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M. Keshavarzi, M. Dehghan, M. Mashinchi (2009)
CLASSIFICATION BASED ON SIMILARITY AND DISSIMILARITY THROUGH EQUIVALENCE CLASSES
A. Tversky (1977)
Features of SimilarityPsychological Review, 84
Petko Valtchev (1999)
Construction automatique de taxonomies pour l'aide à la représentation de connaissances par objets
(1997)
Structural alignment in analogy and similarity
M. Keshavarzi, M. Dehghan, M. Mashinchi (2011)
Classification based on 3-similarityIranian Journal of Mathematical Sciences and Informatics, 6
T. Landauer, S. Dumais (1997)
A Solution to Plato's Problem: The Latent Semantic Analysis Theory of Acquisition, Induction, and Representation of Knowledge.Psychological Review, 104
V. Loia, S. Senatore, M. Sessa (2004)
Combining agent technology and similarity-based reasoning for targeted E-mail servicesFuzzy Sets Syst., 145
B. Schweizer, A. Sklar (2011)
Probabilistic Metric Spaces
R. Shepard (1962)
The analysis of proximities: Multidimensional scaling with an unknown distance function. I.Psychometrika, 27
Levi Larkey, A. Markman (2005)
Processes of Similarity JudgmentCognitive science, 29 6
D. Gentner, A. Markman (1997)
Structure mapping in analogy and similarity.American Psychologist, 52
U. Hahn, N. Chater, L. Richardson (2003)
Similarity as transformationCognition, 87
M. Sessa (2001)
Translations and similarity-based logic programmingSoft Computing, 5
(1983)
Tversky A (1977) Features of similarity
R. Aliguliyev (2008)
EXPERIMENTAL INVESTIGATING THE F-MEASURE AS SIMILARITY MEASURE FOR AUTOMATIC TEXT SUMMARIZATION
R. Shepard (1962)
The analysis of proximities: Multidimensional scaling with an unknown distance function. IIPsychometrika, 27
(1999)
Construction automatique de taxanomies pour l’aide a
H. Rezaei, Masashi Emoto, M. Mukaidono (2006)
New Similarity Measure Between Two Fuzzy SetsJ. Adv. Comput. Intell. Intell. Informatics, 10
(1988)
Algorithm for clustring data
E. Rissland (2006)
AI and SimilarityIEEE Intelligent Systems, 21
L. Zadeh (1996)
Fuzzy sets
F. Klawonn, J. Peña (1995)
Similarity in fuzzy reasoning, 2
Fuzzy Inf. Eng. (2012) 1: 75-91 DOI 10.1007/s12543-012-0102-4 ORIGINAL ARTICLE Applications of Classification Based on Similarities and Dissimilarities M.Keshavarzi· M.A.Dehghan· M.Mashinchi Received: 18 September 2010/ Revised: 10 January 2012/ Accepted: 13 February 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we first review some definitions and propositions regard- ing similarities between two objects (2-similarity), or among three or more objects (3-similarity/n-similarity). Then the notion of 3-dissimilarity is introduced and the relationships and connections between 2-dissimilarities and 3-dissimilarities are stud- ied. Through some examples, the applications regarding the concepts of 3-similarity, 4-similarity, and dissimilarity relations will be brought out. Keywords 3-similarity relations· Equivalence 3-relations· Dissimilarity relations· n-dissimilarity 1. Introduction Similarity is some degree of resemblance between two or more concepts or objects. The notion of similarity refers to approximate (or exact) repetitions of patterns of some sort in the compared items. In many real world applications, designers tend to- wards building classes of objects according to some similarity (dissimilarity) criteria. There are various contexts in which similarities have been widely studied. Similarity- based clustering is described in [2,17]. Rissland studied computational similarity models as a new method for information retrieval [12]. Performance of different sim- ilarity measures in the context of document summarization was investigated in [1] by Alguliev et al. Loia et al in [10] used similarity relations in an internet e-mail application. Rezaei et al in [11] proposed a new similarity measure between fuzzy M.Keshavarzi () · M.Mashinchi () Mathematics and Computer Sciences, Shahid Bahonar University, Kerman, Iran email: mkeshavarzi@mail.vru.ac.ir mashinchi@mail.uk.ac.ir M.A.Dehghan () Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran email: dehghan@mail.vru.ac.ir 76 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) sets and extended it to define two other similarity measures. Another area that sim- ilarities have been widely used is in psychology [3,4,8,9,14,16]. Also, the concepts of similarity, dissimilarity, their relationships, and the functions that convert them to one another have been studied in [5]. In this paper, after the introduction section, the definitions and propositions re- garding the similarity and dissimilarity relations have been brought in Section 2. In Section 3, we will review the classification based on 3-similarities and n-similarities that have been discussed in [6]. We will extend the definitions and propositions to the concept of 3-dissimilarity, and in general n-dissimilarity relations. Also, we will extend the relationship between similarity and dissimilarity to 