Correlation measure for intuitionistic fuzzy multi sets

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 01 | Jan-2014, Available @ http://www.ijret.org 611
CORRELATION MEASURE FOR INTUITIONISTIC FUZZY MULTI SETS
P. Rajarajeswari1
, N. Uma2
1
Department of Mathematics, Chikkanna Arts College, Tirupur, Tamil Nadu, India
2
Department of Mathematics, SNR Sons College, Coimbatore, Tamil Nadu, India
Abstract
In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure
of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis
and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and
only if the two IFMS are the same.
Keywords: Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Correlation measure.
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1. INTRODUCTION
The Intuitionistic Fuzzy sets (IFS) introduced by Krasssimir T.
Atanassov [1, 2] is the generalisation of the Fuzzy set (FS).
The Fuzzy set (FS) proposed by Lofti A. Zadeh [3] allows the
uncertainty belong to a set with a membership degree ( 𝜇)
between 0 and 1. That is, the one and only membership
function (𝜇 ∈ [0,1]) and the non membership function equals
one minus the membership degree. Whereas IFS represent the
uncertainty with respect to both membership (𝜇 ∈ [0,1]) and
non membership ( 𝜗 ∈ [0,1] ) such that 𝜇 + 𝜗 ≤ 1 . The
number 𝜋 = 1 − 𝜇 − 𝜗 is called the hesitiation degree or
intuitionistic index.
Several authors like Murthy and Pal [4] investigated the
correlation between two fuzzy membership functions, Chiang
and Lin [5] studied the correlation of fuzzy sets and Chaudhuri
and Bhattacharya [6] discussed the correlation between two
fuzzy sets on same universal discourse. As the Intuitionistic
fuzzy sets is widely used in various fields like pattern
recognition, medical diagnosis, logic programming, decision
making, market prediction, etc. Correlation Analysis of IFS
plays a vital role in recent research area. Gerstenkorn and
Manko [7] defined and examined the properties the correlation
measure of IFS for finite universe of discourse. Later the
concepts of correlation and the correlation coefficient of IFS
in probability spaces were derived by Hong, Hwang [8] for the
infinite universe of discourse. Hung and Wu [9, 10] proposed
a centroid method to calculate the correlation coefficient of
IFSs, using the positively and negatively correlated values.
The correlation coefficient of IFS in terms of statistical values,
using mean aggregation functions was presented by Mitchell
[11]. Based on geometrical representation of IFSs and three
parameters, a correlation coefficient of IFSs was defined by
Wenyi Zeng and Hongxing Li [12].
The Multi set [13] repeats the occurrences of any element.
And the Fuzzy Multi set (FMS) introduced by R. R. Yager
[14] can occur more than once with the possibly of the same or
the different membership values. Recently, the new concept
Intuitionistic Fuzzy Multi sets (IFMS) was proposed by T.K
Shinoj and Sunil Jacob John [15].
As various distance and similarity methods of IFS are
extended for IFMS distance and similarity measures [16, 17,
18 and 19], this paper is an extension of the correlation
measure of IFS to IFMS. The numerical results of the
examples show that the developed similarity measures are well
suited to use any linguistic variables.
2. PRELIMINARIES
2.1 Definition:
Let X be a nonempty set. A fuzzy set A in X is given by
A = 𝑥, 𝜇 𝐴 𝑥 / 𝑥 ∈ 𝑋 -- (2.1)
where 𝜇 𝐴 : X → [0, 1] is the membership function of the
fuzzy set A (i.e.) 𝜇 𝐴 𝑥 ∈ 0,1 is the membership of 𝑥 ∈ 𝑋
in A. The generalizations of fuzzy sets are the Intuitionistic
fuzzy (IFS) set proposed by Atanassov [1, 2] is with
independent memberships and non memberships.
