INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL ISSN 1841-9836, 12(5), 631-644, October 2017. An Uncertainty Measure for Interval-valued Evidences W. Jiang, S. Wang Wen Jiang*, Shiyu Wang School of Electronics and Information, Northwestern Polytechnical University Xi’an, Shaanxi Province, 710072, China *Corresponding author: jiangwen@nwpu.edu.cn jiangwenpaper@hotmail.com, wangshiyu@mail.nwpu.edu.cn Abstract: Interval-valued belief structure (IBS), as an extension of single-valued belief structures in Dempster-Shafer evidence theory, is gradually applied in many fields. An IBS assigns belief degrees to interval numbers rather than precise numbers, thereby it can handle more complex uncertain information. However, how to measure the uncertainty of an IBS is still an open issue. In this paper, a new method based on Deng entropy denoted as UIV is proposed to measure the uncertainty of the IBS. Moreover, it is proved that UIV meets some desirable axiomatic requirements. Numerical examples are shown in the paper to demonstrate the efficiency of UIV by comparing the proposed UIV with existing approaches. Keywords: Dempster-Shafer theory, interval-valued belief structure, interval evi- dence, uncertainty measure, Deng entropy. 1 Introduction Dempster-Shafer evidence theory, also known as D-S theory was proposed by Dempster [8] and extended by Shafer [45], it has received widespread attention and application in information processing [18,25,40,43,46,52]. As compared with classic probability theory, D-S theory allocates the belief to multi-subset proposition and does not require a priori information. Accordingly, D-S theory is used to process the uncertain information in many fields such as risk assessment [16,24,39,60], decision making [4,7,11,36,38,58], fault diagnosis [20,26,27,41,48,51], information fusion [2,9,12,19,35] and pattern classification [3,42,44,55]. Although the application of D-S theory has made considerable progress, there are still some common issues in urgent need to be solved. For instance, conflict processing should be taken into consideration when the obtained evidence is highly conflicting with each other [28,30,37,53], for we may get the count-intuitive results [29, 59]. In view of this, many scholars have carried out extensive and profound research. Denœux [15] considered the evidence expressed by fuzzy-valued which acquire lots of application [57]. Moreover, the classic D-S theory demands precise belief degrees, yet it is not always available in some cases. For instance, in the decision making, the experts sometimes cannot provide an accurate assessment because of the lack of information. At this time, an interval-valued belief structure (IBS) [56] is more suitable for dealing with the uncertainty problem. About extending the D-S theory to IBS, many scholars have carried out some research such as Denœux [14] put forward a set of concepts about interval-valued belief structure and initially explored the combination and the uncertainty of it. Lee & Zhu [34] proposed the combination of two interval evidence. Wang [54] proposed the approach to combine and standardize the interval evidence in one step. However, it must be noted that there are still many unresolved issues about interval-valued belief structure. One of the crucial issues is uncertainty measurement [10,50]. From the perspective of infor- mation theory, Klir elaborated the inner relationship between uncertainty and information [33]. Copyright © 2006-2017 by CCC Publications 632 W. Jiang, S. Wang Bronevich [5,6] discussed some of the issues and applications of the measurement of the uncer- tainty for imprecise probabilities. However, even how to