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A correlation coefficient of belief functions

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  • 湖南科技大学

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How to manage conflict is still an open issue in Dempster–Shafer (D-S) evidence theory. The conflict coefficient k in D-S evidence theory cannot represent conflict reasonably, especially sometimes two Basic Probability Assignments (BPAs) are identical but k is not zero. Jousselme distance can well measure the similarity or conflict between two BPAs but it becomes invalid in some cases. Some scholars introduced the concept of correlation coefficient to measure the similarity of two BPAs. However, existing correlation coefficients of BPAs are unstable or insensitive to quantify the conflict and sometimes deduce wrong results. In this paper, a novel correlation coefficient is proposed which takes into consideration both the non-intersection and the difference among the focal elements. It satisfies all the requirements for a metric. Some numerical examples and comparisons of this paper demonstrate the effectiveness of the correlation coefficient.
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arXiv:1612.05497v2 [cs.AI] 2 Feb 2017
A correlation coefficient of belief functions
Wen Jianga,
aSchool of Electronics and Information, Northwestern Polytechnical University, Xi’an,
Shannxi, 710072, China
Abstract
How to manage conflict is still an open issue in Dempster-Shafer evidence theory.
The correlation coefficient can be used to measure the similarity of evidence
in Dempster-Shafer evidence theory. However, existing correlation coefficients
of belief functions have some shortcomings. In this paper, a new correlation
coefficient is proposed with many desirable properties. One of its applications is
to measure the conflict degree among belief functions. Some numerical examples
and comparisons demonstrate the effectiveness of the correlation coefficient.
Keywords: Dempster-Shafer evidence theory; belief function; correlation
coefficient; conflict; conflict management; similarity
1. Introduction
Dempster-Shafer evidence theory (D-S theory) [1, 2] is widely used in many
real applications[3–8] due to its advantages in handling uncertain informa-
tion, since decision-relevant information is often uncertain in real systems[9–
11]. However, in D-S theory, the results with Dempster’s combination rule are
counterintuitive[12] when the given evidence highly conflict with each other.
Until now, how to manage conflict is an open issue in D-S theory. In recent
years, hundreds of methods have been proposed to address this issue[6, 13–22].
These solutions are generally divided into two categories: one is to modify the
Corresponding author: School of Electronics and Information, Northwestern Polytechnical
University, Xi’an, Shannxi, 710072, China. Tel:+86 029 88431267; fax:+86 029 88431267. E-
mail address: jiangwen@nwpu.edu.cn; jiangwenpaper@hotmail.com
Preprint submitted to Elsevier February 3, 2017
combination rule and redistribute the conflict; the other is to modify the data
before combination and keep the combination rule unchanged.
Obviously, how to measure the degree of conflict between the evidences is
the first step, since we need to know whether the evidence to be combined is
in conflict before doing anything in conflict management[23–25]. So far there
are no general mechanisms to measure the degree of conflict other than the
classical conflict coefficient k. But since the classical conflict coefficient kis
the mass of the combined belief assigned to the emptyset and ignores the dif-
ference between the focal elements, using kto indicate a conflict between the
evidences may be incorrect. Several conflict measures, such as Jousselme’s ev-
idence distance[26], Liu’s two-dimensional conflict model[27], Song et al.’s con-
flict measurement based on correlation coefficient[28], have been proposed to
measure the conflict in D-S theory. Although some improvements have been
made, there are still some shortcomings in the existing conflict measure meth-
ods. How to measure the the degree of conflict between the evidences is not yet
solved. In D-S theory, the conflict simultaneously contains the non-intersection
and the difference among the focal elements[27]. Only when these two factors
are considered simultaneously, the effective measure of conflict can be realized.
In this paper, we try to measure the conflict from the perspective of the rele-
vance of the evidence, based on simultaneously considering the non-intersection
and the difference among the focal elements.
