The Uncertainty Measure of Hierarchical Quotient Space Structure

Abstract

In the application of fuzzy reasoning, researchers usually choose the membership function optionally in some degree. Even though the membership functions may be different for the same concept, they can generally get the same (or approximate) results. The robustness of the membership function optionally chosen has brought many researchers' attention. At present, many researchers pay attention to the structural interpretation (definition) of a fuzzy concept, and find that a hierarchical quotient space structure may be a better tool than a fuzzy set for characterizing the essential of fuzzy concept in some degree. In this paper, first the uncertainty of a hierarchical quotient space structure is defined, the information entropy sequence of a hierarchical quotient space structure is proposed, the concept of isomorphism between two hierarchical quotient space structures is defined, and the sufficient condition of isomorphism between two hierarchical quotient space structures is discovered and proved also. Then, the relationships among information entropy sequence, hierarchical quotient space structure, fuzzy equivalence relation, and fuzzy similarity relation are analyzed. Finally, a fast method for constructing a hierarchical quotient space structure is presented.

1. Introduction

Since the fuzzy set theory was proposed by Zadeh in 1965 [1], it has been successfully applied to many application areas, such as fuzzy control, fuzzy reasoning, fuzzy clustering analysis, and fuzzy decision. A fuzzy set interprets fuzzy concept with some membership function, and, as a fundamental tool for revealing and analyzing uncertain problem, it has been frequently used in many real-world applications. When people are faced with the interpretation problem of the membership function generally in the applications, though each person has his/her own option about the meaning of the same subjective concept such as “tall” and “big” and he/she always has his/her membership function for the same fuzzy concept such as triangular, trapezoidal, or Gaussian, they can get the same (or approximate) results. The robustness of the optionally chosen membership functions attracts many researches’ attention [2–6]. Lin [2] interpreted memberships as probabilities. Liang and Song [3] regarded the membership function value as an independent and identically distributed random variable and proved that the mean of membership functions exists for all the elements of the universe of discourse. Unfortunately, these results were obtained based on a strong assumption; that is, the membership function value is assumed to be an independent and identically distributed random variable. Lin [4] presented a topological definition of the fuzzy set by using neighborhood systems, discussed the properties of fuzzy set from its structure, and proposed the concept of granular fuzzy set. Lin’s works provide a structural interpretation of member function. Afterward, B. Zhang and L. Zhang [5, 6] developed a structural definition of membership function and found that, for a fuzzy set (concept), it may probably be described by different types of membership functions, as long as the structures of these membership functions are the same, they characterize the fuzzy set (i.e., concept) with the same property. That is to say, although these membership functions are different in appearance, they are the same in essence. The structural description is more essential to a fuzzy concept than the membership function. This structure is called hierarchical quotient space structure in quotient space theory developed by B. Zhang and L. Zhang[6]. As a well-known fact, it is one of the basic characteristics in human problem solving that a person has a kind of ability to conceptualize the world at different granularities and translate from one abstraction level to others easily. Such is a powerful ability of human being to deal with complex problem [5]. According to Zadeh’s viewpoints [7, 8], both rough set theory [9] and quotient space theory can be used to describe a “crisp” granule world, while fuzzy set can be used to describe a “fuzzy” granule world. However, “fuzzy” granule and “crisp” granule are relative; that is, the “fuzzy” granule and “crisp” granule are two kinds of different manifestations of a concept in different granularity levels, and they can be transformed into each other with the changing of granularity [10]. Li et al. [11] presented that uncertainty (fuzzy) and certainty (crisp) are not opposite absolutely, and they can be transformed into each other in some degree. The fuzzy quotient space theory [10] combining fuzzy set theory and quotient space theory was proposed by B. Zhang and L. Zhang. It was a bridge from “fuzzy” granule world to “crisp” granule world and could better uncover the characteristics of human beings dealing with uncertain problems and better interpret the relationship between “fuzziness” and “crispness.” Many important conclusions about the fuzzy quotient space theory could be referred to [10], and the fuzzy quotient space theory for the cut relation of fuzzy equivalence relation with any threshold was discussed by Zhang et al. [12]. In fuzzy quotient space theory, a fuzzy equivalence relation and a hierarchical quotient space structure are one to one, and the hierarchical quotient space structure is a structural description of fuzzy equivalence relation.

The isomorphic fuzzy equivalence relations have the same hierarchical quotient space structure [10]; that is, they have the same classification ability to the objects in domain 𝑋. Therefore, the different fuzzy similarity relations may induce the same fuzzy equivalent relation, and the different fuzzy equivalent relations may have the same hierarchical quotient space structure and classification result as long as they are isomorphic. Recently, as a kind of structural description of a fuzzy concept, the hierarchical quotient space structure attracts many researchers’ attention. The further study about hierarchical quotient space structure of fuzzy equivalence relation with 𝜀-similarity could be referred to [13]. Tsekouras et al. [14] proposed a hierarchical fuzzy clustering approach. Pedrycz and Reformat [15] presented a hierarchical fuzzy C-means (FCM) method in a stepwise discovery of structure in data. Tang et al. [16] discussed the sufficient condition of isomorphism between fuzzy similarity relations and uncovered the relationships between fuzzy similarity relation and fuzzy equivalence relation. Although each person has his/her own membership function for the same concept, and he/she may get the different fuzzy similarity relations, he/she may finally obtain the same (or isomorphic) fuzzy equivalent relation which can produce the same hierarchical quotient space structure and classification of objects in the domain 𝑋. What is the reason that the different fuzzy similarity relations can produce the same hierarchical quotient space structure and the same classification? How can we measure the classification quality of a hierarchical quotient space structure? How much information (knowledge) does a hierarchical quotient space structure contain? How can we measure the uncertainty of a hierarchical quotient space structure? These fundamental issues in hierarchical quotient space theory remain open. In this paper, the fuzzy similarity relation, the fuzzy equivalence relation, the hierarchical quotient space structure, and the entropy sequence of a hierarchical quotient space structure are studied. A fast-constructing hierarchical quotient space structure method is presented. These works uncover the nature of the hierarchical quotient space structure further.

The paper is organized as follows. Some relevant preliminary concepts are reviewed briefly in Section 2. In Section 3, the information entropy sequence of a hierarchical quotient space structure is discussed. In Section 4, a fast-constructing hierarchical quotient space structure method is presented. The paper is concluded in Section 5.

2. Preliminary Concepts

For convenience, some preliminary concepts are reviewed or defined at first. Let 𝑋 be a Cantor set.

Definition 2.1 (see [6]). Let 𝑅 be a fuzzy relation on 𝑋. If it satisfies,(1)for all 𝑥∈𝑋,𝑅(𝑥,𝑥)=1,(2)for all 𝑥,𝑦∈𝑋,𝑅(𝑥,𝑦)=𝑅(𝑦,𝑥),then 𝑅 is called a fuzzy similarity relation on 𝑋.