3-similarity and 3- dissimilarity relations in Section 4. In the fifth section, we have developed some algorithms to be used on real data to support our propositions and ideas. In fact, there are three applications that have been discussed, the first one shows, through an exam- ple, the way that we can select three items, out of a set of objects, that are most similar to one another. There are situations where among a group of items, we are looking for the most dissimilar item(s). In the second application, again through an example, we have shown how this can be achieved, that is, using the computed similarities of the items in Application 1, we have computed their dissimilarities by using one of the functions discussed in [5] to find the most dissimilar items. The last application is an example of 4-similarities and an application related to numerical analysis. At the end, a conclusion is provided in Section 6. 2. Similarity Relation Similarity relation [10] is a mathematical notion that provides a way to manage al- ternative instances of an entity that can be considered “equal” to other entities with a given degree [7,18]. Definition 2.1 [5,10] A 2-similarity on a domain U is a function S : U×U −→ [0, 1] such that the following properties hold: (i) S (x, x) = 1 for any x ∈ U (reflexivity). (ii) S (x, y) = S (y, x) for any x, y ∈ U (symmetry). (iii) S (x, z) ≥ S (x, y)∧S (y, z) (transitivity) for any x, y, z ∈ U, where∧ is a minimum operator. We say that S is strict if the following implication is also verified: (iv) S (x, z) = 1 ⇒ x = z. The following notion ofλ-cut is crucial in our study. Definition 2.2 [5,10] Let S : U × U −→ [0, 1] be a 2-similarity on U. Then for any λ ∈ [0, 1], the relation in U is defined as S,λ x y ⇔ S (x, y) ≥ λ. S,λ The set S = {(x, y)|S (x, y) ≥ λ} is called an λ-cut of S . Note that x y if and λ S,λ only if (x, y) ∈ S . We may simply call the relation as the λ-cut of S . The notion λ S,λ Fuzzy Inf. Eng. (2012) 1: 75-91 77 of allows us to define a 2-similarity by means of suitable family of equivalence S,λ relations according to the following results. Proposition 2.1 [5,10] (a) Let S be a 2-similarity in a domain U, and for any λ ∈ [0, 1], let be the λ-cut of S . Then { } is a family of equivalence S,λ S,λ λ∈[0,1] 2-relations such that: (i) For any μ and λ in [0,1],λ ≤ μ ⇒ ⊇ . S,λ S,μ (ii) For any μ in [0,1], = . S,λ S,μ λ<μ (b) Let { } be a family of equivalence 2-relations on U satisfying conditions (i) λ λ∈[0,1] and (ii) above, then the relation S defined by setting S (x, y) = Sup{λ ∈ [0, 1]|x y} is a 2-similarity whose family of λ-cuts is equal to the family { } . λ λ∈[0,1] Definition 2.3 [6] A 3-similarity on a domain U is a function S : U×U×U −→ [0, 1] such that the following properties hold: (i) S (x, x, x) = 1 for any x ∈ U (reflexivity). (ii) S (x , x , x ) = S (x , x , x ) for any x , x , x ∈ U, where (i , i , i ) is an arbi- 1 2 3 i i i 1 2 3 1 2 3 1 2 3 trary permutation of (1,2,3) (symmetry). (iii) S (x , x , x ) ≥ S (t, x , x )∧ S (x , t, x )∧ S (x , x , t), (transitivity) 1 2 3 2 3 1 3 1 2 for any t, x , x , x ∈ U, where∧ is a minimum operator. We say that S is strict 1 2 3 if the following implication is also verified: (iv) S (x , x , x ) = 1 ⇒ x = x = x . 1 2 3 1 2 3 Definition 2.4 [6] Let U be a set and S : U × U × U −→ [0, 1] be a 3-similarity on U. Then for any λ ∈ [0, 1], the 3-relation in U defined as (x, y, z) ∈ if S,λ S,λ S (x, y, z) ≥ λ. The set is called cut of levelλ of S or λ-cut of S . S,λ Definition 2.5 A subset R of U × U × U, is called an equivalence 3-relation on U if (i) (x, x, x) ∈ R for all x ∈ U. (ii) If (x , x , x ) ∈ R, then (x , x , x ) ∈ R for all permutations (i , i , i ) of 1 2 3 i i i 1 2 3 1 2 3 (1, 2, 3). (iii) (t, y, z) ∈ R, (x, t, z) ∈ R and (x, y, t) ∈ R implies that (x, y, z) ∈ R for all x, y, z, t ∈ U, where by a 3-relation we mean any non-empty subset of U×U×U. Lemma 2.1 [6] Let S : U × U × U −→ [0, 1] be a map and let for any λ ∈ [0, 1], := {(x, y, z) ∈ U× U× U : S (x, y, z) ≥ λ}, theλ-cut of S, be a 3-relation. Then S is S,λ a 3-similarity on U if and only if for any λ ∈ [0, 1], is an equivalence 3-relation. S,λ Theorem 2.1 [6] Let S be a 3-similarity on a set U, and let be theλ-cut of S for S,λ anyλ ∈ [0, 1]. Then{ : λ ∈ [0, 1]} is a family of equivalence 3-relations such that, S,λ 78 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) (i) λ ≤ μ implies that ⊆ for any μ andλ in [0, 1]. S,μ S,λ (ii) = for anyμ in [0, 1]. S,λ S,μ λ<μ Conversely, let { : λ ∈ [0, 1]} be a family of equivalence 3-relations satisfying conditions (i), (ii). Then the relation S defined by setting S (x, y, z) = Sup{λ ∈ [0, 1] : (x, y, z) ∈ } is a 3-similarity whose family ofλ-cuts is equal to the family { } . λ λ λ∈[0,1] For 2-similarity, Sessa in a paper [16] said that in a set like{S (x, y), S (x, z), S (y, z)} two of the members are equal and the third member is either equal or greater than the other two. We generalize this fact for 3-similarity in the following proposition. Proposition 2.2 [6] Let S : U × U −→ [0, 1] be a 2-similarity on U. For all x, y, z ∈ U, define S : U × U × U −→ [0, 1] as follows, S (x, y, z) = min{S (x, y), S (y, z), S (x, z)}. 