A correlation coefficient can quantify some types of correlation relationship
between two or more random variables or observed data values. In D-S theory,
a correlation coefficient is usually used to measure the similarity or relevance
of evidence, which can be applied in conflict management, evidence reliability
analysis, classification, etc[28, 29]. Recently, various types of correlation coeffi-
cient are presented. For example, in [30], a correlation coefficient was introduced
to calculate the similarity of template data and detected data, then the basic
probability assignments (BPAs) were obtained based on classification results for
fault diagnosis. In [29], a correlation coefficient is proposed based on the fuzzy
nearness to characterize the divergence degree between two basic probability
2
assignments (BPAs). In [28], Song et al. defined a correlation coefficient to
measure the conflict degree of evidences. Moreover, some different correlation
coefficients are proposed respectively according to specific applications [30–32].
In this paper, a new correlation coefficient, which takes into consider both the
non-intersection and the difference among the focal elements, is proposed. One
of its applications is to measure the conflict degree among belief functions.
Based on this, a new conflict coefficient is defined. Some numerical examples
illustrate that the proposed correlation coefficient could effectively measure the
conflict degree among belief functions.
The paper is organised as follows. In section 2, the preliminaries D-S the-
ory and the existing conflicting measurement are briefly introduced. Section 3
presents the new correlation coefficient and proofs many desirable properties.
In section 4, some numerical examples are illustrated to show the efficiency of
the proposed coefficient. Finally, a brief conclusion is made in Section 5.
2. Preliminaries
In this section, some preliminaries are briefly introduced.
2.1. Dempster-Shafer evidence theory
D-S theory was introduced by Dempster [1], then developed by Shafer [2].
Owing to its outstanding performance in uncertainty model and process, this
theory is widely used in many fields [33–35].
Definition 2.1. Let Θ = {θ1, θ2,···θi,···, θN}be a finite nonempty set of
mutually exclusive hypothesises, called a discernment frame. The power set of
Θ,2Θ, is indicated as:
2Θ={∅,{θ1},···{θN},{θ1, θ2},···,{θ1, θ2,···θi},···,Θ}(1)
Definition 2.2. A mass function is a mapping mfrom 2Θto [0,1], formally
noted by:
m: 2Θ[0,1] (2)
3
which satisfies the following condition:
m() = 0 and P
A2Θ
m(A) = 1 (3)
When m(A)>0, A is called a focal element of the mass function.
In D-S theory, a mass function is also called a basic probability assignment
(BPA). Given a piece of evidence with a belief between [0,1], noted by m(·), is
assigned to the subset of Θ. The value of 0 means no belief in a hypothesis,
while the value of 1 means a total belief. And a value between [0,1] indicates
partial belief.
Definition 2.3. Evidence combination in D-S theory is noted as . Assume
that there are two BPAs indicated by m1and m2, the evidence combination of
the two BPAs with Dempster’s combination rule [1] is formulated as follows:
m(A) =
0,
1
1kP
BC=A
m1(B)m2(C)
A=
A6=(4)
with
k=X
BC=
m1(B)m2(C) (5)
Where kis a normalization constant, called conflict coefficient because it mea-
sures the degree of conflict between m1and m2.
k= 0 corresponds to the absence of conflict between m1and m2, whereas
k= 1 implies complete contradiction between m1and m2. Note that the Demp-
ster’s rule of combination is only applicable to such two BPAs which satisfy the
condition k < 1.
2.2. Evidence distance
Jousselme et al.[26] proposed a distance measure for evidence.
Definition 2.4. Let m1and m2be two BPAs on the same frame of discernment
Θ, containing N mutually exclusive and exhaustive hypotheses. The distance
4
between m1and m2is represented by:
dBBA(m1, m2) = r1
2(
m1
m2)TD
=(
m1
m2) (6)
Where
m1and
m2are the respective BPAs in vector notation, and D
=is an
2N×2Nmatrix whose elements are D(A, B) = |AB|
|AB|, where A, B 2Θare
derived from m1and m2, respectively.
2.3. Pignistic probability distance
In the transferable belief model(TBM) [36], pignistic probabilities are typi-
cally used to make decisions and pignistic probability distance can be used to
measure the difference between two bodies of evidence.