2.2 Definition:
An Intuitionistic fuzzy set (IFS), A in X is given by
A = 𝑥, 𝜇 𝐴 𝑥 , 𝜗𝐴 𝑥 / 𝑥 ∈ 𝑋 -- (2.2)
where 𝜇 𝐴 : X → [0,1] and 𝜗𝐴 : X → [0,1] with the
condition 0 ≤ 𝜇 𝐴 𝑥 + 𝜗𝐴 𝑥 ≤ 1 , ∀ 𝑥 ∈ 𝑋 Here
𝜇 𝐴 𝑥 𝑎𝑛𝑑 𝜗𝐴 𝑥 ∈ [0,1] denote the membership and the
non membership functions of the fuzzy set A;
For each Intuitionistic fuzzy set in X, 𝜋 𝐴 𝑥 = 1 − 𝜇 𝐴 𝑥 −
1 − 𝜇 𝐴 𝑥 = 0 for all 𝑥 ∈ 𝑋 that is

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𝜋 𝐴 𝑥 = 1 − 𝜇 𝐴 𝑥 − 𝜗𝐴 𝑥 is the hesitancy degree of 𝑥 ∈ 𝑋
in A. Always 0 ≤ 𝜋 𝐴 𝑥 ≤ 1, ∀ 𝑥 ∈ 𝑋.
The complementary set 𝐴 𝑐
of A is defined as
𝐴 𝑐
= 𝑥, 𝜗𝐴 𝑥 , 𝜇 𝐴 𝑥 / 𝑥 ∈ 𝑋 -- (2.3)
2.3 Definition:
Let X be a nonempty set. A Fuzzy Multi set (FMS) A in X is
characterized by the count membership function Mc such that
Mc : X → Q where Q is the set of all crisp multi sets in
[0,1]. Hence, for any𝑥 ∈ 𝑋 , Mc(x) is the crisp multi set from
[0, 1]. The membership sequence is defined as
( 𝜇 𝐴
1
𝑥 , 𝜇 𝐴
2
𝑥 , … … … 𝜇 𝐴
𝑝
𝑥 ) where 𝜇 𝐴
1
𝑥 ≥ 𝜇 𝐴
2
𝑥 ≥
⋯ ≥ 𝜇 𝐴
𝑝
𝑥 .
Therefore, A FMS A is given by
𝐴 = 𝑥, ( 𝜇 𝐴
1
𝑥 , 𝜇 𝐴
2
𝑥 , … … … 𝜇 𝐴
𝑝
𝑥 ) / 𝑥 ∈ 𝑋 2.4)
2.4 Definition:
Let X be a nonempty set. A Intuitionistic Fuzzy Multi set
(IFMS) A in X is characterized by two functions namely
count membership function Mc and count non membership
function NMc such that Mc : X → Q and NMc : X → Q
where Q is the set of all crisp multi sets in [0,1]. Hence, for
any 𝑥 ∈ 𝑋 , Mc(x) is the crisp multi set from [0, 1] whose
membership sequence is defined as
( 𝜇 𝐴
1
𝑥 , 𝜇 𝐴
2
𝑥 , … … … 𝜇 𝐴
𝑝
𝑥 )
Where 𝜇 𝐴
1
𝑥 ≥ 𝜇 𝐴
2
𝑥 ≥ ⋯ ≥ 𝜇 𝐴
𝑝
𝑥 and the corresponding
non membership sequence NMc (x) is defined as (
𝜗𝐴
1
𝑥 , 𝜗𝐴
2
𝑥 , … … … 𝜗𝐴
𝑝
𝑥 ) where the non membership
can be either decreasing or increasing function. such that
0 ≤ 𝜇 𝐴
𝑖
𝑥 + 𝜗𝐴
𝑖
𝑥 ≤ 1 , ∀ 𝑥 ∈ 𝑋 𝑎𝑛𝑑 𝑖 = 1,2, … 𝑝.
Therefore, An IFMS A is given by
𝐴 =
𝑥,
𝜇𝐴1𝑥 , 𝜇𝐴2𝑥, ……… 𝜇𝐴𝑝𝑥 , ( 𝜗𝐴1𝑥 , 𝜗𝐴2𝑥, ……… 𝜗𝐴𝑝𝑥 )
/ 𝑥∈𝑋 (2.5)
Where 𝜇 𝐴
1
𝑥 ≥ 𝜇 𝐴
2
𝑥 ≥ ⋯ ≥ 𝜇 𝐴
𝑝
𝑥
The complementary set 𝐴 𝑐
of A is defined as
𝐴 𝑐
= 𝑥, ( 𝜗𝐴
1
𝑥 , 𝜗𝐴
2
𝑥 , … … … 𝜗𝐴
𝑝
𝑥 ),
𝜇 𝐴
1
𝑥 , 𝜇 𝐴
2
𝑥 , … … … 𝜇 𝐴
𝑝
𝑥 , / 𝑥
∈ 𝑋 − (2.6)
𝑤ℎ𝑒𝑟𝑒 𝜗𝐴
1
𝑥 ≥ 𝜗𝐴
2
𝑥 ≥ ⋯ ≥ 𝜗𝐴
𝑝
𝑥
2.5 Definition:
The Cardinality of the membership function Mc(x) and the
non membership function NMc (x) is the length of an element
x in an IFMS A denoted as 𝜂, defined as η = Mc(x) =
NMc(x)
If A, B and C are the IFMS defined on X, then their cardinality
η = Max { η(A), η(B), η(C) }.
