A New Uncertainty Measure in Belief Entropy Framework

Belief entropy, which represents the uncertainty measure between several pieces of evidence in the DempsterShafer framework, is attracting increasing interest in research. It has been used in many applications and is mainly based on the theory of evidence. To quantify uncertainty, several measures have been proposed in the literature. These measures, sometimes in extended or hybrid forms, use the Shannon entropy principle to determine uncertainty degree. However, the failure to consider the scale of the frame of discernment framework remains an open issue in quantifying uncertainty. In this paper, we propose a new uncertainty measure that takes into account the power set of the frame of discernment. After analysing the different existing methods, we show the performance and effectiveness of our proposed approach. Keywords—Dempster Shafer Theory; Belief entropy; Uncertainty; Information management; Deng entropy

In this work, we are mainly interested in Deng entropy and modified Deng entropy.The Deng entropy, a very effective measure, compared to various measures in some cases, has been used in several fields of application [38], [37].However, one of the main limitations of this uncertainty measure is related to not considering the scale of F OD. To address this limit, some authors, including Zhou et al., have proposed a modified Deng measure [30].Nevertheless, although effective in some cases, the problem related to the scale of F OD is still perceptible.In this paper, we propose a new uncertainty measure by extending the modified Deng entropy.This new measure improves the performance of the measures proposed by Deng and Zhou takes into consideration the power set of the F OD.After analysing the different existing methods, we show the performance and effectiveness of our proposed approach.
The rest of this document is organized as follows : section 2 provides a brief overview of Dempster-Shafer theory and Shannon entropy.Section 3 presents some uncertainty measures in the Dempster-Shafer framework and and some limitations.Section 4 describe the new uncertainty measure.Section 5 presents, using numerical examples, the effectiveness of the new measure.The conclusion and some perspectives related to this work are presented in Section 6.

A. Dempster-Shafer Theory
Dempster-Shafer Theory [10], [11], also known as belief theory or evidence theory, has many advantages for processing uncertain information.We present some basic concepts related to this theory.

1) Formalism:
Let Ω = {ω 1 , . . ., ω N } a set of N mutually exclusive and exhaustive events.Ω represents the frame of discernement F OD.A mass function is defined on the power set of Ω, noted 2 Ω with : In Ω, a mass function assigns to each subset a value between 0 and 1 representing its elementary belief mass defined by : Such as : When m(A) > 0, A is called a focal element.Belief is the amount of trust that supports a hypothesis A of the power set 2 Ω .It is most often called the Body Of Evidence BOE or Basic Probability Assignment BP A or Basic Belief Assignment BBA [39] and is characterized by all the focal elements and their associated mass value (Eq.4): where ℵ represents a subset of the power set 2 Ω and each proposition A ∈ ℵ are focal elements.A BOE can also be represented by its associated belief Bel and plausibility P l functions defined as follows : 2) Combination rules: In Dempster-Shafer theory, two independent mass functions, noted m 1 and m 2 , can be combined with the Dempster's combination rule [40], defined as follows: where k represents the degree of conflict between m 1 and m 2 .k is defined as follows:

B. Shannon entropy
In information theory, Shannon entropy (E s ) [5] is used to measure the volume of information in a system, process or message.This measure determines the expected value of the information contained in a message.The measure is defined as follows : where n represents the quantity or number of basic states, p i represents the probability of the state i with is the basis of the logarithm, it most often takes the value 2.

III. RELATED WORKS
In this section, we present some uncertainty measures in the Dempster-Shafer framework.In these measures, X represents F OD, A and B are the focal elements.| A | refers to the cardinality of A.

B. Problem formulation
In Demspter-Shafer theory, uncertain information should not only be modelled by mass functions, F OD is also a source of uncertainty [38].This paper recall this problem by using the Zhou et al.'s [30] example.Some measures such as Deng entropy (E d ) and Modified Deng entropy (E z ) are calculated.
Example 3.2: Consider two BOEs m 1 and m 2 , representing respectively results of two reliable sensors in a target identification problem as follows : Deng entropy is calculated as follows : Despite the difference in F ODs (i.e.X 1 = {a, b, c, d} et X 2 = {a, b, c}), The results obtained from the Deng measure about the BOE m 1 are similar to the uncertainty measure of BOE m 2 .Intuitively, the uncertainty measure of BOE m 1 should be bigger than that of BOE m 2 .This, because the F OD related to the BOE m 1 contains more elements than the F OD related to the BOE m 2 .At this main limitation, some authors, notably Zhou et al. [30] have proposed the modified Deng entropy.Modified Deng entropy is calculated as follows : As can be seen, the modified Deng entropy gives almost different results for each BOE.In this measure, we can see that the degree of uncertainty calculated from the different BOEs is reduced compared to the Deng measure.However, although the modified Deng entropy takes into account the Deng entropy [37] number of elements in the F OD, this measure often gives counter-intuitive measures.We present another problem in example 3.
Deng entropy is calculated as follows : As can be seen, example 3.3 defines two BOEs m 3 and m 4 where BOE m 3 represents a case of total uncertainty.Intuitively, the uncertainty level of the BOE m 3 must be bigger than that of the BOE m 4 , which is in contradiction with the modified Deng measure (E z ).However, the Deng measure (E d ) better distinguishes total uncertainty with an uncertainty measure of BOE m 3 bigger than that of BOE m 4 .
Thus, after analyzing examples 3.2 and 3.3, it can be seen that example 3.2 presents a case of variable F OD with the same number of elements in the focal elements.And example 3.3 presents a case where the F OD does not vary.In contrast, in example 3.

