# A Dynamic Interval-Valued Intuitionistic Fuzzy Sets Applied to Pattern Recognition

Authors: Zhenhua Zhang; Min Wang; Yong Hu; Jingyu Yang; Youpei Ye; Yafang Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We present dynamic interval-valued intuitionistic fuzzy sets (DIVIFS), which can improve the recognition accuracy when they are applied to pattern recognition. By analyzing the degree of hesitancy, we propose some DIVIFS models from intuitionistic fuzzy sets (IFS) and interval-valued IFS (IVIFS). And then we present a novel ranking condition on the distance of IFS and IVIFS and introduce some distance measures of DIVIFS satisfying the ranking condition. Finally, a pattern recognition example applied to medical diagnosis decision making is given to demonstrate the application of DIVIFS and its distances. The simulation results show that the DIVIFS method is more comprehensive and flexible than the IFS method and the IVIFS method.


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## Body

## 1. Introduction

Professor Zadeh’s paper on fuzzy sets [1] has influenced many researchers and has been applied to many application fields, such as pattern recognition [2], fuzzy reasoning [3], portfolio selection [4], and decision making [5]. In 1986, Atanassov [6] introduced membership function, nonmembership function, and hesitancy function and launched the concept of intuitionistic fuzzy sets (IFS), which generalized the FS theory. In the research field of IFS, Yager [7] discussed its cut set characteristics, some scholars (such as [8–13]) applied it to medical diagnosis, Dengfeng and Chuntian [14] and Li et al. [15] applied it to pattern recognition, and Xu [16–20] and Wei [21] applied it to decision making. In 1989, Atanassov and Gargov [22] proposed interval-valued intuitionistic fuzzy sets (IVIFS) based on the comparative analysis of interval-valued fuzzy sets and IFS. Thereafter, many researchers have investigated IVIFS: some researchers discussed the cut set characteristics of IVIFS [23] based on IFS, some applied it to pattern recognition [16, 19, 20, 24], others applied it to decision making analysis [18, 20, 25–29], and Zhang et al. [29] studied interval-valued intuitionistic fuzzy reasoning. However, there are few references related to the construction of IVIFS from IFS at present. Thus, in this paper, the main objective is to study the construction and application of IVIFS from IFS.First, we introduce a series of definitions and construction methods of DIVIFS and propose some DIVIFS models according to IFS and IVIFS. Then, on the basis of the traditional distance measures, DIVIFS is applied to pattern recognition on medical diagnosis. The simulation results show that the method introduced in this paper is more comprehensive and flexible than the IFS method and the IVIFS method. Thus, this paper can provide valuable conclusion for the application research of IFS and IVIFS. The model of DIVIFS is also useful for the generalization from fuzzy reasoning [3] and intuitionistic fuzzy reasoning to interval-valued intuitionistic fuzzy reasoning [29].


## 2.  Construction of DIVIFS

Definition 1.
An IFSA in universe X is formulated as follows [6, 22]:
(1)A={〈x,μA(x),νA(x)〉∣x∈X},
where μA:X→[0,1] and νA:X→[0,1] with the condition 0≤μA(x)+νA(x)≤1 for each x∈X. μA(x)∈[0,1] and νA(x)∈[0,1] denote a degree of membership and a degree of nonmembership of x to A, respectively. For each IFS in X, we call πA(x)=1-μA(x)-νA(x) a degree of hesitancy of x to A, 0≤πA(x)≤1 for each x∈X.Theorem 2.
LetA be an IFS as mentioned above; then
(2)μA(x)-μB(x)+νA(x)-νB(x)+πA(x)-πB(x)=0.
Based on Definition 1, one has Theorem 2.Definition 3.
An IVIFSA in universe X is stated as follows [22]:
(3)A={〈x,MA(x),NA(x)〉∣x∈X},
where MA(x)=[μA-(x),μA+(x)], NA(x)=[νA-(x),νA+(x)], and HA(x)=[πA-(x),πA+(x)] with the conditions MA(x)⊆[0,1], NA(x)⊆[0,1], and HA(x)⊆[0,1]. The interval MA(x), NA(x), and HA(x)denote a degree range of membership, a degree range of non-membership, and a degree range of hesitancy of x to A, respectively. For each IVIFS in X, we call πA-(x)=1-μA+(x)-νA+(x) lower boundary of x to A and πA+(x)=1-μA-(x)-νA-(x) upper boundary of hesitancy of x to A, respectively, where 0≤πA-(x)≤πA+(x)≤1 for each x∈X.Theorem 4.
LetA be an IVIFS as mentioned above; then
(4)μA+(x)-μB+(x)+νA+(x)-νB+(x)+πA-(x)-πB-(x)=0,μA-(x)-μB-(x)+νA-(x)-νB-(x)+πA+(x)-πB+(x)=0.
Based on Definition 3, one has Theorem 4.Definition 5.
X is a universe of discourse. A DIVIFS A in X is an object having the following form:
(5)A={〈x,MA(x),NA(x)〉∣x∈X},
where MA(x), NA(x), and HA(x) are the same as Definition 3. Let μA+(x)=μA-(x)+αA(x), νA+(x)=νA-(x)+βA(x). Obviously, we have αA(x)+βA(x)=πA+(x)-πA-(x), αA(x)≥0, βA(x)≥0. Thus, DIVIFS is equivalent to IVIFS, and IFS is a special case of DIVIFS when αA(x)=βA(x)=0.
From Definition3, let all sample data be divided into three parts, μA-(x) being the firm support party of event A, νA-(x) representing the firm opposition party of event A, and πA+(x) showing all the absent party. In the absent party, πA-(x) is the firm absent party, and πA+(x)-πA-(x) is the convertible absent party, in which each sample may become either the support party or the opposition party. If there is αA(x) sample supporting event A and βA(x) sample opposing event A, we have DIVIFS based on Definition 5. If the proportion of the absent party converted to the support party is λA1(x) and the proportion of the absent party converted to the opposition party is 1-λA1(x), then the model will become a DIVIFS with single variable, where αA(x)=λA1(x)(πA+(x)-πA-(x)), βA(x)=(1-λA1(x))(πA+(x)-πA-(x)). If the firm absent party is πA-(x)=(1-λA0(x))πA+(x), then the convertible absent party is πA+(x)-πA-(x)=λA0(x)πA+(x), and then we will get the other DIVIFS definition as follows.Definition 6.
X is a universe of discourse. DIVIFS A in X is an object having the following form
(6)A={〈x,MA(x),NA(x)〉∣x∈X},
where MA(x), NA(x), and HA(x) are the same as Definition 5. For each λA0(x)∈[0,1] and for each λA1(x)∈[0,1], we assume that μA+(x)=μA-(x)+λA0(x)λA1(x)πA+(x), νA+(x)=νA-(x)+λA0(x)(1-λA1(x))πA+(x) and that πA-(x)=(1-λA0(x))πA+(x). Evidently, the DIVIFS model is also equivalent to IVIFS. If λA0(x)=0, then DIVIFS is IFS. And if μA-(x)=μA(x), νA-(x)=νA(x), πA+(x)=πA(x), then we have a special DIVIFS which is derived from IFS.Definition 7.
A DIVIFS derived from IFS is given as follows:(7)A={〈x,MA(x),NA(x)〉∣x∈X},
where MA(x), NA(x), and HA(x) are the same as Definition 5 and μA-(x)=μA(x), νA-(x)=νA(x), πA+(x)=πA(x), μA+(x)=μA(x)+λA0(x)λA1(x)πA(x), νA+(x)=νA(x)+λA0(x)(1-λA1(x))πA(x), πA-(x)=(1-λA0(x))πA(x), for each λA0(x)∈[0,1] and for each λA1(x)∈[0,1]. This DIVIFS model is also an extension of the IFS model.


