An improved a-cut approach to transforming fuzzy membership function into basic belief assignment by Ioan Degău

CJA 655 27 June 2016 Chinese Journal of Aeronautics, (2016), xxx(xx): xxx–xxx

No. of Pages 10

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics cja@buaa.edu.cn www.sciencedirect.com

FULL LENGTH ARTICLE

An improved a-cut approach to transforming fuzzy membership function into basic belief assignment

Yang Yi a, X. Rong Li b, Han Deqiang c,*

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SKLSVMS, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China University of New Orleans, New Orleans, LA 70148, USA c Center for Information Engineering Science Research, Xi’an Jiaotong University, Xi’an 710049, China b

Received 26 September 2015; revised 4 November 2015; accepted 7 January 2016

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KEYWORDS

Belief functions; Evidence theory; Fuzzy sets; Membership functions; Uncertainty

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2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

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Abstract In practical applications, pieces of evidence originated from different sources might be modeled by different uncertainty theories. To implement the evidence combination under the Dempster–Shafer evidence theory (DST) framework, transformations from the other type of uncertainty representation into the basic belief assignment are needed. a-Cut is an important approach to transforming a fuzzy membership function into a basic belief assignment, which provides a bridge between the fuzzy set theory and the DST. Some drawbacks of the traditional a-cut approach caused by its normalization step are pointed out in this paper. An improved a-cut approach is proposed, which can counteract the drawbacks of the traditional a-cut approach and has good properties. Illustrative examples, experiments and related analyses are provided to show the rationality of the improved a-cut approach.

1. Introduction Uncertainty modeling and reasoning is a very crucial research field in information fusion. Probability theory, rough set theory,1 fuzzy sets theory,2 and Dempster–Shafer evidence theory (DST)3 are major theories and tools for dealing with various types of uncertainty. * Corresponding author. E-mail address: deqhan@mail.xjtu.edu.cn (D. Han). Peer review under responsibility of Editorial Committee of CJA.

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DST is an important modeling and reasoning tool for uncertainty such as ambiguity4,5 (including non-specificity and discord), and it has been widely used in many applications such as pattern recognition,6,7 information fusion8 and decision-making.9–11 In DST, the basic belief assignment (BBA) is used to model the uncertainty. When multiple BBAs are available, they can be combined to reduce the uncertainty. However, in practical applications, we will usually encounter different types of information sources, where the uncertainty is modeled by different uncertainty theories. In such cases, how to implement the combination or fusion of different types of information under the framework of DST? It needs transformations from other types of uncertainty representation into BBAs.12,13 This paper focus on the transformation from a fuzzy membership function (FMF)2 into a BBA.

http://dx.doi.org/10.1016/j.cja.2016.03.007 1000-9361 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Yang Y et al. An improved a-cut approach to transforming fuzzy membership function into basic belief assignment, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.03.007

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Many approaches6,12–16 to transforming an FMF into a BBA have emerged, where the a-cut approach14 is a simple yet effective and commonly used approach. However, it includes a normalization step. This leads to drawbacks, e.g., in some cases, it has counter-intuitive results. In some extreme cases, it can even not be executed. In this paper, we propose an improved a-cut approach without the problematic normalization. More rational results can be obtained using the improved version. Furthermore, it has some desired properties when compared with the traditional one. Experiments and simulations are provided to illustrate the new a-cut approach and show its efficiency.

2. Basics of DST

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In DST,3 elements in the frame of discernment (FOD) H are mutually exclusive and exhaustive. Assume that 2H denotes the power set of FOD. m:2H ? [0, 1] is called a BBA if it satisfies X mðAÞ ¼ 1; mð£Þ ¼ 0 ð1Þ A#H "A # H, when m(A) > 0, A is called a focal element. A BBA is also called a mass function. Belief function (Bel) and plausibility function (Pl) are defined by Ref. 3 X BelðAÞ ¼ mðBÞ ð2Þ B#A

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PlðAÞ ¼

X A\B–£

mðBÞ

ð3Þ

Suppose that two independent BBAs m1(_s) and m2(_s) have their corresponding focal elements A1, A2, . . ., Ak and B1, B2, . .P ., Bl. If the conflict coefficient K ¼ Ai \Bj ¼£ m1 ðAi Þm2 ðBj Þ < 1, the function m:2H ? [0, 1] given by 8 0 A¼£ > > X > > < m1 ðAi Þm2 ðBj Þ mðAÞ ¼ ð4Þ Ai \Bj ¼A X > A–£ > > 1 m ðA Þm ðB Þ i > 1 2 j : Ai \Bj ¼£

is also a BBA. Eq. (4) is called Dempster’s rule of combination.3 Besides Dempster’s rule of combination, some other alternative combination rules, e.g., robust combination rule (RCR),17 proportional conflict redistribution rule 6 (PCR6),18 and the mean rule19 are given as follows. (1) RCR In RCR,17 the conjunctive rule and the disjunctive rule are jointly used. 8 mRCR ðAÞ ¼ aðKÞmDis ðAÞ þ bðKÞmConj ðAÞ > > X > > < mDis ðAÞ ¼ m1 ðBÞm2 ðCÞ ð5Þ B[C¼A > X > > > m1 ðBÞm2 ðCÞ : mConj ðAÞ ¼ B\C¼A

Here mDis is the BBA obtained by the disjunctive rule, mConj is the BBA obtained by the conjunctive rule, and a(K), b(K) are the weights satisfying aðKÞ þ ð1 KÞbðKÞ ¼ 1