3 2 2 2 Then for every x, y ∈ U, we have (a) S is a 3-similarity on U. (b) S (x, x, y) = S (x, y). 3 2 (c) S (x, x, y) ≥ S (x, y, z). 3 3 (d) S (x, x, y) = S (x, y, y). 3 3 Using the definitions of 2-similarity and 3-similarity, we now can define the n- similarity as follows: Definition 2.6 [6] S : U × U × U··· U −→ [0, 1] is n− similarity, if (i) S (x, x,··· , x) = 1 (reflexivity); (ii) S (x , x ,··· , x ) = S (x , x ,··· , x ) for all permutations (i , i , i ,··· , i ) of 1 2 n i i i 1 2 3 n 1 2 n (1,2,··· , n) (symmetry); (iii) S (x , x ,··· , x ) ≥ min{S (z, x ,··· , x ),··· , S (x , x ,··· , x , z)} (transitiv- 1 2 n 2 n 1 2 n−1 ity) for all x , x ,··· , x , z ∈ U. 1 2 n Now, similarly we can obtain an n-similarity from an (n− 1)-similarity. The proof is similar to the proof of Proposition 2.2. Also, we can prove it by induction. 3. Dissimilarity Relations Dissimilarity relation is a mathematical notion providing a way to manage alternative instances of an entity that can be considered “different” to a given degree [5]. Now, we are going to extend the definitions and the propositions of 3-similarities to 3-dissimilarities. Definition 3.1 A 3-dissimilarity on a domain U is a function S : U×U×U −→ [0, 1] such that the following properties hold: Fuzzy Inf. Eng. (2012) 1: 75-91 79 (i) D(x, x, x) = 0 for any x ∈ U (reflexivity). (ii) D(x , x , x ) = D(x , x , x ) for any x , x , x ∈ U, where (i , i , i ) is an arbi- 1 2 3 i i i 1 2 3 1 2 3 1 2 3 trary permutation of (1,2,3) (symmetry). (iii) D(x , x , x ) ≤ D(t, x , x )∨ D(x , t, x )∨ D(x , x , t) for any t, x , x , x ∈ U, 1 2 3 2 3 1 3 1 2 1 2 3 where∨ is a maximum operator (transitivity). We say that D is strict if the following implication is also verified: (iv) D(x , x , x ) = 0 ⇒ x = x = x . 1 2 3 1 2 3 The following example shows a strict 3-dissimilarity. Example 3.1 Suppose U = {1, 2, 3}. Then we define , if x y or x z, D(x, y, z) = 5 1, if x=y=z. It is obvious that D is a 3-dissimilarity and it satisfies the property (iv) above. There is a correspondance between 3-dissimilarities and equivalence 3-relations. The proof is similar to the proof for the case of 3-similarities. Proposition 3.1 a) Let D be a dissimilarity in a domain U, and for any λ ∈ [0, 1], let be theλ-cut of D. Then,{ } is a family of equivalence relations such D,λ D,λ λ∈[0,1] that: (i) For any μ andλ in [0, 1],λ ≤ μ ⇒ ⊆ . D,λ D,μ (ii) For any μ in [0, 1], = . D,λ D,μ λ≥μ b) Let{ } be a family of equivalence relations satisfying Conditions (i) and (ii) λ λ∈[0,1] above, then the relation D defined by D(x, y) = In f{λ ∈ [0, 1]|x y} is a dissimilarity whose family of λ-cuts is { } . λ λ∈[0,1] Proposition 3.2 Let D be a 3-dissimilarity, a=D(x,y,z), b=D(t,y,z), c=D(x,t,z), and d=D(x,y,t). Then two of the a, b, c and d are equal, and the others are either equal or less than to the first two. proof Using Definition 3.1 (iii), any of the following could be holding: (i) a ≤ b∨ c∨ d, (ii) b ≤ a∨ c∨ d, (iii) c ≤ a∨ b∨ d, (iv) d ≤ a∨ b∨ c. 80 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) Since a, b, c and d have the same role and without loss of generality, consider a ≤ b ≤ c ≤ d. Then c = a∨ b∨ c and by (iv), d ≤ c. Therefore a ≤ b ≤ c = d and the theorem is proved. We can use T -conorms to generalize the concept of 2-dissimilarity to 3-dissimilarity. In the following, after giving the definition of T -conorms [15], we will show how they can be utilized for this purpose. Proposition 3.3 Let D : U × U −→ [0, 1] be a 2-dissimilarity on U. For all x, y, z ∈ U, define D : U × U × U −→ [0, 1] as follows, D (x, y, z) = max{D (x, y), D (y, z), D (x, z)}. 3 2 2 2 Then for every x, y ∈ U, we have (a) D is a 3-dissimilarity on U. (b) D (x, x, y) = D (x, y). 3 2 (c) D (x, x, y) ≤ D (x, y, z). 3 3 (d) D (x, x, y) = D (x, y, y). 3 3 Similarly, we can obtain an n-dissimilarity from an (n− 1)-dissimilarity. The proof is similar to the proof of Proposition 3.3. Also, we can prove it by induction. Definition 3.2 A T-conorm (triangular conorm) is a function P :[0, 1]× [0, 1] −→ [0, 1] which satisfies the following properties for all a, b, c, d ∈ [0, 1]. 1) P(a,b)=P(b,a) (commutativity); 2) P(a, b) ≤ P(c, d) if a ≤ c and b ≤ d (monotonicity); 3) P(a,P(b,c))=P(P(a,b),c) (associativity); 4) P(a,0)=a (boundary condition). Maximum T -conorm, P (a, b) = max{a, b}, is one of the prominent examples of max T -conorms which we will use in Theorem 3.1. Theorem 3.1 Let U be a set, P be a T-conorm of P :[0, 1]× [0, 1] −→ [0, 1], and D be a 2-dissimilarity on U. Then for all x, y, z ∈ U consider, D (x, y, z) = P(P(D (x, y), D (y, z)), D (x, z)). 3 2 2 2 Then D is a 3-dissimilarity on U. Proof D (x, x, x) = P(P(D (x, x), D (x, x)), D (x, x)) = P(P(0, 0), 0) = P(0, 0) = 0. 