Definition 2.5. Let mbe a BPA on the frame of discernment Θ. Its associated
pignistic probability transformation (PPT) BetPmis defined as
BetPm(ω) = X
A2ΘA
1
|A|
m(A)
1m()(7)
where |A|is the cardinality of subset A.
The PPT process transforms basic probability assignments to probability
distributions. Therefore, the pignistic betting distance can be easily obtained
using the PPT.
Definition 2.6. Let m1and m2be two BPAs on the same frame of discernment
Θand let BetPm1and BetPm2be the results of two pignistic transformations
from them respectively. Then the pignistic probability distance between BetPm1
and BetPm2is defined as
difBetP = max
A2Θ(|BetPm1(A)BetPm2(A)|) (8)
2.4. Liu’s conflict model
In [27], Liu noted that the classical conflict coefficient kcannot effectively
measure the degree of conflict between two bodies of evidence. A two-dimensional
conflict model is proposed by Liu [27], in which the pignistic betting distance
and the conflict coefficient kare united to represent the degree of conflict.
5
Definition 2.7. Let m1and m2be two BPAs on the same frame of discernment
Θ. The two-dimensional conflict model is represented by:
cf(m1, m2) = hk , dif BetP i(9)
Where kis the classical conflict coefficient of Dempster combination rule in Eq.
(5), and dif BetP is the pignistic betting distance in Eq. (8). Iff both k > ε and
difBetP > ε,m1and m2are defined as in conflict, where εis the threshold of
conflict tolerance.
Liu’s conflict model simultaneously considers two parameters to realize con-
flict management. To some extent, the two-dimensional conflict model could
effectively discriminate the degree of conflict. But in most cases, an accurate
value, which represents the degree of conflict, is needed for the following process,
such as evaluate the reliability of the evidence with assigning different weights.
2.5. Correlation coefficient of evidence
In [28], Song et al. proposed a correlation coefficient for the relativity be-
tween two BPAs, which can be used to measure the conflict between two BPAs.
Definition 2.8. Let m1and m2be two BPAs on the same frame of discernment
Θ, containing N mutually exclusive and exhaustive hypotheses. Use the Jaccard
matrix D, defined in Eq. (6), to modify the BPA:
m
1=m1D
m
2=m2D
(10)
Then the correlation coefficient between two bodies of evidence is defined as:
cor(m1, m2) = hm
1, m
2i
km
1k · km
2k(11)
where hm
1, m
2iis the inner product of vectors, km
1kis the norm of vector.
Song et al.’s correlation coefficient measures the degree of relevance between
two bodies of evidence: the higher the conflict is, the lower the value of the
correlation coefficient is. In Song et al.’s correlation coefficient, the Jaccard
6
matrix Dis used to modify BPA in order to process BPA including the multi-
element subsets. But this modification will repeatedly allocate the belief value of
BPA , so the modified BPA does not satisfy the condition P
A2Θ
m(A) = 1. Thus,
Song et al.’s correlation coefficient could not satisfy the property cor(m1, m2) =
1m1=m2and sometimes will yield incorrect results.
3. A new correlation coefficient
The nature of conflict between two BPAs is there exists the difference be-
tween the beliefs of two bodies of evidence on the same focal elements, so the
conflict could be quantified by the relevance between two bodies of evidence. If
the value of the relevance between two bodies of evidence is higher, the degree
of the similarity between two bodies of evidence is higher and the degree of
conflict between two bodies of evidence is lower; Conversely, if the value of the
relevance between two bodies of evidence is lower, the degree of the similarity
between two bodies of evidence is lower and the degree of conflict between two
bodies of evidence is higher.
In order to measure the degree of relevance between two bodies of evidence,
a new correlation coefficient, which considers both the non-intersection and
the difference among the focal elements, is proposed. Firstly, some desirable
properties for correlation coefficient are shown as follows.
Definition 3.1. Assume m1, m2are two BPAs on the same discernment frame
Θ,rBP A (m1, m2)is denoted as a correlation coefficient for two BPAs, then
1. rBP A (m1, m2) = rBP A (m2, m1);
2. 0 rB P A(m1, m2)1;
3. if m1=m2,rBP A (m1, m2) = 1;
4. rBP A (m1, m2) = 0 (SAi)T(SAj) = ,Ai, Ajis the focal element of
m1, m2, respectively.
