ∈ red^λ (AT). This shows that red^λ (AT) ⊇ {P_k : k ≤ q}.

(2) Let P ∈ red^λ (AT). Then P ∈ co^λ (AT). By Proposition 4.7, P∩ D^λ (u, v) ≠ ∅ for any $(u, v) \notin R_{AT}^{λ}$ .

Similar to the proof of (1) (ii), we can show that P ∈ {P_k : k ≤ q} .

Thus red^λ (AT) ⊆ {P_k : k ≤ q}.

Hence ${red}^{λ} (AT) = {P_{k} : k \leq q} .$ □ If $\min_{1 \leq k \leq | AT |} \max_{1 \leq i, j \leq | U |} s ((a_{k} (u_{i}), a_{k} (u_{j})) < λ,$ then $R_{AT}^{λ} = \emptyset$ . Thus red^λ (AT) =∅.

In order to ensure that there always exists a λ-reduct of AT, we demand $\min_{1 \leq k \leq | AT |} \max_{1 \leq i, j \leq | U |} s ((a_{k} (u_{i}), a_{k} (u_{j})) \geq λ .$

We can get a feature selection algorithm based on λ-discernibility matrix in an ISVIS (see Algorithm 2).

Below, we analyze the time complexity and space complexity of Algorithm 2. |AT| and |U| are denoted by the the numbers of features and objects, respectively. It should be pointed out that we need to compute such the tolerance relation matrices with respect to each feature that cost O (|AT||U|²) and the λ-discernibility matrix that cost O (|AT| (|U|² - |U|)/2). Now we analyze the time of computing the reducts. Firstly, we use idempotent properties and absorptive properties to reduce conjunctive normal form. In the worst case, it costs O (((|U|² - |U|)/2) ((|U|² - |U|)/2)). Conjunctive normal form can be changed into disjunctive normal form by using the incremental algorithm in [24]. Then, we can call this algorithm to solve the reducts. We ignore the computation time here. Thus, the overall time complexity of Algorithm 2 is O ((|U|⁴ + 2|U|³)/4 + (6|AT||U|² + 1)/4 - |AT||U|/2). The space for all the subsequent matrices and local variables can be reused. So, the space complexity of Algorithm 2 is O (|AT||U|²).

Example 5.8. (Continued from Example 5.6) Pick λ = 0.4.

From Example 5.6, we obtain $Δ^{0.4} (AT) and Δ_{*}^{0.4} (AT)$ .

Thus red^0.4 (AT) = {{a₂, a₄, a₇, a₈} , {a₃, a₄, a₇, a₈} , {a₄, a₅, a₇, a₈} , {a₄, a₆, a₇, a₈}}.

Obviously, core^0.4 (AT) = {a₄, a₇, a₈}.

Based on relationships between λ-information granulation and the proposed feature selection, we can get a feature selection algorithm based on λ-information granulation in an ISVIS (see Algorithm 3).

Based on relationships between λ-information entropy and the proposed feature selection, we can get the following algorithm.

Algorithms 3-4 employ λ-information entropy G^λ and H^λ to determine the optimal feature which is added into the current selected feature subset in each loop. In the worst case, this process is terminated when the whole feature set has been exhausted. The worst search time for a reduct will result in |AT| evaluations. Thus, the overall time complexity of Algorithm 3-4 are O (|AT||U|² + |AT|). Also, at the initial stage, |AT| fuzzy relation matrices of |U| × |U| are computed and stored corresponding to each individual features. Similarly, the space complexity of Algorithm 3 is O (|AT||U|²).

Based on relationships between λ-significance and the proposed feature selection, we can get two feature selection algorithms based on λ-significance in an ISVIS (see Algorithms 5 and 6).

Here, the worst search time for a reduct will result in |AT| (|AT|+1)/2 evaluations by Algorithm 5-6. So the time complexity and space complexity of Algorithm 5-6 are O (|AT||U|² + (|AT|² + |AT|)/2) and O (|AT||U|²), respectively.

Example 5.9. In order to better prove the practical significance of Algorithms 2-6 for incomplete set value data, we select a data set from the paper of Qian [16] to do a numerical experiment. The data sets are described in Table 14.