measure the uncertainty of the mass function in D-S theory is still a considerable issue [21,23]. Dubois & Prade presented weighted Hartley entropy [17] to express the non-specificity of BPA. Klir & Wierman [32] explored five axiomatic requirements for the uncertainty measures including range, probabilistic consistency, set consistency, additivity and subadditivity, respectively. Abellán & Masegosa [1] have extended the axiomatic approach by appending new monotonicity requirement. Among existing uncer- tainty measures, aggregated uncertainty (AU) [22] and ambiguity measure (AM) [31] are two representative measures, yet they have their own shortcomings, such as low sensitivity and high computing complexity. Deng entropy [13] divided the belief for each focal element into all poten- tial subsets. On the other hand, there is not many approaches about the uncertainty measure for interval-valued belief structure. Denoeux [14] proposed a rudiment to measure the uncertainty, yet it was immature and lacked the mathematical proof. Song [49] defined the axiomatic re- quirements for uncertainty measure and presented a new method IU to measure the uncertainty. But IU lost part of the information and may cause the counter-intuitive result because of the transformation from belief structures to probability distributions. Accordingly, how to effectively measure the uncertainty of interval-valued belief structure is still an open issue. In this paper, a new method based on Deng entropy to measure the uncertainty of the interval-valued belief structure and its axiomatic proof is presented as well. Several examples are shown to illustrated the rationality and effectiveness of the method. The remainder of this paper is organized as follows. Section 2 starts with a brief presentation of D-S evidence theory and some other indispensable related concepts. In Section 3, we present a new method to measure the uncertainty of the interval-valued belief structure. Some numerical examples are given to demonstrate the validity of our new method in Section 4. Conclusions are summarized in Section 5. 2 Preliminaries 2.1 Dempster-Shafer evidence theory Dempster-Shafer evidence theory, as introduced by Demster [8] and expanded later by Shafer [45], has been widely used in dealing with uncertainty. Some basic concepts in D-S theory are introduced as follows. Let Θ be a finite set of worlds, which is called a frame of discernment (FOD). Θ consists of some propositions, which are mutually exclusive and exhaustive, and indicated by Θ = {θ1,θ2, . . . ,θi, . . . ,θN}. (1) Let 2Θ be the power set of Θ, namely 2Θ = {∅,θ1,θ2, . . . ,θN,{θ1 ∪θ2}, . . . ,{θ1 ∪θ2 ∪·· ·∪θi}, . . . , Θ}. (2) For a FOD Θ, a mass function is a mapping m : 2Θ → [0, 1], it is also called the basic probability assignment (BPA) or the belief structure. BPA must satisfy the following condition{ ∑ A∈2Θ m(A) = 1, m(∅) = 0. (3) For a BPA, its belief function Bel : 2Θ → [0, 1] is defined as Bel(A) = ∑ B⊆A m(B), (4) An Uncertainty Measure for Interval-valued Evidences 633 the plausibility function Pl : 2Θ → [0, 1] is defined as Pl(A) = 1 −Bel(Ā) = ∑ B∩A6=∅ m(B). (5) Assume there are two BPAs m1 and m2 with the same FOD, it can be combined by Dempster’s combination rule. m(A) = 1 1 −k ∑ B∩C=A m1(B)m2(C), (6) where k = ∑ B∩C=∅ m1(B)m2(C). (7) k is between [0,1], which is called the coefficient of conflict. When k = 1, Dempster’s combination rule will be invalid. 