2.6 Definition:
𝑆 𝑨, 𝑩 is said to be the similarity measure between A and
B, where A, B ∈ X and X is an IFMS, as 𝑆 𝑨, 𝑩 satisfies
the following properties
1. 𝑆 𝑨, 𝑩 ∈ [0,1]
2. 𝑆 𝑨, 𝑩 = 1 if and only if A = B
3. 𝑆 𝑨, 𝑩 = 𝑆 𝑩, 𝑨
3. CORRELATION MEASURE
3.1 Fuzzy Correlation Measure
Let A = { 𝑥𝑖, 𝜇 𝐴 𝑥𝑖 / 𝑥𝑖 𝜖 𝑋 } and B = { 𝑥𝑖, 𝜇 𝐵 𝑥𝑖 /
𝑥𝑖 𝜖 𝑋 } be two FSs on the finite universe of discourse X = {
𝑥1, 𝑥2, … . , 𝑥 𝑛 }, then the correlation coefficient of A and B
[5, 6] is
𝜌 𝐹𝑆 𝐴, 𝐵 =
𝐶 𝐹𝑆 𝐴, 𝐵
𝐶 𝐹𝑆 𝐴, 𝐴 ∗ 𝐶 𝐹𝑆 𝐵, 𝐵
Where 𝐶 𝐹𝑆 𝐴, 𝐵 = 𝜇 𝐴 𝑥𝑖 𝜇 𝐵 𝑥𝑖
𝑛
𝑖=1 and 𝐶 𝐹𝑆 𝐴, 𝐴 =
𝜇 𝐴 𝑥𝑖 𝜇 𝐴 𝑥𝑖
𝑛
𝑖=1
3.2 Intuitionistic Fuzzy Correlation Measure
Let X = { 𝑥1, 𝑥2, … . , 𝑥 𝑛 } be the finite universe of discourse
and A = { 𝑥𝑖, 𝜇 𝐴 𝑥𝑖 , 𝜗𝐴 𝑥𝑖 / 𝑥𝑖 𝜖 𝑋 } , B = { 𝑥𝑖,
𝜇𝐵𝑥𝑖, 𝜗𝐵𝑥𝑖/ 𝑥𝑖 𝜖 𝑋 } be two IFSs then the correlation
coefficient of A and B introduced by Gerstenkorn and
Manko [7] was
𝜌𝐼𝐹𝑆 𝐴, 𝐵 =
𝐶𝐼𝐹𝑆 𝐴, 𝐵
𝐶𝐼𝐹𝑆 𝐴, 𝐴 ∗ 𝐶𝐼𝐹𝑆 𝐵, 𝐵
Where
𝐶𝐼𝐹𝑆 𝐴, 𝐵 = { 𝜇 𝐴 𝑥𝑖 𝜇 𝐵 𝑥𝑖
𝑛
𝑖=1 + 𝜗𝐴 𝑥𝑖 𝜗 𝐵 𝑥𝑖 and
𝐶𝐼𝐹𝑆 𝐴, 𝐵 = {𝜇 𝐴 𝑥𝑖 𝜇 𝐵 𝑥𝑖
𝑛
𝑖=1 + 𝜗𝐴 𝑥𝑖 𝜗 𝐵 𝑥𝑖 }
The correlation coefficient of A and B in X, the infinite
universe of discourse defined by Hong and Hwang [8] is
𝜌𝐼𝐹𝑆 𝐴, 𝐵 =
𝐶𝐼𝐹𝑆 𝐴, 𝐵
𝐶𝐼𝐹𝑆 𝐴, 𝐴 ∗ 𝐶𝐼𝐹𝑆 𝐵, 𝐵

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Where
𝐶𝐼𝐹𝑆 𝐴, 𝐵 = 𝜇 𝐴 𝑥𝑖 𝜇 𝐵 𝑥𝑖 + 𝜗𝐴 𝑥𝑖 𝜗 𝐵 𝑥𝑖 𝑑𝑥 and
𝐶𝐼𝐹𝑆 𝐴, 𝐴 = 𝜇 𝐴 𝑥𝑖 𝜇 𝐴 𝑥𝑖 + 𝜗𝐴 𝑥𝑖 𝜗𝐴 𝑥𝑖 𝑑𝑥
3.3 Proposed Correlation Similarity Measure for
IFMS
3.3.1 Intuitionistic Fuzzy Multi Correlation Measure
Let X = { 𝑥1, 𝑥2, … . , 𝑥 𝑛 } be the finite universe of discourse
and A = { 𝑥𝑖, μA
j
𝑥𝑖 , ϑA
j
𝑥𝑖 / 𝑥𝑖 𝜖 𝑋} , B = {
𝑥𝑖, μB
j
𝑥𝑖 , ϑB
j
𝑥𝑖 / 𝑥𝑖 𝜖 𝑋 } be two IFMSs consisting of the
