Similarity Measures between Temporal Complex Intuitionistic Fuzzy Sets and Application in Pattern Recognition and Medical Diagnosis

This work addresses the issue of similarity measures between two temporal complex Atanassov’s intuitionistic fuzzy sets, many measures of similarity between complex Atanassov’s intuitionistic fuzzy sets. What was proposed before did not consider the abstention group influence, which may lead to counterintuitive results in some cases. A new structure of temporal complex Atanassov’s intuitionistic fuzzy sets is obtained. This set is formally generalized from a conventional Atanassov’s intuitionistic complex fuzzy sets. Here we analyze the limitations of the existing similarity measures.Then, a new similarity measure of temporal complex Atanassov’s intuitionistic fuzzy sets is proposed and several numeric examples are given to demonstrate the validity of the proposed measure. Finally, the proposed similarity measure is applied to pattern recognition and medical diagnosis.


Introduction
Fuzzy set theory was conferred by Zadeh [1] to solve difficulties in dealing with uncertainties. Since then, the theories of fuzzy sets and fuzzy logic have been examined by many researchers to solve many real life problems involving ambiguous and uncertain environment. By adding a new component the idea of the concept of Atanassov's intuitionistic fuzzy set (AIFS) was introduced [2]. Applications of these sets have been broadly studied in other aspects such as image processing [3], multicriteria decision making [4], pattern recognition [5], etc. Buckley [6] and Nguyen et al. [7] combined complex numbers with fuzzy sets. On the other hand, the innovative complex fuzzy set is introduced. The complex fuzzy set is characterized by a membership function, A ( ), whose range is not limited to [0, 1] but extended to the unit circle in the complex plane. Hence, A ( ) is a complex-Valued function that assigns a grade of membership of the form A ( ) A ( ) , = √ −1 to any element in the universe of discourse. The value of ( ) is defined by the two variables, A ( ) and A ( ), both real-valued, with A ( ) ∈ [0, 1]. Complex fuzzy set theory modifies the original concept of fuzzy membership by asserting that, at least in some instances, it is necessary to add a second dimension to the expression of membership. However, this added dimension does not alter the basic concept of fuzziness. Membership in a complex fuzzy set remains "as fuzzy" as membership in a traditional fuzzy set. The fuzziness of membership, i.e., the representation of membership as a value in the range [0, 1], is retained in complex fuzzy sets through the amplitude of the grade of membership, A ( ). The novelty of complex fuzzy sets is manifested in the additional dimension of membership: the phase of the grade of membership, 2 Discrete Dynamics in Nature and Society be visually represented by a three-dimensional graph where the universe of discourse is the third axis. Figure 1 shows the complex fuzzy set. We divide the paper into four main sections. In the first section preliminaries and basic definitions, we provide some details about the complex fuzzy sets. In the second section, detail is given about the complex version of temporal complex intuitionistic fuzzy set, which is an extension of complex intuitionistic fuzzy set by adding the times and studied the correlation coefficient between two temporal complex intuitionistic fuzzy set. In the third section, details is given about similarity measures between other extensions of temporal complex intuitionistic fuzzy set and extend the method proposed by Chaira [12] for intuitionistic fuzzy set based on the Sugeno [13] and Omar [10] intuitionistic fuzzy generator. In the fourth section, we give application in pattern recognition, medical diagnosis, and topology.

Preliminaries and Basic Definitions
Definition 1 (see [8]). A complex fuzzy set (CFS) A defined on a universe is an object of the form A defined on a universe of discourse which is an object of the form Definition 2 (see [2]). A complex intuitionistic fuzzy set (CIFS) A defined on a universe of discourse is an object of the form 2 ], and 0 ≤ ( )+V ( ) ≤ 1.
And A ̸ ⊂ B and A ̸ = B.