2.2 Interval-valued belief structure Uncertainty is sometimes no longer described by a unique belief structure, but by a convex set of belief structures verifying certain constraints. A set of concepts of interval-valued belief structure (IBS) is given as follows [14]. Let Θ be the frame of discernment, F1,F2, . . . ,FN be N subsets of Θ and [ai,bi] be N intervals with 0 6 ai 6 bi 6 1, (i = 1, 2, . . . ,N). An interval-valued belief structure (IBS) m is a belief structure on Θ such that ai 6 m(Fi) 6 bi, (8) where 0 6 ai 6 bi 6 1, i = 1, 2, . . . ,N, (9) N∑ i=1 ai 6 1 and N∑ i=1 bi > 1, (10) m(A) = 0 ∀A /∈{F1,F2, . . . ,FN}. (11) Obviously, m are non-empty imposes certain constraints on the ai and bi. If the singleton m is an IBS with ai = bi = m(Fi) for ∀Fi, m degenerates to a precise belief structure (BS). An IBS means the interval associated to each subset of Θ is [0,1]. It may be interpreted as reflecting “second-order” ignorance, that is, ignorance of what the state of belief of an agent may be. Let m be an interval-valued belief structure, namely ai 6 m(Fi) 6 bi for i = 1, 2, . . . ,N. If ∀k ∈{1, 2, . . . ,N}, ai and bi satisfy N∑ i=1 ai + (bk −ak) 6 1, (12) N∑ i=1 bi − (bk −ak) > 1. (13) Then, m is called a normalized interval-valued belief structure (NIBS) [54]. For a non-normalized interval-valued belief structure m, which violates Eq. (10), it can be normalized by following equations. âi = ai ai + ∑N j=1,j 6=i bj , i = 1, 2, . . . ,N, (14) 634 W. Jiang, S. Wang b̂i = bi bi + ∑N j=1,j 6=i aj , i = 1, 2, . . . ,N. (15) On the other side, if m has already satisfied Eq. (10), but not Eqs. (12) and (13), it can be normalized by following two equations. âi = max  ai, 1 − N∑ j=1,j 6=i bj   , i = 1, 2, . . . ,N, (16) b̂i = min  bi, 1 − N∑ j=1,j 6=i aj   , i = 1, 2, . . . ,N. (17) The concepts of belief function and plausibility function may easily be generalized to an interval-valued belief structure. Since these quantities are linear combinations of belief masses constrained in closed intervals, their ranges are both closed intervals. Let m be a normalized interval-valued belief structure on Θ. For ∀A ∈ Θ, its belief function and plausibility function are defined respectively as Bel(A) =  min ∑ Fi⊆A m(Fi), max ∑ Fi⊆A m(Fi)   , (18) Pl(A) =  min ∑ Fi∩A 6=∅ m(Fi), max ∑ Fi∩A 6=∅ m(Fi)   , (19) where min ∑ Fi⊆A m(Fi) = max   ∑ Fi⊆A ai,  1 − ∑ Fi 6⊂A bi     , (20) max ∑ Fi⊆A m(Fi) = min   ∑ Fi⊆A bi,  1 − ∑ Fi 6⊂A ai     , (21) min ∑ Fi∩A 6=∅ m(Fi) = max   ∑ Fi∩A 6=∅ ai,  1 − ∑ Fi∩A=∅ bi     , (22) max ∑ Fi∩A 6=∅ m(Fi) = min   ∑ Fi∩A 6=∅ bi,  1 − ∑ Fi∩A=∅ ai     . (23) 2.3 Deng entropy Since Shannon entropy [47] was proposed to quantify the expected value of the information volume contained in a message, it has became a significant approach to measure the uncertainty. However, for a mass function in D-S theory, Shannon entropy cannot calculate its uncertainty because the mass function includes multiple subset elements. To measure the uncertainty of the mass function, Deng [13] proposed Deng entropy as follows Ed(m) = − ∑ A⊆Θ m(A) log2 m(A) 2|A| − 1 , (24) An Uncertainty Measure for Interval-valued Evidences 635 where m is a BPA defined on the frame of discernment Θ, A is the focal element of m, and |A| is the cardinality of A. Deng entropy is analogous with the classical Shannon entropy, but the belief for each focal element A is divided by (2|A| − 1) which indicates the potential supports in A. 3 Proposed uncertainty measure for interval-valued belief struc- tures In an interval-valued belief structure, the belief degree for each subset is not a precise value but an interval. So contrasted with single-valued belief structures, an interval-valued belief structure is more vague and more uncertain, since an IBS has the “second-order” ignorance. Thus, how to measure the uncertainty of the IBS is an essential issue. In this paper, A new method to measure the uncertainty of IBS is