Definition 2.2 (see [6]). Let 𝑅 be a fuzzy relation on 𝑋. If it satisfies,(1)for all 𝑥∈𝑋,𝑅(𝑥,𝑥)=1,(2)for all 𝑥,𝑦∈𝑋,𝑅(𝑥,𝑦)=𝑅(𝑦,𝑥),(3)for all 𝑥,𝑦,𝑧∈𝑋,𝑅(𝑥,𝑧)≥sup_𝑦∈𝑋min(𝑅(𝑥,𝑦),𝑅(𝑦,𝑧)),the 𝑅 is called a fuzzy equivalence relation on 𝑋 and is denoted by 𝑅.

Proposition 2.3 (see [17]). Let 𝑅 be a fuzzy similarity relation on 𝑋, and let 𝑅 denote the transitive closure of 𝑅. Then, for all 𝑚≥𝑛, 𝑅=𝑅^𝑚.

According to Proposition 2.3, a fuzzy equivalence relation can be induced from a fuzzy similarity relation by 𝑅→𝑅²→(𝑅²)²→⋯→𝑅^2𝑘=𝑅. Where 𝑘≥log₂𝑛.

Proposition 2.4 (see [17]). Let 𝑅 be a fuzzy equivalence relation on 𝑋, and 𝑅_𝜆={(𝑥,𝑦)∣𝑅(𝑥,𝑦)≥𝜆}(0≤𝜆≤1), then 𝑅_𝜆 is a crisp equivalence relation on 𝑋, 𝑅_𝜆 is called cut-relation of 𝑅.

Proposition 2.4 shows that 𝑅_𝜆 is a crisp equivalence relation on 𝑋, and its corresponding quotient space is denoted by 𝑋(𝜆). Let 𝑅 be a fuzzy equivalence relation on 𝑋, 𝑆={𝑅(𝑥,𝑦)∣𝑥,𝑦∈𝑋}, 𝑆 is called the value domain of 𝑅.

Definition 2.5 (see [6]). Let 𝑅 be a fuzzy equivalence relation on 𝑋, and 𝑆 is the value domain of 𝑋. The set 𝜋_𝑋(𝑅)={𝑋(𝜆)∣𝜆∈𝑆} is called the hierarchical quotient space structure of 𝑅.

Example 2.6. Let 𝑋={𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅}, and 𝑅₁ is a fuzzy equivalence relation on 𝑋, the corresponding relation matrix 𝑀_𝑅1 is defined as follows, 𝑀_𝑅1=10.40.80.50.50.410.40.40.40.80.410.50.50.50.40.510.60.50.40.50.61. (2.1) Its corresponding quotient space sequence is defined as follows, 𝑋𝜆₁=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,where0≤𝜆₁𝑋𝜆≤0.4;₂=𝑥₁,𝑥₃,𝑥₄,𝑥₅,𝑥₂,where0.4<𝜆₂𝑋𝜆≤0.5;₃=𝑥₁,𝑥₃,𝑥₂,𝑥₄,𝑥₅,where0.5<𝜆₃𝑋𝜆≤0.6;₄=𝑥₁,𝑥₃,𝑥₂,𝑥₄,𝑥₅,where0.6<𝜆₄𝑋𝜆≤0.8;₅=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,where0.8<𝜆₅≤1. (2.2) So, a hierarchical quotient space structure induced by the fuzzy equivalence relation 𝑅₁ is 𝜋_𝑋(𝑅₁)={𝑋(𝜆₁),𝑋(𝜆₂),𝑋(𝜆₃),𝑋(𝜆₄),𝑋(𝜆₅)}, which is shown in Figure 1.
In addition, a pyramid model can be established based on the number of block in each layer of the hierarchical quotient space structure, which is shown in Figure 2.
However, the different fuzzy equivalence relations may become into the same hierarchical quotient space structure. In Example 2.6, if the relation matrix 𝑀_𝑅2 of the fuzzy equivalence relation 𝑅₂ is defined as follows: 𝑀_𝑅2=10.20.90.60.60.210.20.20.20.90.210.60.60.60.20.610.70.60.20.60.71 (2.3) then 𝑅₂ can become into the same quotient space sequence and the same hierarchical quotient space structure as 𝑅₁; that is, 𝜋_𝑋(𝑅₁)=𝜋_𝑋(𝑅₂).

Figure 1 

The hierarchical quotient space structure 𝜋𝑋(𝑅1) of 𝑅1.

Figure 2 

The pyramid model of a hierarchical quotient space structure.

Definition 2.7 (see [16]). Let 𝑅₁ and 𝑅₂ be two fuzzy equivalence relations on 𝑋; if they have the same hierarchical quotient space structure, that is, 𝜋_𝑋(𝑅₁)=𝜋_𝑋(𝑅₂), then 𝑅₁ and 𝑅₂ are called isomorphic, denoted by 𝑅₁≅𝑅₂.

Proposition 2.8 (see [13]). Let 𝑅₁ and 𝑅₂ be two fuzzy equivalence relations on 𝑋, for any 𝑥,𝑦,𝑢,𝑣∈𝑋; if 𝑅₁(𝑥,𝑦)<𝑅₁(𝑢,𝑣)⇔𝑅₂(𝑥,𝑦)<𝑅₂(𝑢,𝑣) and 𝑅₁(𝑥,𝑦)=𝑅₁(𝑢,𝑣)⇔𝑅₂(𝑥,𝑦)=𝑅₂(𝑢,𝑣), then 𝑅₁≅𝑅₂.

Definition 2.9 (see [18]). Let 𝑅₁ and 𝑅₂ be two fuzzy equivalence relations on 𝑋; for any 𝑥,𝑦,𝑢,𝑣∈𝑋, if 𝑅₁(𝑥,𝑦)≤𝑅₁(𝑢,𝑣)⇒𝑅₂(𝑥,𝑦)≤𝑅₂(𝑢,𝑣) and 𝑅₂(𝑥,𝑦)<𝑅₂(𝑢,𝑣)⇒𝑅₁(𝑥,𝑦)<𝑅₁(𝑢,𝑣), then 𝑅₁ and 𝑅₂ are called homomorphism, denoted by 𝑅₁≜𝑅₂.

Proposition 2.10 (see [18]). Let 𝑅₁ and 𝑅₂ be two fuzzy equivalence relations on 𝑋; if 𝑅₁≜𝑅₂, then 𝜋_𝑋(𝑅₂)⊆𝜋_𝑋(𝑅₁).

A fuzzy equivalence relation 𝑅 can be induced by the fuzzy similarity relation 𝑅 by computing the transitive closure of 𝑅, then 𝑅 is called a fuzzy equivalence relation derived from 𝑅.

Definition 2.11. Let 𝑅₁ and 𝑅₂ be two fuzzy similarity relations on 𝑋, and 𝑅₁ and 𝑅₂ are fuzzy equivalence relations derived from 𝑅₁ and 𝑅₂, respectively. If 𝑅₁=𝑅₂, then 𝑅₁ and 𝑅₂ are called isomorphism, denoted by 𝑅_1∼𝑅₂. If 𝑅₁≅𝑅₂, then 𝑅₁ and 𝑅₂ are called similarity, denoted by 𝑅₁∼𝑅₂.