In D-S theory, a new correlation coefficient is defined as follows.
7
Definition 3.2. For a discernment frame Θwith Nelements, suppose the mass
of two pieces of evidence denoted by m1,m2. A correlation coefficient is defined
as:
rBP A (m1, m2) = c(m1, m2)
pc(m1, m1)·c(m2, m2)(12)
Where c(m1, m2)is the degree of correlation denoted as:
c(m1, m2) =
2N
X
i=1
2N
X
j=1
m1(Ai)m2(Aj)|AiAj|
|AiAj|(13)
Where i, j = 1, . . . , 2N;Ai,Ajis the focal elements of mass, respectively; and |·|
is the cardinality of a subset.
The correlation coefficient rBP A (m1, m2) measures the relevance between
m1and m2. The larger the correlation coefficient, the high the relevance between
m1and m2.rBP A = 0 corresponds to the absence of relevance between m1and
m2, whereas rBP A = 1 implies m1and m2complete relevant, that is, m1and
m2are identical.
In the following, the mathematical proofs are given to illustrate that the
proposed correlation coefficient satisfies all desirable properties defined in Defi-
nition 3.1. Before the proofs, a lemma is introduced as follows:
Lemma 3.1. For the vector 2-norm, non-zero vector ξ= (ξ1, ξ2,...,ξn)T,
η= (η1, η2,...,ηn)T, the condition for the equality in triangle inequality ||ξ+
η||2≤ ||ξ||2+||η||2is if and only if ξ=.
Proof.
||ξ+η||2
2(||ξ||2+||η||2)2
(ξ1+η1)2+ (ξ2+η2)2+...+ (ξn+ηn)2(qξ2
1+ξ2
2+...+ξ2
n+qη2
1+η2
2+...+η2
n)2
ξ1η1+ξ2η2+...+ξnηnq(ξ2
1+ξ2
2+...+ξ2
n)(η2
1+η2
2+...+η2
n)
From the Cauchy-Buniakowsky-Schwarz Inequality, we can see that the condition
of the equality is if and only if ξ1
η1=ξ2
η2=...=ξn
ηn=k, namely ξ=.
The proofs are details as follows:
8
Proof. Sort the 2Nsubsets in Θas {∅, a, b, . . . , N , ab, ac, . . . , abc, . . .}, then
m1, m2are arranged in column vectors in this order,
m1:x= (m1(), m1(a), m1(b),...,m1(N), m1(ab), m1(ac),...)T,
m2:y= (m2(), m2(a), m2(b),...,m2(N), m2(ab), m2(ac),...)T.
Let D=|AiAj|
|AiAj|, where Ai, Aj2Θand the subsets are arranged in the
same order as described above. Dis positive definite so CR2N×2N
2Nand
satisfies D=CTC. Then it is obvious that c(m1, m1) = xTDx =xTCTCx,
and similarly c(m1, m2) = xTCTCy,c(m2, m2) = yTCTCy.rBP A (m1, m2) =
xTCTCy
xTCTCxyTCTC y .
1. rBP A (m1, m2) = xTCTCy
xTCTCxyTCTC y ,rBP A(m2, m1) = yTCTC x
xTCTCxyTCTC y .
Because xTCTCy is a real number, xTCTCy = (xTCTCy)T=yTCTCx.
Thereby, rBPA (m1, m2) = rBP A (m2, m1).
2. All the elements in x, y, D are non-negative real numbers, so it is clear
that rBP A (m1, m2) = xTDy
xTDxyTDy 0.
Note that xTCTCx = (C x)T(Cx) = ||C x||2
2, then using the trigonometric
inequality on the vector 2-norm for vector Cx, Cy, the following inequal-
ities are obtained.
||C(x+y)||2
2(||Cx||2+||C y||2)2
(x+y)TCTC(x+y)(xTCTCx +pyTCTCy)2
xTCTCx +xTCTC y +yTCTCx +yTCTCy xTCTCx +yTCTCy + 2pxTCTCxyTCTCy
xTCTCy pxTCTC xyTCTCy (xTCTCy =yTCTC x)
Accordingly,
rBP A (m1, m2) = xTCTCy
xTCTCxpyTCTC y 1.