## 3. Conventional Distance Measures of IFS and IVIFS

Derived from the definitions of FS, IFS, and IVIFS, some distance measures are discussed (Figure1). Let d=1-S, Δμ=μA-μB, Δν=νA-νB, and Δπ=πA-πB. According to Definition 1 and Theorem 2, Δμ+Δν+Δπ=0.Distance between each patient and each disease on IFS calculated by Xu’s methods.
(a)
(b)
(c)
(d)According to fuzzy sets [1], the distance measure of FS is composed of membership degree:
(8)dZa(A,B)=(∑i=1nwi|Δμ|p)1/p.In [6], Atanassov introduced a distance measure (9) of IFS from membership degree and non-membership degree:
(9)dAt(A,B)=(∑i=1nwi(|Δμ|p+|Δν|p))1/p.Considering the degree range of hesitancy, Szmidt and Kacprzyk presented a novel distance measure (10) according to Definition 1 and Theorem 2 in 2000 [30]:
(10)dS,K(A,B)=(∑i=1nwi(|Δμ|p+|Δν|p+|Δμ+Δν|p))1/p.In [14], Dengfeng and Chuntian proposed a novel distance model as follows:
(11)dL,C(A,B)=(∑i=1nwi|Δμ-Δν|p)1/p.At the same time, Grzegorzewski [31] and Hung and Yang [32] proposed a Hausdorff distance measure of IFS, respectively:
(12)dP,H(A,B)=∑i=1nwimax(|Δμ|,|Δν|).In 2005, Wang and Xin [33] presented a new distance measure which is made up of dAt and dP,H:
(13)dW(A,B)=∑i=1nwi|Δμ|+|Δν|+2max(|Δμ|,|Δν|)4.In 2005, Liu [34] introduced some similar distance formulas as mentioned above.In Szmidt and Kacprzyk [8–10], the ratio-based measure of similarity is defined as follows:
(14)dRB(A,B)=d(A,B)d(A,BC).
If B is a set (μB(x),νB(x),πB(x)), Szmidt and Kacprzyk describe BC to be a set (νB(x),μB(x),πB(x)). For example, if B is (μB(x)=0.7, νB(x)=0, πB(x)=0.3), then BC is (0, 0.7, 0.3), and so on. And d denotes a kind of distance measure about A and B.In 2010, Xia and Xu [20] presented some weighted distance formulas, where k≥0 and λ≥0 denote the weights:
(15)dDk(A,B)=[∑i=1nwi(|(1-k)Δμ-kΔν|p)]1/p,daFk(16)dFk,λ(A,B)=[0.5∑i=1nwi(|(1-k)Δμ-kΔν|p+|(1-λ)Δν-λΔμ|p+|(1-k-λ)(Δμ+Δν)|p){∑i=1nwi(|(1-k)Δμ-kΔν|p+|(1-λ)Δν-λΔμ|p}]1/p,k+λ≤1,(17)dGk,λ(A,B)=[0.5∑i=1nwi(|kΔμ|p+|λΔν|p+|kΔμ+λΔν|p)]1/p,k+λ≤1,(18)dHk,λ(A,B)=[0.5∑i=1nwi(|kΔμ|p+|λΔμ-(1-λ)Δν|p+|(k-λ)Δμ+(1-λ)Δν|p){∑i=1nwi(|kΔμ|p+|λΔμ-(1-λ)Δν|p}]1/p,k+λ≤1,(19)dHk,λ*(A,B)=[0.5∑i=1nwi(|kΔμ|p+|kλΔμ-(1-λ)Δν|p+|(1-λ)(kΔμ+Δν)|p){∑i=1nwi(|kΔμ|p+|kλΔμ-(1-λ)Δν|p}]1/p,k+λ≤1,(20)dJk,λ(A,B)=[0.5∑i=1nwi(|(1-k)Δμ-kΔν|p+|λΔν|p+|(1-k)Δμ+(λ-k)Δν)|p){∑i=1nwi}]1/p,k+λ≤1,(21)dJk,λ*(A,B)=[0.5∑i=1nwi(|(1-k)Δμ-kλΔν|p+|λΔν|p+|(1-k)(Δμ+λΔν)|p){∑i=1nwi}]1/p,k+λ≤1.LetΔμ-=μA--μB-, Δμ+=μA+-μB+, Δν-=νA--νB-, Δν+=νA+-νB+, Δπ-=πA--πB-, Δπ+=πA+-πB+. According to Definition 3 and Theorem 4, Δμ-+Δν-+Δπ+=0, Δμ++Δν++Δπ-=0. Similar to the distance measures of IFS, some scholars define some distance measures made up of Δμ-, Δμ+, Δν-, Δν+, Δπ-, Δπ+.


