A new distance-based total uncertainty measure in the theory of belief function by Ioan Degău

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A new distance-based total uncertainty measure in the theory of belief functions Yi Yang, Deqiang Han PII: DOI: Reference:

S0950-7051(15)00457-8 10.1016/j.knosys.2015.11.014 KNOSYS 3343

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Knowledge-Based Systems

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27 July 2015 22 October 2015 20 November 2015

Please cite this article as: Yi Yang, Deqiang Han, A new distance-based total uncertainty measure in the theory of belief functions, Knowledge-Based Systems (2015), doi: 10.1016/j.knosys.2015.11.014

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Highlights • A new total uncertainty measure in evidence theory is proposed.

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• The new measure is directly defined in the evidential framework.

• The new measure is not a generalization of those in the probabilistic framework.

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• The belief intervals and distance metric are used for the new measure’s design.

• The new measure has no drawbacks in traditional ones and has desired

properties.

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Yi Yanga , Deqiang Hanb,∗ a

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A new distance-based total uncertainty measure in the theory of belief functions

SKLSVMS, School of Aerospace, Xi’an Jiaotong University, Xi’an, China 710049 Institute of Integrated Automation, Xi’an Jiaotong University, Xi’an, China 710049

Abstract

The theory of belief functions is a very important and effective tool for un-

certainty modeling and reasoning, where measures of uncertainty are very

crucial for evaluating the degree of uncertainty in a body of evidence. Several

uncertainty measures in the theory of belief functions have been proposed.

However, existing measures are generalizations of measures in the proba-

bilistic framework. The inconsistency between different frameworks causes

limitations to existing measures. To avoid these limitations, in this paper,

a new total uncertainty measure is proposed directly in the framework of

belief functions theory without changing the theoretical frameworks. The

average distance between the belief interval of each singleton and the most

uncertain case is used to represent the total uncertainty degree of the given

body of evidence. Numerical examples, simulations, applications and related

analyses are provided to verify the rationality of our new measure.

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Keywords: belief functions, evidence theory, uncertainty measure, belief

interval, distance of interval

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∗

Corresponding author: Deqiang Han Email addresses: jiafeiyy@mail.xjtu.edu.cn (Yi Yang), deqhan@gmail.com (Deqiang Han ) Preprint submitted to Knowledge-Based Systems

November 26, 2015

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1. Introduction Dempster-Shafer evidence theory (DST) [1], also called the theory of be-

lief functions, has been widely used in many applications related to uncer-

tainty modeling and reasoning, e.g., information fusion [2], pattern classifica-

tion [3, 4], clustering analysis [5], fault diagnosis [6], and multiple attribute

decision-making (MADM) [7, 8].

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In the theory of belief functions, there are two types of uncertainty in-

cluding the discord (or conflict or randomness) [9] and the non-specificity

[10], hence the ambiguity [11]. Various kinds of measures for these two types

of uncertainty and the total uncertainty including both two types were pro-

posed. The measures of discord include the discord measure [9], the strife [9],

the confusion [12], etc; the measures of non-specificity include Dubois and

Prade’s definition [10] generalized from the Hartley entropy [13] in the clas-

sical set theory, Yager’s definition [14], and Korner’s definition [15], etc. The

most representative total uncertainty measures are the aggregated measure

(AU ) [16] and the ambiguity measure (AM ) [11].

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In essential, no matter AU or AM , they are the generalization of Shan-

non entropy [17] in probability theory, but not the direct definition in the

framework of the theory of belief functions. That is, they pick up a probabil-

ity according to some criteria or constraints established based on the given

body of evidence (BOE), and then calculate the corresponding Shannon en-

tropy of the probability to indirectly depict the degree of uncertainty for the

given body of evidence. As mentioned in the related references [11][18], AU

and AM have their own limitations. For example, they are insensitive to

the change of BBA. These limitations to some extent are related to the in3

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consistency [19, 20] between the framework of the theory of belief functions

and that of the probability theory. Therefore, to avoid the limitations of

the traditional definitions for the uncertainty measure, in our work, a new

total uncertainty measure is proposed directly in the framework of the theory

of belief functions without the switching between different frameworks. We

analyze the belief interval and conclude that belief intervals carry both the

randomness part and the imprecision part (non-specificity) in the uncertainty

incorporated in a BOE. Thus, it is feasible to define a total uncertainty mea-

sure for a BOE. Given a BOE, the distance between the belief interval of each

singleton and the most uncertain interval [0, 1] is used for constructing the

degree of total uncertainty. The larger the average distance, the smaller the

degree of uncertainty. Since there is no switch of theoretical frameworks, our

new definition has desired properties including the ideal value range and the

monotonicity. Furthermore, the uncertainty measure can provide more ratio-

nal results when compared with the traditional ones, which can be supported

by experimental results and related analyses.

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The rest of this paper is organized as follows. Section 2 provides the brief

introduction of the theory of belief functions. Commonly used uncertainty

measures in the theory of belief functions are introduced in Section 3. Some

drawbacks of the available total uncertainty measures including AM and

AU are also pointed out in Section 3. In Section 4, a new total uncertainty

measure is proposed. Some desired properties and related analyses on the

new proposed definition are also provided. Experiments and simulations are

provided in Section 5 to show the rationality of our proposed total uncertainty

measure. An application of the new total uncertainty measure on feature

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evaluation is provided in Section 6. Section 7 concludes this work.

2. Basics of the theory of belief functions

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The basic conception in the theory of belief functions [1] is the frame

of discernment (FOD), whose elements are mutually exclusive and exhaus-

tive, representing what we concern. m : 2Θ → [0, 1] is called a basic belief

assignment (BBA) defined over an FOD Θ if it satisfies

A⊆Θ

m(A) = 1, m(∅) = 0

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(1)

When m(A) > 0, A is called a focal element. A BBA is also called a mass

function. The set of all the focal elements denoted by F and their corre-

sponding mass assignments constitute a body of evidence (BOE) (F, m). The belief function (Bel) and plausibility function ((P l)) are defined as

Bel(A) = 82

P l(A) =

B⊆A

A∩B6=∅

m(B)

(2)

m(B)

(3)

The belief function Bel(A) represents the justified specific support for the

focal element (or proposition) A, while the plausibility function P l(A) rep-

resents the potential specific support for A. The length of the belief interval

[Bel(A), P l(A)] is used to represent the degree of imprecision for A.