From the viewpoints of both clustering and classification, the isomorphic fuzzy similarity relations can induce the same hierarchical quotient space structure, and have the same ability of clustering (or classification) for objects in 𝑋. However, the different hierarchical quotient space structures maybe have the same clustering (or classification) ability. There is few correlative research works. For discussing the uncertainty of a hierarchical quotient space structure, in this paper, the information entropy sequence of hierarchical quotient space structure is defined, and the relationship between hierarchical quotient space structure and information entropy sequence is analyzed in detail.

3. The Information Entropy Sequence of Hierarchical Quotient Space Structure

A hierarchical quotient space structure can uncover the essential characteristics of a fuzzy concept better than a fuzzy set. However, how to measure uncertainty of the hierarchical quotient space structure is still an open question. Information entropy is a very useful tool for measuring the uncertainty of vague information, and it has been studied based on the rough set and fuzzy set in the literature. For instance, Liang et al. [19, 20] analyzed the uncertainty of the rough set from the perspective of information entropy, conditional entropy, mutual information, and knowledge granule and presented a new rough entropy. Zhang et al. [21] presented a cognition model based on granular computing and analyzed the uncertainty of human cognition with different granularity levels. Fan [22] Miao et al. [23, 24] discussed the relationships among the knowledge granule, knowledge roughness, and information entropy from the viewpoint of both granular computing and information representation. Wang et al. [25] studied fuzzy entropy of rough sets in different granularity levels. However, in the previous studies, the questions of how to measure the information entropy of a hierarchical quotient space structure and what changing regularities the information entropy of a hierarchical quotient space structure have with different granularity are not discussed. In this section, the information entropy sequence of a hierarchical quotient space structure is proposed and discussed in detail.

Definition 3.1 (see [19]). Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a nonempty finite set, 𝑃^′={𝑃^′₁,𝑃^′₂,…,𝑃^′_𝑙} and 𝑃^′′={𝑃₁^′′,𝑃₂^′′,…,𝑃_𝑚^′′} are two partition spaces on 𝑋. If, for all 𝑃^′_𝑖∈𝑃^′,(∃_{𝑃𝑗′′∈𝑃′′}(𝑃^′_𝑖⊆𝑃_𝑗^′′)), then 𝑃^′ is finer than 𝑃^′′, denoted by 𝑃^′≺𝑃^′′.

Definition 3.2 (see [19]). Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a nonempty finite set, 𝑃^′={𝑃^′₁,𝑃^′₂,…,𝑃^′_𝑙} and 𝑃^′′={𝑃^′₁,𝑃₂^′′,…,𝑃_𝑚^′′} are two partition spaces on 𝑋. If 𝑃^′≺𝑃^′′ and there exist 𝑃^′_𝑖∈𝑃^′(∃_{𝑃𝑗′′∈𝑃′′}(𝑃^′_𝑖⊂𝑃_𝑗^′′)), then 𝑃^′ is strictly finer than 𝑃^′′, denoted by 𝑃^′≺𝑃^′′.

Each layer of the hierarchical quotient space structure 𝜋_𝑋(𝑅) is a partition on 𝑋 that is, it is denoted by 𝑋(𝜆). If 0≤𝜆₁<𝜆₂<⋯<𝜆_𝑡≤1, then 𝑋(𝜆_𝑡)≺𝑋(𝜆_𝑡−1)≺⋯≺𝑋(𝜆₁).

Definition 3.3. Let 𝑅 be a fuzzy equivalence relation on 𝑋, let 𝜋_𝑋(𝑅)={𝑋(𝜆₁),𝑋(𝜆₂),…,𝑋(𝜆_𝑡)} be its corresponding hierarchical quotient space structure, and 𝑋(𝜆_𝑡)≺𝑋(𝜆_𝑡−1)≺⋯≺𝑋(𝜆₁). Let 𝐿_𝑖=|𝑋(𝜆_𝑖)|(𝑖=1,2,…,𝑡), the sequence 𝐿(𝜋_𝑋(𝑅))={𝐿₁,𝐿₂,…,𝐿_𝑡} is called the partition sequence of 𝜋_𝑋(𝑅) (where |⋅| denotes the cardinality of a set).

Obviously, any one partition sequence of the hierarchical quotient space structure 𝜋_𝑋(𝑅) is a positive integer sequence, and 𝐿₁<𝐿₂<⋯<𝐿_𝑡. The information entropy [19, 20, 22–25] of a partition space has been studied by many researchers. Here, we discuss the information entropy sequence of a hierarchical quotient space structure based on classical Shannon information entropy.

Definition 3.4. Let 𝑅 be a fuzzy equivalence relation on 𝑋, let 𝜋_𝑋(𝑅)={𝑋(𝜆₁),𝑋(𝜆₂),…,𝑋(𝜆_𝑡)} be its corresponding hierarchical quotient space structure, and let 𝐿(𝜋_𝑋(𝑅))={𝐿₁,𝐿₂,…,𝐿_𝑡} be a partition sequence of the hierarchical quotient space structure 𝜋_𝑋(𝑅). Assume that 𝑋(𝜆_𝑖)={𝑋_𝑖1,𝑋_𝑖2,…,𝑋_𝑖𝐿𝑖} (it is a partition on 𝑋). The information entropy 𝐻_𝑋(𝜆_𝑖) of 𝑋(𝜆_𝑖) is defined as follows: 𝐻_𝑋𝜆_𝑖=−^𝐿𝑖_𝑘=1𝑋_𝑖𝑘|𝑋|𝑋ln^𝑖𝑘|𝑋|. (3.1)

Theorem 3.5. Let 𝑅 be a fuzzy equivalence relation on 𝑋, and let 𝜋_𝑋(𝑅)={𝑋(𝜆₁),𝑋(𝜆₂),…,𝑋(𝜆_𝑛)} be its hierarchical quotient space structure, then the entropy sequence 𝐻(𝜋_𝑋(𝑅))={𝐻_𝑋(𝜆₁),𝐻_𝑋(𝜆₂),…,𝐻_𝑋(𝜆_𝑡)} under Definition 3.4 is a strictly monotonic increasing sequence.