3 2 2 2 Also, it is obvious that by changing the positions of x, y and z, the result would not change. Now, for the transitivity property, we have to show that for all x, y, z and t ∈ U, D (x, y, z)≤{D (t, y, z)∨ D (x, t, z)∨ D (x, y, t)}. 3 3 3 3 Without loss of generality, let Fuzzy Inf. Eng. (2012) 1: 75-91 81 (i) D (t, y, z) ≥ D (x, t, z), and 3 3 (ii) D (t, y, z) ≥ D (x, y, t). 3 3 We have to show D (t, y, z) ≥ D (x, y, z). 3 3 From (i) we have P(P(D (t, y), D (y, z)), D (t, z)) ≥ P(P(D (x, t), D (t, z)), D (x, 2 2 2 2 2 2 z)), hence, D (t, y)] ≥ D (x, t), D (y, z) ≥ D (t, z), and D (t, z) ≥ D (x, z). 2 2 2 2 2 2 From (ii) we have, D (t, y) ≥ D (x, y), D (y, z) ≥ D (y, t), and D (t, z) ≥ D (x, t). 2 2 2 2 2 2 Now, since D (t, y) ≥ D (x, y), and D (t, z) ≥ D (x, z), and D (y, z) ≥ D (y, z), then 2 2 2 2 2 2 P(P(D (t, y), D (y, z)), D (t, z)) ≥ P(P(D (x, y), D (y, z)), D (x, z)). Hence, D (t, y, z) 2 2 2 2 2 2 3 ≥ D (x, y, z). Corollary 3.1 Let D : U×U −→ [0, 1] be a 2-dissimilarity on U. For all x, y, z ∈ U, define D : U × U × U −→ [0, 1] as follows, D (x, y, z) = max{D (x, y), D (y, z), D (x, z)}. 3 2 2 2 Then for every x, y ∈ U, we have (a) D is a 3-dissimilarity on U. (b) D (x, x, y) = D (x, y). 3 2 (c) D (x, x, y) ≤ D (x, y, z). 3 3 (d) D (x, x, y) = D (x, y, y). 3 3 Using the above theorems and propositions, we can convert every 2-dissimilarity to a 3-dissimilarity using T -conorms; the opposite though, may not be true. Naturally, if D be a 3-dissimilarity and D (x, y) = D (x, x, y), by an example we can show that 3 2 3 D is not a 2-dissimilarity. Example 3.2 Let U = {R, M, P}, where the letters in the set stand for: M = mouce, P = pigeon, R = rabit. Define D : U × U × U −→ [0, 1] be a function as in Table 1. Also, let the commutative condition for D holds. Table 1: The level values of 3-dissimilarity D . D (x, y, z)(x, y, z) 0 (P,P,P) (R,R,R) (M,M,M) 0.1 (R,P,P) (R,R,P) 0.2 (R,M,M) (P,M,M) 0.3 (R,R,M) (M,P,P) 0.7 (R,M,P) It is clear that D is a 3-dissimilarity. If we define D : U × U −→ [0, 1] 3 2 as D (x, y) = D (x, x, y), then D (P, M) = D (P, P, M) = 0.3, and D (M, P) = 2 3 2 3 2 82 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) D (M, M, P) = 0.2. Consequently, D (P, M) and D (M, P) are not equal. Hence, 3 2 2 D is not a 2-dissimilarity. Now the question is: under what conditions can we obtain a 2-dissimilarity from a 3-dissimilarity? Similarly, an n-dissimilarity can be obtained from an (n− 1)-dissimilarity. We can prove it by induction. 4. The Relationship between Similarity and Dissimilarity Relations As far as the relationship between similarity and dissimilarity relations is concerned, we should mention that in many theoretical, as well as practical applications, dissim- ilarity, rather than similarity, between two elements of a set needs to be considered. Especially when the elements are very similar to each other, studying the dissimilar- ities (differences) is easier. In [5] functions that convert similarities to dissimilarities have been studied. Application 2, of the following section, shows an example of the way one can obtain dissimilarities from available similarities using the following lemma. Lemma 4.1 Let S : U × U → [0, 1] be a similarity on U. Also let f :[0, 1] → [0, 1] be a decreasing function. Then D : U × U → [0, 1] defined by: ( f (S (x, y))− f (1)) D(x, y) = f (0)− f (1) is a dissimilarity on U. 5. Applications: Algorithms and Working Flows In this section through some applications, we will show how our ideas can be applied in the real world problems. Application 1: In many cases where the similarities of the objects are known, there is a requirement to find similarity among three objects. That is, we need to know what three objects out of the group of objects have the most resemblance to each other, provided that their pairwise similarities are known. For instance, we might be eager to select a three-member team, out of the students of a class for some sort of sports race, or some sort of academic contest. Of course, we want that the selected students have the most similarity to each other, as far as the nature of the contest is concerned. Another example would be the selection of the three (or in general) most politically similar candidates out of a group of candidates for the city’s representatives, say, in parliament, since in that case, we think that they can better serve the city. Example 5.1 Arrangement of residents in a condominium. In a 20-unit condominium, a research has been done on the similarities of the residents on the basis of some criteria, like the number of members in the family, each member’s education and each member’s income as shown in Table 2 and Table 3. The manager of the condominium have decided to choose the three most similar residents for some special purpose. We use Proposition 2.2 to select the three most similar residents. For this purpose, on the basis of Algorithm 1 (Appendix 1), a software has