Temporal Complex Intuitionistic Fuzzy Set
Definition 7. Let be a universe, be a nonempty set of time moments, and A ⊆ . A temporal complex intuitionistic fuzzy set (TCIFS) A defined on a universe of discourse is an object of the form ( , ), and V ( , ) being the degrees of membership and nonmembership, respectively, of the element ∈ at the moment ∈ . And A ( , ), A ( , ) ∈ [0, 2 ] at the moment ∈ , where = √ −1.
The hesitation degree of a TCIFS A is defined by For brevity we will write A instead of A( ) when this does not cause confusions.  Tables 1, 2, and 3 Definition 10. Let A( 1 ) and B( 2 ) be two TCIFSs. Then where Definition 11. We define the following two operators over a TCIFS A: Proof. Suppose that and Therefore, Then (A( )) is TCIFSs. Also, by the same fashion (A( )) are TCIFSs.
Proof. The proof is obvious.
Discrete Dynamics in Nature and Society (2) By the same fashion, one has the following.
Definition 15. Let A and B be two TCIFSs defined on the universe of discourse = { 1 , 2 , 3 , . . . , and the time moments = { 1 , 2 , 3 , . . . , }. The correlation coefficient of A and B is given by where is the correlation of two TCIFSs A and B, and are the information temporal complex intuitionistic energies of A and B, respectively.
The details of a TCIFS A( ) are explained in Table 4, Table 5 explained TCIFS B( ), and Table 6 Table 5: TCIFS B( ).
Proof. (1) Let A and B be two TCIFSs defined on the universe of discourse = { 1 , 2 , 3 , . . . , and the time moments = { 1 , 2 , 3 , . . . , }. The correlation coefficient of A and B is given by (3) We will prove that (A, B) < 1 such that it is evident 0 < (A, B), so suppose that Then Then Then Then Discrete Dynamics in Nature and Society 7 2 (A, B)         are explained in Table 7, Table 8 explained TCIFS B( ), and Table 9

Similarity Measures between Other Extensions of Temporal Complex Intuitionistic Fuzzy Set
The following definition extend the method proposed by Chaira [12] for intuitionistic fuzzy set based on the Sugeno [13] and Omar [10] intuitionistic fuzzy generator.
And (1) = 0, (0) = 1, and by help of the Sugeno [6] intuitionistic fuzzy generator, TCIFS A is given by The hesitation degree of a TCIFS A is Example 22. Suppose that A( ) is TCIFS defined on = { 1 , 2 , 3 } with respect to the time set = { 1 , 2 , 3 }. The details of a TCIFS A( ) are explained in Tables 10, 11, and  12, Table 13 explained TCIFS A 1 when = 1, and Table 14 explained the hesitation degree of a TCIFS A.

Application in Pattern Recognition and Medical Diagnosis.
Let = { 1 , 2 , 3 , 4 , 5 , 6 } be the set of symptoms of the diseases with respect to the time set = { 1 , 2 , 3 , 4 , 5 , 6 } and 1 be the set of diagnoses. By using the similarity measures ( , 1 ) we try to discover that the patient may suffer from one from diseases which have symptoms 1 at the time 1 , and we let be standard case symptoms of one of diseases (Table 18) and 1 be any case (Table 19); Table 20 explained the similarity measures ( , 1 ) between a standard case and any case 1 .
And we define the symptoms of case by Table 19. Then Table 20 explained the similarity measures ( , 1 ) between a standard case and any case 1 . When the similarity measures −1 ≤ ( , 1 ) ≤ 1 are small, then probability that the patient is suffering from the disease at the time is big and the conversely is true.

Complex Intuitionistic Fuzzy Topology
Definition 30. An intuitionistic complex fuzzy topology on is a family of -sets in which satisfies the following properties: (1) 1, 0 ∈ ,  Definition 32. If ( , ) is called complex intuitionistic fuzzy topological space, ⊆ , then the interior of is defined as the union of all -subsets of and it is denoted by ∘ . That is, ∘ is the largest -subset of . The closure of is defined as the intersection of all sets containing and it is denoted by − . That is, − is the smallest -set containing . The implications between these concepts in the following diagram and the converse are not true in general,

Conclusion
In this paper we introduced and studied a temporal complex intuitionistic fuzzy sets as generalization of complex Atanassov's intuitionistic fuzzy sets by taking the time in the moving of the point; a correlation between two temporal complex intuitionistic fuzzy sets is discussed. A similarity between temporal complex intuitionistic fuzzy sets is main points in the paper as a generalization of the similarity introduced by Omar [10] and Sugeno [13]. We calculate the results by the program Maple 7. Finally we give an applications to know if the patient is suffering from the diseases or not and introduce the main building in a topology by using the same the set. In future research, similarity measures