ð6Þ

where K is the conflict coefficient. Robust combination rule can be considered as a weighted sum of the BBAs obtained using the disjunctive rule and the conjunctive rule, respectively. (2) PCR6 Consider s BBAs denoted by m1(_s), m2(_s), . . ., ms(_s). PCR6,18 which redistributes the partial conflicting mass values to the elements involved, is defined as 8 mPCR6 ðXÞ ¼ 0 8X ¼ £ > > > > s s Y X > P > > Y1 \Y2 \...\Ys ¼X m ðXÞ ¼ m ðY Þ þ mi ðXÞ2 > PCR 6 i i > > Y1 ;Y2 ;...;Ys 22H > i¼1 i¼1 > < 0 s 1 1 Y ð7Þ > m ðY Þ ri ðjÞ ri ðjÞ B C > > P B C > > B j¼1 C 8X – £ > þ > \s 1 Yr ðkÞ \X¼£ s 1 B C > k¼1 i Y > A H @ > Y ;Y ;...;Y 22 > ri ð1Þ ri ð2Þ ri ðs 1Þ mi ðXÞþ mri ðjÞ ðYri ðjÞ Þ > : j¼1

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where ri ðjÞ ¼ j if j < i ri ðjÞ ¼ j þ 1 if j P i

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It should be noted that PCR6 coincides with PCR5 when combining two sources,18 but differs from PCR5 when combining more than two sources altogether and PCR6 is considered more efficient than PCR5 because it is compatible with classical frequentist probability estimate.18 (3) Mean rule Mean rule19 aims to find the average of the BBAs to be combined as

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mMean ðAÞ ¼

1 s

s X

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mi ðAÞ

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ð8Þ 122

i¼1

After the combination, we can use the pignistic probability transformation (PPT)20 in Eq. (9): X BetPðhi Þ ¼ mðAÞ=jAj 8A # H ð9Þ fh g2A # H

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to make a probabilistic decision, where |A| denotes the cardinality of the focal element A. Besides the DST, there are also many other theories of uncertainty, where fuzzy set theory and its fuzzy membership function (FMF)2 are widely used in many applications. For the DST, the BBA is essentially defined using random sets, which is a unified framework for almost all the existing uncertainty theories. To combine the information in terms of the BBA and that in terms of the FMF, the FMF should be transformed into a BBA. There are many available transformations of an FMF into a BBA.6,12–16 a-cut approach14 is simple and commonly used.

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3. Traditional a-cut approach

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3.1. Basics of fuzzy sets and fuzzy membership function

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A fuzzy set2 A # H is a set to describe the fuzzy concepts, which are not crisp. A fuzzy set is often defined by an FMF lA ðhÞ : H#½0; 1 , quantifying the degree of membership of

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the element h to the fuzzy set A. Given a 2e [0, 1], an a-cut

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of a fuzzy set A is a crisp set Aa (subset of H) such that

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Aa ¼ fh 2 HjlðhÞ P ag.

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Please cite this article in press as: Yang Y et al. An improved a-cut approach to transforming fuzzy membership function into basic belief assignment, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.03.007

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lA ðhÞ is briefly denoted by l(h) below when no confusion

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arises.

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3.2. a-Cut approach

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Suppose that the FOD is H = {h1, h2, . . ., hn} and the FMF is l(hi), i = 1, 2, . . ., n, the corresponding BBA generated using M a-cuts14 (0 ¼ a0 < a1 < a2 < < aM 6 1), where M 6 jHj ¼ n. ( Bj ¼ hi 2 Hjlðhi Þ P aj ð10Þ a a mðBj Þ ¼ j aMj 1

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where Bj, for j = 1, 2, . . ., M, (M 6 jHj) denotes the focal element. a-Cut approach is illustrated in Example 1.

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3.3. Example 1: Illustrative example of a-cut approach

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A given FMF is l(h1) = 0.2, l(h2) = 0.6, l(h3) = 0.4, l(h4) = 0.8 and a1 = 0.2, a2 = 0.4, a3 = 0.6, a4 = 0.8. According to Eq. (10), the BBA transformed from the FMF is given in Table 1. The details of the transformation are given as follows. Step 1. Since all the lðhi Þ P a1 ¼ 0:2, i e {1, 2, 3, 4}, B1 = {h1, h2, h3, h4}, aM = 0.8, then m(B1) = (a1 a0)/ aM = 0.25. Step 2. Since lðhi Þ P a2 ¼ 0:4, where i e {2, 3, 4}, B2 = {h2, h3, h4}, then, m(B2) = (a2 a1)/aM = 0.25. Step 3. Since lðhi Þ P a3 ¼ 0:6, where i e {2, 4}, B3 = {h2, h4}, then, m(B3) = (a3 a2)/aM = 0.25. Step 4. Since lðhi Þ P a4 ¼ 0:8, where i = 4, B4 = {h4}, then, m(B4) = (a4 a3)/aM = 0.25.

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The procedure can also be illustrated in Fig. 1. Note that a could be unequal to the given FMF. For example, when a1 = 0.3, a2 = 0.5, the BBA obtained from the given FMF using the a-cut approach is shown in Table 2. There are less focal elements, since the length of a is small. When a1 = 0.1, a2 = 0.3, a3 = 0.5, a4 = 0.8, the BBA obtained using the a-cut approach is shown in Table 3. Given different a’s, the corresponding BBAs obtained from a given FMF using the a-cut approach might be different. When using a with smaller length, the BBA obtained is simpler, which can be used as an approximation.13 Note that in our work, the focus is the mechanism of transformation from an FMF to a BBA when a is given. As we can see, there involves a normalization at each step of a-cut approach. This can assure the unity of the BBA obtained; however, it may lead to counter-intuitive results in some cases, as illustrated later.

Table 1

BBA obtained in Example 1 using a-cut approach.

Focal element

Mass value

B1 = {h1, h2, h3, h4} B2 = {h2, h3, h4} B3 = {h2, h4} B4 = {h4}

0.25 0.25 0.25 0.25

Fig. 1

Illustration of Example 1 using a-cut approach.