Table 14
The tolerance class of each object under $R_{a}^{λ}$ (a = a₅, a₆, λ = 0.4)

$R_{a_{5}}^{λ} (u)$ $R_{a_{6}}^{λ} (u)$

u ₁ U - {a₂, a₆, a₁₀, a₁₁} U - {a₂, a₃, a₉, a₁₀}

u ₂ {a₂, a₆, a₇, a₁₀, a₁₁} {a₂, a₃, a₇, a₉, a₁₀, a₁₁}

u ₃ U - {a₂, a₆, a₁₀, a₁₁} {a₂, a₃, a₇, a₉, a₁₀, a₁₁}

u ₄ U - {a₂, a₆, a₁₀, a₁₁} U - {a₂, a₃, a₉, a₁₀}

u ₅ U - {a₂, a₆, a₁₀, a₁₁} U - {a₂, a₃, a₉, a₁₀}

u ₆ {a₂, a₆, a₇, a₁₀, a₁₁} U - {a₂, a₃, a₉, a₁₀}

u ₇ U U - {a₁₁}

u ₈ U - {a₂, a₆, a₁₀, a₁₁} U - {a₂, a₃, a₉, a₁₀}

u ₉ U - {a₂, a₆, a₁₀, a₁₁} {a₂, a₃, a₇, a₉, a₁₀, a₁₁}

u ₁₀ {a₂, a₆, a₇, a₁₀, a₁₁} {a₂, a₃, a₇, a₉, a₁₀, a₁₁}

u ₁₁ {a₂, a₆, a₇, a₁₀, a₁₁} U - {a₇}

u ₁₂ U - {a₂, a₆, a₁₀, a₁₁} U - {a₂, a₃, a₉, a₁₀}

	$R_{a_{5}}^{λ} (u)$	$R_{a_{6}}^{λ} (u)$
u ₁	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₂	{a₂, a₆, a₇, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₃	U - {a₂, a₆, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₄	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₅	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₆	{a₂, a₆, a₇, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₇	U	U - {a₁₁}
u ₈	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}
u ₉	U - {a₂, a₆, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₁₀	{a₂, a₆, a₇, a₁₀, a₁₁}	{a₂, a₃, a₇, a₉, a₁₀, a₁₁}
u ₁₁	{a₂, a₆, a₇, a₁₀, a₁₁}	U - {a₇}
u ₁₂	U - {a₂, a₆, a₁₀, a₁₁}	U - {a₂, a₃, a₉, a₁₀}

In fact, the data set is a complete data. However, we need to study incomplete set value data. Therefore, We delete some data from raw data set to get the missing data set. In my paper, we randomly missed at a probability of 0.6 in |U||A|. In summary, some incomplete set value data are gained by this way.

Based on the above preparation, we take the new data to do the experiment.

(1) λ-feature selection algorithm:

We randomly generate ten incomplete set-valued data sets and then use λ-feature selection algorithm to reduce. The result is as follows in Table 15. Table 16 gives reduction set by λ-feature selection algorithms.

From Table 15, no reduction is one time, 1 reduction is one time, 2 reductions are four times and 3 reductions are four times in the 10 tests. The average size of the reduct is 2.95.

Table 15

A practical conjunctive set-valued information system from the test for foreign language ability in Shanxi University

Students	Audition (a₁)	Spokenlanguage (a₂)	Reading (a₃)	Writing (a₄)
u ₁	{E}	{E}	{F, G}	{F, G}
u ₂	{E, F, G}	{E, F, G}	{F, G}	{E, F, G}
u ₃	{F, G}	{F}	{F, G}	{F, G}
u ₄	{E, F}	{E, G}	{F, G}	{F}
u ₅	{F, G}	{F, G}	{F, G}	{F}
u ₆	{F}	{F}	{E, F}	{E, F}
u ₇	{E, F, G}	{E, F, G}	{E, G}	{E, F, G}
u ₈	{F, G}	{F}	{F, G}	{F, G}
u ₉	{E, G}	{G}	{F, G}	{F, G}
u ₁₀	{E, F}	{E, G}	{F, G}	{E, F}
⋮	⋮	⋮	⋮	⋮
u ₄₀	{F}	{E, F}	{F}	{G}
u ₄₁	{E, F, G}	{F, G}	{F}	{E, F, G}
u ₄₂	{E, F, G}	{E, F, G}	{E, F, G}	{E, G}
u ₄₃	{E, F}	{F}	{E, F, G}	{F}
u ₄₄	{F}	{F}	{F, G}	{E, F}
u ₄₅	{E, F, G}	{E, F, G}	{F, G}	{E, F, G}
u ₄₆	{E, G}	{E, G}	{GF}	{F, G}
u ₄₇	{E, F, G}	{E, F, G}	{E, F, G}	{E, G}
u ₄₈	{E, F}	{E, G}	{F, G}	{F}
u ₄₉	{E, F, G}	{F, G}	{E, F, G}	{E, G}
u ₅₀	{F, G}	{F}	{F, G}	{F, G}