proposed. Definition 1. Let m be a normalized interval-valued belief structure on the frame of discernment Θ = {F1,F2, . . . ,FN}, and it satisfies ai 6 m(Fi) 6 bi, which means the accurate belief m(Fi) ∈ [ai,bi]. Then the uncertainty measure of the IBS m is as follows UIV (m) = 2N∑ i=1 [ min m(Fi)∈[ai,bi] Ẽd(Fi), max m(Fi)∈[ai,bi] Ẽd(Fi)], (25) where Ẽd(Fi) = −m(Fi) log2 m(Fi) 2|Fi|−1 , (26) and |Fi| is the cardinality of Fi. The new measurement method we proposed is based on Deng entropy, not Shannon entropy, so our method is more suitable to handle the proposition of multi-subsets. For Deng entropy, the belief of the focal element m(Fi) is divided by the number of potential subsets 2|Fi| − 1 that demonstrates the non-specificity of the evidence. The more single elements are contained in focal elements, it is obvious that the greater the uncertainty. The term −m(Fi) log2 m(Fi) is analogous to Shannon entropy and is the measure of discord of the evidence. Thereby, it is also appropriate to quantify the uncertainty of interval-valued belief structure. Obviously, UIV is an interval number. Its value embodies the belief distribution of different proposition in IBS, and its interval length reflects the ambiguity generated by the belief expressed in intervals. Song [49] proposed the axiomatic requirements for a measure of uncertainty for a normalized interval-valued belief structure m. Theorem 2. Let U be a measure of uncertainty for a normalized interval-valued belief structure m on the FOD Θ = {θ1,θ2, . . . ,θN}, then U must content the following condition. 1. Whenever the NIBS defines a precise probability distribution, U degenerates to Shannon entropy. 2. When the NIBS assigned to all subsets of Θ are completely unknown, its uncertainty is maximum. Thus, U reaches its maximum value. 3. If the NIBS assigns to a certain singleton of Θ is 1, the uncertainty of it is 0. Therefore, U gets its minimum value 0. It will be shown that our new method satisfies the above-mentioned axiomatic requirement. Proof: 636 W. Jiang, S. Wang 1. If the NIBS m defines a precise probability distribution on Θ = {F1,F2, . . . ,FN}, UIV (m) = 2N∑ i=1 [Ẽd(Fi), Ẽd(Fi)] = N∑ i=1 −m(Fi) log2 m(Fi) 2|1| − 1 = − N∑ i=1 m(Fi) log2 m(Fi). From the above equation, we can see that when m defines a precise belief structure on Θ, UIV degenerates to Deng entropy. Moreover, when m defines a precise probability distribution, UIV degenerates to Shannon entropy. 2. When the NIBS assigned to all subsets of Θ are completely unknown, that is for ∀Fi ∈ 2Θ, [ai,bi] = [0, 1]. It is apparent that UIV (m) = 2N∑ i=1 [ min m(Fi)∈[0,1] Ẽd(Fi), max m(Fi)∈[0,1] Ẽd(Fi) where Ẽd(Fi) = −m(Fi) log2 m(Fi) 2|Fi|−1 , and it can be seen as a function of Fi, now the independent variable Fi is ∈ [0, 1]. Therefore, the minimum value of Ẽd(Fi) is 0 and the maximum value may be mutative with the change of |Fi| yet it can always get its maximum value for any Fi, that is max m(Fi)∈[0,1] Ẽd(Fi) = max Ẽd(Fi) So, UIV (m) = [0, 2N∑ i=1 max Ẽd(Fi)]. In this case, the value and the interval length of UIV are both the maximum value, which indicates that m is totally uncertain, that is, its uncertainty is maximum. 