Two independent BBAs m1 (·) and m2 (·) can be combined using Demp-

ster’s rule of combination as [1]     0, AP= ∅ m1 (Ai )m2 (Bj ) m(A) = Ai ∩Bj =A  P , A 6= ∅  m1 (Ai )m2 (Bj )  1− Ai ∩Bj =∅

(4)

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There are still some other alternative combination rules. See [21] for details.

The theory of belief functions has been criticized for its validity [19, 20, 22,

23, 24, 25]. It is not a successful generalization of the probability theory, i.e.,

there exists inconsistency [20] between the framework of the theory of belief

functions and that of the probability theory.

3. Uncertainty measures in the theory of belief functions

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In the theory of belief functions, there are two types of uncertainty includ-

ing the discord (or the conflict or the randomness) and the non-specificity,

hence ambiguity [11].

3.1. Measure of discord in belief function

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Measures of discord are for describing the randomness (or discord or con-

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flict) in a BOE [11]. Available definitions are listed below. Although with

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various names, they are all for the discord part of the uncertainty in a BOE.

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1) Confusion measure [12]

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m(A) log2 (Bel(A))

(5)

m(A) log2 (P l(A))

(6)

A⊆Θ

2) Dissonance measure [14]

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Conf (m) = −

Diss(m) = −

A⊆Θ

3) Discord measure [26] Disc(m) = −

m(A) log2

A⊆Θ

|B − A| 1− m(B) |B| B⊆Θ

(7)

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4) Strife measure [9] Stri(m) = −

m(A) log2

A⊆Θ

|A − B| 1− m(B) |A| B⊆Θ

(8)

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Note that all these definitions can be considered as Shannon entropy-alike

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measures. More detailed information on these measures can be found in [9].

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3.2. Measures for non-specificity in belief function

Non-specificity [27, 15] means two or more alternatives are left unspecified

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and represents an imprecision degree. It only focuses on those focal elements

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with cardinality larger than one. Non-specificity is a special uncertainty

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type in the framework of belief functions theory when compared with the

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probabilistic framework. Some non-specificity measures [10, 14, 15] were

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proposed. The most commonly used non-specificity definition is [10]:

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N S(m) =

A⊆Θ

m(A) log2 |A|

(9)

It can be regarded as a generalized Hartley measure [13] from the classical

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set theory. When the BBA m(·) is a Bayesian BBA, i.e., it only has singleton

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focal elements, it reaches the minimum value 0. When BBA m(·) is a vacuous

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BBA, i.e., m(Θ) = 1, it reaches the maximum value log2 (|Θ|). This definition

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was proved to have the uniqueness by Ramer [28] and it satisfies all the

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requirements of the non-specificity measure.

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3.3. Measures for total uncertainty in belief functions theory 1) Aggregated Uncertainty (AU ) [16] P AU (m) = max − θ∈Θ pθ log2 pθ   pθ ∈ [0, 1], ∀θ ∈ Θ     P pθ = 1 s.t. θ∈Θ   P   ¯ ∀A ⊆ Θ  Bel(A) ≤ pθ ≤ 1 − Bel(A),

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θ∈A

(10)

In fact for AU , given a BBA, the probability with the maximum Shannon

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entropy under the constraints established using the given BBA is selected

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and its corresponding Shannon entropy value is defined as the value for AU .

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Therefore, it is also called as ”upper entropy” [29]. It is an aggregated total

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uncertainty (ATU) measure, which captures both non-specificity and discord.

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AU satisfies all the requirements [29] for uncertainty measure including prob-

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ability consistency, set consistency, value range, monotonicity, sub-additivity

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and additivity for the joint BBA in Cartesian space.

2) Ambiguity Measure (AM ) [11] X BetPm (θ) log2 (BetPm (θ)) AM (m) = − P

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(11)

θ∈Θ

m(B)/ |B| is the pignistic probability [30] of a

where BetPm (θ) =

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BBA. In fact AM uses the Shannon entropy of the pignistic probability of a

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given BBA to represent the uncertainty.

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3.4. Drawbacks of available total uncertainty measures for belief functions

θ∈B⊆Θ

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The traditional total uncertainty measures have their own drawbacks.

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AM cannot satisfy the sub-additivity (for joint BBA in Cartesian space)

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which has been pointed out by Klir [18]. Moreover, AM is criticized due to

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its logical non-monotonicity [29] as shown in Example 1. 8

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3.4.1. Example 1 Suppose that the FOD is {θ1 , θ2 , θ3 }. There are two BBAs as follows. m1 ({θ1 , θ2 }) = 1/3, m2 ({θ1 , θ2 , θ3 }) = 1/3,

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m1 ({θ1 , θ3 }) = 1/2, m2 ({θ1 , θ3 }) = 1/2,

Obviously, from the two BBAs, there exists Bel2 (A) 6 Bel1 (A),

m1 ({θ2 , θ3 }) = 1/6;

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m2 ({θ2 , θ3 }) = 1/6.

P l1 (A) 6 P l2 (A), ∀A ⊆ Θ.

That is to say, all the belief intervals of m1 can be contained by those

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corresponding belief intervals of m2 , which means that m2 has a higher level

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of uncertainty. We can calculate their corresponding AM and AU as follows.

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1.5546 = AM (m1 ) > AM (m2 ) = 1.5100; AU (m1 ) = AU (m2 ) = 1.5850 = log2 3.

