A belief combination rule for a large number of sources
11 A belief combination rule for a large number ofsources
Kuang Zhou a , Arnaud Martin b , and Quan Pan a a. Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China.b. DRUID, IRISA, University of Rennes 1, Rue E. Branly, 22300 Lannion, France Abstract —The theory of belief functions is widely used for datafrom multiple sources. Different evidence combination rules havebeen proposed in this framework according to the propertiesof the sources to combine. However, most of these combinationrules are not efficient when there are a large number of sources.This is due to either the complexity or the existence of anabsorbing element such as the total conflict mass function for theconjunctive based rules when applied on unreliable evidence. Inthis paper, based on the assumption that the majority of sourcesare reliable, a combination rule for a large number of sourcesis proposed using a simple idea: the more common ideas thesources share, the more reliable these sources are supposed tobe. This rule is adaptable for aggregating a large number ofsources which may not all be reliable. It will keep the spirit ofthe conjunctive rule to reinforce the belief on the focal elementswith which the sources are in agreement. The mass on the emptyset will be kept as an indicator of the conflict.The proposed rule, called LNS-CR (Conjunctive combinationRule for a Large Number of Sources), is evaluated on syntheticmass functions. The experimental results verify that the rule canbe effectively used to combine a large number of mass functionsand to elicit the major opinion.
Index Terms —Theory of belief functions, big data, combina-tion, large number of sources, reliability
I. I
NTRODUCTION I N recent years, Dempster–Shafer Theory (DST), also calledthe theory of belief functions, has gained increasing at-tention in the scientific community as it allows to the dealwith the imprecise and uncertain information. It has beenapplied in various domains, such as data classification [2, 3],data clustering [4, 5], social network analysis [6], etc. Incomplex environment, multiple stake-holders attempt to reacha decision by combining several sources of information and ag-gregating their points of view by stressing common agreement.The theory of belief functions, which has provided many rulesto combine information represented by mass functions [7], arewidely used for decision making. In real applications, thereare usually a large number of sources. Most of the existingcombination rules are not applicable in this case, and cannotbe used to find the major opinion from many participants.One of the most famous combination rule in belief functionframework is the Dempster’s rule [7]. Smets [8] proposeda modification of Dempster’s rule, often called “conjunctiverule”, where the empty set can be assigned with a non-nullmass under the Transferable Belief Model (TBM) [9]. In fact,the conjunctive rule is equivalent to the Dempster rule without
This paper is an extension and revision of [1]. the normalization process. It has a fast and clear convergencetowards a solution. But this rule has a strong assumption thatall the sources are reliable. In real applications, it is difficult tobe either satisfied or verified. Moreover, the more sources thereare, the more chance that there is some unreliable evidence.Smets [8] reasoned that the mass on the empty set canplay the role of alarm. When the global conflict (the massassigned to the empty set) is high, it indicates that there isstrong disagreement among the sources of mass functions tocombine. However, as observed in [10, 11, 12], the mass on theempty set is not sufficient to exactly describe the conflict sinceit includes an amount of auto-conflict [13]. Sometimes whenthere is only a small amount of concordant evidence, the totalconflict mass function, i.e. m ( ∅ ) = 1 will be an absorbingelement. Consequently, when combining a large number of(incompatible) mass functions using the conjunctive rule, theglobal conflict may tend to 1. This makes it impossible toreveal the cause of high global conflict. We do not knowwhether it is due to the sources to fuse or caused by theabsorption power of the empty set [10, 14]. In other words,even the combined mass function by the conjunctive ruleis m ( ∅ ) ≈ , the proposition that the sources are highlyconflicting may be incorrect.In order to rectify the drawbacks of the classical Dempster’srule and Smets’ conjunctive rule, many approaches