## 4. Construction of Distance Measures on DIVIFS

First, we will define a novel essential ranking property of the distance measure on IFS and IVIFS.Definition 8.
Assume thatA and B are IFSs, and then we have the following: A=B if and only if μA(x)=μB(x), νA(x)=νB(x), πA(x)=πB(x).A⊆B if and only if μA(x)≤μB(x), νA(x)≥νB(x).A⊂B if and only if μA(x)≤μB(x) and νA(x)>νB(x), or μA(x)<μB(x) and νA(x)≥νB(x).Definition 9.
Assume thatA and B are IVIFSs, and then we obtain the following: A=B if and only if μA-(x)=μB-(x), μA+(x)=μB+(x), vA-(x)=vB-(x), vA+(x)=vB+(x), πA-(x)=πB-(x), πA+(x)=πB+(x).A⊆B if and only if μA-(x)≤μB-(x), μA+(x)≤μB+(x), vA-(x)≥vB-(x), vA+(x)≥vB+(x).A⊂B if and only if A⊆B and A≠B.
In general, a distance measure should conform to the following Theorem10 according to its properties.Theorem 10.
A, B, and C are IFSs; the distances among them must satisfy the following:(P1)
d(A,B)≥0 and d(A,B)=0 if and only if A=B.(P2)
d(A,B)=d(B,A).(P3)
IfC⊆B⊆A, then d(A,C)≥d(A,B) and d(A,C≥d(B,C).IfC⊂B⊂A, then d(A,C)>d(A,B) and d(A,C)>d(B,C).(P4)
IfD⊆C⊆B⊆A, then d(A,D)≥d(B,C), and d(A,D)=d(B,C) if and only if A=B and C=D.IfD⊂C⊂B⊂A, then d(A,D)>d(B,C).
In fact, property(P4) is derived from property (P3). According to (P3), one has the following:IfD⊆C⊆B⊆A, then d(A,D)≥d(A,C)≥d(B,C), d(A,D)≥d(B,D)≥d(B,C)→d(A,D)≥d(B,C).IfD⊂C⊂B⊂A, then d(A,D)>d(A,C)>d(B,C), d(A,D)>d(B,D)>d(B,C)→d(A,D)>d(B,C).
Obviously, for all the distance formulas above,(P1), (P2), and (P3) can be obtained. However, most of them do not satisfy the ranking property (P4).Example 11.
LetA=(0.5,0.4,0.1), B=(0.5,0.5,0), C=(0.4,0.4,0.2), and D=(0.4,0.5,0.1). For IFS A, we obtain μA(x)=0.5, νA(x)=0.4, and πA(x)=0.1. In the following, we denote μA=μA(x)=0.5, νA=νA(x)=0.4, and πA=πA(x)=0.1.
According to Definition3, if μ-(x)=μ+(x)=μ(x), ν-(x)=ν+(x)=ν(x), π-(x)=π+(x)=π(x), then IVIFS is equivalent to IFS. Correspondingly, we can define some IVIFSs as follows, respectively:
(22)A=〈x,[0.5,0.5],[0.4,0.4]〉,B=〈x,[0.5,0.5],[0.5,0.5]〉,C=〈x,[0.4,0.4],[0.4,0.4]〉,D=〈x,[0.4,0.4],[0.5,0.5]〉.
If X={x1,x2,…,xn} is a universe of discourse, we define
(23)A={〈x1,[0.5,0.5],[0.4,0.4]〉,…,〈xn,[0.5,0.5],[0.4,0.4]〉};B={〈x1,[0.5,0.5],[0.5,0.5]〉,…,〈xn,[0.5,0.5],[0.5,0.5]〉};C={〈x1,[0.4,0.4],[0.4,0.4]〉,…,〈xn,[0.4,0.4],[0.4,0.4]〉};D={〈x1,[0.4,0.4],[0.5,0.5]〉,…,〈xn,[0.4,0.4],[0.5,0.5]〉}.
Thus we haveμA(x)=μB(x)>μC(x)=μD(x), νA(x)=νC(x)<νB(x)=νD(x). According to Definitions 8 and 9, we have D⊂C⊂B⊂A. Therefore, we obtain the necessary condition for distance measure on IFS and IVIFS: d(A,D)>d(B,C).Obviously, according toA, B, C, and D, we obtain
(24)Δμ(A,D)=μA-μD=0.1,Δν(A,D)=νA-νD=-0.1,Δπ(A,D)=πA-πD=0,Δμ(B,C)=μB-μC=0.1,Δν(B,C)=νB-νC=0.1,Δπ(B,C)=πB-πC=0.2.From formulas (8)–(14), we have
(25)dZa(A,D)=(∑i=1nwi0.1p)1/p=dZa(B,C),dAt(A,D)=(∑i=1nwi(0.1p+0.1p))1/p=dAt(B,C),dS,K(A,D)=(∑i=1nwi(0.1p+0.1p))1/p<(∑i=1nwi(0.1p+0.1p+0.2p))1/p=dS,K(B,C),dL,C(A,D)=(∑i=1nwi0.2p)1/p>0=dL,C(B,C),dP,H(A,D)=∑i=1nwimax(0.1,0.1)=dP,H(B,C),dW(A,D)=∑i=1nwi0.1+0.1+2max(0.1,0.1)4=dW(B,C),dRB(A,D)=+∞>1=dRB(B,C).
Only (11) and (14) satisfy property (P4).Considering (15), we get
(26)dDk(A,D)=(∑i=1nwi(0.1p))1/p≥(∑i=1nwi(|1-2k|0.1)p)1/p=dDk(B,C).
If k≠0 and k≠1, then we have dDk(A,D)>dDk(B,C); otherwise, we have dDk(A,D)=dDk(B,C).From (16), we get
(27)dFk,λ(A,D)=(0.5∑i=1nwi(0.1p+0.1p))1/p,dFk,λ(B,C)=(0.5∑i=1nwi0.1p(|1-2k|p+|1-2λ|p+|1-k-λ|p))1/p,k+λ≤1.
If p=1, then we have k+λ<1/2→dDk(A,D)<dDk(B,C); k+λ≥1/2→dDk(A,D)≥dDk(B,C).From (17), we get
(28)dGk,λ(A,D)=(0.5∑i=1nwi(|0.1k|p+|0.1λ|p+|0.1(k-λ)|p))1/p≤(0.5∑i=1nwi(|0.1k|p+|0.1λ|p+|0.1(k+λ)|p))1/p=dGk,λ(B,C).