3. rBP A (m1, m2) = 1, and now xTCTCy =pxTCTCxyTCTC y, that is
||C(x+y)||2=||Cx||2+||C y||2. We can get Cx =kCy from Lemma 3.1.
Cis an invertible matrix, so x=ky. The vectors xand yeach represent
a BPA, their length are both 1, thus x=y,m1=m2.
9
4. If rBP A (m1, m2) = c(m1,m2)
c(m1,m1)×c(m2,m2)= 0, then
c(m1, m2) =
2N
X
i=1
2N
X
j=1
m1(Ai)m2(Aj)|AiAj|
|AiAj|= 0.
The above formula shows if m1(Ai)m2(Aj)6= 0, namely Ai, Ajare the
focal elements of m1, m2, respectively, |AiAj|= 0 must occur. In other
words, Ai∩ ∀Aj=when Ai, Ajare the focal elements of m1, m2,
respectively. Thereby, (SAi)T(SAj) = and vice versa.
In summary, the new correlation coefficient satisfies all the desirable prop-
erties and could measure the relevance between two bodies of evidence. Based
on this, a new conflict coefficient is proposed.
Definition 3.3. For a discernment frame Θwith Nelements, suppose the mass
of two pieces of evidence denoted by m1,m2. A new conflict coefficient between
two bodies of evidence kris defined with the proposed correlation coefficient as:
kr(m1, m2) = 1 rBP A (m1, m2)
= 1 c(m1,m2)
c(m1,m1)·c(m2,m2)
(14)
The conflict coefficient kr(m1, m2) measures the degree of conflict between
m1and m2. The larger the conflict coefficient, the high the degree of conflict
between m1and m2.kr= 0 corresponds to the absence of conflict between
m1and m2, that is, m1and m2are identical, whereas kr= 1 implies complete
contradiction between m1and m2.
4. Numerical examples
In this section, we use some numerical examples to demonstrate the effec-
tiveness of the proposed conflict coefficient.
Example 1. Suppose the discernment frame is Θ = {A1, A2, A3, A4}, two
bodies of evidence are defined as following:
m1:m1(A1, A2) = 0.9, m1(A3) = 0.1, m1(A4) = 0.0
m2:m2(A1, A2) = 0.0, m2(A3) = 0.1, m2(A4) = 0.9
10
The various conflict measure value are calculated as follows:
The classical conflict coefficient [1] k= 0.99.
Jousselme’s evidence distance [26] dB P A = 0.9.
Song et al.’s correlation coefficient [28] cor = 0.3668.
The proposed correlation coefficient rBP A = 0.0122.
The proposed conflict coefficient kr= 1 rBP A = 0.9878.
In this example, the first BPA is almost certain that the true hypothesis is
either A1or A2, whilst the second BPA is almost certain that the true hypothesis
is A4. Hence these two BPAs largely contradict with each other, that is, the
two BPAs are in high conflict. The results of the classical conflict coefficient k,
Jousselme’s evidence distance dB P A and the proposed conflict coefficient krare
all consistent with the fact, but Song et al.’s correlation coefficient cor = 0.3668,
which means there are a relatively great correlation between these two BPAs.
The value of Song et al.’s correlation coefficient is unreasonable.
Let us discuss the condition when two BPAs totally contradict. In Example
1, if we revise the two BPAs as:
m1:m
1(A1, A2) = 1.0, m
1(A3) = 0.0, m
1(A4) = 0.0
m2:m
2(A1, A2) = 0.0, m
2(A3) = 0.0, m
2(A4) = 1.0
then these two BPAs totally contradict with each other. This is the situation
when the maximal conflict occurs. The results are shown as follows:
The classical conflict coefficient [1] k= 1.0.
Jousselme’s evidence distance [26] dB P A = 1.0.
Song et al.’s correlation coefficient [28] cor = 0.3229.