membership and non membership functions, then the
correlation coefficient of A and B
𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐵
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐴 ∗ 𝐶𝐼𝐹𝑀𝑆 𝐵, 𝐵
Where 𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐵
𝑗
𝑥𝑖 +𝑛
𝑖=1
𝜂
𝑗=1
𝜗𝐴𝑗𝑥𝑖 𝜗𝐵𝑗𝑥𝑖 and
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐴 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐴
𝑗
𝑥𝑖 +𝑛
𝑖=1
𝜂
𝑗=1
𝜗𝐴𝑗𝑥𝑖 𝜗𝐴𝑗𝑥𝑖
Let A = { 𝑥𝑖, μA
j
𝑥𝑖 , ϑA
j
𝑥𝑖 , πA
j
𝑥𝑖 / 𝑥𝑖 𝜖 𝑋} and B = {
𝑥𝑖, μB
j
𝑥𝑖 , ϑB
j
𝑥𝑖 , πA
j
𝑥𝑖 / 𝑥𝑖 𝜖 𝑋 } be two IFMSs
consisting of the membership, non membership functions and
the hesitation functions, then the correlation coefficient of A
and B
𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐵
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐴 ∗ 𝐶𝐼𝐹𝑀𝑆 𝐵, 𝐵
Where 𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐵
𝑗
𝑥𝑖 +𝑛
𝑖=1
𝜂
𝑗=1
𝜗𝐴𝑗𝑥𝑖 𝜗𝐵𝑗𝑥𝑖 + 𝜋𝐴𝑗𝑥𝑖 𝜋𝐵𝑗𝑥𝑖
and 𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐴 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐴
𝑗
𝑥𝑖 +𝑛
𝑖=1
𝜂
𝑗 =1
𝜗𝐴𝑗𝑥𝑖 𝜗𝐴𝑗𝑥𝑖 + 𝜋𝐴𝑗𝑥𝑖 𝜋𝐵𝑗𝑥𝑖
3.4 Proposition
The defined Similarity measure 𝝆 𝑰𝑭𝑴𝑺 𝑨, 𝑩 between IFMS A
and B satisfies the following properties
D1. 0 ≤ 𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 ≤ 1
D2. 𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 = 1 if and only if A = B
D3. 𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 = 𝜌𝐼𝐹𝑀𝑆 𝐵, 𝐴
Proof
D1. 𝟎 ≤ 𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 ≤ 𝟏
As the membership and the non membership functions of the
IFMSs lies between 0 and 1, 𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 also lies between 0
and 1.