Obviously, we get dDk(A,D)≤dDk(B,C) for each k≥0, for each λ≥0, and for each P>0.For (18), we have
(29)dHk,λ(A,D)=(0.5∑i=1nwi(|0.1k|p+|0.1|p+|0.1(1-k)|p))1/p,dHk,λ(B,C)=(0.5∑i=1nwi(|0.1k|p+|0.1(1-2λ)|p+|0.1(1+k-2λ)|p){∑i=1nwi})1/p.
If p=1 and k+λ≤1, then we have 0≤λ<1/3, 2λ<k→dHk,λ(A,D)<dHk,λ(B,C); otherwise, dDk(A,D)≥dDk(B,C).For example, letp=1, λ=0.2, and k=0.5, and then dHk,λ(A,D)=0.5∑i=1nwi0.2<0.5∑i=1nwi0.22=dHk,λ(B,C).Similar to the result from (18), for (20), we have
(30)dJk,λ(A,D)=(0.5∑i=1nwi(|0.1|p+|0.1λ|p+|0.1(1-λ)|p))1/p,dJk,λ(B,C)=(0.5∑i=1nwi(|0.1(1-2k)|p+|0.1λ|p+|0.1(1+λ-2k)|p){∑i=1nwi})1/p.
If p=1 and k+λ≤1, then we have 0≤k<1/3, 2k<λ→dHk,λ(A,D)<dHk,λ(B,C); otherwise, dDk(A,D)≥dDk(B,C).For example, letp=1, k=0.2, and λ=0.5, and then dHk,λ(A,D)=0.5∑i=1nwi0.2<0.5∑i=1nwi0.22=dHk,λ(B,C).For (19), we obtain
(31)dHk,λ*(A,D)=(0.5∑i=1nwi(|0.1k|p+|0.1(1+kλ-λ)|p+|0.1(1-λ)(1-k)|p){∑i=1nwi})1/p,dHk,λ*(B,C)=(0.5∑i=1nwi(|0.1k|p+|0.1(1-kλ-λ)|p+|0.1(1-λ)(1+k)|p){∑i=1nwi})1/p,k+λ≤1.Ifp=1 and k+λ≤1, then we have 0≤λ<1/2→dHk,λ(A,D)<dHk,λ(B,C); otherwise, dDk(A,D)≥dDk(B,C).Similarly, we have the same results for (21), which will be as follows:ifp=1 and k+λ≤1, then 0≤k<1/2→dHk,λ(A,D)<dHk,λ(B,C); otherwise, dDk(A,D)≥dDk(B,C).From the analysis to the formulas from Xia and Xu above, only (15) basically meets the requirement of (P4); the others only meet the requirement of (P4) in part. Similarly, most of the conventional distance measures of IVIFS do not satisfy property (P4). Therefore, it is necessary to find a distance measure satisfying the necessary ranking property. Considering the component elements of the distance formula of IFS and IVIFS, we can define some appropriate distance measures of DIVIFS (Figure 2).Distance between each patient and each disease on DIVIFS.
(a)
(b)
(c)
(d)According to Definition7 and Example 11, we haveμA-(x)=0.5, vA-(x)=0.4, πA+(x)=0.1, μA+(x)=0.5+λ0λ10.1, vA+(x)=0.4+λ0(1-λ1)0.1, πA-(x)=(1-λ0)0.1,μB-(x)=0.5, vB-(x)=0.5, πB-(x)=0, μB+(x)=0.5, vB+(x)=0.5, πB+(x)=0,μC-(x)=0.4, vC-(x)=0.4, πC+(x)=0.2, μC+(x)=0.4+λ0λ10.2, vC+(x)=0.4+λ0(1-λ1)0.2, πC-(x)=(1-λ0)0.2,μD-(x)=0.4, vD-(x)=0.5, πD+(x)=0.1, μD+(x)=0.4+λ0λ10.1, vD+(x)=0.5+λ0(1-λ1)0.1, πD-(x)=(1-λ0)0.1.Letμ-=μ-(x), μ+=μ+(x), ν-=ν-(x), ν+=ν+(x), π-=π-(x), π+=π+(x), and we obtain(1)
Δπ-(A,D)=πA--πD-=0, Δπ-(B,C)=πB--πC-=-(1-λ0)0.2→|Δπ-(A,D)|≤|Δπ-(B,C)|;(2)
Δπ+(A,D)=πA+-πD+=0<πB+-πC+=Δπ+(B,C)=0.2→|Δπ+(A,D)|<|Δπ+(B,C)|;(3)
Δμ-(A,D)=μA--μD-=0.1=μB--μC-=Δμ-(B,C)→|Δμ-(A,D)|=|Δμ-(B,C)|;(4)
Δν-(A,D)=νA--νD-=-0.1=-1×(νB--νC-)=-Δν-(B,C)→|Δν-(A,D)|=|Δν-(B,C)|;(5)
Δμ-(A,D)-Δν-(A,D)=0.2>0=Δμ-(B,C)-Δν-(B,C)→|Δμ-(A,D)-Δν-(A,D)|>|Δμ-(B,C)-Δν-(B,C)|;(6)
Δμ+(A,D)=μA+-μD+=0.1, Δμ+(B,C)=μB+-μC+=0.1(1-2λ0λ1)→|Δμ+(A,D)|≥|Δμ+(B,C)|;(7)
Δν+(A,D)=νA+-νD+=-0.1, Δν+(B,C)=νB+-νC+=0.1(1-2λ0(1-λ1))→|Δμ+(A,D)|≥|Δμ+(B,C)|;(8)
Δμ+(A,D)-Δν+(A,D)=0.2, Δμ+(B,C)-Δν+(B,C)=0.2λ0(1-2λ1), →|Δμ+(A,D)-Δν+(A,D)|≥|Δμ+(B,C)-Δν+(B,C)|.Considering the necessary condition (P4), we draw a conclusion as follows.Δπ+ does not meet the requirement of the ranking property, and if λ0≠1, Δπ- does not satisfy (P4) too. Δu--Δν- meets the necessary condition (P4), and Δμ+, Δv+, and Δμ+-Δv+ do so if λk∈(0,1) (k=1,2). All the calculated results are shown in (32) and (33):
