The Uncertainty Measure of Hierarchical Quotient Space Structure

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.


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 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.

Preliminary Concepts
For convenience, some preliminary concepts are reviewed or defined at first.Let X be a Cantor set.
Proposition 2.4 see 17 .Let R be a fuzzy equivalence relation on X, and 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 {x 1 , x 2 , x 3 , x 4 , x 5 }, and R 1 is a fuzzy equivalence relation on X, the corresponding relation matrix M R 1 is defined as follows, Its corresponding quotient space sequence is defined as follows,

2.2
So, a hierarchical quotient space structure induced by the fuzzy equivalence relation , 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 R 2 of the fuzzy equivalence relation R 2 is defined as follows: 1 0.2 0.9 0.6 0.6 0.2 1 0.2 0.2 0.2 0.9 0.2 1 0.6 0.6 0.6 0.2 0.6 1 0.7 0.6 0.2 0.6 0.7 1 then R 2 can become into the same quotient space sequence and the same hierarchical quotient space structure as R 1 ; that is, π X R 1 π X R 2 .

Figure 1:
The hierarchical quotient space structure π X R 1 of R 1 .
The pyramid model of a hierarchical quotient space structure.
Definition 2.7 see 16 .Let R 1 and R 2 be two fuzzy equivalence relations on X; if they have the same hierarchical quotient space structure, that is, Proposition 2.8 see 13 .Let R 1 and R 2 be two fuzzy equivalence relations on X, for any Definition 2.9 see 18 .Let R 1 and R 2 be two fuzzy equivalence relations on X; for any Proposition 2.10 see 18 .Let R 1 and R 2 be two fuzzy equivalence relations on 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 R 1 and R 2 be two fuzzy similarity relations on X, and R 1 and R 2 are fuzzy equivalence relations derived from R 1 and R 2 , respectively.If R 1 R 2 , then R 1 and R 2 are called isomorphism, denoted by 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.

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.} are two partition spaces on X.If P ≺ P and there exist P i ∈ P ∃ P j ∈P P i ⊂ P j , then P is strictly finer than P , denoted by P ≺ P .
Each layer of the hierarchical quotient space structure . ., X λ t } be its corresponding hierarchical quotient space structure, and Obviously, any one partition sequence of the hierarchical quotient space structure π X R is a positive integer sequence, and 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 λ 1 , X λ 2 , . . ., X λ t } be its corresponding hierarchical quotient space structure, and let L π X R {L 1 , L 2 , . . ., 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 iL i } it is a partition on X .The information entropy H X λ i of X λ i is defined as follows: Let R be a fuzzy equivalence relation on X, and let π X R {X λ 1 , X λ 2 , . . ., X λ n } be its hierarchical quotient space structure, then the entropy sequence H π X R {H X λ 1 , H X λ 2 , . . ., 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 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

Mathematical Problems in Engineering
Because 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 {x 1 , x 2 , . . ., x n } be a nonempty finite set, and let P k X {X 1 , X 2 , . . ., X k } be a partition space on X, P k X is called k-order partition on X where Obviously, two different k-order partitions on X may have the same subblock sequence I k .Definition 3.8.Let X {x 1 , x 2 , . . ., x n } be a nonempty finite set, let P k X {X 1 , X 2 , . . ., X k } be a partition space on X, and let I 1 k {a 1 , a 2 , . . ., a k } be a subblock sequence of the partition P k X .I 2 k {a 1 , a 2 , . . ., a p 1, . . ., a q − 1, . . ., a k } is called a σ-displacement of I 1 k , where p < q, a p 1 < a q − 1, and I 2 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.

Theorem 3.10. If a subblock sequence
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, Therefore,