The proposed correlation coefficient rBP A = 0.0.
The proposed conflict coefficient kr= 1 rBP A = 1.0.
In this case, the values of k,dBP A , and Krare all equal to 1.0, which
indicates that these two BPAs are in total conflict, whilst cor = 0.3229, which
means there are some correlation between the two BPAs. The value of Song et
al.’s correlation coefficient is unreasonable.
Let us continue to discuss the following two pairs of BPAs.
11
Example 2. Suppose the discernment frame is Θ = {A1, A2, A3, A4, A5, A6},
two pairs of BPAs are defined as following:
m1:m1(A1) = 0.5, m1(A2) = 0.5, m1(A3) = 0.0, m1(A4) = 0.0
m2:m2(A1) = 0.0, m2(A2) = 0.0, m2(A3) = 0.5, m2(A4) = 0.5
m3:m3(A1) = 1/3, m3(A2) = 1/3, m3(A3) = 1/3, m3(A4) = 0.0, m3(A5) = 0.0, m3(A6) = 0.0
m4:m4(A1) = 0.0, m4(A2) = 0.0, m4(A3) = 0.0, m4(A4) = 1/3, m4(A5) = 1/3, m4(A6) = 1/3
The summary of k,dBP A ,cor, and krvalues of the two pairs of BPAs is
shown in Table 1.
Obviously, these two pairs of BPAs both totally contradict with each other.
k, and krvalues are equal to 1.0, which is consistent with the fact, whilst dBP A ,
and cor values show that there is some similarity or relevance between the two
BPAs, that is, these two BPAs are not in total conflict, especially cor = 0.5606
for m3, m4shows that there is high relevance between m3and m4.dBP A , and
cor values are unreasonable. In summary, Jousselme’s evidence distance and
Song et al.’s correlation coefficient could not always give us the correct conflict
measurement.
Table 1: Comparison of k,dBP A ,cor , and krvalues of the two pairs of BPAs in Example 2
BPAs kr= 1 rB P A k dBP A cor
m1,m21.0 1.0 0.7071 0.3990
m3,m41.0 1.0 0.5774 0.5606
On the other hand, the total absence of conflict occurs when two BPAs
are identical. In this situation, whatever supported by one BPA is equally
supported by the other BPA and there is no slightest difference in their beliefs.
The following example is two identical BPAs.
Example 3. Suppose the discernment frame is Θ = {A1, A2, A3, A4, A5},
two bodies of evidence are defined as following:
m1:m1(A1) = 0.2, m1(A2) = 0.2, m1(A3) = 0.2, m1(A4) = 0.2, m1(A5) = 0.2
m2:m2(A1) = 0.2, m2(A2) = 0.2, m2(A3) = 0.2, m2(A4) = 0.2, m2(A5) = 0.2
12
In this case, kr= 0, dB P A = 0, and cor = 1 all indicates that the two BPAs are
identical, which are consistent with the fact. But the classical conflict coefficient
k= 0.8, which indicates that these two BPAs are in high conflict. Obviously,
the value of k= 0.8 is incorrect. With regard to this example, using kas a
quantitative measure of conflict is not always suitable.
According the above examples, it can be concluded that the proposed cor-
relation coefficient and the proposed conflict coefficient always give the correct
quantitative measure of relevance and conflict between two bodies of evidence.
In order to further verify the effectiveness of the proposed method, consider the
following example:
Example 4. Suppose the number of elements in the discernment frame is
20, such as Θ = {1,2,3,4, ..., 20}, two bodies of evidence are defined as following:
m1:m1(2,3,4) = 0.05, m1(7) = 0.05, m1) = 0.1, m1(A) = 0.8
m2:m2(1,2,3,4,5) = 1 where the A is a variable set taking values as follow:
{1},{1,2} {1,2,3},{1,2,3,4},. . . ,{1,2,3,4,. . . ,19,2 0}. In terms of conflict analysis,
the comparative behavior of the two BPAs are shown in Table 2 and Fig. 1.