(32)|Δπ-(A,D)|≤|Δπ-(B,C)|;|Δπ-(A,D)|=|Δπ-(B,C)|⟺λ0=1;|Δπ+(A,D)|=|Δπ+(B,C)|;|Δμ-(A,D)|=|Δμ-(B,C)|;|Δν-(A,D)|=|Δν-(B,C)|;|Δμ-(A,D)-Δν-(A,D)|>|Δμ-(B,C)-Δν-(B,C)|;|Δμ+(A,D)|≥|Δμ+(B,C)|;|Δμ+(A,D)|=|Δμ+(B,C)|⟺λ0=0,orλ1=0,orλ0=λ1=1;|Δν+(A,D)|≥|Δν+(B,C)|;|Δν+(A,D)|=|Δν+(B,C)|⟺λ0=0,orλ1=1,orλ0=1,λ1=0;(33)|Δμ+(A,D)-Δν+(A,D)|≥|Δμ+(B,C)-Δν+(B,C)|;|Δμ+(A,D)-Δν+(A,D)|=|Δμ+(B,C)-Δν+(B,C)|⟺λ0=1,λ1=0,orλ0=1,λ1=1.In all, in order to meet the needs of necessary condition (P4), we conclude that the distance measure of DIVIFS should be composed ofΔμ--Δv-, Δμ+, Δv+, and Δμ+-Δv+ and that Δμ- and Δν- can be involved in the distance formula too. However, considering the contradiction between Δπ and (P4), Δπ+ and Δπ- are not suitable for new distance measures.In Xu’s papers [16, 17, 19, 20],  many weighted similarity measures are proposed according to standardized distance measures based on the IFS theory and the IVIFS theory. And Xu presented them by defining the appropriate weights of membership function, non-membership function, and hesitancy function. According to Xu, we can also define the corresponding weighted distance measures on IVIFS as follows:
(34)d(A,B)={∑i=1nwi[C1|Δμ-(A(xi),B(xi))|p+C2|Δμ+(A(xi),B(xi))|p+C3|Δν-(A(xi),B(xi))|p+C4|Δν+(A(xi),B(xi))|p+|C5Δμ-(A(xi),B(xi))-C7Δν-(A(xi),B(xi))|p+|C6Δμ+(A(xi),B(xi))-C8Δν+(A(xi),B(xi))|p]{∑i=1nwi}}1/p;∑i=1nwi=1,wi≥0;Cj≥0,j=1,2,3,4,5,6,7,8.It is easy to prove that (11) is equivalent to a special case of (15) when k=0.5. And (15) is also a special case of (34) when C1=C2=C3=C4=C6=C8=0 and C5=C7=1. To simplifiy the formula, let C5=C7 and C6=C8. Assume that A and B are two DIVIFS, and we have
(35)d(A,B)={∑i=1nwi[C1|μA-(xi)-μB-(xi)|p+C2|νA-(xi)-νB-(xi)|p+C3|{(πA+(xi)-πB+(xi))}μA-(xi)-μB-(xi)+λ0λ1(πA+(xi)-πB+(xi))|p+C4|νA-(xi)-νB-(xi)+λ0(1-λ1)(πA+(xi)-πB+(xi))|p+C5|(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))|p+C6|(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))+λ0(2λ1-1)(πA+(xi)-πB+(xi))|p]{∑i=1nwi}}1/p,∑i=1nwi=1,wi≥0;Cj≥0,j=1,2,3,4,5,6.Comparing (15)–(21) with (34)-(35), we conclude that Xu’s methods are different from the DIVIFS method. Xu defined the measures by constructing the weights of membership function, non-membership function, and hesitancy function, respectively. However, we concentrate on the detachment of hesitancy function and present the DIVIFS method. If we use Xu’s methods solely, we can define the weights of certainty and uncertainty, but we cannot analyze the variation of the absent party. Moreover, in Xu’s research, and even in all the conventional research, there is no research on the ranking condition (P4). Therefore, the distance measures on DIVIFS above are an important supplement to Xu’s methods and the others’.In order to simplify the calculation, we define the following distance measure:(36)d(A,B)=∑i=1nwi[{|(πA+(xi)-πB+(xi))|p}C1|μA-(xi)-μB-(xi)|p+C2|νA-(xi)-νB-(xi)|p+C3|{(πA+(xi)-πB+(xi))}μA-(xi)-μB-(xi)+λ0λ1(πA+(xi)-πB+(xi))|p+C4|{(πA+(xi)-πB+(xi))}νA-(xi)-νB-(xi)+λ0(1-λ1)(πA+(xi)-πB+(xi))|p+C5|(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))|p+C6|(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))+λ0(2λ1-1)(πA+(xi)-πB+(xi))|p],∑i=1nwi=1,wi≥0;Cj≥0,j=1,2,3,4,5,6.Ifp=1, the distance measure is the Hamming distance; if p=2, it is the Euclidean distance. In this paper, the Euclidean distance on DIVIFS will be used in the simulation of pattern recognition:
(37)d(A,B)=∑i=1n{(πA+(xi)-πB+(xi))[μA-(xi)-μB-(xi)]2+[νA-(xi)-νB-(xi)]2+[μA-(xi)-μB-(xi)+λ0λ1(πA+(xi)-πB+(xi))]2+[{(πA+(xi)-πB+(xi))}νA-(xi)-νB-(xi)+λ0(1-λ1)(πA+(xi)-πB+(xi))]2},(38)d(A,B)=∑i=1n{(πA+(xi)-πB+(xi))[(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))]2+[(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))+λ0(2λ1-1)(πA+(xi)-πB+(xi))]2},(39)d(A,B)=∑i=1n{(πA+(xi)-πB+(xi)){[(νA-(xi)-νB-(xi))]2}[μA-(xi)-μB-(xi)]2+[νA-(xi)-νB-(xi)]2+[μA-(xi)-μB-(xi)+λ0λ1(πA+(xi)-πB+(xi))]2+[{(πA+(xi)-πB+(xi))}νA-(xi)-νB-(xi)+λ0(1-λ1)(πA+(xi)-πB+(xi))]2+[(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))]2+[(μA-(xi)-μB-(xi))-(νA-(xi)-νB-(xi))+λ0(2λ1-1)(πA+(xi)-πB+(xi))]2}.