We can see that the results of AM are counter-intuitive, because AM violates

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the monotonicity, i.e., AM decreases the total quantity of uncertainty in this

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example where a clear increment of uncertainty of m2 compared with that

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of m1 . Although AU does not bring out a decrease of the quantity of uncer-

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tainty, it generates two equal values, which is also counter-intuitive, because

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m2 should have higher level of uncertainty. We provide further analyses on

this example. The belief function and plausibility function of m1 and m2 are Bel1 ({θ1 }) = 0.0000, P l1 ({θ1 }) = 0.8333;

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Bel1 ({θ2 }) = 0.0000,

P l1 ({θ2 }) = 0.5000;

Bel1 ({θ1 , θ2 }) = 0.3333,

P l1 ({θ1 , θ2 }) = 1.0000;

Bel1 ({θ3 }) = 0.0000,

P l1 ({θ2 }) = 0.6667;

Bel1 ({θ1 , θ3 }) = 0.5000,

P l1 ({θ1 , θ3 }) = 1.0000;

Bel1 ({θ2 , θ3 }) = 0.1667,

P l1 ({θ1 , θ3 }) = 1.0000;

Bel1 (Θ) = 1.0000,

P l1 (Θ) = 1.0000; 9

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Bel2 ({θ1 }) = 0.0000,

P l2 ({θ1 }) = 0.8333;

Bel2 ({θ2 }) = 0.0000,

P l2 ({θ2 }) = 0.5000;

Bel2 ({θ3 }) = 0.0000,

P l2 ({θ1 , θ2 }) = 1.0000;

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Bel2 ({θ1 , θ2 }) = 0.0000,

P l2 ({θ2 }) = 0.6667;

Bel2 ({θ1 , θ3 }) = 0.5000,

P l2 ({θ1 , θ3 }) = 1.0000;

Bel2 ({θ2 , θ3 }) = 0.1667,

P l2 ({θ1 , θ3 }) = 1.0000;

Bel2 (Θ) = 1.0000,

P l2 (Θ) = 1.0000;

Since AU tries to find a probability maximizing the Shannon entropy,

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and the uniform p.m.f.(probability mass function) P (θ1 ) = P (θ2 ) = P (θ3 ) =

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1/3, which has the maximum Shannon entropy, satisfies all the constraints

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(shown in Eq. (10)) established using Bel1 , P l1 and those using Bel2 and P l2 .

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Therefore, both AU1 and AU2 reach the maximum value log2 3 = 1.5850.

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3.4.2. Example 2

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Suppose that the FOD Θ = {θ1 , θ2 }. A BBA over Θ is m({θ1 }) = a,

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m({θ2 }) = b, m({Θ}) = 1 − a − b, where a, b ∈ [0, 0.5]. Here, we calculate

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AU , AM values corresponding to different values of a and b. The change of

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AU and AM values with the change of a and b are illustrated in Figure 1.

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As shown in Figure 1, AM reaches its maximum when a = b, because when a = b (no matter a or b’s value is large or small), the pignistic probabil-

ity is uniformly distributed. This is counter-intuitive, because AM cannot

discern the effect of total set’s mass assignment values to the uncertainty degree. For example, m1 ({θ1 }) = m1 ({θ2 }) = 0.5 and m2 ({θ1 }) = m2 ({θ2 }) = 10

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Figure 1: The change of total uncertainty measures in Example 2.

0.25, m2 (Θ) = 0.5, there exists AM (m1 ) = AM (m2 ). Obviously, it is ir-

rational to say that m1 and m2 have the same degree of total uncertainty. AU never changes with the change of a and b. The value of AU is always

log2 2 = 1. The reason is analyzed as follows. The constraints for calculating AU in this example are

Bel({θ1 }) = a 6 P (θ1 ) 6 1 − b = P l({θ1 });

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Bel({θ2 }) = b 6 P (θ2 ) 6 1 − a = P l({θ2 }); Bel(Θ) = 1 − a − b 6 P (θ1 ) 6 1 = P l(Θ);

Since AU tries to find a p.m.f. with the maximum Shannon entropy, and

the uniformly distributed P (θ1 ) = P (θ2 ) = 0.5 always satisfies the constraints

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above (because a, b ∈ [0, 0.5]), no matter how a and b change, P (θ1 ) =

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P (θ2 ) = 0.5 is always picked up when calculating AU and thus, AU always

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equals to log2 2 = 1. However, intuitively, the degree of uncertainty should

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change with the change of a and b. So, AU cannot well describe the degree

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of total uncertainty here. As we can see above, both AM and AU borrow the uncertainty measure

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(Shannon entropy) in the probability theory framework and they both have

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drawbacks. However, as aforementioned, the belief functions theory is not a

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successful generalization of the probability theory. There exists inconsistency

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between the two frameworks. We do not prefer such a switch between differ-

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ent frameworks, which might cause problems in representing the uncertainty

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in belief functions. Therefore, in our work, we design the total uncertainty

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measure directly in the framework of belief functions theory as introduced in

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the next section.

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4. Belief interval distance-based total uncertainty measure

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4.1. Belief interval and uncertainty

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Case I: The most uncertain case is Bel(A) = 0 and P l(A) = 1, i.e., the belief interval is [0,1].

Case II: The clearest case is Bel(A) = 1 and P l(A) = 1 (A is assure to occur) or Bel(A) = 0 and P l(A) = 0 (A is assure to never occur). Case III: When Bel(A) = a, P l(A) = b, ∀a, b ∈ (0, 1), the belief interval

is [a, b]. For A, the degree of imprecision can be represented by b − a, the

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m can represent its uncertainty degree as analyzed below.

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The belief interval [1] [Bel(A, P l(A)] of a given focal elements A in a BBA

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probability of whether A occurs or not (randomness) is large than a and less

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than b.

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Case IV: When Bel(A) = P l(A) = a, ∀a ∈ (0, 1), the belief interval

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is [a, a]. It means that A has no imprecision and whether A occurs or not 12

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cannot be clearly determined (it is with the randomness, i.e., the probability

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a). Therefore, the information related to the uncertainty carried by a belief

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interval both include the randomness part and the imprecision part (non-

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specificity). Thus, given a BBA, we can fully utilize the information of the

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belief intervals to measure the total uncertainty of the BBA. Then, how to

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use the information of belief intervals?