have beenmade through the modification of the combination rule. Someauthors tried to find alternative repartitions of the conflict. Aplethora of combination rules have been brought forward inthis way. For example, Yager [15] and Dubois and Prade [16]suggested assigning the highly conflicting mass to the wholeset or a particular set. The Proportional Conflict Redistribution(PCR) rule, which can distribute the partial conflicts amongthe involved focal elements rather than to their union, isdeveloped in [13, 17]. Apart from these approaches workingdirectly on the combination rule, some studies manage theconflict through evidence discounting, where the reliabilityof sources is automatically and adaptively taken into account[10, 16, 18, 19].Most of the existing combination rules are not efficient whenapplied on a large number of sources due to the ineffective wayto handle conflict or the high complexity of the computation.Orponen [20] proved that the complexity of the conjunctiverule is NP-hard, but the complexity depends on the way toprogram the belief functions [21]. Some rules can manageefficiently the conflict but have large complexity [13, 16, 22,23], making them infeasible when applied to combine a large a r X i v : . [ c s . A I] S e p number of mass functions.In this paper, a conjunctive-based combination rule, namedLNS-CR (Large Number of Sources), is proposed to aggregatea large number of mass functions. Our perspective on belieffunction combination is that combining mass functions fromdifferent sources is similar to combining opinions from multi-ple stake-holders in group decision-making [24], i.e. the moreone’s opinion is consistent with the other experts, the morereliable the source is. We assume that all the mass functionsavailable are separable mass functions, which means they canbe expressed by a group of simple support mass functions. Inmany applications, the mass assignments are directly in theform of Simple Support Functions (SSF) [25]. The advantageof SSFs is that we can group the mass functions in such away that sources in the same group share the same viewpoint.Mass functions in each small group are first fused and thendiscounted according to the proportions. After that the numberof mass functions participating the next global combinationprocess is independent of the number of sources, but onlydepends on the number of classes. As a result, the problembrought by the absorbing element (the empty set) using theconjunctive rule can be avoided. Moreover, an approximationmethod when the number of mass functions is large enough ispresented. The main contributions of this paper are as follows: • A new conjunctive-based combination rule, namedLNS-CR rule, is brought froward. The property to re-inforce the belief on the focal elements with which mostof the sources agree is preserved in the proposed rule; • The assumption of the LNS-CR rule on the reliability ofthe sources is more relaxed, as it does not require all thesources are reliable, but only at least half of them arereliable. • LNS-CR can be used to combine mass functions from alarge number of sources, especially can be used to elicitthe major opinion; • Derivation that the LNS-CR rule is within acceptablecomplexity.The rest of this paper is organized as follows. In Section2, some basic knowledge of belief function theory is brieflyintroduced. The proposed evidence combination approach ispresented in detail in Section 3. Numerical examples areemployed to compare different combination rules and showthe effectiveness of LNS-CR rule in Section 4. Finally, Section5 concludes the paper.II. B
ACKGROUND
A. Basic knowledge of belief function theory
Let
Θ = { θ , θ , . . . , θ n } be the discernment frame. A massfunction is defined on the power set Θ = { A : A ⊆ Θ } . Themass function m : 2 Θ → [0 , is said to be a Basic BeliefAssignment (bba) on 2 Θ , if it satisfies: (cid:88) A ⊆ Θ m ( A ) = 1 . (1)Every A ∈ Θ such that m ( A ) > is calleda focal element, and the set of focal elements is de-noted by F . In a practical way of programming, the element of Θ can be arranged by natural order [26]: θ , θ , { θ , θ } , θ , · · · , { θ , θ , θ } , θ , · · · , Θ . The frame of discernment can also be a focal element.If Θ is a focal element, the mass function is called non-dogmatic. The mass assigned to the frame of discernment, m (Θ) , is interpreted as a degree of ignorance. In the caseof total ignorance, m (Θ) = 1 . This type of mass assign-ment is vacuous. If there is only