D2. 𝝆 𝑰𝑭𝑴𝑺 𝑨, 𝑩 = 1 if and only if A = B
(i) Let the two IFMS A and B be equal (i.e.) A = B. Hence for
any 𝜇 𝐴
𝑗
𝑥𝑖 = 𝜇 𝐵
𝑗
𝑥𝑖 and 𝜗𝐴
𝑗
𝑥𝑖 = 𝜗 𝐵
𝑗
𝑥𝑖 then
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐴 = 𝐶𝐼𝐹𝑀𝑆 𝐵, 𝐵 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐴
𝑗
𝑥𝑖 + 𝜗𝐴
𝑗
𝑥𝑖 𝜗𝐴
𝑗
𝑥𝑖
𝑛
𝑖=1
𝜂
𝑗 =1
and 𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐵
𝑗
𝑥𝑖 +𝑛
𝑖=1
𝜂
𝑗=1
𝜗𝐴
𝑗
𝑥𝑖 𝜗 𝐵
𝑗
𝑥𝑖 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐴
𝑗
𝑥𝑖 + 𝜗𝐴
𝑗
𝑥𝑖 𝜗𝐴
𝑗
𝑥𝑖
𝑛
𝑖=1
𝜂
𝑗 =1 =
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐴
Hence 𝝆 𝑰𝑭𝑴𝑺 𝑨, 𝑩 =
𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐵
𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐴 ∗𝐶 𝐼𝐹𝑀𝑆 𝐵,𝐵
=
𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐴
𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐴 ∗𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐴
= 1
(ii) Let the 𝝆 𝑰𝑭𝑴𝑺 𝑨, 𝑩 = 1
The unit measure is possible only if
= 1
this refers that 𝜇 𝐴
𝑗
𝑥𝑖 = 𝜇 𝐵
𝑗
𝑥𝑖 and 𝜗𝐴
𝑗
𝑥𝑖 = 𝜗 𝐵
𝑗
𝑥𝑖 for all
i, j values. Hence A = B.
D3. 𝝆 𝑰𝑭𝑴𝑺 𝑨, 𝑩 = 𝝆 𝑰𝑭𝑴𝑺 𝑩, 𝑨
It is obvious that 𝜌𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
=
𝐶 𝐼𝐹𝑀𝑆 𝐵,𝐴
𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐴 ∗𝐶 𝐼𝐹𝑀𝑆 𝐴,𝐴
= 𝜌𝐼𝐹𝑀𝑆 𝐵, 𝐴 as
𝐶𝐼𝐹𝑀𝑆 𝐴, 𝐵 =
1
𝜂
𝜇 𝐴
𝑗
𝑥𝑖 𝜇 𝐵
𝑗
𝑥𝑖 +𝑛
𝑖=1
𝜂
𝑗=1
𝜗𝐴𝑗𝑥𝑖 𝜗𝐵𝑗𝑥𝑖
=
1
𝜂
𝜇 𝐵
𝑗
𝑥𝑖 𝜇 𝐴
𝑗
𝑥𝑖 + 𝜗 𝐵
𝑗
𝑥𝑖 𝜗𝐴
𝑗
𝑥𝑖
𝑛
𝑖=1
𝜂
𝑗=1 =
𝐶𝐼𝐹𝑀𝑆 𝐵, 𝐴
3.5 Numerical Evaluation
3.5.1 Example:
Let X = {A1, A2, A3, A4........ An } with A = { A1, A2, A3, A4,
A5} and B ={ A6, A7, A8, A9, A10} are the IFMS defined as
A={
𝐴1 ∶ 0.6,0.4 , 0.5, 0.5 , 𝐴2 ∶ 0.5,0.3 , 0.4, 0.5 ,
𝐴3 , 0.5, 0.2 , 0.4, 0.4 , 𝐴4 ∶ 0.3,0.2 , 0.3, 0.2 ,
𝐴5 ∶ 0.2,0.1 , 0.2, 0.2 }
B={
𝐴6 ∶ 0.8,0.1 , 0.4, 0.6 ,
𝐴7 ∶ 0.7,0.3 , 0.4, 0.2 , 𝐴8 , 0.4, 0.5 , 0.3, 0.3
𝐴9 ∶ 0.2,0.7 , 0.1, 0.8 , 𝐴10 ∶ 0.2,0.6 , 0, 0.6 }
Here, the cardinality η = 5 as Mc(A) = NMc(A ) = 5 and
Mc(B) = NMc(B) = 5 and the Correlation IFMS
similarity measure is = 0.8147

__________________________________________________________________________________________
3.5.2 Example:
Let X = {A1, A2, A3, A4........ An } with A = { A1, A2 } and B
={ A9, A10} are the IFMS defined as
A = { 𝐴1 ∶ 0.1,0.2 , 𝐴2 ∶ 0.3,0.3 } , B = { 𝐴9 ∶
:0.1,0.2,𝐴10 :0.2,0.3 }
Mc(B) = NMc(B) = 2 and the Correlation IFMS
similarity measure is 0.9827
3.5.3 Example:
Let X = {A1, A2, A3, A4........ An } with A = { A1, A2 } and B
={ A3, A4 } are the IFMS defined as
A = { 𝐴1 0.4,0.2,0.1 , 0.3, 0.1, 0.2 , 0.2,0.1, 0.2 , 0.1, 0.4, 0.3 ,
𝐴2 0.6,0.3,0 , 0.4, 0.5, 0.1 , 0.4, 0.3, 0.2 , 0.2, 0.6, 0.2 }
B= { 𝐴3 ∶ 0. 5,0.2,0.3 , 0.4, 0.2, 0.3 , 0.4, 0.1, 0.2 , 0.1, 0.1, 0.6
𝐴4 ∶ 0.4,0.6,0.2 , 0.4, 0.5, 0 , 0.3,0.4, 0.2 , 0.2, 0.4, 0.1 }