From which we can find the proposed conflict coefficient kradopts the similar
behavior as Jousselme’s evidence distance, that is, when set A tends to the
set {1,2,3,4,5}, both the values of krand dB P A tend to their minimum. On
the contrary, the two values will increase when the set A departs from the set
{1,2,3,4,5}. Fig. 1 shows that the trends of kr, and dBP A value are consistent
with intuitive analysis, when the size of set A changes, whilst the classical
conflict coefficient kfails to differentiate the changes of evidence. So both the
proposed conflict coefficient and Jousselme’s evidence distance are appropriate
to measure the conflict degree of evidence in this example.
However, the major drawback of dB P A is its inability to full consider the
non-intersection among the focal elements of BPAs. In Example 2, Jousselme’s
evidence distance can not give us the correct conflict measurement. As to the
classical conflict coefficient k, it only takes into consider the non-intersection of
the focal elements, but not the difference among the focal elements. Because
of the lack of information, kis not sufficient as the quantitative measure of
13
conflict between two BPAs. Compared with Jousselme’s evidence distance and
the classical conflict coefficient, the proposed correlation coefficient takes into
consider both the non-intersection and the difference among the focal elements
of BPAs. Therefore, to some extent, the proposed correlation coefficient combine
the classical conflict coefficient and Jousselme’s evidence distance, and overcome
their respective demerits. Thus, the proposed method can measure the degree
of relevance and conflict between belief functions correctly and effectively.
Table 2: Comparison of kr,dBP A , and kvalues in Example 4
Pkr= 1 rB P A dB P A k
{1}0.7348 0.7858 0.05
{1,2}0.5483 0.6866 0.05
{1,2,3}0.3690 0.5705 0.05
{1,2,3,4}0.1964 0.4237 0.05
{1,2,3,4,5}0.0094 0.1323 0.05
{1,2,. . . ,6}0.1639 0.3884 0.05
{1,2,. . . ,7}0.2808 0.5029 0.05
{1,2,. . . ,8}0.3637 0.5705 0.05
{1,2,. . . ,9}0.4288 0.6187 0.05
{1,2,. . . ,10}0.4770 0.6554 0.05
{1,2,. . . ,11}0.5202 0.6844 0.05
{1,2,. . . ,12}0.5565 0.7082 0.05
{1,2,. . . ,13}0.5872 0.7281 0.05
{1,2,. . . ,14}0.6137 0.7451 0.05
{1,2,. . . ,15}0.6367 0.7599 0.05
{1,2,. . . ,16}0.6569 0.7730 0.05
{1,2,. . . ,17}0.6748 0.7846 0.05
{1,2,. . . ,18}0.6907 0.7951 0.05
{1,2,. . . ,19}0.7050 0.8046 0.05
{1,2,. . . ,20}0.7178 0.8133 0.05
14
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
kr
dBPA
k
Figure 1: Comparison of correlation degree
5. Conclusions
In D-S theory, it is necessary to measure the conflicts of belief functions.
A correlation coefficient provides a promising way to address the issue. In
this paper, a new correlation coefficient of belief functions is presented. It
can overcome the drawbacks of the existing methods. Numerical examples in
conflicting management are illustrated to show the efficiency of the proposed
correlation coefficient of belief functions.
Acknowledgment
The work is partially supported by National Natural Science Foundation of
China (Grant No. 61671384), Natural Science Basic Research Plan in Shaanxi
Province of China (Program No. 2016JM6018), the Fund of Shanghai Aerospace
Science and Technology (Program No. SAST2016083).
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19
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... Since the reliability of m 1 is relative, in this example, we use m * and m 2 (where m 2 (x 2 , x 3 ) = 1) to determine the upper and lower bounds of m 1 . For comparative analysis, we employ Liu's dissimilarity measurement [31], Jiang's correlation coefficient [32], Lefevre's adapted conflict [45], and Jousselme et al.'s distance [44] to compare with the proposed reliability calculation method. The specific results are shown in Fig. 3. From Fig. 3, with the uniform increase of η, the reliability of m 1 also steadily increases, and the evidence source, after the RPS transformation, exhibits an approximately linear change in reliability. ...
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