Proof. It follows from the definition of the hierarchical quotient space structure that 𝑋(𝜆_𝑡)≺𝑋(𝜆_𝑡−1)≺⋯≺𝑋(𝜆₁); that is, 𝑋(𝜆_𝑖−1) is a quotient space of 𝑋(𝜆_𝑖). Let 𝑋(𝜆_𝑖)={𝑋_𝑖1,𝑋_𝑖2,…,𝑋_𝑖𝐿𝑖} and 𝑋(𝜆_𝑖−1)={𝑋_𝑖−1,1,𝑋_𝑖−1,2,…,𝑋_{𝑖−1,𝐿𝑖−1}}. Each subblock 𝑋_{𝑖−1,𝑗}(1≤𝑗≤𝐿_𝑖−1) in 𝑋(𝜆_𝑖−1) is a combination of one or more subblocks in 𝑋(𝜆_𝑖). For simplicity, and without any loss of generality, we assume that only a subblock 𝑋_{𝑖−1,𝑗} in 𝑋(𝜆_𝑖−1) is a combination of two subblocks in 𝑋(𝜆_𝑖); that is, 𝑋_{𝑖−1,𝑗}=𝑋_𝑖,𝑝∪𝑋_𝑖,𝑞, other subblocks in 𝑋(𝜆_𝑖−1) are equal to the corresponding rest subblocks in 𝑋(𝜆_𝑖), respectively. Thus 𝐻_𝑋𝜆_𝑖−1=−^𝐿𝑖−1_𝑘=1𝑋_{𝑖−1,𝑘}|𝑋|𝑋ln^{𝑖−1,𝑘}|𝑋|=−^𝑗−1_𝑘=1𝑋_{𝑖−1,𝑘}|𝑋|𝑋ln^{𝑖−1,𝑘}|𝑋|−^𝐿𝑖−1_𝑘=𝑗+1𝑋_{𝑖−1,𝑘}|𝑋|𝑋ln^{𝑖−1,𝑘}|𝑋|−𝑋_{𝑖−1,𝑗}|𝑋|𝑋ln^{𝑖−1,𝑗}|𝑋|=−^𝑗−1_𝑘=1𝑋_𝑖,𝑘|𝑋|𝑋ln^𝑖,𝑘|𝑋|−^𝐿𝑖_𝑘=𝑗+1𝑋_𝑖,𝑘|𝑋|𝑋ln^𝑖,𝑘|𝑋|−𝑋_𝑖,𝑝+𝑋_𝑖,𝑞|𝑋|𝑋ln^𝑖,𝑝+𝑋_𝑖,𝑞|𝑋|. (3.2) Because 𝑋_𝑖,𝑝+𝑋_𝑖,𝑞|𝑋|𝑋ln^𝑖,𝑝+𝑋_𝑖,𝑞|𝑋|=𝑋_𝑖,𝑝|𝑋|𝑋ln^𝑖,𝑝+𝑋_𝑖,𝑞|𝑋|+𝑋_𝑖,𝑞|𝑋|𝑋ln^𝑖,𝑝+𝑋_𝑖,𝑞|𝑋|>𝑋_𝑖,𝑝|𝑋|𝑋ln^𝑖,𝑝|𝑋|+𝑋_𝑖,𝑞|𝑋|𝑋ln^𝑖,𝑞|𝑋|, (3.3) so 𝐻_𝑋𝜆_𝑖−1<−^𝑗−1_𝑘=1𝑋_𝑖,𝑘|𝑋|𝑋ln^𝑖,𝑘|𝑋|−^𝐿𝑖_𝑘=𝑗+1𝑋_𝑖,𝑘|𝑋|𝑋ln^𝑖,𝑘|𝑋|−𝑋_𝑖,𝑝|𝑋|𝑋ln^𝑖,𝑝|𝑋|−𝑋_𝑖,𝑞|𝑋|𝑋ln^𝑖,𝑞|𝑋|=𝐻_𝑋𝜆_𝑖(2≤𝑖≤𝑛), (3.4) the sequence {𝐻_𝑋(𝜆₁),𝐻_𝑋(𝜆₂),…,𝐻_𝑋(𝜆_𝑡)} is a strictly monotonic increasing sequence; that is, 𝐻_𝑋(𝜆₁)<𝐻_𝑋(𝜆₂)<⋯<𝐻_𝑋(𝜆_𝑡).

Theorem 3.5 shows that the information entropy sequence of a hierarchical quotient space structure is a monotonic increasing sequence with the partition (or quotient space) becoming finer; that is, the finer the partition in a hierarchical quotient space structure, the bigger its information entropy.

Definition 3.6. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a nonempty finite set, and let 𝑃_𝑘(𝑋)={𝑋₁,𝑋₂,…,𝑋_𝑘} be a partition space on 𝑋, 𝑃_𝑘(𝑋) is called 𝐤-order partition on 𝑋 (where |𝑃_𝑘(𝑋)|=𝑘).

Definition 3.7. Assume that 𝑃_𝑘(𝑋)={𝑋₁,𝑋₂,…,𝑋_𝑘} is a 𝐤-order partition on 𝑋, and let |𝑋₁|=𝑎^′₁,|𝑋₂|=𝑎^′₂,…, and |𝑋_𝑘|=𝑎^′_𝑘. Arranging the sequence {𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑘} in increasing order, we obtain a new increasing sequence {𝑎₁,𝑎₂,…,𝑎_𝑘} denoted by 𝐼(𝑘). 𝐼(𝑘) is called a subblock sequence of the partition 𝑃_𝑘(𝑋).
Obviously, two different 𝐤-order partitions on 𝑋 may have the same subblock sequence 𝐼(𝑘).

Definition 3.8. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a nonempty finite set, let 𝑃_𝑘(𝑋)={𝑋₁,𝑋₂,…,𝑋_𝑘} be a partition space on 𝑋, and let 𝐼₁(𝑘)={𝑎₁,𝑎₂,…,𝑎_𝑘} be a subblock sequence of the partition 𝑃_𝑘(𝑋). 𝐼₂(𝑘)={𝑎₁,𝑎₂,…,𝑎_𝑝+1,…,𝑎_𝑞−1,…,𝑎_𝑘} is called a 𝜎-displacement of 𝐼₁(𝑘), where 𝑝<𝑞, 𝑎_𝑝+1<𝑎_𝑞−1, and 𝐼₂(𝑘) keeps its increasing property.
A 𝜎-displacement is equal to minus one from some “bigger” element and add one to some “smaller” element, and the sequence still keeps monotonically increasing in a subblock sequence.

Example 3.9. Let 𝐼₁(4)={1,2,4,5} be a subblock sequence, then 𝐼₂(4)={1,3,4,4} is derived from 𝐼₁(4) through the 𝜎-displacement once.

Theorem 3.10. If a subblock sequence 𝐼₂(𝑘) is derived from 𝐼₁(𝑘) through the 𝜎-displacement once, then 𝐻(𝐼₁(𝑘))<𝐻(𝐼₂(𝑘)).