## 5. Application to Medical Diagnosis Based on Pattern Recognition

To make a proper diagnosisD for a patient with given values of symptoms S, a medical knowledge base is necessary that involves elements described in terms of IFS. We adopt the same data as those in [8, 9]: the set of disease diagnoses is D={Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}, and the set of symptoms is a universe of discourse S={temperature, headache, stomach pain, cough, chest pain}. The data are given in Table 1, where each symptom is described as follows: membership function μA(x), non-membership function νA(x), and hesitancy function πA(x). For example, for malaria the temperature is high (μA(x)=0.7, νA(x)=0, πA(x)=0.3), while for chest problem the temperature is low (μA(x)=0.1, νA(x)=0.8, πA(x)=0.1), and so forth.Table 1
Symptoms characteristic of the diagnoses under consideration.
Disease symptoms
Temperature
Headache
Stomach pain
Cough
Chest pain
Viral fever
(
0.4,0.0,0.6
)
(
0.3,0.5,0.2
)
(
0.1,0.7,0.2
)
(
0.4,0.3,0.3
)
(
0.1,0.7,0.2
)
Malaria
(
0.7,0.0,0.3
)
(
0.2,0.6,0.2
)
(
0.0,0.9,0.1
)
(
0.7,0.0,0.3
)
(
0.1,0.8,0.1
)
Typhoid
(
0.3,0.3,0.4
)
(
0.6,0.1,0.3
)
(
0.2,0.7,0.1
)
(
0.2,0.6,0.2
)
(
0.1,0.9,0.0
)
Stomach problem
(
0.1,0.7,0.2
)
(
0.2,0.4,0.4
)
(
0.8,0.0,0.2
)
(
0.2,0.7,0.1
)
(
0.2,0.7,0.1
)
Chest problem
(
0.1,0.8,0.1
)
(
0.0,0.8,0.2
)
(
0.2,0.8,0.0
)
(
0.2,0.8,0.0
)
(
0.8,0.1,0.1
)The set of patients isP={AI, Bob, Joe, Ted}. The symptoms are given in Table 2, where we need all three parameters (μA(x),νA(x),πA(x)) to describe each symptom (see [8, 9]) as before. We seek a diagnosis for each patient pj, j=1,2,3,4. In [8, 9], Szmidt and Kacprzyk [8, 9] proposed to solve the problem in the following way: (1) to calculate a distance from a set of symptoms si(i=1,2,3,4,5) (using the normalized Hamming distance, the Euclidean distance, and ratio-based measure of similarity) between each patient pj (j=1,2,3,4, Table 2) and each diagnosis dk, (k=1,2,3,4,5, Table 1); (2) to determine that the shortest distance should be a proper diagnosis.Table 2
Symptoms characteristic of the patients under observation.
Patient symptoms
Temperature
Headache
Stomach pain
Cough
Chest pain
AI
(
0.8,0.1,0.1
)
(
0.6,0.1,0.3
)
(
0.2,0.8,0.0
)
(
0.6,0.1,0.3
)
(
0.1,0.6,0.3
)
Bob
(
0.0,0.8,0.2
)
(
0.4,0.4,0.2
)
(
0.6,0.1,0.3
)
(
0.1,0.7,0.2
)
(
0.1,0.8,0.1
)
Joe
(
0.8,0.1,0.1
)
(
0.8,0.1,0.1
)
(
0.0,0.6,0.4
)
(
0.2,0.7,0.1
)
(
0.0,0.5,0.5
)
Ted
(
0.6,0.1,0.3
)
(
0.5,0.4,0.1
)
(
0.3,0.4,0.3
)
(
0.7,0.2,0.1
)
(
0.3,0.4,0.3
)For (10), the normalized Hamming distance for all symptoms of patient jth from diagnosis kth is (40), and the normalized Euclidean distance for all symptoms of patient jth from diagnosis kth is (41). From (14), the ratio-based measure of similarity for all symptoms of patient jth from diagnosis kth is (42):
(40)dH(pj,dk)=∑i=15|μpj-(si)-μdk-(si)|+|νpj-(si)-νdk-(si)|+(πpj+(si)-πdk+(si))10,(41)dE(pj,dk)=110∑i=15(|μpj-(si)-μdk-(si)|2+|νpj-(si)-νdk-(si)|2+|πpj+(si)-πdk+(si)|2),(42)dRB(pj,dk)=dH(pj,dk)dH(pj,dkC).
Szmidt and Kacprzyk describe dkC to be a set (νA(x),μA(x),πA(x)). For example, for malaria the temperature dk is high (μA(x)=0.7, νA(x)=0, πA(x)=0.3), whereas dkC is low (μA(x)=0, νA(x)=0.7, πA(x)=0.3), and so on.From the obtained distance measures and similarity measures, Szmidt and Kacprzyk get the results as Table3. Three types of distances given in Table 3 are described as follows: the first data is the Hamming distance as in (40), the second is the Euclidean distance as in (41), the third is the ratio-based measure as in (42). For example, dH(AI, viral fever), dE(AI, viral fever), and dRB(AI, viral fever) are 0.28, 0.29, and 0.75 respectively. Similarly, we can get the other results. Obviously, Bob suffers from stomach problemand Joefrom typhoid based on (40)–(42) above. AI suffers from Malaria andTedfrom Viral fever calculated by the Hamming distance and the Euclidean distance, while AI suffers from Viral fever andTedfrom Malaria calculated by the ratio-based measure.Table 3
Distance description between each patient and each disease based on IFS.
Patient/disease
Viral fever
Malaria
Typhoid
Stomach problem
Chest problem
AI
(
0.28,0.29
,
0.75
)
(
0.24
,
0.25
,
1.19
)
(
0.28,0.32,1.31
)
(
0.54,0.53
,
3.27
)
(
0.56,0.58
,
∞
)
Bob
(
0.40,0.43,2.1
)
(
0.50,0.56,3.73
)
(
0.31,0.33,1.10
)
(
0.14,0.14,0.35
)
(
0.42,0.46
,
∞
)
Joe
(
0.38,0.36,0.87
)
(
0.44,0.41,1.52
)
(
0.32,0.32,0.46
)
(
0.50,0.52,2.61
)
(
0.55,0.57
,
∞
)
Ted
(
0.28
,
0.25
,
0.95
)
(
0.30,0.29
,
0.77
)
(
0.38,0.35,1.67
)
(
0.44,0.43
,
∞
)
(
0.54,0.50,2.56
)Letp=5 and wi=1/n for each i, and k and λ are constants. Calculated by (15)–(21), Table 4 can be obtained.Table 4
Distance between each patient and each disease on IFS calculated by Xu’s methods.
Distance between patient and disease
D
D
(
k
=
0.2
)
D
F
(
k
=
0.2
,
λ
=
0.1
)
D
G
(
k
=
0
.
7
,
λ
=
0.8
)
D
H
(
k
=
0.8
,
λ
=
0.3
)
D
H
*
(
k
=
0.9
,
λ
=
0.2
)
D
J
(
k
=
0.1
,
λ
=
0.6
)
D
J
*
(
k
=
0.2
,
λ
=
0.7
)
D
(
AI
,
viral  fever
)
0.26143
0.29072
0.25963
0.26478
0.29445
0.29111
0.27069
D
(
AI
,
milaria
)
0.30533
0.33445
0.26197
0.30561
0.31294
0.27027
0.27086
D
(
AI
,
typhoid
)
0.35839
0.36029
0.28616
0.3306
0.34773
0.3355
0.31541
D
(
AI
,
stomach  problem
)
0.56074
0.59068
0.45948
0.54019
0.55532
0.49994
0.4901
D
(
AI
,
chest  problem
)
0.60507
0.61828
0.47761
0.5666
0.58378
0.54117
0.52311
D
(
Bob
,
viral  fever
)
0.42194
0.52601
0.43156
0.47154
0.48979
0.38652
0.40853
D
(
Bob
,
milaria
)
0.60123
0.64839
0.50394
0.59342
0.60684
0.53092
0.52867
D
(
Bob
,
typhoid
)
0.33646
0.40929
0.33026
0.3696
0.38103
0.30293
0.31836
D
(
Bob
,
stomach  problem
)
0.14469
0.13263
0.11231
0.12354
0.13796
0.14745
0.13245
D
(
Bob
,
chest  problem
)
0.52473
0.55204
0.42999
0.50446
0.51855
0.46688
0.45823
D
(
Joe
,
viral  fever
)
0.35629
0.35212
0.29083
0.32246
0.3462
0.34447
0.32153
D
(
Joe
,
milaria