The cardinality η = 2 as Mc(A) = NMc(A ) = Hc(A) = 2
and Mc(B) = NMc(B) = Hc(B) = 2. Here, the
Correlation IFMS similarity is 0.8939
3.5.4 Example:
Let X = {A1, A2, A3, A4..... An } with A = { A1, A2 } and B =
{A6} such that the IFMS A and B are
A = { 𝐴1 ∶ 0.6,0.2,0.2 , 0.4, 0.3, 0.3 , 0.1, 0.7, 0.2 ,
𝐴2 ∶ 0.7,0.1,0.2 , 0.3, 0.6, 0.1 , 0.2, 0.7, 0.1 }
B = { 𝐴6 ∶ 0.8,0.1,0.1 , 0.2, 0.7, 0.1 , 0.3, 0.5, 0.2 }
As Mc(A) = NMc(A ) = Hc(A) =2 and Mc(B) =
NMc(B) = Hc(B) =1,
their cardinality η = Max { η(A), η(B) } = max {2,1} = 2. The
Correlation IFMS measure is 0.9214
4. MEDICAL DIAGNOSIS USING IFMS –
CORRELATION MEASURE
As Medical diagnosis contains lots of uncertainties, they are
the most interesting and fruitful areas of application for fuzzy
set theory. A symptom is an uncertain indication of a disease
and hence the uncertainty characterizes a relation between
symptoms and diseases. Thus working with the uncertainties
leads us to accurate decision making in medicine. In most of
the medical diagnosis problems, there exist some patterns, and
the experts make decision based on the similarity between
unknown sample and the base patterns.
In some situations, terms of membership function (fuzzy set
theory) alone is not adequate. Hence, the terms like
membership and non membership function (Intuitionistic
fuzzy set theory) is considered to be the better one. Due to the
increased volume of information available to physicians from
new medical technologies, the process of classifying different
set of symptoms under a single name of disease becomes
difficult. Recently, there are various models of medical
diagnosis under the general framework of fuzzy sets are
proposed. In some practical situations, there is the possibility
of each element having different membership and non
membership functions. The distance and similarity measure
among the Patients Vs Symptoms and Symptoms Vs diseases
gives the proper medical diagnosis. Here, the proposed
correlation measure point out the proper diagnosis by the
highest similarity measure.
The unique feature of this proposed method is that it considers
multi membership and non membership. By taking one time
inspection, there may be error in diagnosis. Hence, this multi
time inspection, by taking the samples of the same patient at
different times gives best diagnosis.
Let P = { P1, P2, P3, P4 } be a set of Patients.
D = { Fever, Tuberculosis, Typhoid, Throat disease } be the
set of diseases
and S = { Temperature, Cough, Throat pain, Headache, Body
pain } be the set of symptoms.
Our solution is to examine the patient at different time
intervals (three times a day), which in turn give arise to
different membership and non membership function for each
patient.