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Given the belief interval of A, i.e., [Bel(A), P l(A)], if the belief interval is

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farther from the most uncertain case [0,1], then A has smaller uncertainty; if

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the belief interval is nearer to the most uncertain case [0,1], then A has larger

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uncertainty. Then, how to describe the distance between belief intervals?

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Here we use the distance between interval numbers [31, 32] as follows. Given two interval numbers [a1 , b1 ] and [a2 , b2 ], a strict distance is defined as s 2 2 a1 + b 1 a2 + b 2 1 b 1 − a1 b 2 − a2 I d ([a1 , b1 ], [a2 , b2 ]) = − − + 2 2 3 2 2 (12)

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The belief interval [Bel(A), P l(A)] can be considered as an interval num-

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ber. If we replace [a1 , b1 ] by [Bel(A), P l(A)] and [a2 , b2 ] by [0, 1], then,

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the distance obtained can be used to represent how far it is from A to

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the most uncertain case. As shown in Eq. (12), in dI , the proportion of

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[(Bel({θi }) + P l({θi })) /2 − 1]2 and [(P l({θi }) − Bel({θi })) /2 − 1]2 is 1/3.

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Generally speaking, an interval number is a kind of fuzzy data [31]. To define

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the distance between two fuzzy data, the value of such a proportion depends

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on the shape [31] (could be the symmetric triangular, normal, parabolic,

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square root fuzzy data, and interval number) of the fuzzy data model. Here,

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we use the belief intervals (interval numbers), therefore, there is a rational

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and natural assumption that the interval is with a uniform distribution. Un-

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der such an assumption, the proportion should be 1/3 and dI corresponds

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to Mallows’ distance between two distributions, hence a strict distance met-

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ric for interval numbers. By using such a strict distance metric for interval

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numbers, we can further define a total uncertainty measure for a BOE as

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follows.

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4.2. New total uncertainty measure based on belief intervals

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Suppose that m is a BBA over the FOD Θ = {θ1 , ..., θn }. First, calcu-

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late the belief interval [Bel({θi }), P l({θi )}] , i = 1, ..., n for each singleton

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θi .Then, calculate the distance between each [Bel({θi }), P l({θi )}] and [0, 1],

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which represents the degree of departure from the most uncertain case for

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each singleton. Since larger distance (larger departure from the most uncer-

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tain case) means smaller total uncertainty, one minus the normalized and

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averaged belief interval distances (for all singletons) is used as the total un-

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certainty measure for the BBA m as shown in Eq. (13):

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233

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(13)

√ 3 is the normalization factor 1 .

The newly proposed T U I has no drawbacks as those of AU and AM

pointed out in Examples 1 and 2. See the Examples 3 and 4 in Section 5 for

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where

n 1 √ X I · 3· d ([Bel({θi }), P l({θi })], [0, 1]) n i=1

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T U I (m) = 1 −

details. 1

The distance dI ([Bel({θi }), P l({θi })], [0, 1]) reaches its maximum value when the be-

lief interval is [0,0] or [1,1]. Therefore, the normalization factor is 1/dI ([0, 0], [0, 1]) = √ 1/dI ([1, 1], [0, 1]) = 3.

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Here, we provide an illustrative example to show how T U I is calculated.

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Suppose that FOD is Θ = {θ1 , θ2 , θ3 }. A BBA over Θ is m({θ1 }) =

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0.3, m({θ2 , θ3 }) = 0.5, m(Θ) = 0.2.

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First calculate the belief function and plausibility function for singletons: Bel({θ1 }) = m({θ1 }) = 0.3, Bel({θ2 }) = m({θ2 }) = 0.0, Bel({θ3 }) = m({θ3 }) = 0.0,

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P l({θ1 }) = m({θ1 }) + m(Θ) = 0.5,

P l({θ2 }) = m({θ2 , θ3 }) + m(Θ) = 0.7, P l({θ3 }) = m({θ2 , θ3 }) + m(Θ) = 0.7.

Then, calculate the distance between each belief interval of singleton and

the interval [0, 1]:

dI ([Bel({θ1 }), P l({θ1 })], [0, 1]) = dI ([0.3, 0.5], [0, 1]) q 2 1 0.5−0.3 1−0 2 0.3+0.5 = − 0+1 − 2 +3 = 0.2517; 2 2 2

dI ([Bel({θ2 }), P l({θ2 })], [0, 1]) = dI ([0, 0.7], [0, 1]) q 0+1 2 1 0.7−0 1−0 2 0+0.7 = − + − = 0.1732; 2 2 3 2 2

Then T U I = 1 −

√ 3/3 · (0.2517 + 0.1732 + 0.1732) = 0.6547.

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dI ([Bel({θ3 }), P l({θ3 })], [0, 1]) = dI ([0, 0.7], [0, 1]) q 2 1 0.7−0 1−0 2 0+0.7 = − 0+1 +3 2 − 2 = 0.1732; 2 2

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4.3. Desired properties of the new total uncertainty measure

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4.3.1. Range

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The range of T U I is [0, 1], which is desired for practical use. For a vacuous

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BBA m(Θ) = 1, all the belief intervals for singletons are [0, 1]. Therefore, by 15

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using Eq. (13), its degree of total uncertainty is 1, which is the maximum.

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For a categorical BBA m(θi ) = 1, the belief interval for θi is [1, 1] and those

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of other singletons are [0, 0]. Therefore, by using Eq. (13), its degree of total

247

uncertainty is 0, which is the minimum.

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Proof of the range: For a vacuous BBA, when using Eq. (13), T U I = 1.

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On the other hand, if T U I = 1, i.e., all the belief intervals for singletons are

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[0, 1], the corresponding BBA must be a vacuous one. So the maximum value

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is unique. If a BBA is not a vacuous one, then some singletons’ belief intervals

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can be strictly contained by [0, 1]. Therefore, according to Eq. (13), the

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corresponding T U I will be smaller than 1. Therefore, the unique maximum

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value of T U I is 1.