been developed that works according to the following. Fuzzy Inf. Eng. (2012) 1: 75-91 83 Table 2: The similarity values of families F1-F20 to families F1-F10. family F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F1 1.0 0.4 0.4 0.1 0.1 0.1 0.0 0.2 0.2 0.4 F2 0.4 1.0 0.4 0.1 0.1 0.1 0.0 0.2 0.2 0.4 F3 0.4 0.4 1.0 0.1 0.1 0.1 0.0 0.2 0.2 0.4 F4 0.1 0.1 0.1 1.0 0.2 0.8 0.0 0.1 0.1 0.1 F5 0.1 0.1 0.1 0.2 1.0 0.2 0.0 0.1 0.1 0.1 F6 0.1 0.1 0.1 0.8 0.2 1.0 0.0 0.1 0.1 0.1 F7 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 F8 0.2 0.2 0.2 0.1 0.1 0.1 0.0 1.0 0.4 0.2 F9 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.4 1.0 0.2 F10 0.4 0.4 0.4 0.1 0.1 0.1 0.0 0.2 0.2 1.0 F11 0.4 0.4 0.4 0.1 0.1 0.1 0.0 0.2 0.2 0.8 F12 0.4 0.4 0.4 0.1 0.1 0.1 0.0 0.2 0.1 0.4 F13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 F14 0.4 0.4 0.4 0.1 0.1 0.1 0.0 0.2 0.2 0.4 F15 0.4 0.6 0.4 0.1 0.1 0.1 0.0 0.2 0.2 0.4 F16 0.4 0.8 0.4 0.1 0.1 0.1 0.0 0.2 0.2 0.4 F17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 F18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 F19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 F20 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Working flow: Fig.1 shows the snapshots of the user interface panels that have been used in our software. On the left side, there is a set of buttons for rapid commands. The following describes the steps that needs to be taken from entering data to get the results. Entering the names: First of all, in order to set a data base of items, family names in our example, the user clicks on the Enter Family Names button (left side of Fig.1). A window appears for entering the names. The user enters F1, F2,··· , F20 (for simplicity purposes) as the names of the families. Entering Similarities values: There are two buttons on the panel, Enter Similarities ONLINE and Read in Data from File. If the user clicks on the first button, then a window opens for the user to enter similarity values interactively (subjectively). If the user clicks on the other one, the program first asks for the name of the file, then it reads the similarity values from that file (assuming that the user has prepared a file of similarity values previously). In this example as we said before, the similarity values have been computed according to the number of members in the family, their education, their income, that have been stored in a file previously. Clicking on the Show Similarity Values Table, button will cause the similarity values appear at the right side of the panel. 84 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) Table 3: The similarity values of families F1-F20 to families F11-F20. family F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F1 0.4 0.4 0.0 0.4 0.4 0.4 0.0 0.0 0.0 0.0 F2 0.4 0.4 0.0 0.4 0.6 0.8 0.0 0.0 0.0 0.0 F3 0.4 0.4 0.0 0.4 0.4 0.4 0.0 0.0 0.0 0.0 F4 0.1 0.1 0.0 0.1 0.1 0.1 0.0 0.0 0.0 0.0 F5 0.1 0.1 0.0 0.1 0.1 0.1 0.0 0.0 0.0 0.0 F6 0.1 0.1 0.0 0.1 0.1 0.1 0.0 0.0 0.0 0.0 F7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 F8 0.2 0.2 0.0 0.2 0.2 0.2 0.0 0.0 0.0 0.0 F9 0.2 0.2 0.0 0.2 0.2 0.2 0.0 0.0 0.0 0.0 F10 0.8 0.4 0.0 0.4 0.4 0.4 0.0 0.0 0.0 0.0 F11 1.0 0.4 0.0 0.4 0.4 0.4 0.0 0.0 0.0 0.0 F12 0.4 1.0 0.0 0.8 0.4 0.4 0.0 0.0 0.0 0.0 F13 0.0 0.0 1.0 0.0 0.0 0.0 0.4 0.4 0.4 0.4 F14 0.4 0.8 0.0 1.0 0.4 0.4 0.0 0.0 0.0 0.0 F15 0.4 0.4 0.0 0.4 1.0 0.6 0.0 0.0 0.0 0.0 F16 0.4 0.4 0.0 0.4 0.6 1.0 0.0 0.0 0.0 0.0 F17 0.0 0.0 0.4 0.0 0.0 0.0 1.0 0.4 0.6 0.4 F18 0.0 0.0 0.4 0.0 0.0 0.0 0.4 1.0 0.4 0.8 F19 0.0 0.0 0.4 0.0 0.0 0.0 0.6 0.4 1.0 0.4 F20 0.0 0.0 0.4 0.0 0.0 0.0 0.4 0.8 0.4 1.0 Extracting the most 3-similar families: By clicking on the Show the most 3-similar Families button, another panel (Fig.2) appears. In this panel, at first the user must enter the similarity level, which would be in the range of 0.0 to 1.0, having clicked on the Show the Most 3-similar Families button, the user observes the results. In this example, we have entered 0.6 for similarity level. As it can be seen from the results panel, families of F2, F15, and F16 have been chosen. This shows what the management was looking for, i.e., the three most similar families among all, who are F2, F15, and F16. Application 2: This is an application concerning the usage of dissimilarities from similarities. In Lemma 4.1, various functions can be chosen to compute dissim- ilarities from similarities. In the following example, we have used the function f (t) = 1 − t . Depending on the convexity or concavity of the function used, the sum of similarity and dissimilarity would be either less or more than 1 [5]. Depend- ing on the nature of problem, other convex or concave functions could be used as well. Fuzzy Inf. Eng. (2012) 1: 75-91 85 Fig. 1 Snapshot of user interface data entry panel Fig. 2 Snapshot of user interface results panel Example 5.2 Residents classification on the basis of their dissimilarities. Let families F1, F2,··· , F20 and their