Proof. Let 𝐼₁(𝑘)={𝑎₁,𝑎₂,…,𝑎_𝑘}, 𝐼₂(𝑘)={𝑎₁,𝑎₂,…,𝑎_𝑖+1,…,𝑎_𝑗−1,…,𝑎_𝑘}, 𝑎₁+𝑎₂+⋯+𝑎_𝑘=𝑛, then 𝐻𝐼₂(𝑘)=−^𝑘_𝑖=1𝑎_𝑙𝑛𝑎ln^𝑙𝑛+𝑎_𝑝𝑛𝑎ln^𝑝𝑛+𝑎_𝑞𝑛𝑎ln^𝑞𝑛−𝑎_𝑝+1𝑛𝑎ln^𝑝+1𝑛−𝑎_𝑞−1𝑛𝑎ln^𝑞−1𝑛. (3.5)
Let 𝑓(𝑥)=−(𝑥/𝑛)ln(𝑥/𝑛)−((𝑚−𝑥)/𝑛)ln((𝑚−𝑥)/𝑛) (where 𝑚=𝑎_𝑝+𝑎_𝑞), then 𝑓^′(𝑥)=(1/𝑛)ln((𝑚−𝑥)/𝑥).
Let 𝑓^′(𝑥)=0, we have a solution; that is, 𝑥=𝑚/2. In addition, because 𝑓^′′(𝑥)=(−𝑚/𝑛(𝑚−𝑥)𝑥)<0, when 0≤𝑥≤(𝑚/2), 𝑓(𝑥) is monotonic increasing function.
Let 𝑥=𝑎_𝑝 and 𝑦=𝑎_𝑝+1, where 𝑎_𝑝+1<𝑎_𝑞−1; that is, 𝑥<𝑦≤(𝑚/2)=(𝑎_𝑝+𝑎_𝑞)/2. Due to the monotonicity of 𝑓(𝑥), 𝑓(𝑦)−𝑓(𝑥)>0; that is, 𝑎_𝑝𝑛𝑎ln^𝑝𝑛+𝑎_𝑞𝑛𝑎ln^𝑞𝑛−𝑎_𝑝+1𝑛𝑎ln^𝑝+1𝑛−𝑎_𝑞−1𝑛𝑎ln^𝑞−1𝑛>0. (3.6)
Therefore, 𝐻𝐼₂(𝑘)=−^𝑘_𝑖=1𝑎_𝑙𝑛𝑎ln^𝑙𝑛+𝑎_𝑝𝑛𝑎ln^𝑝𝑛+𝑎_𝑞𝑛𝑎ln^𝑞𝑛−𝑎_𝑝+1𝑛𝑎ln^𝑝+1𝑛+𝑎_𝑞−1𝑛𝑎ln^𝑞−1𝑛𝑎>−^𝑝+1𝑛𝑎ln^𝑝+1𝑛+𝑎_𝑞−1𝑛𝑎ln^𝑞−1𝑛>−^𝑘_𝑖=1𝑎_𝑙𝑛𝑎ln^𝑙𝑛=𝐻𝐼₁(𝑘). (3.7)

Theorem 3.10 shows that through the 𝜎-displacement once, the information entropy of a subblock sequence will increase. In the case of nonconfusion, the information entropy of a partition and the information entropy of its corresponding subblock sequence can be deemed as the same value. If two different partitions 𝑃^′_𝑘(𝑋) and 𝑃_𝑘^′′(𝑋) on 𝑋 have the same subblock sequence 𝐼(𝑘), then they should have the same information entropy sequence. So, from viewpoint of the entropy, they contain the same information. However, if two subblock sequences 𝐼₁(𝑘)≠𝐼₂(𝑘) on 𝑋, then 𝐻(𝐼₁(𝑘))≠𝐻(𝐼₂(𝑘)), which is discussed in Theorem 3.11 as follows.

Theorem 3.11. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a nonempty finite set. For any two k-order partitions 𝑃^′_𝑘(𝑋) and 𝑃_𝑘^′′(𝑋) based on 𝑋, their corresponding subblock sequences are 𝐼₁(𝑘) and 𝐼₂(𝑘), respectively. If 𝐼₁(𝑘)≠𝐼₂(𝑘), then 𝐻(𝐼₁(𝑘))≠𝐻(𝐼₂(𝑘)), where 𝐻(𝐼₁(𝑘)) and 𝐻(𝐼₂(𝑘)) denote the information entropy of partitions 𝑃^′_𝑘(𝑋) and 𝑃_𝑘^′′(𝑋) based on Definition 3.4, respectively (2≤𝑘≤𝑛).

Proof. Firstly we prove that the conclusion in Theorem 3.11 holds in the case of 𝑘=2. Assume that 𝑃(𝑋)={𝑋₁,𝑋₂}, let |𝑋₁|=𝑥 and |𝑋₂|=𝑛−𝑥, then a 2-order subblock sequence is obtained; that is, 𝐼(2)={𝑥,𝑛−𝑥}. Supposing that 𝑥<(𝑛/2) (if 𝑛 is even) or 𝑥<((𝑛−1)/2) (if 𝑛 is odd). We have 𝐻(𝑃(𝑋))=−(𝑥/𝑛)ln(𝑥/𝑛)−((𝑛−𝑥)/𝑛)ln((𝑛−𝑥)/𝑛). Let 𝑓(𝑥)=−(𝑥/𝑛)ln(𝑥/𝑛)−((𝑛−𝑥)/𝑛)ln((𝑛−𝑥)/𝑛), then 𝑓^′(𝑥)=−(1/𝑛)ln(𝑥/𝑛)−(1/𝑛)+(1/𝑛)ln((𝑛−𝑥)/𝑛)+(1/𝑛)=(1/𝑛)ln((𝑛−𝑥)/𝑥). Letting 𝑓^′(𝑥)=0, we can obtain the solution 𝑥=𝑛/2. Since 𝑓^′′(𝑥)=(−1/(𝑛−𝑥)𝑥)<0, when 𝑥=𝑛/2, 𝑓(𝑥) achieves its maximum value. Thus, when 𝑥<𝑛/2 (𝑛 is even) or 𝑥<(𝑛−1)/2 (𝑛 is odd), 𝑓(𝑥) is a monotonic increasing function; that is, information entropy 𝐻(𝑃(𝑋)) is a monotonic increasing function. If 𝑥>𝑛/2 (𝑛 is even) or 𝑥>(𝑛+1)/2 (𝑛 is odd), the 2-order subblock sequence 𝐼(2)={𝑥,𝑛−𝑥} becomes 𝐼(2)={𝑛−𝑥,𝑥}; it is the same as that when 𝑥<𝑛/2 (𝑛 is even) or 𝑥<(𝑛−1)/2 (𝑛 is odd). Therefore, when 𝑘=2 and 𝐼₁(𝑘)≠𝐼₂(𝑘), 𝐻(𝐼₁(𝑘))≠𝐻(𝐼₂(𝑘)) holds.
When 𝑘≥3, the idea to prove them is the same as 𝑘=2. For the limitation of paper length, we only discuss the case of 𝑘=3 (similar to other situations) in this paper. Assume that there are two different 3-order partitions 𝑃^′₃(𝑋) and 𝑃₃^′′(𝑋) and their corresponding subblock sequences 𝐼₁(3) and 𝐼₂(3) are different. Let 𝐼₁(3)={𝑎₁,𝑎₂,𝑎₃}, 𝐼₂(3)={𝑏₁,𝑏₂,𝑏₃}, and 𝐼₁(3)≠𝐼₂(3). If only one element in 𝐼₁(3) is equal to the element of 𝐼₂(3) (e.g., let 𝑎₁=𝑏₁, then 𝑎₂≠𝑏₂ and 𝑎₃≠𝑏₃), due to the former conclusion in case of 𝑘=2, 𝐻(𝑃^′₃(𝑋))≠𝐻(𝑃₃^′′(𝑋)) is held. If all elements in 𝐼₁(3) are not equal to that in 𝐼₂(3), then we get the conclusion from the following three cases (other cases can be transformed into one of the after-mentioned three conditions):(1)𝑎₁>𝑏₁, 𝑎₂>𝑏₂, and 𝑎₃<𝑏₃. 𝐼₂(3) can be replaced by 𝐼₁(3) through many times of 𝜎-displacements. By Theorem 3.10, 𝐻(𝐼₁(3))≠𝐻(𝐼₂(3)).(2)𝑎₁>𝑏₁, 𝑎₂<𝑏₂, and 𝑎₃<𝑏₃. 𝐼₂(3) can be also replaced by 𝐼₁(3) through many times of 𝜎-displacements. By Theorem 3.10, 𝐻(𝐼₁(3))≠𝐻(𝐼₂(3)).(3)𝑎₁>𝑏₁, 𝑎₂<𝑏₂, and 𝑎₃>𝑏₃. 𝐼₂(3) cannot be replaced by 𝐼₁(3) directly through the 𝜎-displacement once, but 𝐼₁(3) can be replaced by 𝐼₃(3)={𝑎₁,𝑎₂+(𝑎₃−𝑏₃),𝑏₃} through the 𝜎-displacement once. By Theorem 3.10, 𝐻(𝐼₁(3))<𝐻(𝐼₃(3))<𝐻(𝐼₂(3)) holds. So 𝐻(𝐼₁(3))≠𝐻(𝐼₂(3)) holds.