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For a categorical BBA m(θi ) = 1, when using Eq. (13), T U I = 0. On

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the other hand, if T U I = 0, the only possible belief interval for singletons

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are either [0, 0] or [1, 1]. For a BBA, one and only one belief interval for

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singleton could be [1, 1]. So, the corresponding BBA should be m({θi }) =

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1, m({θj }) = 0, ∀j 6= i, which represents the clearest case. If a BBA is not

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m({θi }) = 1, m({θj }) = 0, ∀j 6= i, some singletons’ belief intervals are neither

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[0, 0] nor [1, 1]. Therefore, according to Eq. (13), the corresponding T U I will

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be larger than 0. So, the minimum value of T U I is 0.

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End of Proof

4.3.2. Monotonicity

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Given an uncertainty measure U N and two BBAs m1 and m2 , if ∀A ∈ P(Θ) : P l1 (A) 6 P l2 (A), Bel1 (A) > Bel2 (A) or ∀A ∈ P(Θ) : [Bel1 (A), P l1 (A)] ⊆ [Bel2 (A), P l2 (A)] 16

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there exists U N (m1 ) 6 U N (m2 ), then it is said that U N satisfies the mono-

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tonicity [29]. The physical meaning of the monotonicity is that an uncertainty

267

measure in the theory of belief functions must not decrease the total quan-

268

tity of uncertainty in situations where there is a clear decrease in information

269

(increment of uncertainty).

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Proof of the monotonicity: For two BBAs defined over the same FOD,

271

if it holds that ∀A ∈ P(Θ) : [Bel1 (A), P l1 (A)] ⊆ [Bel2 (A), P l2 (A)], then,

272

there exists ∀θi ∈ Θ : [Bel1 ({θi }), P l1 ({θi })] ⊆ [Bel2 ({θi }), P l2 ({θi })]. Ac-

273

cording to the property of the distance of interval numbers in Eq. (12), there

274

exists dI ([Bel1 ({θi }), P l1 ({θi })], [0, 1]) > dI ([Bel2 ({θi }), P l2 ({θi })], [0, 1]).

277

278

279

280

holds. End of Proof 4.4. On other properties

276

Then, according to Eq. (13), T U I (m1 ) 6 T U I (m2 ), i.e., the monotonicity

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It is declared in [29, 11] that AU and AM satisfy the probability consistency and set consistency.

Probability consistency: When m is a Bayesian BBA (all focal ele-

ments are singletons), AU reduces to Shannon entropy in probability theory: AU (m) = AM (m) = −

m({θ})log2 (m({θ}))

θ∈Θ

Set consistency: When m focuses on a single set (i.e., m(A) = 1, ∀A ⊆

Θ), AU reduces to a Hartley measure in classical theory: AU (m) = AM (m) = log2 (|A|)

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It should be noted that it is meaningless to apply these two requirements

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or properties to our new measure. First, our new T U I is designed directly

283

in the framework of belief functions theory, but not a generalization of the

284

measures in probability framework or those in classical set theory. Second,

285

the theory of belief functions is not a successful generalization of the prob-

286

ability theory. It cannot degenerate back to the probability framework. For

287

example, a Bayesian BBA is not a probability but still a special BBA, which

288

cannot satisfy the additivity for mutually exclusive events and cannot satisfy

289

the unity for the total set in Kolmogorov probability axioms [33]. Therefore,

290

the validity and strictness of such a consistency is doubtful.

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Furthermore, AU also satisfies the subadditivity and additivity, which

292

defined for the joint BBA in Cartesian space. In our work, we do not con-

293

cern the joint BBA in Cartesian space, therefore, we will not analyze the

294

subadditivity and additivity in this paper. We will further verify our new total uncertainty measure by examples and

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291

simulations in the next section.

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5. Experiments and simulations

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5.1. Example 3

This Example 3 is a revisiting of Example 1. Using Eq. (13), we can

obtain that T U I (m1 ) = 0.6667 and T U I (m2 ) = 0.7778, i.e., m2 has a higher

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299

296

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level of uncertainty, which is intuitive. It can be proved that T U I satisfy the

302

monotonicity. The proof can be found at Section 4.3.2.

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5.2. Example 4 This Example 4 is a revisiting of Example 2. Our new total uncertainty

305

measure T U I , non-specificity measure N S, dissonance measure Diss are also

306

calculated. The results are shown in Figure 2.

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TU I

0.95 0.9 0.85 0 0.5

0.8 0.5

0.5 a

0 0

0.8

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0.6

0.5

0 0

Diss

0.4 0.5

0.5 b

1 0.8

0.6

0.5

0 0.5

0 0

0 0.5

0.5 a

0 0

0.2 0

0.5

0.4

Figure 2: The change of total uncertainty measures in Example 4.

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AU never changes as analyzed in Example 2, which is counter-intuitive.

309

AM values never changes, so far as a = b holds, which is also counter-intuitive

310

as aforementioned in Example 2. As we can see in Figure 2, T U I brings out 19

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rational results. It reaches the maximum when a = b = 0, i.e., m(Θ) = 1.

312

When a + b is fixed, the maximum values are reached if a = b holds. For

313

example, suppose that a + b = 0.3. The T U I values are shown in Figure 3.

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This makes sense, because m(Θ) = 1 − (a + b) is fixed, the non-specificity

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part is fixed. When a = b, the conflict part reaches its maximum value.

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0.9

TU I

0.8

X: 0.15 Y: 0.15 Z: 0.85

0.7 0.6

0.5 0.6

0.4

a+b=0.3

0.2

0.1

0.3

0.4

0.5

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Figure 3: T U I for those with fixed a + b

Note that N S reaches 0 (minimum), when a = b = 0.5 (m(Θ) = 0); it reaches

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1 (maximum), when a = b = 0 (m(Θ) = 1). Diss reaches 1 (maximum),

319

when a = b = 0.5; it reaches 0 ( minimum), when a = 0 or b = 0 (no conflict).