Table 2 BBA obtained in Example 1 using a-cut approach (a1 = 0.3, a2 = 0.5). Focal element

Mass value

B1 = {h2, h3, h4} B2 = {h2, h4}

(0.3 0)/0.5 = 0.6 (0.5 0.3)/0.5 = 0.4

Table 3 BBA obtained in Example 1 using a-cut approach (a1 = 0.1, a2 = 0.3, a3 = 0.5, a4 = 0.8). Focal element

Mass value

B1 = {h1, h2, h3, h4} B2 = {h2, h3, h4} B3 = {h2, h4} B4 = {h4}

(0.1 0)/0.8 = 1/8 (0.3 0.1)/0.8 = 1/4 (0.5 0.3)/0.8 = 1/4 (0.8 0.5)/0.8 = 3/8

4. Drawbacks of traditional a-cut approach Using the traditional a-cut approach with normalization might lead to counter-intuitive results shown as follows. 4.1. Example 2: Invariance of fixed ratio FMFs Suppose that the FOD is H = {h1, h2, h3}. Three FMFs are l1(h1) = 0.01, l1(h2) = 0.03, l1(h3) = 0.04; l2(h1) = 0.10, l2(h2) = 0.30, l2(h3) = 0.40; and l3(h1) = 0.25, l3(h2) = 0.75, l3(h3) = 1. For simplicity, we sort each li(_s) in ascending order to generate their corresponding a values. For l1(_s), a1 = 0.01, a2 = 0.03, a3 = 0.04. For l2(_s) a1 = 0.10, a2 = 0.30, a3 = 0.40. For l3( s_ ), a1 = 0.25, a2 = 0.70, a3 = 1. Then, their corresponding BBAs using acut approach are m1(h3) = 0.25, m1(h2, h3) = 0.50, m1(H) = 0.50, m2(h3) = 0.25, m2(h2, h3) = 0.50, m2(H) = 0.50, and m3(h3) = 0.25, m3(h2, h3) = 0.50, m3(H) = 0.50. As we can see, three different FMFs correspond to the same BBA. Note that as far as the ratios between different l(hi) are fixed, the a-cut transformed BBAs are always the same due to the normalization involved. For l1(_s), the membership degree of h3 is very small (l1(h3) = 0.04, i.e., a close to zero possibility) while for l3(_s), the membership degree of h3 is much greater (l3(h3) = 1, i.e., almost sure) compared with l1(h3). However, m1({h3}) = m3({h3}) = 0.25, and BetP1(h3)

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= BetP3(h3) = 0.5833. For l1(_s), h3 has a very low possibility, while it has a possibility greater than 0.5 after the transformation into BBA followed by the PPT, which is counter-intuitive.

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4.2. Example 3: Inability to handle singular cases

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For an FMF l(hi) = 0, "i = 1, 2, . . ., n, traditional a-cut approach cannot be executed due to the normalization step (aM = 0). This is a limitation of a-cut, that is, it is unable to handle the singular case (all-zero case).

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4.3. Example 4: Lack of discriminability

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So, as far as a/b > c/d, m({h1}) > m({h2}). That is, the decision result is determined by the ratio of alternatives’ membership degree when using the traditional a-cut approach. This is caused by the normalization. In the above examples, all the counter-intuitive results or drawbacks are caused by the normalization in the traditional a-cut approach. This approach should be modified, if we want to get rid of these drawbacks. 5. A novel improved a-cut approach

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For an FMF l(_s) defined on H = {h1, h2, . . ., hn}, l(hi) = a e [0, 1], and l(hj) = 0, "j – i, where i, j e {1, 2, . . ., n}. By using the traditional a-cut approach, the BBA obtained is always m ({hi}) = 1. For example, for the two FMFs defined on H = {h1, h2}: l1(h1) = 0.01, l1(h2) = 0; l2(h1) = 0.99, l2(h2) = 0. Their corresponding BBAs are the same: m({h1}) = 1, although l1(_s) and l2(_s) are very different. So m1({h1}) and m2({h1}) cannot be discriminated. This is counter-intuitive.

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4.4. Example 5: Inability to reflect the magnitude of FMF

where Bj, j = 1, 2, . . ., M, represents the focal element. Then, the summation of m(Bj) is

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Suppose that the FOD is H = {h1, h2}. Two FMFs are l1 ð Þ : l1 ðh1 Þ ¼ 0:010; l1 ðh2 Þ ¼ 0:012; l2 ð Þ : l2 ðh1 Þ ¼ 0:450; l2 ðh2 Þ ¼ 0:400:

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Using the traditional a-cut approach, their corresponding BBAs are m1 ð Þ : m1 ðfh2 gÞ ¼ 0:1667; m1 ðfh1 ; h2 gÞ ¼ 0:8333; m2 ð Þ : m2 ðfh1 gÞ ¼ 0:1111; m2 ðfh1 ; h2 gÞ ¼ 0:8889: Using Dempster’s rule of combination, the combined BBA

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is mðfh1 gÞ ¼ 0:0943; mðfh2 gÞ ¼ 0:1509; mðfh1 ; h2 gÞ ¼ 0:7547: Using PPT, BetP(h2) = 0.5283 is the maximum one; thus, the decision results is h2. However, l1(h1) is only slightly smaller than l1(h2) (the additive difference is 0.002) and l2(h1) is larger than l2(h2) with the additive difference 0.05. When using the averaging rule for the two FMFs, l12(h1) = 0.23, l12(h2) = 0.206, then, h1 is more preferred. Due to the normalization, the a-cut approach only emphasizes the relative values of the FMF between different alternatives. That is, the absolute magnitude of the values of FMF is not informative by the a-cut approach. For more general case, suppose that FOD is {h1, h2}. Two FMFs are l1(h1) = a, l1(h2) = b and l2(h1) = d, l2(h2) = c, where b > a, d > c. Using the traditional a-cut approach, the two corresponding BBAs are m1({h2}) = 1 a/b; m1({h1, h2}) = a/b and m2({h1}) = 1 c/d; m2({h1, h2}) = c/d. Using Dempster’s rule, m( ) = m1( ) m2( ) is obtained. Suppose the conflict between m1( ) and m2( ) is K. Then, m ({h1}) = ((a/b) (1 c/d))/K, m({h2}) = ((c/d) (1 a/b))/K. Their additive difference is mðfh1 gÞ mðfh2 gÞ ¼

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a=b ac=bd c=d þ ac=bd a=b c=d ¼ K K