Given a hierarchical quotient space structure 𝜋_𝑋(𝑅), we can easily obtain its corresponding information entropy sequence {𝐻(𝐼(𝑛₁)),𝐻(𝐼(𝑛₂)),…,𝐻(𝐼(𝑛_𝑠)),…} (where {𝑛₁,𝑛₂,…,𝑛_𝑠,…} is the subsequence of natural number sequence {1,2,3,…,𝑛,…}). Theorem 3.11 shows that the different subblock sequences can have different information entropy sequences. According to Theorem 3.5 and Theorem 3.11, the different hierarchical quotient space structures which have the same subblock sequence should have the same information entropy sequence {𝐻(𝐼(𝑛₁)),𝐻(𝐼(𝑛₂)),…,𝐻(𝐼(𝑛_𝑠)),…}, the same information, and the same ability for classification (clustering). Both the information entropy sequence and the subblock sequence of a hierarchical quotient space structure are one to one. The information entropy sequence {𝐻(𝐼(𝑛₁)),𝐻(𝐼(𝑛₂)),…,𝐻(𝐼(𝑛_𝑠)),…} uncovers the uncertainty of a hierarchical quotient space structure. Therefore, we can analyze the information (uncertainty) of a hierarchical quotient space structure with its information entropy sequence.

Definition 3.12. Let 𝜋_𝑋(𝑅₁) and 𝜋_𝑋(𝑅₂) be two hierarchical quotient space structures derived from the fuzzy equivalence relations 𝑅₁ and 𝑅₂ on 𝑋, and their partition sequences are 𝐿(𝜋_𝑋(𝑅₁))={𝐿₁₁,𝐿₁₂,…,𝐿_1𝑟} and 𝐿(𝜋_𝑋(𝑅₂))={𝐿₂₁,𝐿₂₂,…,𝐿_2𝑠}, respectively. Let {𝐼(𝐿₁₁),𝐼(𝐿₁₂),…,𝐼(𝐿_1𝑟)} and {𝐼(𝐿₂₁),𝐼(𝐿₂₂),…,𝐼(𝐿_2𝑠)} denote their corresponding subblock sequences of each layer in 𝜋_𝑋(𝑅₁) and 𝜋_𝑋(𝑅₂), respectively. If 𝐿(𝜋_𝑋(𝑅₁))=𝐿(𝜋_𝑋(𝑅₂)) (obviously, where 𝑟=𝑠 and 𝐼(𝐿_1𝑖)=𝐼(𝐿_2𝑖), 𝑖=1,2,…,𝑟), then 𝜋_𝑋(𝑅₁) and 𝜋_𝑋(𝑅₂) are called isomorphism, denoted by 𝜋_𝑋(𝑅₁)≃𝜋_𝑋(𝑅₂).

Example 3.13. Assume that 𝑋={1,2,3,4,5,6}, and there are two hierarchical quotient space structures 𝜋_𝑋(𝑅₁) and 𝜋_𝑋(𝑅₂) on 𝑋 as follows: 𝜋_𝑋𝑅₁=},{;},{3},{;1},{2},{3},{,𝜋{{{1,2,34,5,6}}{{1,24,5,6}}{{4,5,6}}}_𝑋𝑅₂={{{1,5,6},{2,3,4}};{{1,5},{6},{2,3,4}};{{1},{5},{6},{2,3,4}}}, (3.8) namely, },{↓},{3},{↓1},{2},{3},{,𝜋{{1,2,34,5,6}}{{1,24,5,6}}{{4,5,6}}_𝑋𝑅₁},{↓},{6},{↓1},{5},{6},{.𝜋{{1,5,62,3,4}}{{1,52,3,4}}{{2,3,4}}_𝑋𝑅₂ (3.9) Obviously, 𝐿(𝜋_𝑋(𝑅₁))=𝐿(𝜋_𝑋(𝑅₂))={2,3,4}, and 𝐼(𝐿₁₁)=𝐼(𝐿₂₁)={3,3}, 𝐼(𝐿₁₂)=𝐼(𝐿₂₂)={1,2,3}, and 𝐼(𝐿₁₃)=𝐼(𝐿₂₃)={1,1,1,3}. Therefore, 𝜋_𝑋(𝑅₁)≃𝜋_𝑋(𝑅₂).

From the viewpoint of classification (clustering) analysis, if two hierarchical quotient space structures are isomorphic, they have the same classification abilities of the objects in the set of 𝑋. From the viewpoint of information entropy, if two hierarchical quotient space structures are isomorphic, they have the same information entropies. Based on Theorem 3.11, Theorem 3.14 is given as follows.

Theorem 3.14. Let 𝜋_𝑋(𝑅₁) and 𝜋_𝑋(𝑅₂) be two hierarchical quotient space structures derived from fuzzy equivalence relations 𝑅₁ and 𝑅₂ on 𝑋, respectively. If 𝐿(𝜋_𝑋(𝑅₁))=𝐿(𝜋_𝑋(𝑅₂)) and 𝐻(𝜋_𝑋(𝑅₁))=𝐻(𝜋_𝑋(𝑅₂)), then 𝜋_𝑋(𝑅₁)≃𝜋_𝑋(𝑅₂).

Proof. Since 𝐿(𝜋_𝑋(𝑅₁))=𝐿(𝜋_𝑋(𝑅₂)), 𝜋_𝑋(𝑅₁) has the same number of layer as 𝜋_𝑋(𝑅₂), each layers of both 𝜋_𝑋(𝑅₁) and 𝜋_𝑋(𝑅₂) has the same number of the subblock. Let 𝐿(𝜋_𝑋(𝑅₁))=𝐿(𝜋_𝑋(𝑅₂))={𝑘₁,𝑘₂,…,𝑘_𝑟}. For any 𝑘_𝑖(1≤𝑖≤𝑟), according to Theorem 3.11, if 𝐼₁(𝑘_𝑖)≠𝐼₂(𝑘_𝑖), 𝐻(𝐼₁(𝑘_𝑖))≠𝐻(𝐼₂(𝑘_𝑖))(1≤𝑖≤𝑟). Therefore, if 𝐻(𝜋_𝑋(𝑅₁))=𝐻(𝜋_𝑋(𝑅₂)), 𝜋_𝑋(𝑅₁)≃𝜋_𝑋(𝑅₂) is held.