Table 16

Reduction set by λ-feature selection algorithm

No .	Reduction set
1	{a₁, a₃}, {a₁, a₂, a₄}, {a₂, a₃, a₄}
2	{a₁, a₃, a₄}, {a₂, a₃, a₄}
3	{a₁, a₂, a₄}, {a₂, a₃, a₄}
4	{a₁, a₂, a₃}, {a₁, a₃, a₄}, {a₂, a₃, a₄}
5	{a₁, a₂, a₃, a₄}
6	{a₁, a₂, a₄}, {a₁, a₃, a₄}, {a₂, a₃, a₄}
7	{a₁, a₂, a₃}, {a₁, a₃, a₄}
8	{a₁, a₂, a₄}
9	{a₁, a₂, a₃}, {a₁, a₃, a₄}, {a₁, a₂, a₄}
10	{a₁, a₄}, {a₂, a₃, a₄}

Fig. 2 shows the number of features for each reduction with λ-red algorithm.

Fig. 2

Number of features by λ-feature selection algorithm.

(2) G^λ-feature selection algorithm:

We randomly generate ten incomplete set-valued data sets. Here we run G^λ-red algorithm on each data set and get ten reduction results. The average size of the reduct is 3.6. Fig. 3 shows the average of the number of selected features on each data set in 10 tests.

Fig. 3

Number of features by G^λ-red algorithm.

Fig. 4

Number of features by H^λ-red algorithm.

Fig. 5

Number of features by G^λ-sig algorithm.

(3) H^λ-feature selection algorithm:

We randomly generate ten incomplete set-valued data sets. Here we run H^λ-red algorithm on each data set and get ten reduction results. The average size of the reduct is 3.4. Fig.4 shows the average of the number of selected features on each data set in 10 tests.

(4) G^λ-significance feature selection algorithm:

We randomly generate ten incomplete set-valued data sets. Here we run G^λ-sig algorithm on each data set and get ten reduction results. The average size of the reduct is 3.2. Fig.5 shows the average of the number of selected features on each data set in 10 tests.

(5) H^λ-significance feature selection algorithm:

We randomly generate ten incomplete set-valued data sets. Here we run H^λ-sig algorithm on each data set and get ten reduction results. The average size of the reduct is 3.1. Fig.6 shows the average of the number of selected features on each data set in 10 tests.

Fig. 6

Number of features by H^λ-sig algorithm.

Fig.7 compares the average reduct size of the five algorithms.

Fig. 7

Average size of features by each feature selection algorithm.

6 Conclusions

This paper has given feature selection for an ISVIS by using the combination of similarity measurement and uncertainty measurement. Based on λ-reduction, features in an ISVIS have been divided into three categories according to their importance. Connections between the proposed feature selection and uncertainty measurement have been researched. Feature selection algorithms based on λ-discernibility matrix, λ-information granulation, λ-information entropy and λ-significance in an ISVIS have been proposed. Time complexity and space complexity of these algorithms have been presented. Feature selection is studied from two aspects of relation reduction and information entropy reduction. In this way, two methods of feature selection are formed. Theorem 4.23 proves that relation reduction and information entropy reduction for incomplete set-valued data are essentially the same. Thus, these two methods are actually the same. The proposed algorithms has been compared by means of an example. Due to the lack of research on feature selection for incomplete set-valued data, we did not compare the proposed algorithms with other algorithms, this is the deficiency of this paper. Before long, we will study applications of feature selection for incomplete set-valued data.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions, which have helped immensely in improving the quality of the paper. This work is supported by the Project of Improving the Basic Scientific Research Ability of Young and Middle-aged Teachers in Guangxi Universities (2018KY0470).

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