M X

mðBj Þ ¼ a1 a0 þ a2 a1 þ þ aM aM 1 ¼ aM

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ð12Þ 294

i¼1

Since aM 6 1, the remaining mass value is 1 aM. Add the remaining mass values 1 aM to m(H). mðHÞ :¼ mðHÞ þ ð1 aM Þ

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We propose an improved a-cut approach (denoted by a0 -cut) without the normalization. To sum up to unity, the remaining mass values are assigned to the total set H. The implementation of the a0 -cut is as follows. Suppose that the FOD is H = {h1, h2, . . ., hn} and the FMF is l(hi), i = 1, 2, . . ., n. First, use M a-cuts (0 ¼ a0 < a1 < a2 < < aM 6 1), where M 6 jHj ¼ n: ( Bj ¼ hi 2 Hjlðhi Þ P aj ð11Þ mðBj Þ ¼ aj aj 1

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ð13Þ

Then, m( ) is the BBA obtained using the a0 -cut approach. Obviously, when aM = 1, there is no difference between the a0 -cut and the traditional a-cut.

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5.1. Example 1 revisited

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First, according to Eq. (11), we can obtain m({h4}) = 0.2, m ({h2, h4}) = 0.2, m({h2, h3, h4}) = 0.2, m(H) = 0.2. Then according to Eq. (13), m(H) is modified as m(H): = m (H) + 1 aM = 0.2 + 1 0.8 = 0.4. The BBA obtained using the a0 -cut approach is given in Table 4. When a1 = 0.3, a2 = 0.5, the BBA obtained from the FMF using the a0 -cut approach is given in Table 5. Note that in Table 2, the BBA obtained using traditional acut approach has no focal element of H = {h1, h2, h3, h4}, while the BBA obtained using a0 -cut approach has the focal element of H = {h1, h2, h3, h4}, since the remaining mass value is assigned to H. When a1 = 0.1, a2 = 0.3, a3 = 0.5, a4 = 0.8, the BBA obtained using the a0 -cut approach is given in Table 6. Given different a’s, the BBAs obtained using the a0 -cut approach from a given FMF might be different.

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5.2. Example 2 revisited

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Using the a0 -cut approach, the BBAs obtained in Example 2 are m1(h3) = 0.01, m1(h2, h3) = 0.02, m1(H) = 0.97, m2(h3) = 0.10, m2(h2, h3) = 0.20, m2(H) = 0.70, and m3(h3) = 0.25, m3(h2, h3) = 0.50, m3(H) = 0.50.

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An improved a-cut approach to transforming fuzzy membership function into basic belief assignment Table 4

BBA obtained in Example 1 using a0 -cut approach.

Focal element

Mass value

B1 = {h1, h2, h3, h4} B2 = {h2, h3, h4} B3 = {h2, h4} B4 = {h4}

0.4 0.2 0.2 0.2

Table 5 BBA obtained in Example 1 using a0 -cut approach (a1 = 0.3, a2 = 0.5). Focal element

Mass value

B1 = {h2, h3, h4} B2 = {h2, h4} B3 = {h1, h2, h3, h4}

0.3 0 = 0.3 0.5 0.3 = 0.2 1.0 0.5 = 0.5

Using the a0 -cut approach, their corresponding BBAs are m1( ):m1({h2}) = 0.0020, m1(H) = 0.9980; m2( ):m2({h1}) = 0.0500, m2(H) = 0.9500. As we can see, since the normalization is removed, the magnitude of the FMF is informative by using the a0 -cut approach. After using Dempster’s rule of combination, the combined BBA is m({h1}) = 0.0499, m({h2}) = 0.0019, m(H) = 0.9482. Using PPT, BetP(h1) = 0.5240 is the maximum one. Therefore, the decision results is h1, which is intuitive and consistent with the result obtained using the averaging rule of FMFs. For a more general case, suppose that FOD = {h1, h2}. Two FMFs are l1(h1) = a, l1(h2) = b; l2(h1) = d, l2(h2) = c, where b > a, d > c. Using the a0 -cut approach, the two corresponding BBAs are m2 ðfh1 gÞ ¼ d c; m2 ðfh1 ; h2 gÞ ¼ 1 ðd cÞ

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5.3. Example 3 revisited

where K denotes the conflict between m1(_s) and m2(_s). The difference between h1 and h2 is

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For an FMF l(hi) = 0, "i = 1, 2, . . ., n, the BBA obtained using the a0 -cut approach is m(H) = 1, which represents a totally unknown state. Note that the traditional a-cut approach cannot be applied to this case.

mðfh1 gÞ mðfh2 gÞ ððd cÞ ðb aÞðd cÞ ðb aÞ þ ðb aÞðd cÞÞ ¼ K ðd cÞ ðb aÞ ¼ K

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5.4. Example 4 revisited

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For an FMF l( ) defined on H = {h1, h2, . . ., hn}, l(hi) = a 2 [0, 1], and l(hj) = 0, "j – i, where i, j 2 {1, 2, . . ., n}. By using the proposed a0 -cut approach, the BBA obtained is m({hi}) = a, m(H) = 1 a. For example, for the two FMFs defined on H = {h1, h2}: l1(h1) = 0.01, l1(h2) = 0; l2(h1) = 0.99, l2(h2) = 0. Their corresponding BBAs are m1({h1}) = 0.01, m1(H) = 0.99; m2({h1}) = 0.99, m2(H) = 0.01. l1( ) and l2( ) are very different, and m1({h1}) and m2({h1}) are also very different. This is more intuitive than the results obtained using the traditional a-cut approach.

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5.5. Example 5 revisited

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Two FMFs are l1( ):l1(h1) = 0.010, l1(h2) = 0.012; l2( ): l2(h1) = 0.450, l2(h2) = 0.400.

