Application of interval-valued intuitionistic fuzzy sets of second type in pattern recognition

After the introduction of intuitionistic fuzzy sets (IFS), many researchers have confirmed the resourcefulness of IFSs in decision making problems like pattern recognition, machine learning, medical diagnosis, electoral system, career determination, market prediction, and so on. In this paper, we propose the new distance measures on interval-valued Intuitionistic Fuzzy Sets of Second Type and the application of interval-valued Intuitionistic Fuzzy Sets of Second Type in Pattern Recognition by using normalized hamming distance measure and compare the result with the existing IVIFS.


Introduction
An Intuitionistic Fuzzy Set (IFS) for a given underlying set X was introduced by K. T. Atanassov [1] as a generalization of ordinary Fuzzy Sets (FS, see [9]), and he introduced the theory of Interval-Valued Intuitionistic Fuzzy Sets (IVIFS) and established some of their properties. Many researchers have contributed their research work in the application of IVIFS in pattern recognition. Most of the applications of IFSs and their extensions are carried out using distance measures approach [4,5]. Distance measure between intuitionistic fuzzy sets is an important concept in fuzzy mathematics because of its wide applications in real life situations like pattern recognition, machine learning, medical diagnosis, electoral system, career determination, market prediction, and so on as in [7,8]. In this paper we present the application of IVIFSST in Pattern Recognition by using normalized hamming distance measure and compare the result with the existing IVIFS.
The rest of the paper is designed as follows: In Section 2, we give some basic definitions. In Section 3, we propose the various distance measures on interval-valued intuitionistic fuzzy sets of second type with suitable examples. In Section 4, we give the application of IVIFSST in pattern recognition and analyzed the results. This paper is concluded in Section 5.

Preliminaries
In this section, we give some basic definitions.
Let X be a non-empty set. An Intuitionistic Fuzzy Set (IFS) A in X is defined as an object of the following form where the functions µ A : X → [0, 1] and ν A : X → [0, 1] denote the degree of membership and the degree of non-membership of the element x ∈ X, respectively, and for every x ∈ X : An Intuitionistic Fuzzy Sets of Second Type (IFSST) A in X is defined as an object of the following form where the functions µ A : X → [0, 1] and ν A : X → [0, 1] denote the degree of membership and the degree of non-membership of the element x ∈ X, respectively, and for every x ∈ X : where M A : X → [0, 1], M A (x) denote the degree of membership of the element x to the set A.
An Interval-Valued Intuitionistic Fuzzy Set (IVIFS) A in X is given by The intervals M A (x) and N A (x) denote the degree of membership and the degree of non-membership of the element Definition 2.5.
[6] An Interval-Valued Intuitionistic Fuzzy Sets of Second Type (IVIFSST) A in X is given by Definition 2.6.
[7] Let X be nonempty such that IVIFS A, B, C ∈ X. Then the distance measure between IVIFS A and B is a mapping d : ) satisfies the following axioms:

Normalized Hamming Distance between A and B is
Euclidean Distance between A and B is d EIV IF S (A, B) Normalized Euclidean Distance between A and B is

Some distance measures over IVIFSST
In this section, we introduce the various distance measures on IVIFSST with example.   The Normalised Hamming distance between A and B is equal to 0.08 (iii) The Euclidean distance between A and B is equal to 0.37 (iv) The Normalised Euclidean distance between A and B is equal to 0.26

Normalized Euclidean Distance between A and B is
From the above results, we infer that the normalized Hamming distance gives the best distance measure between A and B. This is because the distance is the shortest or smallest. For this reason, we shall make use of normalized hamming distance in the applications for its high rate of confidence in terms of accuracy.

Application of IVIFSST in pattern recognition
In this process, a sets of patterns and another unknown pattern called sample is given (IVIFSST in nature). Both the set of the pattern and that of the sample are within the same feature space or attributes n. The aim is to find the distance between each of the patterns and the sample. The smallest or shortest distance between any of the patterns and the sample shows that, the sample belongs to that pattern. Assume that there exist n patterns given by,

Results and discussion
Using normalized hamming distance measure for IVIFS, we have the following results: From these results, we see that, the distance between A 6 and B is the smallest, and the distance between A 3 and B is the greatest. Since A 6 approaches B, we say that the unknown pattern B belongs to A 6 .
Again using normalized hamming distance measure for IVIFSST we have the following results: From these results, we see that, in both the cases the distance between A 6 and B is the smallest, and the distance between A 3 and B is the greatest since A 6 approaches B, we say that the unknown pattern B belongs to A 6 .
We conclude that, IVIFSST plays a vital role in obtaining the shortest distance when compared with IVIFS.

Conclusion
In this paper, we have introduced the various distance measures on IVIFSST in particular, we have applied the normalized hamming distance measure on IVIFSST for calculating the shortest distance. Also we have compared the results with existing IVIFS. It is observed that IVIFSST is the good tool for finding the shortest distance.