3. If the NIBS assigns to a certain singleton of Θ is 1, there is no harm in supposing that for singleton Fk, m(Fk) = 1, and the belief degree of all the rest subsets is 0. Then UIV (m) = 2N∑ i=1 [ min m(Fi)∈[ai,bi] Ẽd(Fi), max m(Fi)∈[ai,bi] Ẽd(Fi)] = [ min m(Fi)∈[1,1] Ẽd(Fk), max m(Fi)∈[1,1] Ẽd(Fk)] + 2N∑ i=1 i 6=k [ min m(Fi)∈[0,0] Ẽd(Fi), max m(Fi)∈[0,0] Ẽd(Fi)] = −1 × log2 1 21 − 1 − 2N∑ i=1 i 6=k (0 × log2 0 2|Fi| − 1 ) = 0 In fact, the UIV at this time is not 0, but [0,0]. This result thoroughly explains the m under this circumstance is totally definite, and it is also in line with intuition. An Uncertainty Measure for Interval-valued Evidences 637 Table 1: NIBSs in Example 3 {F1} {F2} {F3} {F1,F3} m1 [0.2,0.3] [0.1,0.35] [0.4,0.6] [0,0] m2 [0.2,0.3] [0.1,0.35] [0.35,0.7] [0,0] m3 [0.2,0.3] [0.1,0.35] [0,0] [0.4,0.6] m4 [0.2,0.3] [0.1,0.35] [0.2,0.3] [0.2,0.3] the number of the NIBS 1 2 3 4 U IV 0 0.5 1 1.5 2 2.5 3 Figure 1: The UIV of each NIBS in Example 3 2 4 Numerical examples In this section, several examples are given to demonstrate the effectiveness of UIV . Example 3. Assume a frame of discernment Θ = {F1,F2,F3}, and consider four NIBSs defined as shown in Table 1. We can calculate the UIV of the NIBSs as follows UIV (m1) = [1.239, 1.580] UIV (m2) = [1.157, 1.583] UIV (m3) = [1.959, 2.444] UIV (m4) = [2.042, 2.569] and they are also graphically shown in Fig. 1. The yellow portion represents the endpoint of the interval of the UIV . The range of UIV (m2) is larger than UIV (m1) from the figure, since m2(F3) is more uncertain than m1(F3). However, the value of UIV (m2) is close to UIV (m1) because the belief distribution in m1 and m2 are about the same. Considering UIV (m3) and UIV (m1), it is obvious that both the length and the value of UIV (m3) are bigger since the 638 W. Jiang, S. Wang Table 2: UIV in Example 4 Cases UIV A={1} [2.080,3.803] A={1,2} [3.216,4.886] A={1,2,3} [3.949,5.864] A={1,2,. . . ,4} [4.609,6.743] A={1,2,. . . ,5} [5.238,7.581] A={1,2,. . . ,6} [5.851,8.400] A={1,2,. . . ,7} [6.458,9.209] A={1,2,. . . ,8} [7.062,10.013] A={1,2,. . . ,9} [7.663,10.816] multi-element can take along more uncertainty than single element even though in the same interval. It is worth noting that compared with UIV (m3), UIV (m4) is close but slightly larger. Although a great deal of belief are assigned on the multi-element in m3 and it conveys illegibility, the allocation form which distributes the belief to more subsets is more excursive and this result is we take for granted. Example 4. Suppose that we have a frame of discernment Θ = {1, 2, . . . , 10}. A NIBS m is shown as follows. m(2, 3) = [0.1, 0.25], m(A) = [0.6, 0.8], m(Θ) = [0.1, 0.2] where A is a varying subset of Θ. A starts at A = {1}, increases one more element every time and ending with A = {1, 2, . . . , 9}. The UIV of m are shown in Table 2 and Fig. 2. The yellow portion represents the endpoint of the interval of the UIV . From Fig. 2, the result shows that UIV increases monotonically with the number of elements in A. This is rational because the more elements contained in a subset, the more uncertain it is. From the example it can be seen that UIV is capable of reflecting such a feature. In the first two examples, some superior properties are demonstrated. Then an example from Song [49] are used to illustrate our proposed UIV and contrast it with Song’s uncertainty measure IU. The formula of Song’s measurement are shown as follows. Definition 5. Let m be a normalized interval-valued belief structure on the FOD Θ = {F1,F2, . . . ,FN}, and it satisfies ai 6 m(Fi) 6 bi. Then IU of the IBS m is as follows IU(m) = N∑ i=1 (− ai + bi 2 log2 ai + bi 2 + bi −ai 2 ) (27) Example 6. The example Song used in the paper is shown in Table 3, and to make a comparison with Song’s method, the consequents of IU and our new method UIV are both demonstrated in Table 4. For the NIBSs from m1 to m5, we can see their belief intervals are completely consistent, merely the corresponding subsets are disparate. The uncertainty degree IU proposed by Song, are so similar that it is difficult to measure the uncertainty accurately. Moreover, the belief assignment of m1 and m5 are entirely