one focal element, i.e. m ( A ) = 1 , A ⊂ Θ , the mass function is categorical. Anotherspecial case of assignment is named consonant mass functions,where the focal elements include each other as a subset, i.e. if A, B ∈ F , A ⊂ B or B ⊂ A .The credibility and plausibility functions are derived froma bba m as in Eqs. (2) and (3): Bel ( A ) = (cid:88) B ⊆ A,B (cid:54) = ∅ m ( B ) , ∀ A ⊆ Θ , (2) P l ( A ) = (cid:88) B ∩ A (cid:54) = ∅ m ( B ) , ∀ A ⊆ Θ . (3)Each quantity Bel ( A ) measures the minimal belief on A justified by available information on B ( B ⊆ A ) , while P l ( A ) is the maximal belief on A justified by informationon B which are not contradictory with A ( A ∩ B (cid:54) = ∅ ). Thecommonality function q and the implicability function b aredefined respectively as q ( A ) = (cid:88) A ⊆ B m ( B ) , ∀ A ⊆ Θ (4)and b ( A ) = Bel ( A ) + m ( ∅ ) , ∀ A ⊆ Θ . (5)A bba m can be recovered from any of these functions. Forinstance, m ( A ) = (cid:88) B ⊇ A ( − | B |−| A | q ( B ) , ∀ A ⊆ Θ (6)and m ( A ) = (cid:88) B ⊆ A ( − | A |−| B | b ( B ) , ∀ A ⊆ Θ . (7)Belief functions can be transformed into a probabilityfunction by Smets’ method [27], where each mass of belief m ( A ) is equally distributed among the elements of A . Thisleads to the concept of pignistic probability, BetP . For all θ i ∈ Θ , we have BetP( θ i ) = (cid:88) A ⊆ Θ | θ i ∈ A m ( A ) | A | (1 − m ( ∅ )) , (8)where | A | is the cardinality of set A (number of elements of Θ in A ). Pignistic probabilities can help make a decision. B. Consistency of mass assignments
The consistency between two bbas can be defined in twodifferent ways. Suppose the sets of focal elements for m and m are F and F respectively. Mass functions m and m are called strong consistent if and only if ∩ E ∈{F ∪F } (cid:54) = ∅ . (9) Meanwhile, bbas m and m are called weak consistent if andonly if ∀ A ∈ F , B ∈ F , A ∩ B (cid:54) = ∅ . (10)Strong consistent evidence means that there is at least oneelement that is common to all subsets [28]. It is easy to seethat, when m and m are strong consistent, they are sure to beweak consistent. This is the definition of consistency betweenbelief functions. The inconsistency within an individual massassignment can be defined similarly [12]. C. Reliability-based discounting
When the sources of evidence are not completely reliable,the discounting operation proposed by Shafer [25] and justifiedby Smets [29] could be applied. Denote the reliability degreeof mass function m by α ∈ [0 , , then the discountingoperation can be defined as: m (cid:48) ( A ) = (cid:40) α × m ( A ) ∀ A ⊂ Θ , − α + α × m (Θ) if A = Θ . (11)If α = 1 , the evidence is completely reliable and the bba willremain unchanged. On the contrary, if α = 0 , the evidenceis completely unreliable. In this case the so-called vacuousbelief function, m (Θ) = 1 , could be got. It describes the totalignorance.Before evoking the discounting process, the reliability ofeach sources should be known. One possible way to estimatethe reliability is to use confusion matrices [30]. Generally,the goal of discounting is to reduce global conflict beforecombination. One can assume that the conflict comes fromthe unreliability of the sources. Therefore, the source reliabilityestimation is to some extent linked to the estimation of conflictbetween sources.Hence, Martin et al. [10] proposed to use a conflict measureto evaluate the relative reliability of experts. Once the degreeof conflict is computed, the relative reliability of the sourcecan be computed accordingly. Suppose there are S sources, S = { s , s , · · · , s S } , the reliability discounting factor α j ofsource s j can be defined as follows: α j = f (Conf ( s j , S )) , (12)where Conf ( s j , S ) quantifies the degree that source s j con-flicts with the other sources in S , and f is a decreasingfunction. The following function is suggested by the authors: α j = (cid:16) − Conf ( s j , S ) λ (cid:17) λ , (13)where λ > .In [31], the authors considered to use those two possibleconflict origins, extrinsic measure and intrinsic measure, toestimate reliability. In their opinion, conflict may not onlycome from the source’s contradiction (extrinsic measure), butalso from the confusion rate of a source (intrinsic measure).The reliability discounting factor, called Generic DiscountingFactor (GDF), is then suggested to be a weighted sum of thetwo items: α = kδ + lβk + l , (14) where k > , l > are the weight factors. In the aboveequation, δ denotes the internal