Table 6 BBA obtained in Example 1 using a0 -cut approach (a1 = 0.1, a2 = 0.3, a3 = 0.5, a4 = 0.8). Focal element

Mass value

B1 = {h1, h2, h3, h4} B2 = {h2, h3, h4} B3 = {h2, h4} B4 = {h4}

0.1 + 1.0 0.8 = 0.3 0.3 0.1 = 0.2 0.5 0.3 = 0.2 0.8 0.5 = 0.3

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m1 ðfh2 gÞ ¼ b a; m1 ðfh1 ; h2 gÞ ¼ 1 ðb aÞ;

Three different FMFs correspond to three different BBAs, respectively, although the ratio between different l(hi) is fixed. In BBAs obtained, the mass of H are large due to the reassignment of 1 aM. H is always non-informative in evidence combination, so, the large mass of H does not matter for the later combination.

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Using Dempster’s rule of combination, m(_s) = m1(_s) m2(_s), then mðfh1 gÞ ¼ ððb aÞ ð1 ðd cÞÞÞ=K; mðfh2 gÞ ¼ ððd cÞ ð1 ðb aÞÞÞ=K

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So, when (d c) > (b a), m({h1}) > m({h2}). That is, the decision result is determined by the absolute difference of alternatives’ membership degree when using the a0 -cut approach, which appears more intuitive. 6. Experiments on pattern classification

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To verify the advantages of the proposed a0 -cut approach compared with the traditional a-cut approach, experimental results on pattern classification are provided below. Here we use one of the commonly used iris dataset from UCI database.21 Iris dataset has three classes, where each class has 50 samples. Suppose that h1, h2, h3 represent Class 1, Class 2 and Class 3, respectively. Each sample has four feature dimensions. On each experimental run, samples of each class are separated into two parts with an equal probability: training samples and test samples. So, on each run, totally 75 samples are for training and 75 samples are for testing. On each run, the 75 training samples are used to generate each class’s triangular fuzzy number [lbj(hi), mej(hi), ubj(hi)], where lbj(hi) is the minimum value, mej(hi) is the mean value, and ubj(hi) is the maximum value of j th dimension of the training samples in the class hi. Using [lbj(hi), mej(hi), ubj(hi)], each class’ FMF can be obtained using Eq. (14) as illustrated in Fig. 2. 8 j < j xðjÞ j j lb ðhi Þ j 8xðjÞ 2 ½lb j ðhi Þ;me j ðhi Þ me ðhi Þ lb ðhi Þ me ðhi Þ lb ðhi Þ j l ðxðjÞ;hi Þ ¼ j xðjÞ ðhi Þ j : j þ ub j ðhubi Þ me j ðh Þ 8xðjÞ 2 ½me ðhi Þ;ub ðhi Þ ub j ðhi Þ me j ðhi Þ i ð14Þ

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where x(j) is the j th feature value of sample x. Given a query sample xq, we use each dimension xq(i) to generate an FMF li( ) (i = 1, 2, 3, 4, four FMFs in total) based on Eq. (14). Then, using the traditional a-cut approach, we can generate four BBAs (denoted by mai ð Þ, i = 1, 2, 3, 4 corresponding to four dimensions) of xq, and using the proposed a0 -cut approach, four other BBAs can be obtained 0 (denoted by mai ð Þ, i = 1, 2, 3, 4). Four mai ð Þ are combined using Dempster’s rule, RCR, PCR 6, and the mean rule, respectively, to obtain four combined BBAs. Then, use the PPT to make a classification decision. The same operation is 0 executed for mai ð Þ. Calculate the classification accuracies based on the a-cut and a0 -cut approaches, respectively, for all the training samples on the current run. Repeat the run 100 times and then calculate the average classification accuracy. The experimental results are listed in Table 7. As shown in Table 7, using the proposed a0 -cut approach, classification accuracies are higher than those using the traditional a-cut approach judging from all four combination rules. We make a detailed analysis below. Some query samples are picked out, whose classification are correct using the a0 -cut and incorrect using the a-cut approach. (1) Case 1 The test sample is xq = [6.1000, 2.6000, 5.6000, 1.4000] belonging to Class 3. The triangular fuzzy number of each dimension obtained based on the training samples are listed in Table 8. Based on Eq. (14), we can use these triangular fuzzy numbers to generate each class’s FMF for each dimension of xq below, also as shown in Fig. 3. For xq ð1Þ ¼ 6:1; l1 ðh1 Þ ¼ 0; l1 ðh2 Þ ¼ 0:8824; l1 ðh3 Þ ¼ 0:4682: For xq ð2Þ ¼ 2:6; l2 ðh1 Þ ¼ 0:2679; l2 ðh2 Þ ¼ 0:8000; l2 ðh3 Þ ¼ 0:5348: For xq ð3Þ ¼ 5:6; l3 ðh1 Þ ¼ 0; l3 ðh2 Þ ¼ 0; l3 ðh3 Þ ¼ 0:9569: For xq ð4Þ ¼ 1:4; l4 ðh1 Þ ¼ 0; l4 ðh2 Þ ¼ 0:6757; l4 ðh3 Þ ¼ 0: Then, by using the traditional a-cut approach, we can generate four BBAs (denoted by mai ð Þ, i = 1, 2, 3, , 4 corresponding to four dimensions) of xq: ma1 ðfh2 gÞ ¼ 0:4694; ma1 ðfh2 ; h3 gÞ ¼ 0:5036; ma2 ðfh2 gÞ ¼ 0:3316; ma2 ðfh2 ; h3 gÞ ¼ 0:3336;

441

ma2 ðHÞ ¼ 0:3348; ma3 ðfh3 gÞ ¼ 1; ma4 ðfh2 gÞ ¼ 1:

Table 7

a0 -cut

Combination rule

a-cut

Dempster

Class 1 Class 2 Class 3 Total

80.62% 76.58% 82.80% 80.00%

Class 1 Class 2 Class 3 Total

96.42% 79.82% 89.82% 88.69%

RCR

Class 1 Class 2 Class 3 Total

81.08% 87.06% 83.56% 83.90%

Class 1 Class 2 Class 3 Total

99.88% 88.84% 84.94% 91.22%

PCR6

Class 1 Class 2 Class 3 Total

85.38% 85.58% 86.06% 85.67%

Class 1 Class 2 Class 3 Total

97.82% 82.30% 89.48% 89.87%

ERmean

Class 1 Class 2 Class 3 Total

85.38% 87.58% 86.12% 86.36%

Class 1 Class 2 Class 3 Total

98.64% 88.60% 92.50% 93.25%

Due to the total conflict between ma3 and ma4 , Dempster’s rule cannot be executed. Using RCR, PCR6, and the mean rule, the combination could be executed; however, the classifications are incorrect based on the pignistic probability transformation, where the classification results are class 2 as shown below:

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BetPRCR ðh1 Þ ¼ 0:0502; BetPRCR ðh2 Þ ¼ 0:6689; BetPRCR ðh3 Þ ¼ 0:2809; BetPPCR6 ðh1 Þ ¼ 0:000; BetPPCR6 ðh2 Þ ¼ 0:6807; BetPPCR6 ðh3 Þ ¼ 0:3193; BetPMean ðh1 Þ ¼ 0:0140; BetPMean ðh2 Þ ¼ 0:6681; BetPMean ðh3 Þ ¼ 0:3180:

where h2 always has the maximum pignistic probability. Using the proposed a0 -cut approach, four other BBAs can 0 be obtained (denoted by mai ð Þ, i = 1, 2, 3, 4). 0 ma1 ðfh2 gÞ 0 ma2 ðfh2 gÞ

¼ 0:4142; ¼ 0:2652;

0 ma1 ðfh2 ; h3 gÞ 0 ma2 ðfh2 ; h3 gÞ

¼ 0:4682; ¼ 0:2669;

0 ma1 ðHÞ 0 ma1 ðHÞ

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¼ 0:1176; ¼ 0:4679;

ma3 ðfh3 gÞ ¼ 0:9569; ma3 ðHÞ ¼ 0:0431; 0 0 ma4 ðfh2 gÞ ¼ 0:6757; ma4 ðHÞ ¼ 0:3243:

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Dempster’s rule of combination, RCR, PCR6 and Mean rule could all be executed when using the a0 -cut approach. After the PPT, their corresponding pignistic probabilities are

Table 8

Fig. 2

Classification accuracy comparison.

Triangular fuzzy number model.

Class

Class 1

Class 2

Class 3

1st dimensions 2nd dimensions 3rd dimensions 4th dimensions

[4.4000, 4.9640, 5.7000] [2.3000, 3.4200, 4.2000] [1.0000, 1.4720, 1.9000] [0.1000, 0.2600, 0.6000]

[5.0000, 5.9800, 7.0000] [2.2000, 2.7000, 3.2000] [3.3000, 4.3240, 5.1000] [1.0000, 1.3040, 1.6000]

[5.6000, 6.6680, 7.7000] [2.2000, 2.9480, 3.8000] [4.8000, 5.6360, 6.9000] [1.5000, 2.0360, 2.5000]

Triangular fuzzy member-based FMF.

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Fig. 3

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FMF generation using triangular fuzzy member for Case 1.

BetPDS ðh1 Þ ¼ 0:0015; BetPDS ðh2 Þ ¼ 0:2260; BetPDS ðh3 Þ ¼ 0:7725;

ma1 ðfh2 gÞ ¼ 0:8689; ma1 ðfh2 ; h3 gÞ ¼ 0:1297; ma1 ðHÞ ¼ 0:0014; ma2 ð Þ : Cannot be generated due to all zero values of FMF;

BetPRCR ðh1 Þ ¼ 0:1712; BetPRCR ðh2 Þ ¼ 0:4827; BetPRCR ðh3 Þ ¼ 0:3462; BetPPCR6 ðh1 Þ ¼ 0:0003; BetPPCR6 ðh2 Þ ¼ 0:4840; BetPPCR6 ðh3 Þ ¼ 0:5157; BetPMean ðh1 Þ ¼ 0:0820; BetPMean ðh2 Þ ¼ 0:5507;

ma3 ðfh1 gÞ ¼ 1; ma4 ðfh1 gÞ ¼ 1:

478

ma2 ð Þ

479

BetPMean ðh3 Þ ¼ 0:3672 where Dempster’s rule, PCR6 has correct classification results (their pignistic probability of h3 is the maximum). (2) Case 2 The test sample is xq = [5.7000, 4.4000, 1.5000, 0.4000] belonging to Class 3. According to the triangular fuzzy number in Table 8 and Eq. (10), the FMFs for xq(j), j = 1, 2, 3, 4 are given below, as shown in Fig. 4. For xq ð1Þ ¼ 5:7000; l1 ðh1 Þ ¼ 0:0010; l1 ðh2 Þ ¼ 0:7143; l1 ðh3 Þ ¼ 0:0936: For xq ð2Þ ¼ 4:4000; l2 ðh1 Þ ¼ 0; l2 ðh2 Þ ¼ 0; l2 ðh3 Þ ¼ 0:

Since does not exist, evidence combination cannot be executed based on any combination rules. Using the proposed a0 -cut approach, four other BBAs can 0 be obtained (denoted by mai ð Þ, i = 1, 2, 3, 4). 0 ma1 ðfh2 gÞ 0 ma1 ðHÞ ¼ 0 ma3 ðfh1 gÞ 0

ma4 ðfh2 gÞ

0 ¼ 0:6207; ma1 ðfh2 ; h3 gÞ ¼ 0:0926; 0 0:2867; ma2 ðHÞ ¼ 1; 0 ¼ 0:9346; ma3 ðHÞ ¼ 0:0654; 0 ¼ 0:5882; ma4 ðHÞ ¼ 0:4118:

Using the Dempster’s rule, RCR, PCR 6, and mean rule followed by the PPT, their pignistic probabilities are

For xq ð3Þ ¼ 1:5000; l3 ðh1 Þ ¼ 0:9346; l3 ðh2 Þ ¼ 0; l3 ðh3 Þ ¼ 0: For xq ð4Þ ¼ 0:4000; l4 ðh1 Þ ¼ 0:5882; l4 ðh2 Þ ¼ 0; l4 ðh3 Þ ¼ 0:

BetPDS ðh1 Þ ¼ 0:9204; BetPDS ðh2 Þ ¼ 0:0671; BetPDS ðh3 Þ ¼ 0:0125; BetPRCR ðh1 Þ ¼ 0:3755; BetPRCR ðh2 Þ ¼ 0:3142; BetPRCR ðh3 Þ ¼ 0:3103; BetPPCR6 ðh1 Þ ¼ 0:6647; BetPPCR6 ðh2 Þ ¼ 0:2380;

By using the traditional a-cut approach, we can generate four BBAs (denoted by mai ð Þ, i = 1, 2, 3, 4 corresponding to four dimensions) of xq.