similarity measures be as in Example 5.1. Suppose the manager of the condominium of Example 5.1 wants to find out which family is the most dissimilar to others, i.e., the family that can not get along with any other family. Probably, they want to know who is the cause of the daily disturbances in the condominium. Our program will find that out. On the basis of Proposition 3.3 and the above mentioned function, we have obtained all 2-dissimilarity measures for every pair of families using their previously computed similarities. The results are in Table 4 and Table 5. We use Proposition 2.2 to classify 20 family members according to their dissimilarity measures by some desired level (λ-level). Algorithm 2 has been used for this purpose. The algorithm starts from the partition Q = U/ , corresponding to the cut n λ,n of level 0, where every element stands in a class by itself. At any iterative step, the program provides the new class in Q as union of classes in the previous partition Q i−1 i by increasing the dissimilarity value level, that is, by a more significant-levelλ>λ i−1 between elements in this class. The outcome is as follows: U/ = {{F1},{F2},{F3},{F4},{F5},{F6},{F7},{F8},{F9},{F10},{F11}, D,0 {F12},{F13},{F14},{F15},{F16},{F17},{F18},{F19},{F20}}, 86 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) U/ = {{F1},{F2, F16},{F3},{F4, F6},{F5},{F7},{F8},{F9}, D,0.5 {F10, F11},{F12, F14},{F13},{F15},{F17},{F18, F20},{F19}}, U/ = {{F1},{F2, F15, F16},{F3},{F4, F6},{F5},{F7},{F8},{F9}, D,0.7 {F10, F11},{F12, F14},{F13},{F15},{F17, F19},{F18, F20}}, U/ = {{F1, F2, F3, F8, F9, F10, F11, F12, F13, F14, F15, F16, F17, D,0.9 F18, F19, F20},{F4, F6},{F5},{F7}}, U/ = {{F1, F2, F3, F4, F5, F6, F7, F8, F9, F10, D,1.0 F11, F12, F13, F14, F15, F16, F17, F18, F19, F20}}. Table 4: The dissimilarity values of families F1-F20 to families F1-F10. family F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F1 0.00 0.84 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.84 F2 0.84 0.00 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.84 F3 0.84 0.84 0.00 0.99 0.99 0.99 1.00 0.96 0.96 0.84 F4 0.99 0.99 0.99 0.00 0.96 0.36 1.00 0.99 0.99 0.99 F5 0.99 0.99 0.99 0.96 0.00 0.96 1.00 0.99 0.99 0.99 F6 0.99 0.99 0.99 0.36 0.96 0.00 1.00 0.99 0.99 0.99 F7 1.00 1.00 1.00 1.00 1.00 1.00 0.00 1.00 1.00 1.00 F8 0.96 0.96 0.96 0.99 0.99 0.99 1.00 0.00 0.84 0.96 F9 0.96 0.96 0.96 0.99 0.99 0.99 1.00 0.84 0.00 0.96 F10 0.84 0.84 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.00 F11 0.84 0.84 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.36 F12 0.84 0.84 0.84 0.99 0.99 0.99 1.00 0.96 0.99 0.84 F13 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F14 0.84 0.84 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.84 F15 0.84 0.64 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.84 F16 0.84 0.36 0.84 0.99 0.99 0.99 1.00 0.96 0.96 0.84 F17 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F18 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F20 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 As it can be seen, the program has partitioned the families according to the λ- levels of dissimilarities. The interpretation of the above classification is as follows. In zero level (λ-level of 0), no item is dissimilar with another, in other words, they are completely similar. So, every item (family) resides in a class by itself. For the level of 0.5, families F2 and F16 are dissimilar, the same for F4 and F6, and the same for (F10, F11), (F12, F14), (F17, F19), and finally (F18, F20). For the level of 0.9, the families F1, F2, F3, F8, F9, F10, F11, F12, F13, F14, F15, F16, and F17 all share the same class, meaning that they share the same degree of dissimilarity. F4 and F6 are in another class, F5 is in a class by itself, and finally F7 resides in a class by itself too. For the level of 1.0, all items reside in the same class, meaning that, in this level, Fuzzy Inf. Eng. (2012) 1: 75-91 87 they are all completely dissimilar to each other in level 1. In other words, we can not find any point of similarity between any two of them. Table 5: The dissimilarity values of families F1-F20 to families F11-F20. family F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F1 0.84 0.84 1.00 0.84 0.84 0.84 1.00 1.00 1.00 1.00 F2 0.84 0.84 1.00 0.84 0.64 0.36 1.00 1.00 1.00 1.00 F3 0.84 0.84 1.00 0.84 0.84 0.84 1.00 1.00 1.00 1.00 F4 0.99 0.99 1.00 0.99 0.99 0.99 1.00 1.00 1.00 1.00 F5 0.99 0.99 1.00 0.99 0.99 0.99 1.00 1.00 1.00 1.00 F6 0.99 0.99 1.00 0.99 0.99 0.99 1.00 1.00 1.00 1.00 F7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F8 0.96 0.96 1.00 0.96 0.96 0.96 1.00 1.00 1.00 1.00 F9 0.96 0.96 1.00 0.96 0.96 0.96 1.00 1.00 1.00 1.00 F10 0.36 0.84 1.00 0.84 0.84 0.84 1.00 1.00 1.00 1.00 F11 0.00 0.84 1.00 0.84 0.84 0.84 1.00 1.00 1.00 1.00 F12 0.84 0.00 1.00 0.36 0.84 0.84 1.00 1.00 1.00 1.00 F13 1.00 1.00 0.00 1.00 1.00 1.00 0.84 0.84 0.84 0.84 F14 0.84 0.36 1.00 0.00 0.84 0.84 1.00 