conflict measure of thetreated source indicating its confusion rate while β is theaverage distance between the treated sources s i and s j where j ∈ S , j (cid:54) = i . Different intrinsic and extrinsic conflict measurescan be adopted here.There are some other methods to estimate the reliability.In [32], the authors proposed to estimate the reliability ofsources based on a degree of falsity. The bbas are sequentiallyand incrementally discounted until the mass assigned to theempty set is smaller than a given threshold k . After thatthe discounted mass functions can be combined using theconjunctive rule since there is little global conflict at this time.In [33], the source reliability is obtained by minimizing thedistance between the pignistic probabilities computed from thediscounted beliefs and the actual value of the data. In Sametet al. [34], the authors proposed two different versions ofgeneric discounting approaches: weighted GDA and exponentGDA. A new degree of disagreement is proposed by Yang et al.[35], where the reliability discounting factor can be generated.Klein and Colot [36] viewed the degree of conflict as afunction of discounting rates and introduced a new criterionassessing bbas’ reliability. These reliability estimation methodseither consider the distance (or dissimilarity) between eachpair of bbas, or the mass assigned to the empty set after theconjunctive combination. However, these methods are of highcomplexity and not suitable for large data applications. D. Simple support function
Suppose m is a bba defined on the frame of discernment Θ .If there exists a subset A ⊆ Θ such that m could be expressedin the following form: m ( X ) = w X = Θ , − w X = A, otherwise . (15)where w ∈ [0 , , then the belief function related to bba m is called a Simple Support Function (SSF) (also called simplemass function) [25] focused on A . Such a SSF can be denotedby A w ( · ) where the exponent w of the focal element A is thebasic belief mass (bbm) given to the frame of discernment Θ , m (Θ) . The complement of w to 1, i.e. − w , is the bbmallocated to A [37]. If w = 1 the mass function represents thetotal ignorance, if w = 0 the mass function is a categoricalbba on A .A belief function is separable if it is a SSF or if it is theconjunctive combination of some SSFs [38]. In the work of[38], this kind of separable masses is called u-separable where“u” stands for “unnormalized”, indicating the conjunctive ruleis the unnormalized version of Dempster-Shafer rule. The setof separable mass functions is not obvious to obtain. It is easyto see consonant mass functions (the focal element are nested)are separable [39]. Smets [37] defined the Generalized SimpleSupport Function (GSSF) by relaxing the weight w to [0 , ∞ ) .Those GSSFs with w ∈ (1 , ∞ ) are called Inverse SimpleSupport Functions (ISSF). Smets proved all non-dogmaticmass functions are separable if one uses GSSFs. For any non-dogmatic belief function m , the canonical decompositionmethod proposed by Smets is as follows. First, calculate thecommonality number for all focal elements, which is given by Q ( X ) = (cid:88) B ⊇ X m ( B ) . (16)Secondly for any A ⊆ Θ , calculate w A value as follows: w A = (cid:89) X ⊇ A Q ( X ) ( − | X |−| A | +1 . (17)Then the belief function m can be represented by the con-junctive combination of all the functions A w A , i.e. m = ∩ (cid:13) A ⊆ Θ A w A , (18)where ∩ (cid:13) denotes the conjunctive combination rule. For fastcomputation, the Fast M¨obius Transform (FMT) method [40]can be evoked. E. Some combination rules
How to combine efficiently several bbas coming fromdistinct sources is a major information fusion problem in thebelief function framework. Many rules have been proposed forsuch a task. Here we just briefly recall how some most popularrules are mathematically defined.When information sources are reliable, the used fusionoperators can be based on the conjunctive combination. If bbas m j , j = 1 , , · · · , S describing S distinct items of evidence on Θ , the included result of the conjunctive rule [9] is definedas m conj ( X ) = ( ∩ (cid:13) j =1 , ··· ,S m j )( X ) = (cid:88) Y ∩···∩ Y S = X S (cid:89) j =1 m j ( Y j ) , (19)where m j ( Y j ) is the mass allocated to Y j by expert j . To applythis rule, the sources are assumed reliable and cognitivelyindependent.Another kind of conjunctive combination is Dempster’srule [41]. Assuming that m conj ( ∅ ) (cid:54) = 1 , the result of thecombination by Dempster’s rule is m Dempster ( X ) = (cid:40) if X = ∅ , m conj ( X )1 − m conj ( ∅ ) otherwise . (20)The item κ (cid:44) m conj ( ∅ ) = (cid:88) Y ∩···∩ Y S = ∅ S (cid:89) j =1 m j ( Y j ) is generally called Dempster’s degree of conflict of the com-bination or the inconsistency of the combination. As the con-junctive rule is not idempotent, m conj ( ∅ ) includes an amountof auto-conflict [42], and it is called global conflict to makethe difference.The conjunctive rule can be applied only if all the expertsare reliable. In the other case, the disjunctive rule [43], whichonly assumes that at least one of the sources is reliable, can be used. The disjunctive combination of S sources can be definedas m disj ( X ) = (cid:18) ∪ (cid:13) j =1 , ··· ,S m j (cid:19) ( X ) = (cid:88) Y ∪···∪ Y S = X S (cid:89) j =1 m j ( Y j ) . (21)The conjunctive and disjunctive rules can be convenientlyexpressed by means of the commonality function q (Eq. (4))and the implacability function b (Eq. (5)) [43]. Let q i and b i bethe commonality function and implacability function respec-tively (associated with m i ), then the commonality function ofthe conjunctive combination of S bbas is q conj ( A ) = S (cid:89) i =1 q i ( A ) , ∀ A ⊆ Θ (22)while the implacability function of the disjunctive combinationof S bbas is b disj ( A ) = S (cid:89) i =1 b i ( A ) , ∀ A ⊆ Θ . (23)Since functions m , q and b (as well as bel and pl ) areequivalent representations, the mass function m can be recov-ered using the Fast M¨obius Transform (FMT) method giventhe functions q and b . The conversion can be done in timeproportional to n n [44] ∗ . For the conjunctive combination of S sources, the S bbas should be converted into commonalityfunctions first. After calculating the product of S commonalityfunctions, another transformation from m to q should beevoked. Overall the total complexity is O ( Sn n + S n + n n ) ,and the time needed is proportional to Sn n [44, 45].The conflict could be redistributed on partial ignorance likein the Dubois and Prade rule ( DP rule ) [16], which can beseen as a mixed conjunctive and disjunctive rule. For all X ⊆ Θ , X (cid:54) = ∅ : m DP ( X ) = (cid:88) Y ∩···∩ Y S = X S (cid:89) j =1 m j ( Y j )+ (cid:88) Y ∪ · · · ∪ YS = XY ∩ · · · ∩ YS = ∅ S (cid:89) j =1 m j ( Y j ) , (24)where m j is the mass function delivered by expert j . In ageneral case, this rule cannot be programmed with the FastM¨obius Transform method because all the partial conflict mustbe considered. If the implementation is made like that inRef. [46], it takes much more time than the conjunctive rule.Denœux [38] proposed a family of conjunctive and disjunc-tive rules using triangular norms. The cautious rule [47, 48]belongs to that family and could be used to combine massfunctions for which independence assumption is not verified.Cautious combination of S non-dogmatic mass functions ∗ This is based on the assumption that the mass functions are arranged innatural order. If not, the complexity is proportional to n n . The complexityanalysis in this work all assumes that the bbas to be combined are encodedusing the natural order. m j , j = 1 , , · · · , S is defined by the bba with the followingweight function: w ( A ) = S ∧ j =1 w j ( A ) , A ∈ Θ \ Θ . (25)We thus have m Cautious ( X ) = ∩ (cid:13) A (cid:40) Θ A S ∧ j =1 w j ( A ) , (26)where A w j ( A ) is the simple support function focused on A with weight function w j ( A ) issued from the canonical decom-position of m j . Note also that ∧ is the min operator. The timeconsumption of the cautious rule includes the canonical de-composition of non-dogmatic mass functions and is thereforebigger than the conjunctive rule. If this rule is implemented inFast M¨obius Transform method, the complexity is proportionalto Sn n .Murphy [49] presented the average combination rule andproposed to utilize the mean of the basic belief assignmentsas the fusion of evidence. Therefore, for each focal element X ∈ Θ of S mass functions, the combined one is defined asfollows: m Ave ( X ) = 1 S S (cid:88) j =1 m j ( X ) , ∀ X ⊆ Θ . (27)The complexity of the average is proportional to S n .A family of fusion rules based on new Proportional ConflictRedistributions (PCR) for the combination of uncertaintyand conflicting information have been developed in Dezert–Smarandache Theory (DSmT) framework [50]. Among them,the fusion rule called PCR6 proposed by Martin and Osswald[13] is one of the most popular one among the PCR rules. Forthe combination of