BetPPCR6 ðh3 Þ ¼ 0:0974; BetPMean ðh1 Þ ¼ 0:5277; BetPMean ðh2 Þ ¼ 0:3137; BetPMean ðh3 Þ ¼ 0:1586:

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Fig. 4

FMF generation using triangular fuzzy member for Case 2.

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The pignistic probability of h1 is always the maximum using any rule here, so xq = [5.7000, 4.4000, 1.5000, 0.4000] can all be correctly classified.

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7. Property of order perseverance for uncertainty degree

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Suppose that there are s FMFs l1( ), l2( ), . . ., ls( ). s BBAs m1( ), m2( ) . . ., ms( ) can be obtained using some transformation Tr. We can calculate the uncertainty degree of the s FMFs denoted by UnF1 , UnF2 , . . ., UnFs , and the uncertainty degree of the s BBAs denoted by UnB1 , UnB2 , . . ., UnBs . By sorting UnF1 , UnF2 . . ., UnFs , the ranking KF can be obtained, and then by sorting UnB1 ; UnB2 ; . . . ; UnBs , the ranking KB can also be obtained. If the ranking KF is close to KB, the transformation Tr loses less information. Such a desired property is called the order perseverance for the uncertainty degree. The new proposed a0 -cut approach has better order perseverance for the uncertainty degree, which is verified by the simulation below. On each run, Step 1. Suppose that the size of FOD is n. Randomly generate an integer s 2 [3, 20]. Then, randomly generate s FMFs and calculate their fuzzy entropy22 according to n X UnF ðlð ÞÞ ¼ C ½lðhi Þlog2 lðhi Þ þ ð1 lðhi ÞÞlog2 ð1 lðhi ÞÞ i¼1

ð15Þ

513 514

where C is a normalization constant.

Sort all the fuzzy entropy values to generate a ranking KF = [r(F1), r(F2), . . ., r(Fi), . . ., r(Fs)] where r(Fi), i = 1, 2, . . ., s denotes the ranking position of Fi. Step 2. Generate s BBAs with the traditional a-cut approach denoted by ma1 ð Þ; ma2 ð Þ; . . . ; mas ð Þ. Calculate their corresponding ambiguity measures (AM)4 AMa1 ; AMa2 ; . . . ; AMas according to X AMðmð ÞÞ ¼ BetPðhÞlog2 BetPðhÞ ð16Þ h2H

where BetP( ) is the pignistic probability defined in Eq. (9). AM has been criticized for the reason that it cannot satisfy the sub-additivity23; however, in our work, there is no problem for the joint BBA (Cartesian space). Therefore, there is no sub-additivity problem and AM can be used as an uncertainty measure for BBA here. Besides AM, aggregated uncertainty (AU) measure24 is also a total uncertainty measure for belief functions; however, it has some limitations. First, AU is optimization-based. It will cause large computational cost. Second, it is insensitive to the change of BBA,4 which is negative for the definition of the order perseverance for uncertainty. The non-specificity25 is also an uncertainty measure in the DST, which means two or more alternatives are left unspecified and represents an imprecision degree. However, it is not a total uncertainty measure, which only describes the non-specificity part of uncertainty in a BBA. In summary, here we choose AM to calculate the uncertainty measure.

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Step 3. Also generate s BBAs with the proposed a0 -cut 0 0 0 approach denoted by ma1 ð Þ; ma2 ð Þ; . . . mas ð Þ and calculate 0 0 a0 their corresponding AM measures AM1 ; AMa2 ; . . . ; AMas . 0 0 0 Sort AMa1 ; AMa2 ; . . . ; AMas and AMa1 ; AMa2 ; . . . ; AMas , 0 respectively, to generate the rankings KaAM and KaAM . Step 4. Calculate the Spearman distance26 a a0 F a F a0 dAM ¼ qðK ; KAM Þ, dAM ¼ qðK ; KAM Þ using Eq. (17) P 2 6 N i¼1 ðK1 ðiÞ K2 ðiÞÞ ð17Þ qðK1 ; K2 Þ ¼ sðs2 1Þ Clearly, q 2 [0, 2]. q = 0 means a total positive correlation between rankings and q = 2 means a total negative one. To be more comprehensive, we can also use another ranking distance, i.e., Kendall distance(Kd)27 defined in Eq. (18) to calculate the distance KdaAM ¼ KenðKF ; KaAM Þ and a0 F a0 KdAM ¼ KenðK ; KAM Þ: 1 X KenðK1 ; K2 Þ ¼ 2 K ðK1 ; K2 Þ ð18Þ Cs i;j2f1;2;::;sg i;j where K i;j ðK1 ;K2 Þ ¼¼

0 If Ii ; Ij are in the same order in K1 and K2 1 If Ii ; Ij are in the inverse order in K1 and K2

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The ranking KaAM = [3, 2, 1]. Using the a0 -cut approach, three corresponding BBAs are 0 0 ma1 ðfh3 gÞ ¼ 0:2543; ma1 ðfh1 ; h3 gÞ ¼ 0:4899; 0 a0 m1 ðfh1 ; h3 ; h4 gÞ ¼ 0:0261; ma1 ðHÞ ¼ 0:2297; 0