In Theorem 3.14, the condition 𝐿(𝜋_𝑋(𝑅₁))=𝐿(𝜋_𝑋(𝑅₂)) is very important. If this condition is not held, then Theorem 3.14 is not held. Theorem 3.14 reveals the importance of information entropy sequence. The uncertainty of a hierarchical quotient space structure may be characterized by the information entropy sequence, the partition sequence, and the subblock sequence. The different fuzzy similarity relations may induce the isomorphic fuzzy equivalence relations on 𝑋, and the isomorphic fuzzy equivalence relations may produce the same or isomorphic hierarchical quotient space structures which have the same uncertainty (or information entropy sequence).

Definition 3.15. Let 𝑅₁ and 𝑅₂ be two fuzzy similarity relations on 𝑋, and let 𝑅₁ and 𝑅₂ be two fuzzy equivalence relations derived from 𝑅₁ and 𝑅₂, respectively. If 𝜋_𝑋(𝑅₁)≃𝜋_𝑋(𝑅₂), then 𝑅₁ and 𝑅₂ are called isogenous fuzzy similarity relations, denoted by 𝑅₁≐𝑅₂. If 𝑅₁ and 𝑅₂ are not isomorphic, but 𝜋_𝑋(𝑅₁)≃𝜋_𝑋(𝑅₂), then 𝑅₁ and 𝑅₂ are called similar fuzzy equivalence relation, denoted by 𝑅₁∼𝑅₂.

According to Definitions 2.7, 2.11, and 3.15, the relationships among the information entropy sequence, the hierarchical quotient space structure, the fuzzy equivalence relation and the fuzzy similarity relation are shown in (3.10) and Table 1.𝑅⟶₁^induce𝜋_𝑋𝑅₁𝑅⟶₂^induce𝜋_𝑋𝑅₂⟹𝑖𝑓𝜋_𝑋𝑅₁=𝜋_𝑋𝑅₂,𝑡ℎ𝑒𝑛𝑅₁≅𝑅₂,𝑖𝑓𝜋_𝑋𝑅₁≃𝜋_𝑋𝑅₂,𝑡ℎ𝑒𝑛𝑅₁∼𝑅₂,𝑅⟶₁^induce𝑅₁𝑅⟶₂^induce𝑅₂⟹𝑖𝑓𝑅₁=𝑅₂,𝑡ℎ𝑒𝑛𝑅₁∼𝑅₂,𝑖𝑓𝑅₁≅𝑅₂,𝑡ℎ𝑒𝑛𝑅₁∼𝑅₂,𝑖𝑓𝑅₁∼𝑅₂,𝑡ℎ𝑒𝑛𝑅₁≐𝑅₂, (3.10)

Table 1 

The relationships between fuzzy relations and hierarchical quotient space structure.

Note 3. 𝑅₁ and 𝑅₂ are two fuzzy equivalence relations derived from two fuzzy similarity relations 𝑅₁ and 𝑅₂ on 𝑋, respectively.

4. A Fast Constructing Hierarchical Quotient Space Structure Method

In Section 3, in order to obtain a classification (clustering) result (or a hierarchical quotient space structure), first we should establish a fuzzy similarity relation 𝑅 on 𝑋. Then a fuzzy equivalence relation 𝑅 is derived from the fuzzy similarity relation 𝑅 by the transitive closure operation. Finally, a kind of classification (or clustering) result is obtained. Many literature studies [26] have discussed about how to efficiently transform a fuzzy similarity relation into a fuzzy equivalence relation, and many important results have been obtained. In spite of these previous studies, the efficiency of obtaining classification (or clustering) result can be further improved through directly constructing a hierarchical quotient space structure from a fuzzy similarity relation. In this section, a fast constructing hierarchical quotient space structure method is discussed, and this method can effectively improve the efficiency of the classification (or clustering).

Assume that 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛}, 𝑅 is a fuzzy similarity relation on 𝑋, and its corresponding relation matrix is defined by 𝑀(𝑅)=(𝑟_𝑖𝑗)_𝑛×𝑛. Let 𝑟̃_𝑖𝑗=𝑅(𝑥_𝑖,𝑥_𝑗) stand for the similarity degree between objects 𝑥_𝑖 and 𝑥_𝑗 according to some similarity degree criterion. 𝑀(𝑅) is a 𝐧×𝐧 matrix as follows:𝑀𝑅=𝑟̃₁₁𝑟̃₁₂⋯𝑟̃_1𝑛𝑟̃₂₁𝑟̃₂₂⋯𝑟̃_2𝑛𝑟̃⋮⋮⋱⋮_𝑛1𝑟̃_𝑛2⋯𝑟̃_𝑛𝑛. (4.1)

Because 𝑀(𝑅) is reflexive and symmetric, that is, 𝑟̃_𝑖𝑗=𝑟̃_𝑗𝑖 and 𝑟̃_𝑖𝑖=1(𝑖=1,2,…,𝑛), the matrix 𝑀(𝑅) may be constructed by elements 𝑟̃₁₂,𝑟̃₁₃,…,𝑟̃_1𝑛,𝑟̃₂₃,𝑟̃₂₄,…,𝑟̃_2𝑛,𝑟̃₃₄,…,𝑟̃_3𝑛,…,𝑟̃_{𝑛−1,𝑛}. We reorder the sequence 𝑟̃₁₂,𝑟̃₁₃,…,𝑟̃_1𝑛,𝑟̃₂₃,𝑟̃₂₄,…,𝑟̃_2𝑛,𝑟̃₃₄,…,𝑟̃_3𝑛,…,𝑟̃_{𝑛−1,𝑛} in degressive order. If the same element occurs, just leaves one of them, and discards others, then we can obtain a new degressive sequence denoted by ={𝜆₁,𝜆₂,…,𝜆_𝑡}(𝑡≤𝑛(𝑛−1)/2), where 1≥𝜆₁>𝜆₂>⋯>𝜆_𝑡>0. Because the maximal number of layers in hierarchical quotient space structure is 𝑛, only anterior 𝑛 numbers in the sequence 𝜆={𝜆₁,𝜆₂,…,𝜆_𝑡} is useful for constructing a hierarchical quotient space structure, and other values are ignored. In some layers (it is a quotient space on 𝑋) of the hierarchical quotient space structure, if 𝑥_𝑖 and 𝑥_𝑗 are indiscernible, then they must be located in the same subblock.

Theorem 4.1. Let the relation matrix of the fuzzy similarity relation 𝑅 be 𝑀(𝑅)=(𝑟̃_𝑖𝑗)_𝑛×𝑛, let 𝑅 be fuzzy equivalence relation derived from 𝑅, and its matrix is denoted by 𝑀(𝑅)=(𝑟_𝑖𝑗)_𝑛×𝑛. If 𝑟̃_𝑖𝑗≥𝜆, then 𝑥_𝑖 and 𝑥_𝑗 are located in the same subblock in quotient space 𝑋(𝜆) of the hierarchical quotient space structure 𝜋_𝑋(𝑅).