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5.3. Example 5 Suppose that the FOD is Θ= {θ1 , θ2 ,θ3 } and a BBA over Θ is m(A) =

322

1, ∀A = Θ. We make changes to the BBA step by step. In each step, m(Θ)

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has a decrease of ∆ = 0.05 and each singleton mass m({θi }), i = 1, 2, 3 has an

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increase of ∆/3. Finally, m(Θ) reaches 0 and m({θi }) = 1/3. We calculate

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total uncertainty measures including T U I , AM , AU , non-specificity N S, and

326

dissonance (Diss) at each step. The changes of uncertainty values are shown

327

in Figure 4.

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Change of Uncertainty values 1.6

NS AU AM

1 0.8 0.6

0.4

TU I Diss

1.2

Uncertainty values

1.4

0.2

0 0

Steps

Figure 4: The change of total uncertainty measures in Example 5.

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As we can see in Figure 4, with the change of the BBA in each step, the

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non-specificity changes from the maximum to zero. This is intuitive, because 21

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the mass assignments are transferred from the focal elements with larger

332

cardinality to those with smaller cardinality. The measure for conflict, i.e.,

333

Diss here, becomes larger and reaches the maximum in the final. This is

334

also intuitive, because the BBA changes from a vacuous BBA (no conflict)

335

to a Bayesian BBA (with uniform distribution on all singletons) in the fi-

336

nal. The traditional total uncertainty measures AU and AM never change

337

and they are always at the maximum value. This is counter-intuitive be-

338

cause the total uncertainty degree of a vacuous BBA should be the largest,

339

and thus, it should be larger than that of a Bayesian BBA (even if, it is

340

a Bayesian BBA with uniform distribution on all singletons). The reasons

341

for the irrational results of AU and AM are analyzed as follows. They are

342

both defined based on some probabilistic transformation from BBAs. In this

343

example, for the probabilistic transformation used in AM and AU , the re-

344

sults are always a uniformly distributed probability mass function (p.m.f.):

345

P (Î¸i ) = 1/3, i = 1, 2, 3, therefore, AM and AU will never change here. Our

346

new measure T U I provides rational results. It becomes smaller and smaller

347

when the BBA changes from a vacuous one to a Bayesian one as shown in

348

Figure 2. Therefore, according to our opinion, it is inappropriate to define

349

total uncertainty measure in the theory of belief functions by indirectly using

350

the uncertainty measure in probability framework, i.e., Shannon entropy. It

351

should be better not to switch the framework but to directly design in the

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framework of belief functions theory. This is also the motivation of our work

353

in this paper.

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5.4. Example 6 The FOD is Θ = {θ1 , θ2 , θ3 }. A BBA over Θ is m(A) = 1, ∀A = Θ. We

356

make changes to the BBA step by step. In each step, m(Θ) has a decrease

357

of ∆ = 0.05 and mass assignment of one singleton m({θ1 }) has an increase

358

of ∆ = 0.05. In the final step, m(Θ) = 0 and m({θ1 }) = 1. We calculate

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total uncertainty measures inlcuding T U I , AM , AU , non-specificity measure

360

N S, and dissonance (Diss) of the BBA at each step. The changes of these

361

uncertainty values are shown in Figure 5.

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Change of Uncertainty values

1.6

NS AU AM

1.4

0.8

0.4

0.2

Uncertainty values

0.6

TU Diss

1.2

−0.2 0

Steps

Figure 5: The change of total uncertainty measures in Example 6.

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The BBA changes from a vacuous one to a categorical one. As shown in

364

Figure 5, non-specificity N S becomes smaller and smaller, which is intuitive. 23

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The conflict part Diss is always 0, which is also intuitive, because at most

366

two focal elements are available for the BBA in each step (including {θ1 } and

Θ), therefore, there is no conflict. AU , AM , T U I all decrease intuitively.

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However, AU is not sensitive to the BBA’s change at the beginning stage

369

(nearly unchanged). This has already been criticized by Jousselme [11].

370

5.5. Example 7

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Examples 7-9 are used for reference from [11]. Suppose that the FOD is

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Θ = {θ1 , ..., θ8 }. A BBA over Θ is m(A) = 1, ∀A = Θ. We make changes

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to the BBA step by step. In each step, m(Θ) has a decrease of ∆ = 0.05,

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and the mass assignment for one focal element B with cardinality s < 8 has

375

an increase of ∆ = 0.05. In the final step, m(B) = 1. Here, we can set

376

s = 2, 4, 6, respectively. Given an s value, we repeat the whole procedure

377

of BBA change. Under the different s, the changes of values for uncertainty

378

measures including T U I , AM , AU , N S, and Diss are shown in Figure 6. 3

|B| = s = 2

3 2.5

1.5

0.5

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|B| = s = 4

2.5

Uncertainty values

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0 0

Steps

0 0

|B| = s = 6

Diss NS AU AM TU I

Figure 6: The change of total uncertainty measures in Example 7.

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As shown in Figure 6, when the mass assignments are transferred to a smaller

381

size focal element, the non-specificity N S intuitively decreases. The total

382

uncertainty AU , AM , T U I all decrease. If s has a smaller value, the total

383

uncertainty and the non-specificity will decrease faster, because the mass

384

assignments are transferred to a smaller size focal element. Comparatively,

385

AU is not sensitive to the change of the BBA. The conflict part (Diss) is

386

always 0. The reason is analyzed as follows. At the beginning, the BBA is a

387

vacuous one; in the middle steps, the BBA has two focal elements including

388

the total sets; in the final step, it has only one focal element. Therefore,

389

there is no conflict.