590 591 592

ma2 ðfh3 gÞ ¼ 0:1691; ma2 ðfh1 ; h3 gÞ ¼ 0:5900; 0 0 ma2 ðfh1 ; h3 ; h4 gÞ ¼ 0:0841; ma2 ðHÞ ¼ 0:1568; ma3 ðfh1 gÞ ¼ 0:3032; ma3 ðfh1 ; h4 gÞ ¼ 0:1820; 0 0 ma3 ðfh1 ; h2 ; h4 gÞ ¼ 0:2528; ma3 ðHÞ ¼ 0:2620:

594 a0

Their corresponding AM measures are AM ðm1 ð ÞÞ ¼ 0 0 1:4849; AMa ðm2 ð ÞÞ ¼ 1:4605; AMa ðm3 ð ÞÞ ¼ 1:6402. 0 The ranking KaAM = [2, 1, 3]. 0 Spearman distance values are daAM ¼ 1:5000; daAM ¼ 0; a a0 Kendall distance values are KdAM ¼ 0:6667; KdAM ¼ 0: In our simulation, the cardinality of the FOD is set to be 5. The averaging results of the 10,000 times run are 0 0 d aAM ¼ 0:5598; d aAM ¼ 0:5026 and KdaAM ¼ 0:5598; KdaAM ¼ 0 0:5026. It means that the a -cut approach is more preferred in terms of the order perseverance for the uncertainty degree when using two different ranking distances. 8. Conclusions

595 596 597 598 599 600 601 602 603 604 605

606

ð19Þ

565 566

Ii, Ij are two different items in a ranking and C2s ¼ sðs 1Þ=2 is a normalization factor. The steps above are repeated for multiple times. Average 0 0 values of daAM , daAM , KdaAM and KdaAM are obtained. A smaller distance value is preferred within a given ranking distance. It means that the corresponding transformation has better order perseverance for uncertainty degree. Here, we provide an example to illustrate the procedure over a single run. Suppose that the size of FOD n = 4. Three generated FMFs are l1 ðh1 Þ ¼ 0:7407; l1 ðh2 Þ ¼ 0:2247; l1 ðh3 Þ ¼ 0:9951; l1 ðh4 Þ ¼ 0:2508; l2 ðh1 Þ ¼ 0:8288; l2 ðh2 Þ ¼ 0:1547; l2 ðh3 Þ ¼ 0:9979; l2 ðh4 Þ ¼ 0:2388; l3 ðh1 Þ ¼ 0:7837; l3 ðh2 Þ ¼ 0:2984; l3 ðh3 Þ ¼ 0:0456; l3 ðh4 Þ ¼ 0:4804: Their corresponding fuzzy entropy are UnF(l1( )) = 0.6129, UnF(l2( )) = 0.5242, UnF(l3( )) = 0.7248. The ranking KF = [2, 1, 3] (in ascending order, the same holds in the sequel). Then using the a-cut approach, three corresponding BBAs are ma1 ðfh3 gÞ ¼ 0:2556; ma1 ðfh1 ; h3 gÞ ¼ 0:4924; ma1 ðfh1 ; h3 ; h4 gÞ ¼ 0:0263; ma1 ðHÞ ¼ 0:2257; ma2 ðfh3 gÞ ¼ 0:1694; ma2 ðfh1 ; h3 gÞ ¼ 0:5912; ma2 ðfh1 ; h3 ; h4 gÞ ¼ 0:0844; ma2 ðHÞ ¼ 0:1550; ma3 ðfh1 gÞ ¼ 0:3869; ma3 ðfh1 ; h4 gÞ ¼ 0:2322; ma3 ðfh1 ; h2 ; h4 gÞ ¼ 0:3226; ma3 ðHÞ ¼ 0:0583: a

Their corresponding AM measures are AM (m1( )) = 1.4792, AMa(m2( )) = 1.4578, AMa(m3( )) = 1.3759.

In this paper, an improved a-cut approach is proposed, which is free from some drawbacks of the traditional one. Furthermore, the robustness and accuracy for pattern classification can be improved using the new proposed approach. It also has better order perseverance for the uncertainty degree, which means the relation between the BBAs obtained is closer to the relation between the original FMFs. These have been verified by the numerical examples and simulation results provided in this paper. In our work, the criterion for evaluating different approaches is the intuition or the rationality, which are qualitative. Quantitative evaluation criteria are needed for a more objective evaluation or design of better approaches. The order perseverance for the uncertainty degree used in this paper is our attempt on the quantitative evaluation. In our future work, we will try to design more solid performance evaluation approaches or criterion.

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Acknowledgements

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This work was supported by the Grant for State Key Program for Basic Research of China (No. 2013CB329405), National Natural Science Foundation of China (No. 61573275), Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 61221063), Science and Technology Project of Shaanxi Province (No. 2013KJXX-46), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120201120036), and Fundamental Research Funds for the Central Universities of China (No. xjj2014122).

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X. Rong Li received the B.S. and M.S. degrees from Zhejiang University, Hangzhou, Zhejiang, PRC, in 1982 and 1984, respectively, and the M.S. and Ph.D. degrees from the University of Connecticut, Storrs, USA, in 1990 and 1992, respectively. He joined the Department of Electrical Engineering, University of New Orleans, LA, USA, in 1994, where he is now Chancellor’s University Research Professor. He has authored or coauthored 4 books, 10 book chapters, and more than 300 journal and conference proceedings papers. His current research interests include estimation and decision, signal and data processing, information fusion, target information processing, performance evaluation, statistical inference, and stochastic systems. Dr. Li was elected President of the International Society of Information Fusion in 2003 and a member of Board of Directors (1998–2009); served as General Chair for several international conferences; served as Editor (1996– 2003) of the IEEE Transactions on Aerospace and Electronic Systems; received a CAREER award and an RIA award from the U.S. National Science Foundation. He has given more than 150 invited seminars and taught short courses in North America, Europe, Asia, and Australia. He won several outstanding paper awards and consulted for several companies.

719

Han Deqiang is an associate professor and Ph.D. supervisor at the school of electronic and information engineering, Xi’an Jiaotong University, China. He received the Ph.D. degree from the same university in 2008. His current research interests are evidence theory, pattern classification and information fusion.

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