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 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 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 P 3 X and their corresponding subblock sequences I 1 3 and I 2 3 are different.Let I 1 3 {a 1 , a 2 , a 3 }, I 2 3 {b 1 , b 2 , b 3 }, and I 1 3 / I 2 3 .If only one element in I 1 3 is equal to the element of I 2 3 e.g., let a 1 b 1 , then a 2 / b 2 and a 3 / b 3 , due to the former conclusion in case of k 2, H P 3 X / H P 3 X is held.If all elements in I 1 3 are not equal to that in I 2 3 , then we get the conclusion from the following three cases other cases can be transformed into one of the after-mentioned three conditions : Given a hierarchical quotient space structure π X R , we can easily obtain its corresponding information entropy sequence {H I n 1 , H I n 2 , . . ., H I n s , . ..}where {n 1 , n 2 , . . ., 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 n 1 , H I n 2 , . . ., 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 n 1 , H I n 2 , . . ., 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 R 1 and π X R 2 be two hierarchical quotient space structures derived from the fuzzy equivalence relations R 1 and R 2 on X, and their partition sequences are L π X R 1 {L 11 , L 12 , . . ., L 1r } and L π X R 2 {L 21 , L 22 , . . ., L 2s }, respectively.Let {I L 11 , I L 12 , . . ., I L 1r } and {I L 21 , I L 22 , . . ., I L 2s } denote their corresponding subblock sequences of each layer in π X R 1 and π X R 2 , respectively.If L π X R 1 Mathematical Problems in Engineering 11 L π X R 2 obviously, where r s and I L 1i I L 2i , i 1, 2, . . ., r , then π X R 1 and π X R 2 are called isomorphism, denoted by π X R 1 π X R 2 .
Example 3.13.Assume that X {1, 2, 3, 4, 5, 6}, and there are two hierarchical quotient space structures π X R 1 and π X R 2 on X as follows: 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 R 1 and π X R 2 be two hierarchical quotient space structures derived from fuzzy equivalence relations R 1 and R 2 on X, respectively.
In Theorem 3.14, the condition L π X R 1 L π X R 2 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 .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.

3.10
Note.R 1 and R 2 are two fuzzy equivalence relations derived from two fuzzy similarity relations R 1 and R 2 on X, respectively.

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 {x 1 , x 2 , . . ., 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 r 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: Because M R is reflexive and symmetric, that is, r ij r ji and r ii 1 i 1, 2, . . ., n , the matrix M R may be constructed by elements r 12 , r 13 , . . ., r 1n , r 23 , r 24 , . . ., r 2n , r 34 , . . ., r 3n , . . ., r n−1,n .We reorder the sequence r 12 , r 13 , . . ., r 1n , r 23 , r 24 , . . ., r 2n , r 34 , . . ., r 3n , . . ., r 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 Because the maximal number of layers in hierarchical quotient space structure is n, only anterior n numbers in the sequence λ {λ 1 , λ 2 , . . ., 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 r 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 r 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 r ij ≥ λ, r 2 ij denotes the element of matrix M R 2 ; according to Proposition 2.3, we have r 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 r 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 r ij ≥ λ and r 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 There is an example to illuminate Algorithm 1.
When 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

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.
and a 3 > b 3 .I 2 3 cannot be replaced by I 1 3 directly through the σdisplacement once, but I 1 3 can be replaced by I 3 3 {a 1 , a 2 a 3 −b 3 , b 3 } through the σ-displacement once.By Theorem 3.10, H I 1 3 < H I 3 3 < H I 2 3 holds.So H I 1 3 / H I 2 3 holds.
Definition 3.2 see 19 .Let X {x 1 , x 2 , . . ., x n } be a nonempty finite set, P {P 1 , P 2 , . . ., P l } and P {P 1 , P 2 , . . ., P m 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 {x 1 , x 2 , . . ., x n } be a nonempty finite set, P {P 1 , P 2 , . . ., P l } and P {P 1 , P 2 , . . ., P m } are two partition spaces on X. If, for all P i ∈ P , ∃ P j ∈P P i ⊆ P j , then P is finer than P , denoted by P ≺ P .
partition on X, and let |X 1 | a 1 , |X 2 | 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 {a 1 , a 2 , . . ., a k } denoted by I k .I k is called a subblock sequence of the partition P k X .
which is discussed in Theorem 3.11 as follows.Let X {x 1 , x 2 , . . ., 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 I 1 k and I 2 k , respectively.If I 1 k / I 2 k , then H I 1 k / H I 2 k , where H I 1 k and H I 2 k denote the information entropy of partitions P k X and P k X based on Definition 3.4, respectively 2 ≤ k ≤ n .
1 , X 2 }, let |X 1 | x and |X 2 | 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

Table 1 :
The relationships between fuzzy relations and hierarchical quotient space structure.R 1 and R 2 R 1 and R 2 π X R 1 and π X R 2 H π X R1 and H π X R 2 Isogeny but not Similarity .Let R 1 and R 2 be two fuzzy similarity relations on X, and let R 1 and R 2 be two fuzzy equivalence relations derived from R 1 and R 2 , respectively.If π X R 1 π X R 2 , then R 1 and R 2 are called isogenous fuzzy similarity relations, denoted by R 1 .R 2 .If R 1 and R 2 are not isomorphic, but π X R 1 π X R 2 , then R 1 and R 2 are called similar fuzzy equivalence relation, denoted by R 1 ∼ R 2 .