)
0.46742
0.49471
0.38926
0.45165
0.46599
0.41851
0.41061
D
(
Joe
,
typhoid
)
0.31951
0.31011
0.2846
0.28115
0.31803
0.31742
0.29265
D
(
Joe
,
stomach  problem
)
0.61639
0.57475
0.43227
0.53105
0.55428
0.55892
0.52068
D
(
Joe
,
chest  problem
)
0.66501
0.63332
0.48018
0.58484
0.60883
0.60414
0.56562
D
(
Ted
,
viral  fever
)
0.218
0.23438
0.19472
0.2132
0.22744
0.21415
0.20328
D
(
Ted
,
milaria
)
0.26922
0.33348
0.27321
0.29927
0.31105
0.24789
0.2599
D
(
Ted
,
typhoid
)
0.35577
0.36789
0.29537
0.33285
0.34867
0.32407
0.31266
D
(
Ted
,
stomach  problem
)
0.45321
0.46272
0.35531
0.42507
0.43656
0.40408
0.39108
D
(
Ted
,
chest  problem
)
0.48203
0.51701
0.40614
0.47177
0.48604
0.43213
0.42575From Table4, we conclude that Bob suffers from stomach problem, Joe from typhoid, Ted from viral fever, and AI from viral fever according to most of Xu’s methods. But on the basis of DJ0.1,0.6, AI can be diagnosed as a malaria sufferer. Using Xu’s methods of IVIFS from [20], we can get the same results as Table 4. The results above are also the same as those from [35] by Wei et al.LetμA-(x)=μA(x), νA-(x)=νA(x), and πA+(x)=πA(x). Using (37), we will get Table 5.Table 5
Distance description between each patient and each disease based on DIVIFS.
(AI, Bob, Joe, Ted)
constant
λ
0
λ
0
λ
1
λ
0
2
λ
0
2
λ
1
(
1
-
λ
1
)
Viral fever
(
1.06
,
3.38
,
1.70
,
0.92
)
(
-
0
.
24
,
-
0.86
,
-
0.34
,
-
0.12
)
(
-
0
.
14
,
1.34
,
-
0.18
,
-
0.08
)
(
0.31
,
0.19
,
0.43,0.16
)
(
-
0
.
62
,
-
0.38
,
-
0.86
,
-
0.32
)
Malaria
(
1.08
,
6.12
,
3.12,1.46
)
(
-
0
.
20
,
-
0.62
,
-
0.64
,
-
0.40
)
(
0.20
,
1.12
,
0.60,0.54
)
(
0.10
,
0.06
,
0.34,0.13
)
(
-
0
.
20
,
-
0.12
,
-
0.68
,
-
0.26
)
Typhoid
(
1.60,2.04
,
1.12
,
2.06
)
(
-
0
.
18
,
-
0.52
,
-
0.36
,
-
0.42
)
(
-
0
.
04,0.84
,
-
0.24
,
0.46
)
(
0.20,0.10
,
0.48
,
0.19
)
(
-
0
.
40
,
-
0.20
,
-
0.96
,
-
0.38
)
Stomach problem
(
5.28
,
0.28
,
4.76,3.42
)
(
-
0
.
42
,
0.02
,
0.38
,
-
0.16
)
(
0.56
,
-
0.16
,
-
1.36,0.02
)
(
0.14
,
0.06
,
0.30,0.15
)
(
-
0
.
28
,
-
0.12
,
-
0.60
,
-
0.30
)
Chest problem
(
6.44,3.96
,
6.00,4.54
)
(
-
0
.
36
,
-
0.46
,
0.28
,
-
0.44
)
(
0.44,0.64
,
-
1.24,0.50
)
(
0.14,0.14
,
0.34,0.19
)
(
-
0.28
,
-
0.28
,
-
0.68
,
-
0.38
)The distances given in Table5 are described as follows: for example, from the first data of viral fever and malaria, we can describe the distance measure between AI and viral fever, and the distance measure between AI and malariafrom (37), respectively. Similarly, we can get all the results (43), and obtain Table 5 from all (43):
(43)d(AI,Viralfever)=1.06-0.24λ0-0.14λ0λ1+0.31λ02-0.62λ02λ1(1-λ1),d(AI,Malaria)=1.08-0.2λ0+0.2λ0λ1+0.1λ02-0.2λ02λ1(1-λ1),d(AI,Typhoid)=1.6-0.18λ0-0.04λ0λ1+0.2λ02-0.4λ02λ1(1-λ1),d(AI,Stomachproblem)=5.28-0.42λ0+0.56λ0λ1+0.14λ02-0.28λ02λ1(1-λ1),d(AI,Chestproblem)=6.44-0.36λ0+0.44λ0λ1+0.14λ02-0.28λ02λ1(1-λ1),d(Bob,Viralfever)=3.38-0.86λ0+1.34λ0λ1+0.19λ02-0.38λ02λ1(1-λ1),d(Bob,Malaria)=6.12-0.62λ0+1.12λ0λ1+0.06λ02-0.12λ02λ1(1-λ1),d(Bob,Typhoid)=2.04-0.52λ0+0.84λ0λ1+0.1λ02-0.2λ02λ1(1-λ1),d(Bob,Stomachproblem)=0.28+0.02λ0-0.16λ0λ1+0.06λ02-0.12λ02λ1(1-λ1),d(Bob,Chestproblem)=3.96-0.46λ0+0.64λ0λ1+0.14λ02-0.28λ02λ1(1-λ1),d(Joe,Viralfever)=1.7-0.34λ0-0.18λ0λ1+0.43λ02-0.86λ02λ1(1-λ1),d(Joe,Malaria)=3.12-0.64λ0+0.6λ0λ1+0.34λ02-0.68λ02λ1(1-λ1),d(Joe,Typhoid)=1.12-0.36λ0-0.24λ0λ1+0.48λ02-0.96λ02λ1(1-λ1),d(Joe,Stomachproblem)=4.76+0.38λ0-1.36λ0λ1+0.3λ02-0.6λ02λ1(1-λ1),d(Joe,Chestproblem)=6+0.28λ0-1.24λ0λ1+0.34λ02-0.68λ02λ1(1-λ1),d(Ted,Viralfever)=0.92-0.12λ0-0.08λ0λ1+0.16λ02-0.32λ02λ1(1-λ1),d(Ted,Malaria)=1.46-0.4λ0+0.54λ0λ1+0.13λ02-0.26λ02λ1(1-λ1),d(Ted,Typhoid)=2.06-0.42λ0+0.46λ0λ1+0.19λ02-0.38λ02λ1(1-λ1),d(Ted,Stomachproblem)=3.42-0.16λ0+0.02λ0λ1+0.15λ02-0.3λ02λ1(1-λ1),d(Ted,Chestproblem)=4.54-0.44λ0+0.5λ0λ1+0.19λ02-0.38λ02λ1(1-λ1).
We aim to seek the minimum distance among d(AI,disease)s. Therefore, the minimum among all these d(AI,disease)s is the pattern we need. From Table 5, it is clear that if  λ0=0, AI, Bob, Joe, and Ted are identified as having viral fever, stomach problem, typhoid, and viral fever, respectively; if λ0>0, Bob, Joe, and Ted are also identified as having stomach problem, typhoid, and viral fever, respectively, which is the same as  Xu’s results in [20] and Wei’s results in [35], but AI is different from the above. When d(AI,viralfever)>d(AI,malaria), AI is identified as malaria; otherwise, AI is viral fever. All in all, we have the following results: if 0.02+0.04λ0+0.34λ0λ1-0.21λ02+0.42λ02λ1(1-λ1)<0, we can infer that AI suffers from malaria, and on the contrary, AI from viral fever. From Table 5, we conclude that AI suffers from either malaria or viral fever, Bob from stomach problem, Joe from typhoid, and Ted suffers from viral fever. The pattern recognition results are somewhat different from those of the dynamic intuitionistic fuzzy sets (DIFS) method in [26–28], in which we only proposed a DIFS method derived from IFS. However, in this paper, we obtain a series of DIVIFS methods derived from IFS and IVIFS, which are the further study in deep from [26–28].The experimental results above show that there is a slight difference between the recognition results of DIVIFS and those of IFS. Conventional IFS method is simple, but its recognition results are fixed on the basis of the conventional distance formulas. Therefore, it is difficult to reveal the potential law from all available information when the IFS method is used, whereas the results of the DIVIFS method are flexible, which can be adjusted to sample data with the variables. Furthermore, if the fixed pattern recognition results are different from those of the sample data, the IFS method will fail. However, when using the DIVIFS method, we can meet the requirements of the sample data by adjusting variables to appropriate values. All the results above show that the DIVIFS method is more comprehensive and flexible than the IFS method.