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5.6. Example 8

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Suppose that the FOD is Θ = {θ1 , ..., θ10 }. A BBA over Θ is m(Θ) = 1−a,

and m(A) = a, |A| = s 6 10. Given an a, and the initial |A| = 1, we make

393

changes to the BBA step by step. |A| increases by 1 in each step. In the

394

final, the BBA becomes a vacuous one. We set a = 0.3, 0.5, 0.8, respectively.

395

Given an a value, we repeat the whole procedure of BBA change. Under the

396

different a, the changes of values for uncertainty measures including T U I ,

397

AM , AU , N S, and Diss are shown in Figure 7.

As shown in Figure 7, non-specificity N S, total uncertainty measures in-

398

392

cluding AU , AM , and our T U I all intuitively increase with the change of

400

BBA at each step. If a has a larger value, they will increase faster, because

401

relatively more mass assignments are transferred to a larger size focal ele-

402

ment. Comparatively, AU is not sensitive to the BBA’s change. The conflict

403

part (Diss) is zero and never changes. The reason is analyzed as follows.

404

Before the final step, the BBA has two focal elements including the total

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sets; in the final step, it has only one focal element. Therefore, there is no

406

conflict. 3.5

a=0.5

3.5

2.5

1.5

0.5

0 0

a=0.8

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a=0.3

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Uncertainty values

3.5

Diss NS AU AM TU I

0.5

Steps

0 0

Figure 7: The change of total uncertainty measures in Example 8.

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5.7. Example 9

Suppose that the size of an FOD is |Θ| = 5. Randomly generate 10 BBAs

with k, 1 6 k 6 25 − 1 focal elements using the algorithm [11] in Table 1.

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407

In this simulation, the number of focal elements is set to 15. Those

412

generated BBAs are then combined one by one using Dempster’s rule of

413

combination in Eq. (4). At each instant t of the evidence combination, the

414

values of N S, AU , AM , Diss, and T U I are calculated. The whole procedure

411

415

is repeated for 100 times and the averaging values of these different uncer-

416

tainty measures at different t are shown in Figure 8. It is shown that all

417

the uncertainty measures, except for Diss, decrease with the increase of the

418

combination steps. This makes sense, because it is intuitive that the total 26

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Table 1: Algorithm 1: Random generation of BBA

Input: Θ: FOD; Nmax : Maximum number of focal elements;

Generate the power set of Θ: P(Θ);

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Output: m: a BBA

Generate a random permutation of P(Θ) → R(Θ); Generate a integer between 1 and Nmax → k; FOReach First k elements of R(Θ) do

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Generate a value within [0, 1] → mv(i), i = 1, ..., k; END

Normalize the vector mv = [mv(1), ..., mv(2)] → mv 0 m(Aj ) = mv 0 (j), j = 1, ..., k.

uncertainty decreases in the information fusion procedure such as the evide-

419

420

421

nce combination. The non-specificity measure drops faster and more sig-

423

nificantly than the total uncertainty measures. This is because that in the

424

evidence combination based on Dempster’s rule, the focal elements are split

425

into focal elements with smaller cardinality. Also, since the focal elements

426

are split into smaller size focal elements, the conflict in the BBA will sig-

422

427

nificantly increase at the beginning steps. At the final steps, since there is

428

less further split of focal elements and the consensus effect in combination

429

becomes more dominant, the conflict in the BBA decreases slightly as shown

430

in Figure 8. 27

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Change of Uncertainty values

2.5

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Uncertainty values

1.5

NS AU AM

TU I Diss

0 0

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0.5

Steps

Figure 8: The change of total uncertainty measures in Example 8.

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6. Application of the new total uncertainty measure Here we use our new total uncertainty measure in feature evaluation for

431

pattern classification to further show its rationality. Three classes of samples are artificially generated. There are 200 samples

435

in each class. Each sample has four dimensions. In each class, each dimen-

436

sion of samples is Gaussian distributed with different mean and standard

437

deviation (Std for short) as shown in Figure 9 and Table 2.

438

434

As shown in Figure 9 and Table 2, since the three Gaussian probability

density functions (PDFs) in feature 2 are quite well separated, the class

440

discriminability of feature 2 is the best; the class discriminability of feature

441

4 is the worst (totally overlapped), and that of the feature 3 is also very bad.

442

The class discriminability of feature 1 is in the middle. This can also be

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Figure 9: Probability density functions (PDFs) of different features of the samples in three

classes.

Table 2: Gaussian distribution parameters of the samples.

Features

Feature 1

Feature 2

Feature 3

Feature 4

Class1

Class2 Class3

Mean

5.5

Std

1.2

Mean

Std

1.2

Mean

4.8

5.2

Std

1.2

Mean

Std

1.2

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verified by using the discrimination criterion [34] as follows.

tr(Sw ) tr(Sb )

(14)

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444

where tr denotes the trace of a matrix. Suppose that there are C classes and

445

each class Ci has Ni samples. The degree of inner-class cohesion Sw and the degree of inter-class separability Sb are as follows [34] "  T # C P P P    P (Ci )E X − N1i X X − N1i X  Sw = i=1 X∈Ci X∈Ci T C  P P P  1 1  X −M X −M P (Ci ) Ni  Sb = Ni i=1

447

X∈Ci

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(15)

X∈Ci

where X is a feature (vector) of a sample and

C 1 X M= C i=1

1 X X Ni X∈C i

(16)

is the mean of all the classes’ centroids. If J of some feature (or set of

449

features) in Eq. (14) is smaller, then such a feature (or set of features) is

450

more crisp and discriminable.

448

For the four dimensions of the artificially generated samples, there exists

451

J(1) = 0.2666, J(2) = 0.1336, J(3) = 488.9156, J(4) = 981.3525 It means that in terms of discriminability, feature 2 is the best; feature 4

is the worst; feature 3 is also very bad, and feature 1 is in the middle, which

453

is accordant to the judgement based on the Gaussian Parameters.