IV. NEW UNCERTAINTY MEASURE
In the Dempster-Shafer framework, the new uncertainty measure (E N m ) we propose is as follows: where m is the mass function defined on X.A is the focal element of X and | A | represents the cardinality of A.The particularity of this measure is that it takes into account the number of elements of the power set represented by 2 |X| .
A simple transformation of the new entropy is as follows: As can be seen in this expression, the first two terms refer to Deng entropy [37].These are respectively the measure of the total non-specificity in the mass function m, and the measure of the discord of the mass function between focal elements.The third term, the exponential factor, e 3 ) The results of the different uncertainty measures of the two (BOEs) m 1 and m 2 are summarized in the table II.II, provides the distance d(m 1 , m 2 ) between the measures of BOEs m 1 and m 2 to determine the observed information loss between these BOEs.Finally, the BOE m1 measure is added to the calculated distance.In this case, the new measure compared to the Zhou et al. measure, takes into account the loss of perceived information in the Deng measure.
2.5559 2.3154 2.5559 Thus, using the example 3.3, the new measure of the different BOEs m 3 and m 4 is as follows : The results of the (BOEs) m 3 and m 4 are summarized in the table IV.In this table, the new entropy (E N m ) is also represented.And as can be seen, only the entropy proposed by Zhou et al. gives counter-intuitive results.In this case of example where the F OD does not vary, the new measure is close to the merits of the Deng measure.
Thus, the proposed new measure responds to the limitations of Deng and Zhou et al. by taking a more generic character and an efficient quantification of uncertainty.

V. PROOF AND DISCUSSIONS
In this section, we first present some fundamental properties of the new uncertainty measure.Then, using numerical examples, we show the concordance of the new entropy with some basic entropy including Shannon's entropy (E s ), Deng entropy (E d ) and the entropy proposed by Zhou et al. (E z ).Finally, we discuss the superiority of the new entropy compared to the above-mentioned entropies.

A. Concordance with Shanon entropy
The proposed entropy measure is identical to the basic entropy, the Shannon entropy (Eq.17 In addition, another proof of concordance of the new entropy with different uncertainty measures is provided in the case where the (F OD) has only one element (ı.e. total uncertainty case) as shown in the following example.
Example 5.1 : Consider an information processing system in which information I reported by a sensor has a belief equal to one hundred percent.In X = {I}, the mass function can be noted as m({I}) = 1.The calculation of the entropies of Shannon (E s ), Deng (E d ), Zhou et al (E z ) and the new entropy measure (E N m ) are defined as follows:

B. Superiority of the new uncertainty measure
To show the superiority of the new uncertainty measure, recall the example mentioned in [37].As can be seen in Figure 1, the new and modified entropy of Deng proposed by Zhou et al. increase monotonously with increasing size of the subset E. However, the values of the new entropy are significantly bigger than that of Zhou et al. measure.As shown in example 3.3.the measure proposed by Zhou et al. records losses of information especially in such a case where F OD that does not vary.Moreover, the new entropy gives results almost identical to the Deng measurement (figure 2), hence the effectiveness of the approach when the F OD does not change.
Moreover, the new entropy gives results almost identical to the Deng measure (figure 2), hence the effectiveness of the new approach when the F OD does not change.The new measure does not differ from the merits of the Deng measure.George & Pal's conflict measure [36].In this figure, we can observe that only the entropies of Dubois & Prade [32], Deng [37], Zhou [30], and the new entropy increase monotonously with the increase in the size of subset E. Also with the increase in size of E, there is either a declination or a change in the pace of other uncertainty measures.Hence the effectiveness of the new measure.

VI. CONCLUSION
Quantifying uncertainty in information systems is very important for evaluating the quality of information.Several methods based on entropy of beliefs have been proposed in the literature, but these give counter-intuitive results, particularly in the case of variable F OD with BOE.In this paper, we have proposed a new measure to address these deficiencies.This measure extends the measures proposed by Deng and Zhou et al.From numerical examples and mathematical properties, we have shown the effectiveness of the new measure which gives more information in the power set of the F OD. Our future studies will focus on the actual application of the news in several areas including decision making, fault diagnosis and detection, and so on.

3 . 3 . 3 :
Example Consider the BOEs m 3 and m 4 in the F OD X = {a, b} as follows :m 3 : m 3 ({a, b}) = 1 m 4 : m 4 ({a}) = m 4 ({b}) = 0.5Modified Deng entropy is calculated as follows : 3, the measure proposed by Deng has better results compared to the Zhou et al. results.Therefore, how to quantify optimally uncertainty by taking into account the limits observed in the measures proposed by Deng and Zhou et al.? To solve this problem, we propose a new uncertainty measure by extending the measures proposed by Deng and Zhou et al.

|A|− 1 2
|X| , is the main factor in this contribution.The choice of this factor is based on the exponential factor (i.e. e |A|−1 |X| ) proposed by Zhou et al. [30], which represents the measure of uncertain information in a BOE.Compared to the Zhou et al. measure, the new measure takes into account the number of elements in the power set represented by 2 |X| .Thus, the new measure is intended to be more generic in resolving the limitations of the measures proposed by Deng and Zhou et al. in examples 3.2 and 3.3 respectively.Let's go back to example 3.2, the new entropy is as follows: ), when we have a Bayesian mass function (i.e. a single element in the BOE or | A |≡ 1) as follows.

TABLE II
In this table, like Zhou et al.'s proposed measure, the new measure gives different measures for each of BOEs m 1 and m 2 .However, the new measure gives bigger values compared to those of Zhou et al.Table III takes the measures from table