## 6. Simulation to Medical Diagnosis

In the following, we will compare the DIVIFS method with the conventional method in medical diagnosis. We use a sample of 77 women of embryos transferred, which is from Laboratory of Reproductive Medicine, The First Affiliated Hospital, Southern Medical University. And the set of disease diagnoses isD={Not successful pregnancy and no symptoms of OHSS, Successful pregnancy and no symptoms of OHSS, Not successful pregnancy and symptoms of OHSS, Successful pregnancy and symptoms of OHSS}, and the set of symptoms is a universe of discourse S={Basic Hormone FSH, Basic Hormone P, Basic Hormone E2, Basic Hormone T, BMI, Day of HCG E2, Day of HCG endometrial thickness}, where each symptom is described as follows: membership function μA(x), non-membership function νA(x), and hesitancy function πA(x).Applying the conventional distance measures on IFS and DIVIFS to medical diagnosis, we have the results in Table6. From Table 6, we conclude that the most accurate results of medical diagnosis are from Xu’s equation (15) when k=0.5, which is also only a special case of (35) when C1=C2=C3=C4=C6=C8=0 and C5=C7=1. And in 77 individuals, 74% of all the women can be classified correctly. However, using (38), the recognition rate will be 83%. Some results of medical diagnosis have been shown in Table 7.Table 6
The optimal results of medical diagnosis based on the distances of IFS and DIVIFS.
Equation
Right classification
Wrong classification
Correct recognition rate
Equation (15),k=0.5
57
20
74.026%
Equation (38), λ0≥0.6,λ1=0.5
64
13
83.117%Table 7
Comparison between classification of DIVIFS and existing classification calculated by similarity (S=1-D, where D denotes the standardized distance).
i
A
~

1
(
x
i
)  Not successful pregnancy and no symptoms of OHSS
A
~

2
(
x
i
)  Successful pregnancy and no symptoms of OHSS
A
~

3
(
x
i
)  Not successful pregnancy and symptoms of OHSS
A
~

4
(
x
i
)  Successful pregnancy and symptoms of OHSS
DIVIFSclassification results
D sets existing classification
1
0.572945016
0.490904912
0.245052718
0.258999945
1
2
2
0.903658468
0.866369075
0.500009121
0.575267057
1
1
3
0.532136793
0.668331297
0.402139818
0.437292429
2
2
4
0.569281981
0.351763437
0.369988299
0.317016112
1
1
5
0.75309088
0.866822163
0.523453221
0.565023619
2
2
6
0.603807663
0.620439622
0.585688802
0.585218663
2
2
7
0.833143565
0.629507422
0.278184995
0.343708098
1
1
8
0.692742485
0.978491415
0.654192881
0.398168922
2
2
9
0.742052675
0.592646858
0.289624287
0.326558445
1
1
10
0.62781314
0.365138462
0.485198477
0.283729817
1
2
⋮
⋮
⋮
⋮
⋮
⋮
⋮
70
0.367412382
0.590387248
0.692746959
0.62396661
3
3
71
0.479555791
0.630052699
0.795747193
0.654957234
3
3
72
0.318797905
0.290232035
0.376480055
0.324328504
3
3
73
0.418720261
0.451631428
0.77908381
0.227788025
3
3
74
0.29301287
0.321279436
0.545408461
0.366476514
3
3
75
0.385226266
0.413714892
0.658865834
0.669631158
4
4
76
0.545467491
0.539314756
0.760187187
0.786090732
4
3
77
0.383687857
0.392311966
0.742113723
0.49908387
3
3


## 7. Conclusion

We propose a method for the evaluation of a degree of agreement in a group of individuals by calculating distances between DIVIFS. By analyzing the variation on the degree of hesitancy, we introduce a novel DIVIFS method derived from IVIFS, which is an effective method for us to construct dynamic IVIFS model from IFS and IVIFS. And then we propose a novel ranking condition for the distance measures on IFS, IVIFS, and DIVIFS. According to the new condition, we show the construction of distance measures on DIVIFS in theory. Finally, we apply some distances on DIVIFS to medical diagnosis based on pattern recognition. The experimental results show that the DIVIFS method and its distance measures are more comprehensive and more effective than the conventional distance measures on IFS and IVIFS.



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*Source: 408012-2013-06-26.xml*