1.00 1.00 1.00 F15 0.84 0.84 1.00 0.84 0.00 0.64 1.00 1.00 1.00 1.00 F16 0.84 0.84 1.00 0.84 0.64 0.00 1.00 1.00 1.00 1.00 F17 1.00 1.00 0.84 1.00 1.00 1.00 0.00 0.84 0.64 0.84 F18 1.00 1.00 0.84 1.00 1.00 1.00 0.84 0.00 0.84 0.36 F19 1.00 1.00 0.84 1.00 1.00 1.00 0.64 0.84 0.00 0.84 F20 1.00 1.00 0.84 1.00 1.00 1.00 0.84 0.36 0.84 0.00 Working flow: The working flow of this application is pretty much like the one for Example 1, that is, it first asks for the names and pairwise similarity values. If the data are not stored into the system previously, they can be entered interactively. The values of dissimilarities will be computed automatically (using entered similarity values). Then, the program asks for the level of classification. The user enters 0.9 for the level, meaning that he wants to see the names of those families who have highest degree of dissimilarity with the others. At this level, the classification is as shown above for U/ . As it can be seen, F5 and F7 do not share their classes with any D,0.9 one, so we infer that they most probably are the causes of the disturbances. So, the manager can make a proper decision, either to ask them to leave the condominium, or somehow separates them from the others. Application 3: In the fields like pure mathematics, numerical analysis, differential equations, and coding theory, similarity and dissimilarity relations have many ap- plications. Here in this part, we study through an example, a numerical analysis application to obtain some sort of similarity among a set of integer numbers. 88 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) According to Algorithm 3, a computer program does the computations of the fol- lowing example which is related to Application 3. Example 5.3 4-similar integer numbers. Let m , m ,··· , m ∈ Z, such that m ≡ (i mod n), where 0 ≤ i ≤ n − 1, j=1, 1 2 n j j j 2,··· , n. Set A = {i , i , i ,··· , i }. Define 1 2 3 n n−|A| S (m , m ,··· , m ) = , 1 2 n n− 1 then S is an n-similarity. Note that,|A| denotes the cardinality of A. For the case of n = 4, that is, 4-similarity, we have A ⊆{0, 1, 2, 3}. Each digit in the set shows the possible remainder of the division of a given integer number into 4. All the integer numbers can be partitioned to Z , Z , Z and Z classes, where 0 1 2 3 Z = {z ∈ Z : z ≡ (i mod 4)} for i = 0, 1, 2, 3. Since|A|∈{0, 1, 2, 3}, the values of S could be either 0, 0.33, 0.66, or 1. According to Definition 2.4 and Lemma 2.1, for every λ ∈ [0, 1], an equivalence 4-relation on {Z× Z× Z× Z} have been obtained as follows: = {Z ×Z ×Z ×Z Z ×Z ×Z ×Z Z ×Z ×Z ×Z Z ×Z ×Z ×Z }. S,1 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 = {Z × Z × Z × Z : |{i , i , i , i }| = 2} if 0.66 ≤λ< 1. S,λ i i i i 1 2 3 4 1 2 3 4 = {Z × Z × Z × Z : |{i , i , i , i }| = 3} if 0.33 ≤λ< 0.66. S,λ i i i i 1 2 3 4 1 2 3 4 = {Z × Z × Z × Z : |{i , i , i , i }| = 4} if 0 ≤λ< 0.33. S,λ i i i i 1 2 3 4 1 2 3 4 The following are the classes corresponding to the similarity levels{λ ,λ ,λ , 0 0.2 0.5 λ } ={0, 0.2, 0.5, 0.7}. 0.7 U/ = {Z ∪ Z ∪ Z ∪ Z }, S,0 1 2 3 4 U/ = {Z ∪ Z ∪ Z :1 ≤ i, j, k ≤ 4}, S,0.2 i j k U/ = {Z ∪ Z , Z ∪ Z , Z ∪ Z , Z ∪ Z , Z ∪ Z , Z ∪ Z }, S,0.5 1 2 1 3 1 4 2 3 2 4 3 4 U/ = {Z , Z , Z , Z }. S,0.7 1 2 3 4 Working flow: The program asks the user to enter four integer numbers, then the program reports the similarity level of these numbers which is either 0, 0.33, 0.66, or 1.0. If the level is 0, that means our numbers are completely similar to each other, 0.33 means the level of similarity among them is 0.33 (i.e., almost 30%), 0.66 means their level of similarity is 0.66 (almost 60%), and 1.0 means the numbers are completely dissimilar. As in [6], whether the above equivalence relations can lead to forming the equivalence classes on a set or not have been posed as an open question. 6. Conclusion In this paper, by using some examples, we have shown how our ideas and propositions posed in [12] and in [13] can be applied to the real world problems. Lots of work has to be done in the future to answer the open questions that have been posed in the above mentioned papers. There is a long way yet to go to solve the problems of the concepts of similarity and dissimilarity and n-similarity relations. How nice it would be, if through these concepts, let’s say, the couples could find out about their similarities and their dissimilarities before getting married, in that case, their Fuzzy Inf. Eng. (2012) 1: 75-91 89 problems after marriage would probably lean to minimum. This is just one example, among thousands, about the concept of similarities and dissimilarities, that If one can find the proper and accurate model for them, perhaps many of the problems that we face today would be solved. Acknowledgements We wish to express our special gratitude and thanks to reviewers for their invaluable guidance for this paper. References 1. Alguliev R, Aliguliyev R (2007) Experimental investigating the F-measure as