Abstract
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.
1. 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, , whose range is not limited to but extended to the unit circle in the complex plane. Hence, is a complex-Valued function that assigns a grade of membership of the form , to any element in the universe of discourse. The value of is defined by the two variables, and , both real-valued, with . 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 , is retained in complex fuzzy sets through the amplitude of the grade of membership, . The novelty of complex fuzzy sets is manifested in the additional dimension of membership: the phase of the grade of membership, . The properties of membership phase are discussed at length in this section. Ramot et al. [8, 9] extended the range of membership to “unit circle in the complex plane”, unlike others who limited the range to . Omar [10] studied similarity measures between temporal intuitionistic fuzzy sets. As the complex fuzzy membership grade is two-dimensional (amplitude and phase), a complex fuzzy set can 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.
2. Preliminaries and Basic Definitions
Definition 1 (see [8]). A complex fuzzy set (CFS) defined on a universe is an object of the form defined on a universe of discourse which is an object of the form where , , and
Definition 2 (see [2]). A complex intuitionistic fuzzy set (CIFS) defined on a universe of discourse is an object of the form where , , , and +
Definition 3 (see [14]). Let and be two CIFSs in , where Then, is given aswhere
Definition 4 (see [15]). Let and be two CIF-sets in , where Then, for all ,
(1) if and only if ; For amplitude terms and , for phase terms.
(2) if and only if ; For amplitude terms and , for phase terms.
Definition 5 (see [16]). Let and be two CIFSs in , where Then, for all ,
(1)
where(2) The complex fuzzy complement of , denoted by , is specified by a function(3) and .
Example 6. Consider . Let be a CIF-subset of , as given by Then And and .
3. Temporal Complex Intuitionistic Fuzzy Set
Definition 7. Let be a universe, be a nonempty set of time moments, and A temporal complex intuitionistic fuzzy set (TCIFS) defined on a universe of discourse is an object of the form where and such that +, , and being the degrees of membership and nonmembership, respectively, of the element at the moment . And at the moment , where .
The hesitation degree of a TCIFS is defined by such that for each For brevity we will write instead of when this does not cause confusions.
Example 8. Suppose that is a universal, with respect to the time set , and . Then, TCIFSs are defined by
Example 9. Suppose with respect to the time set . Then, the details of a TCIFS are explained in Tables 1, 2, and 3
Definition 10. Let and be two TCIFSs. Then where
Definition 11. We define the following two operators over a TCIFS :
Theorem 12. and are TCIFSs.
Proof. Suppose thatand Therefore, Then is TCIFSs. Also, by the same fashion are TCIFSs.
Theorem 13. For every TCIFS ,(1);(2);(3);(4)
Proof. The proof is obvious.
Theorem 14. For every TCIFS ,(1);(2)
Proof. (1) (2) By the same fashion, one has the following.
Definition 15. Let and be two TCIFSs defined on the universe of discourse and the time moments . The correlation coefficient of and is given bywhere is the correlation of two TCIFSs and , and are the information temporal complex intuitionistic energies of and , respectively.
Example 16. Suppose that with respect to the time set . The details of a are explained in Table 4, Table 5 explained , and Table 6 explained the correlation coefficient between TCIFS and TCIFS .
Proposition 17. Let and be two TCIFS. Then (1);(2)If and , then, ;(3)
Proof. Let From Definition 10, , , , , and then
(1) (2) If and (3)
Theorem 18. Suppose that and are TCIFSs in the universal with respect to the time set . Then(1)if and , then ;(2);(3)
Proof. (1) Let and be two TCIFSs defined on the universe of discourse and the time moments . The correlation coefficient of and is given byIf , then Then from Proposition 17 (2), one has the following.
If and , then ; then (2) From Proposition 17 (3), Then (3) We will prove that such that it is evident , so suppose that Then ThenThen Then But Hence , and then
Definition 19. Let be a function, and let TCIFSs in the universal with respect to the time set . Then is said to be the similarity degree between and , satisfying the following statements:(1).(2) if .(3).(4)If . Then Now we can have the following degrees of the similarity between and satisfy the conditions from to . Let where ; . ThenFrom a comparison between similarity measures , we give the following example.
Example 20. Suppose that is TCIFSs defined on with respect to the time set The details of a are explained in Table 7, Table 8 explained , and Table 9 explained a comparison between similarity measures , , and .
4. 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.
Definition 21. If is the degrees of membership function of the element at the moment , then nonmembership function , where And , , and by help of the Sugeno [6] intuitionistic fuzzy generator, TCIFS is given by The hesitation degree of a TCIFS is
Example 22. Suppose that is TCIFS defined on with respect to the time set . The details of a are explained in Tables 10, 11, and 12, Table 13 explained when , and Table 14 explained the hesitation degree of a TCIFS .
If , then one has the following (see Tables 13 and 14).
Definition 23. Suppose that is TCIFSs in the universal with respect to the time set . Then a cosine similarity measure between is proposed as follows:where ;
Theorem 24. Suppose that and are TCIFSs in the universal with respect to the time set . Then(1) if and ,(2),(3),(4)if , then