__________________________________________________________________________________________
Table: 1 – IFMs Q : The Relation between Patient and Symptoms
Q Temperature Cough Throat Pain Head Ache Body Pain
P1
(0.6, 0.2)
(0.7, 0.1)
(0.5, 0.4)
(0.4, 0.3)
(0.3, 0.6)
(0.4, 0.4)
(0.1, 0.7)
(0.2, 0.7)
(0, 0.8)
(0.5, 0.4)
(0.6, 0.3)
(0.7, 0.2)
(0.2, 0.6)
(0.3, 0.4)
(0.4, 0.4)
P2
(0.4, 0.5)
(0.3, 0.4)
(0.5, 0.4)
(0.7, 0.2)
(0.6, 0.2)
(0.8, 0.1)
(0.6, 0.3)
(0.5, 0.3)
(0.4, 0.4)
(0.3, 0.7)
(0.6, 0.3)
(0.2, 0.7)
(0.8, 0.1)
(0.7, 0.2)
(0.5, 0.3)
P3
(0.1, 0.7)
(0.2, 0.6)
(0.1, 0.9)
(0.3, 0.6)
(0.2, 0 )
(0.1, 0.7)
(0.8, 0)
(0.7, 0.1 )
(0.8, 0.1)
(0.3, 0.6)
(0.2, 0.7)
(0.2, 0.6)
(0.4, 0.4)
(0.3, 0.7)
(0.2, 0.7)
P4
(0.5, 0.4)
(0.4, 0.4)
(0.5, 0.3)
(0.4, 0.5)
(0.3, 0.3)
(0.1, 0.7)
(0.2, 0.7)
(0.1, 0.6)
(0, 0.7)
(0.5, 0.4)
(0.6, 0.3)
(0.3, 0.6)
(0.4, 0.6)
(0.5, 0.4)
(0.4, 0.3)
Let the samples be taken at three different timings in a day (morning, noon and night)
Table: 2 – IFMs R: The Relation among Symptoms and Diseases
R Viral Fever Tuberculosis Typhoid Throat disease
Temperature (0.8, 0.1) (0.2, 0.7) (0.5, 0.3) (0.1, 0.7)
Cough (0.2, 0,7) (0.9, 0) (0.3, 0,5) (0.3, 0,6)
Throat Pain (0.3, 0.5) (0.7, 0.2) (0.2, 0.7) (0.8, 0.1)
Head ache (0.5, 0.3) (0.6, 0.3) (0.2, 0.6) (0.1, 0.8)
Body ache (0.5, 0.4) (0.7, 0.2) (0.4, 0.4) (0.1, 0.8)
Table: 3 – The Correlation Measure between IFMs Q and R :
Correlation
Measure
Viral Fever Tuberculosis Typhoid Throat disease
P1 0.9117 0.6835 0.9096 0.5767
P2 0.7824 0.9140 0.8199 0.7000
P3 0.6289 0.7319 0.7143 0.9349
P4 0.8795 0.6638 0.9437 0.6663
The highest similarity measure from the table 4.3 gives the proper medical diagnosis.
Patient P1 suffers from Viral Fever, Patient P2 suffers from Tuberculosis, Patient P3 suffers from Throat disease and
the Patient P4 suffers from Typhoid.
5. PATTERN RECOGNISION OF IFMS
CORRELATION SIMILARITY MEASURE
5.1 Example
Let X = {A1, A2, A3, A4........ An } with A = { A1, A2, A3, A4,
A5} and B ={ A2, A5, A7, A8, A9} are the IFMS defined as
Pattern I = { 𝐴1 ∶ 0.6,0.4 , 0.5, 0.5 , 𝐴2 ∶
:0.5,0.3 , 0.4, 0.5 , 𝐴3 , 0.5, 0.2 , 0.4, 0.4 ,
𝐴4 ∶ 0.3,0.2 , 0.3, 0.2 , 𝐴5 ∶ 0.2,0.1 , 0.2, 0.2 }
Pattern II = { 𝐴2 ∶ 0.5,0.3 , 0.4, 0.5 , 𝐴5 ∶
0.2,0.1 , 0.2, 0.2
𝐴7 ∶ 0.7,0.3 , 0.4, 0.2 , 𝐴8 , 0.4, 0.5 , 0.3, 0.3 ,
𝐴9 ∶ 0.2,0.7 , 0.1, 0.8 }
Then the testing IFMS Pattern III be { A6, A7, A8, A9, A10}
such that { 𝐴6 ∶ 0.8, 0.1 , 0.4, 0.6 ,
𝐴7 ∶ 0.7,0.3 , 0.4, 0.2 , 𝐴8 , 0.4, 0.5 , 0.3, 0.3 , 𝐴9 ∶
:0.2,0.7, 0.1, 0.8 , 𝐴10 :0.2,0.6, 0, 0.6 }

__________________________________________________________________________________________
Here, the cardinality η = 5 as Mc(A) = NMc(A ) =
5 and Mc(B) = NMc(B) = 5
then the Correlation Similarity measure between Pattern (I,