different, yet their IU are almost identical. For UIV , m5 with more belief assigned to multiple elements has a higher uncertainty, m2 and m3 take An Uncertainty Measure for Interval-valued Evidences 639 the size of A 1 2 3 4 5 6 7 8 9 U IV 0 2 4 6 8 10 12 Figure 2: UIV in Example 4 Table 3: NIBSs in Song’s example (Θ = {F1,F2,F3}) {F1} {F2} {F3} {F1,F2} {F1,F3} {F2,F3} {F1,F2,F3} m1 [0.2,0.4] [0.1,0.3] [0.3,0.6] [0,0.1] [0,0] [0,0] [0,0] m2 [0.2,0.4] [0,0] [0,0] [0,0.1] [0.3,0.6] [0,0] [0.1,0.3] m3 [0,0] [0.1,0.3] [0,0] [0,0.1] [0.2,0.4] [0.3,0.6] [0,0] m4 [0,0] [0,0] [0.3,0.6] [0,0.1] [0.3,0.6] [0.1,0.3] [0.2,0.4] m5 [0,0] [0,0] [0,0] [0,0.1] [0.3,0.6] [0.1,0.3] [0.2,0.4] m6 [0,1] [0,1] [0,1] [0,0] [0,0] [0,0] [0,0] m7 [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,1] Table 4: IU and UIV of the NIBSs IU UIV m1 1.930 [1.239,2.073] m2 1.609 [2.074,3.778] m3 1.889 [2.110,3.568] m4 1.575 [1.714,3.181] m5 1.939 [2.513,4.532] m6 3 [0,1.592] m7 1.793 2.807 640 W. Jiang, S. Wang Table 5: NIBSs and their IU and UIV in Example 7 {F1} {F2} {F1,F2} IU UIV m1 [0.1,0.25] [0.3,0.4] [0.4,0.5] 1.163 [2.016,2.323] m2 [0.2,0.3] [0.4,0.45] [0.2,0.4] 1.163 [1.764,2.213] second place, as well as m1 is the most precise of these NIBSs. Furthermore, UIV is suitable for measurement for the reason that the difference in calculated values is significant and thus has a degree of discrimination. Another detail of concern is m6 and m7. The uncertainty of m7 is low, while the maximum uncertainty degree occurs on m6. The cause of this consequence as Song said in [49], “This is caused by the transformation from belief structures to Bayesian belief structures, which will cause information loss.” UIV (m6) is comparatively small because m6 only distribute the belief to singleton. In addition, m7 actually is not a normalized interval-valued belief structure. It turns into a NIBS m7({F1,F2,F3}) = 1 by Eqs. (16) and (17). After standardization, UIV (m7) is a precise number and its uncertainty can be effectively measured. Example 7. Let a frame of discernment be Θ = {F1,F2}. Two NIBSs, their IU and UIV are shown in Table 5. We can calculate that both two Bayesian belief structures of m1 and m2 are m(a) = [0.3, 0.5], m(b) = [0.5, 0.65], and IU is not competent to measure the uncertainty in this sit- uation. Because for two unrelated NIBSs with significant differences in the degree of uncertainty, their IU are equivalent. Through the above analysis, it is found that UIV is more reasonable to measure the uncertainty of the interval-valued belief structures. 5 Conclusion D-S theory has been widely used in information processing and information fusion. In many applications, we can only obtain an interval-valued belief structure instead of a basic probability assignment defined on single values, due to lack of information and some other reasons. It is indispensable to measure the uncertainty of the IBS, there is still an open issue. The main contribution of this paper is a new method based on Deng entropy, UIV is proposed to measure the uncertainty of an IBS. It is proved that UIV meets some axiomatic properties. Numerical examples are illustrated to show the effectiveness of UIV and discuss its characteristic. Moreover, it is found that UIV is more reasonable and sensitive in comparison with existing methods. Acknowledgments 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), Aviation Science Foundation (Program No. 20165553036), Fundamental Research Funds for the Central Universities (Program No. 3102017OQD020). An Uncertainty Measure for Interval-valued Evidences 641 Bibliography [1] Abellán J., Masegosa A. 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