similarity measure for automatic text summarization. Applied and Computational Mathematics 6(2): 278-287 2. Jain A, Dubes R (1988) Algorithm for clustring data. Prentice Hall 3. Gentner D, Markman A B (1997) Structural alignment in analogy and similarity. American Psychol- ogist 52(1): 45-56 4. Hahn U, Chater N, Richardson L B (2003) Similarity as transformation. Cognition 87: 1-32 5. Keshavarzi M, Dehghan M A, Mashinchi M (2009) Classification based on similarity and dissimilar- ity through equivalence classes. Applied and Computational Mathematics 8(2): 203-215 6. Keshavarzi M, Dehghan M A, Mashinchi M (2011) Classification based on 3-similarity. Iranian Journal of Mathematical Sciences and Informatics 6(1): 7-21 7. Klawonn F, Castro J L (1995) Similarity in fuzzy reasoning. Mathware and Soft Computing 3: 197- 8. Landauer T K, Dumais S T (1997) A solution to Plato’s problem: The latent semantic analysis theory of acquisition, induction, and representation of knowledge. Psychological Review (104)2: 211-240 9. Larkey L B, Markman A B (2005) Processes of similarity judgment. Cognitive Science 29: 1061- 10. Loia V, Senatore S, Sessa M (2004) Combining agent technology and similarity-based reasoning for targeted e-mail services. Fuzzy Sets and Systems 145: 29-56 11. Rezaei H, Emoto M, Mukaidono M (2006) New similarity measure between two fuzzy sets. Journal of Advanced Computational Intelligence and Intelligent Informatics 10(6): 946-953 12. Rissland E L (2006) AI and similarity. IEEE Intelligent Systems 21(3): 39-49 13. Sessa M I (2001) Translation and similarity-based logic programming. Soft Computing 5: 160-170 14. Shepard R N (1962) The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika 27(2): 125-140 15. Schweitzer B, Sklar A(1983) Probabilistic metric spaces. North-Holland, New York 16. Tversky A (1977) Features of similarity. Psychological Review 84(4): 327-352 17. Valtchev P (1999) Construction automatique de taxanomies pour l’aide a’ la representaion de con- naissance par objects. These de doctorat, Universite Joseph Fourier, Grenoble I 18. Zadeh L A (1965) Fuzzy Sets. Information Control: 338-353 90 M.Keshavarzi · M.A.Dehghan· M.Mashinchi (2012) Appendix: List of Algorithms Algorithm 1: Arrangement of residents in a condominium. 1) SET size=20 /* size is the number of items(families) under study. */ 2) FOR i= 1 to size READ names[size] /* Read in the names of items under study (families in this example:F1-F20).*/ 3) ENDFOR 4) READ choice /*Read in the user’s choice of entering similarity values interactively(subjectively), or reading them from a file.*/ 5) FOR i=1 to size FOR j=i+1 to size SET sim[i,j]=simvalue /* Read similarity values into array sim of size by size to hold similarities values, for every pair of families, either from file, or interactively depending on user’s value of chice. simvalue is a real number between 0.0 and 1.0, computed previously according to some predefined criteria. */ 6) ENDFOR 7) FOR i=1 to size FOR j=i+1 to size FOR k=j+1 to size /* Compute the 3-similarity values according to our proposition.*/ READ level /* Read in the value of the similarity levels. */ WRITE report /* Report the most 3-similar families according to the values of the level. */ 8) ENDFOR 9) END /* of algorithm */ Algorithm 2: Residents classification on the basis of their dissimilarities. Note: An algorithm similar to the following have been used in [10]. Input: A domain U = {a ,··· , a }, and the ordered setΛ= {λ ,··· ,λ }, 1 m 0 n of dissimilarity levels, with 0 = λ <λ <···<λ = 1. n n−1 0 i i Output: The quotient sets Q = {C ,··· , C },0 ≤ i ≤ n, associated 1 k withλ− cuts in the family{ } . D,λ 0≤i≤n 1) SET k = 1 i i 2) SET Q = {C ,··· , C },0 ≤ i ≤ n 1 k 3) FOR i=n down to 1 SET Q = Q i i−1 FOR j=1 to k i−1 Fuzzy Inf. Eng. (2012) 1: 75-91 91 i−1 IF C is singleton THEN continue ELSE i i READ another input /* For the C ,··· , C j j 1 h i−1 i such that: C = ∪ C , and 1≤r≤h j j i i C ∩ C = 0, for any p, q∈{ j ,··· , j } */ 1 h p q i−1 i i SET Q = (Q − C ) {C ,··· , C } i i j j j 1 h ENDFOR 4) ENDFOR 5) END /* of algorithm */ Algorithm 3: Integer numbers 4-similarity. 1) FOR i=1 to 4 2) READ z[i] /* Read in four integer numbers into array z*/ 3) SET simset[i]=z[i] mod 4 /* compute the remainders of each of the integer numbers, and store them into array simset assigned for this purpose.*/ 4) ENDFOR 5) CALL FUNCTION cardin=function(simset) /* The function gets the array sim- set and computes the cardinality of the set on the basis of remainders of the integer numbers int 4, and returns the value of cardin.*/ 6) SET simil4=(4.0-cardin)/3.0 /* Computing 4-similarity value of these numbers according to the proposed formula */ 7) END /* of algorithm */
Fuzzy Information and Engineering – Taylor & Francis
Published: Mar 1, 2012
Keywords: 3-similarity relations; Equivalence 3-relations; Dissimilarity relations; n -dissimilarity
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