Proof. If 𝑟̃_𝑖𝑗≥𝜆, 𝑟̃_𝑖𝑗⁽²⁾ denotes the element of matrix 𝑀(𝑅²); according to Proposition 2.3, we have 𝑟⁽²⁾_𝑖𝑗=⋁_{1≤𝑘≤𝑛}(𝑟_𝑖𝑘∧𝑟_𝑘𝑗). Since 𝑟_𝑖𝑖=1, when 𝑘=𝑖, we have 𝑟⁽²⁾_𝑖𝑗≥𝑟̃_𝑖𝑗. 𝑟_𝑖𝑗≥𝑟̃⁽²⁾_𝑖𝑗≥𝑟̃_𝑖𝑗≥𝜆, that is, 𝑅(𝑥_𝑖,𝑥_𝑗)≥𝜆 and (𝑥_𝑖,𝑥_𝑗)∈𝑅_𝜆. So, if 𝑟̃_𝑖𝑗≥𝜆 in the quotient space 𝑋(𝜆) (the partition 𝑋(𝜆) is induced by the equivalence relation 𝑅_𝜆) of the hierarchical quotient space structure 𝜋_𝑋(𝑅), 𝑥_𝑖 and 𝑥_𝑗 are located in the same subblock.

Corollary 4.2. Let the matrix of fuzzy similarity relation 𝑅 be 𝑀(𝑅)=(𝑟̃_𝑖𝑗)_𝑛×𝑛, let 𝑅 be a fuzzy equivalence relation derived from 𝑅, and let its matrix be 𝑀(𝑅)=(𝑟_𝑖𝑗)_𝑛×𝑛. If 𝑟̃_𝑖𝑗≥𝜆 and 𝑟̃_𝑗𝑘≥𝜆, then 𝑥_𝑖, 𝑥_𝑗, and 𝑥_𝑘 are located in the same subblock in the quotient space 𝑋(𝜆) of the hierarchical quotient space structure 𝜋_𝑋(𝑅).

Based on Theorem 4.1 and Corollary 4.2, a fast constructing hierarchical quotient space structure method is presented in Algorithm 1

Let 𝑋={𝑥1,𝑥2,…,𝑥𝑛}. Input: A fuzzy similarity relation matrix 𝑀(𝑅)=(𝑟̃𝑖𝑗)𝑛×𝑛 Output: A hierarchical quotient space structure 𝜋𝑋(𝑅) (Where 𝑅 is a fuzzy equivalence relation derived from 𝑅) Step  1 Using the matrix 𝑀(𝑅)=(𝑟̃𝑖𝑗)𝑛×𝑛, a similarity degree sequence 𝜆={𝜆1,𝜆2,…,𝜆𝑡} can be obtained. Step  2 For 𝑘=1 to 𝑛    {If 𝑟̃𝑖𝑗≥𝜆𝑘, 𝑥𝑖 and 𝑥𝑗 are located in the same subblock, else 𝑥𝑖 and 𝑥𝑗 are located    in the different subblock respectively. A quotient space 𝑋(𝜆𝑘) is constructed}    {if 𝑋(𝜆𝑘)={𝑋}, go to Step 𝟑} Step  3 End Step  4 A hierarchical quotient space structure 𝜋𝑋(𝑅) is obtained.

Algorithm 1 

There is an example to illuminate Algorithm 1.

Example 4.3. Let 𝑋={𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆}; according to some similarity degree criterion, a similarity relation matrix can be established; that is, 𝑀𝑅=𝑟̃_{𝑖𝑗𝑛×𝑛}=10.9280.8880.8380.7880.7580.92810.9550.9020.8480.8150.8880.95510.9410.8860.8520.8380.9020.94110.9400.9400.7880.8480.8860.94010.9650.7580.8150.8520.9400.9651. (4.2)

Step 1. Because the similarity relation matrix 𝑀(𝑅)=(𝑟̃_𝑖𝑗)_𝑛×𝑛 is reflexive and symmetrical, we only need to consider its upper triangular matrix. A similarity degree sequence 𝜆={0.965,0.955,0.941,0.940,0.928,0.902} is obtained from (𝑅).

Step 2. When 𝜆₁=0.965, 𝑟̃₅₆=0.965, the first quotient space 𝑋(0.965)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆ (4.3) is constructed, which is the first layer in 𝜋_𝑋(𝑅).
When 𝜆₂=0.955, 𝑟̃₅₆>0.955 and 𝑟̃₂₃=0.955, the second quotient space 𝑋(0.955)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆ (4.4) is constructed, which is the second layer in 𝜋_𝑋(𝑅).
When 𝜆₃=0.941, 𝑟̃₅₆>0.941, 𝑟̃₂₃>0.941 and 𝑟̃₃₄=0.941, the third quotient space 𝑋(0.941)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆ (4.5) is constructed, which is the third layer in 𝜋_𝑋(𝑅).
When 𝜆₄=0.940, 𝑟̃₅₆>0.940, 𝑟̃₂₃>0.940, 𝑟̃₃₄>0.940, 𝑟̃₄₅=0.940 and 𝑟̃₄₆=0.940, the fourth quotient space 𝑋(0.940)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆ (4.6) is constructed, which is the fourth layer in 𝜋_𝑋(𝑅).
When 𝜆₅=0.928, 𝑟̃₅₆>0.928, 𝑟̃₂₃>0.928, 𝑟̃₃₄>0.928, 𝑟̃₄₅>0.928, 𝑟̃₄₆>0.928 and 𝑟̃₁₂=0.928, the fifth quotient space 𝑋(0.928)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆ (4.7) is constructed, which is the fifth layer in 𝜋_𝑋(𝑅).
Since 𝑋(0.928)={𝑋}, the whole hierarchical quotient space structure is successfully constructed, and it is 𝜋_𝑋(𝑅)={𝑋(0.965),𝑋(0.955),𝑋(0.941),𝑋(0.940),𝑋(0.928)}. (4.8)

5. Conclusions

Due to human various subjective ideas and evaluation criteria, the same fuzzy concept might have different memberships which leads to different fuzzy similarity relations. However, if these fuzzy similarity relations are isomorphic, then they will induce to the same hierarchical quotient space structure. If different hierarchical quotient space structures have the same information entropy sequence and the same subblock sequence, then they have the same classification (clustering) ability and contain the same information. Therefore, the information entropy sequence is a very useful attribute for measuring the uncertainty of a hierarchical quotient space structure. In this paper, the information entropy sequence of a hierarchical quotient space structure is discussed, through the analysis of the relationships among information entropy sequence, hierarchical quotient space structure, fuzzy equivalence relation, and fuzzy similarity relation. A fast-constructing hierarchical quotient space structure method is presented. This further reveals the essential of a hierarchical quotient space structure. In real world, “fuzziness” and “crispness” are relative and can be transformed into each other with different granularity. The hierarchical quotient space structure is just a bridge between the fuzzy granule world and the clear granule world, and the research on uncertainty of the hierarchical quotient space structure will contribute to develop both granular computing theory and information entropy theory.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 61073146 and no. 60773113), Science & Technology Research Program of the Municipal Education Committee of Chongqing of China (no. KJ060517), Natural Science Foundation Project of CQ CSTC (no. 2008BA2017), and Doctor Foundation of CQUPT (no. 2010-06).