452

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Now, we use different total uncertainty measures (including AM, AU, and

455

our new T U I ) for the feature evaluation. Intuitively, the feature with smaller

456

total uncertainty measure should be better (higher discriminability). 30

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458

First, we generate BBA from each sample xq in the three-class artificial samples on different feature i ∈ {1, 2, 3} according to [35] mixq (Aj )

−2/(β−1)

|Aj |−α/(β−1) dqj

Ak 6=∅

459

−2/(β−1)

|Ak |−α/(β−1) dqk

(17)

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+ δ −2/(β−1)

where parameters α = 1, β = 2 as suggested in [34]. Here we give an

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illustrative example as in Figure 10

460 461

In Figure 10, three different colors represent three different classes. c1

denotes the centroid of samples in class 1; c1,2 denotes the centroid of samples

462

Figure 10: Illustration of BBA generation.

463

in class 1 and class 2; c1,2,3 denotes the centroid of samples in classes 1, 2,

464

and 3. Calculate the distance d between xq and those centroids of single

465

classes and compound classes. Then according to Eq. (17), the BBA can be

466

generated. 31

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Second, we calculate AU, AM, and our proposed T U I for all the generated

468

BBAs (corresponding to each sample on different features). Then, calculate

469

the average values of AU, AM and T U I for different features. The results

470

are shown in Table 3.

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Table 3: Average total uncertainty measures

Feature1

Featue 2

Feature 3

Feature 4

1.4022

1.1968

1.5850

1.1855

0.9860

1.5844

1.5847

TU I

0.4496

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Measures (Ave)

0.3787

0.6218

0.6221

We consider the average total uncertainty measures as scores for different

472

features. The feature with smaller score is better (with high discriminability)

473

As shown in Table 3 the feature evaluation based on the average T U I

474

values is accordant to the results obtained using discrimination criterion.

475

That is, feature 2 is the best; feature 4 is the worst; feature 3 is also very

476

bad, and feature 1 is in the middle.

471

AM can also provide the correct feature evaluation result. Note that

478

feature 3 is more ambiguous than feature 4. The difference of average T U I

479

for features 3 and 4 is 0.0003; The difference of average AM for features 3

480

and 4 is also 0.0003. However, the value range of T U I is [0, 1], while the value

481

range of AM is [0, log2 3]. Therefore, our proposed T U I is more sensitive

482

when measuring the ambiguity in different features, which agrees with the

483

analysis in above examples.

477

484

485

The average AU values for both features 3 and 4 are equal. They are both the maximum value log2 3 = 1.5850. The reason is analyzed below. 32

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According to the BBA generation approach used, because in features 3 and 4, all the centroids (7 centroids, 3 singleton classes, and 4 compound

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classes) are very close, all the distances between xq and those centroids are almost equal when generating BBAs for xq on features 3 and 4. Therefore, each focal element of the BBA generated for xq on features 3 and 4 has the

mass value approximating to 1/7. According to all the BBAs generated, for feature 3, m(A) = (1/7) ± 0.010; for feature 4, m(A) = (1/7) ± 0.007, A ⊆

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{c1 , c2 , c3 }. According to Eq. (10), since AU is calculated based on the maximization of Shannon entropy given the constraints established using belief functions as

1 4 ≈ m({c1 }) ≤ P (c1 ) ≤ m({c1 }) + m({c1 , c2 }) + m({c1 , c3 }) + m({c1 , c2 , c3 }) ≈ 7 7} | {z } | {z Bel({c1 })

P l({c1 })

1 4 ≈ m({c2 }) ≤ P (c2 ) ≤ m({c2 }) + m({c1 , c2 }) + m({c2 , c3 }) + m({c1 , c2 , c3 }) ≈ 7 7} | {z } | {z Bel({c2 })

P l({c2 })

Bel({c3 })

4 1 ≈ m({c3 }) ≤ P (c3 ) ≤ m({c3 }) + m({c1 , c3 }) + m({c2 , c3 }) + m({c1 , c2 , c3 }) ≈ 7 7} | {z } {z | P l({c3 })

P (c1 ) = P (c2 ) = P (c3 ) = 1/3, which corresponds to the maximum Shannon

487

entropy, always satisfies the constraints, so, the AU value is fixed to log2 3 =

488

1.5850. Based on AU, we cannot judge which one is better for features 3 and

489

In summary, our proposed T U I can be well used in feature evaluation for

pattern classification, and it performs better than AM and AU.

491

490

486

492

7. Conclusions

493

In this paper, a new total uncertainty measure based on the distance

494

of belief intervals is proposed in the framework of belief functions theory. 33

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There is no switch between frameworks in the new measure. It is based on a

496

new perspective that for a body of evidence, the farther it is from the most

497

uncertain case, the less uncertainty it carries. It has been experimentally

498

shown that our new measure can rationally represent the total uncertainty

499

in BOEs. The new measure also has some desired properties and behaves

500

more rational than traditional measures like AU and AM in some cases as

501

shown in our experiments and simulations. Furthermore, our new measure

502

can provide more rational results in practical applications such as the feature

503

evaluation.

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Till now, the theory of belief functions is still not so solid and needs fur-

505

ther refining. The available works mainly focus on the uncertainty modeling

506

and reasoning in belief functions theory and there have emerged many valu-

507

able results. However, the performance evaluation in belief functions theory

508

is far from mature, which has become its bottle neck for the further develop-

509

ment. In future work, we will focus on performance evaluation in the theory

510

of belief functions and try to use our new total uncertainty measure to im-

511

plement the theoretical evaluation and the application-based evaluation in

512

the theory of belief functions.

513

Acknowledgements

This work was supported by the Grant for State Key Program for Ba-

514

504

515

sic Research of China (973) (No. 2013CB329405), National Natural Science

516

Foundation (Nos. 61203222, 61573275), Foundation for Innovative Research

517

Groups of the National Natural Science Foundation of China (No. 61221063),

518

Science & technology project of Shaanxi Province (No.2013KJXX-46), Spe34

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cialized Research Fund for the Doctoral Program of Higher Education (No.

520

20120201120036), and Fundamental Research Funds for the Central Univer-

521

sities (No. xjj2014122).

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