III) is 0.8147, Pattern (II, III) is 0.8852
The testing Pattern III belongs to Pattern II type
5.2 Example:
Let X = {A1, A2, A3, A4........ An } with A = { A1, A2 }; B ={
A4, A6}; C = { A1, A10} ; D = { A4, A6} ; E = { A4,
A6}are the IFMS defined as
A = { 𝐴1 ∶ 0.1,0.2 , 𝐴2 ∶ 0.3, 0.3 } ; B = { 𝐴4 ∶
:0.2,0.2 , 𝐴6 :0.3, 0.2 } ;
C = { 𝐴1 ∶ 0.1,0.2 , 𝐴10 ∶ 0.2, 0.3 } ; D = { 𝐴3 ∶
:0.2,0.1 , 𝐴4 :0.3, 0.2 } ;
E = { 𝐴1 ∶ 0.5,0.4 , 𝐴4 ∶ 0.8, 0.1 }
The IFMS Pattern Y = { 𝐴1 ∶ 0.1, 0.2 , 𝐴10 ∶
:0.2, 0.3 }
Mc(B) = NMc(B) = 2,
then the Correlation measure between the Patten (A, Y) =
0.9829, Patten (B, Y) = 0.9258, Patten (C, Y) = 1, Patten (D,
Y) = 0.8889, Patten (E, Y) = 0.7325 and the testing Pattern
Y belongs to Pattern C type
5.3 Example:
Let X = {A1, A2, A3, A4........ An } with X1 = { A1, A2 }; X2
={ A3, A4 }; X3 = { A1, A4 } are the IFMS defined as
A = { 𝐴1 ∶ 0.4,0.2,0.1 , 0.3, 0.1, 0.2 , 0.2, 0.1, 0.2 , 0.1, 0.4, 0.3 ,
𝐴2 ∶ 0.6,0.3,0 , 0.4, 0.5, 0.1 , 0.4, 0.3, 0.2 , 0.2, 0.6, 0.2 }
B = { 𝐴3 ∶ 0. 5,0.2,0.3 , 0.4,0.2, 0.3 , 0.4, 0.1, 0.2 , 0.1, 0.1, 0.6
𝐴4 ∶ 0.4,0.6,0.2 , 0.4, 0.5, 0 , 0.3, 0.4, 0.2 , 0.2, 0.4, 0.1 }
C = { 𝐴1 ∶ 0.4,0.2,0.1 , 0.3, 0.1, 0.2 , 0.2, 0.1, 0.2 , 0.1, 0.4, 0.3 ,
𝐴4 ∶ 0.4,0.6,0.2 , 0.4, 0.5, 0 , 0.3, 0.4, 0.2 , 0.2, 0.4, 0.1 }
then the Pattern D of IFMS referred as {
𝐴5 ∶ 0.4,0.6,0.2 , 0.4, 0.5, 0 , 0.3, 0.4, 0.2 , 0.2, 0.4, 0.1 ,
𝐴6 ∶ 0.4,0.2,0.2 , 0.5, 0.5, 0 , 0.2,0.4, 0.2 , 0.2, 0.5, 0.1 }
The cardinality η = 2 as Mc(A) = NMc(A ) = Hc(A) =
2 and Mc(B) = NMc(B) = Hc(B) = 2, then the
Proposed Correlation Similarity measure between the Pattern
(A, D) is 0.8628 ; the Pattern (B, D) is 0.8175 and the Pattern
(C, D) is 0.8597.
Hence, the testing Pattern D belongs to Pattern A type
6. CONCLUSIONS
The Correlation similarity measure of IFMS from IFS theory
is derived in this paper. The prominent characteristic of this
method is that the Correlation measure of any two IFMSs
equals to one if and only if the two IFMSs are the same,
referred in the example 5:2 of pattern recognition. From the
numerical evaluation, it is clear that this proposed method can
be applied to decision making problems. The example 3.5.1,
3.5.2 of numerical evaluation shows that the new measure
perform well in the case of membership and non membership
function and example 3.5.3, 3.5.4 of numerical evaluation
depicts that the proposed measure is effective with three
representatives of IFMS – membership, non membership and
hesitation functions. Finally, the medical diagnosis has been
given to show the efficiency of the developed Correlation
similarity measure of IFMS.
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Correlation measure for intuitionistic fuzzy multi sets

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