S > sources, the fused mass is given by m PCR6 ( ∅ ) = 0 , and for X (cid:54) = ∅ in Θ m PCR6 ( X ) = m conj ( X ) + S (cid:88) i =1 (cid:40) ( m i ( X )) × (cid:88) (cid:84) S − k =1 Y σ i ( k ) ∩ X ≡ ∅ (cid:0) Y σ i (1) , · · · , Y σ i ( S − (cid:1) ∈ (cid:0) Θ (cid:1) S − S − (cid:81) j =1 m σi ( j ) ( Y σi ( j ) ) m i ( X )+ S − (cid:80) j =1 m σi ( j ) ( Y σi ( j ) ) (cid:41) , (28)where σ i counts from 1 to S avoiding i : (cid:40) σ i ( j ) = j if j < i,σ i ( j ) = j + 1 if j ≥ i. (29)As Y i is a focal element of expert/source i , we have m ( Y i ) > . Then m i ( X ) + S − (cid:88) j =1 m σ i ( j ) (cid:0) Y σ i ( j ) (cid:1) (cid:54) = 0 . In Eq. (28), m conj is the conjunctive rule given by Eq. (19).Here again, the Fast M¨obius Transform method to program thebelief functions is not generally the best way. If the implemen-tation is made like that in Ref. [46], the time consumption isvery high. III. A COMBINATION RULE FOR A LARGE NUMBER OFMASS FUNCTIONS
The main idea of the conjunctive combination rule is toreinforce the belief on the focal elements with which most ofthe sources agree. Martin et al. [10] showed that the mass onthe empty set, which is an absorbing element, tends quicklyto 1 with the number of sources when combining inconsistentbbas. Consequently, when using Dempster rule (Eq. (20)), thegap between κ and 1 may rapidly exceed machine precision,even if the combination is valid theoretically. In that case thefused bba by the conjunctive rules (normalized or not) and thepignistic probability are inefficient. Moreover, the assumptionthat all the sources are reliable for the conjunctive combinationrule is difficult to reach in real applications. The more sourcesthere are, the less chance that this assumption is valid.The principle of the conjunctive rule with the reinforcementof belief and the role of the empty set as an alarm are essentialin the theory of belief functions. In order to propose a rulewhich can be adapted to the combination of a large number ofmass functions and keep the previous behavior, the followingassumptions are made: • The majority of sources are reliable; • The larger extent one source is consistent with others,the more reliable the source is; • The sources are cognitively independent [43].These assumptions seem reasonable if we consider combingmass functions as some kind of group decision making prob-lems. As a result, the proposed rule will give more importanceto the groups of mass functions that are in a domain, andit is without auto-conflict [13, 14]. In order to take intoaccount this effect, this rule will discount the mass functionsaccording to the number of sources giving bbas with the samefocal elements. The discounting factor is directly given by theproportion of mass functions with the same focal elements.This procedure is for the elicitation of the majority opinion.The simple support mass functions are considered here. Inthis case, the mass functions can be grouped in the light oftheir focal elements (except the frame Θ ). To make the ruleapplicable on separable mass functions, the decompositionprocess should be performed to decompose each bba intosimple support mass functions. In most of applications, thebasic belief can be defined using separable mass functions,such as simple support functions [2] and consonant massfunctions [51, 52].Hereafter we describe the proposed LNS-CR rule for sim-ple support functions, and then an approximation calculationmethod of LNS-CR rule is suggested. A. LNS-CR rule for simple support functions
Suppose that each evidence is represented by a SSF. Thenall the bbas can be divided into at most n groups (where n = | Θ | ). It is easy to see that there is no conflict at allin each group because of consistency. The focal elements ofthe SSF are singletons and Θ itself. For the combination ofbbas inside each group, the conjunctive rule can be employeddirectly. Then the fused bbas are discounted according to thenumber of mass functions in each group. Finally, the global combination of the bbas of different groups is preformed alsousing the conjunctive rule. Suppose that all bbas are defined onthe frame of discernment Θ = { θ , θ , · · · , θ n } , and denotedby m j = ( A i ) w j , j = 1 , · · · , S and i = 1 , , · · · , c , where c ≤ n . The detailed process of the combination is listed asfollows. Our proposed rule called LNS-CR for Large Numberof Sources rule is composed of the four following steps:1) Cluster the simple bbas into c groups based on their focalelement A i . For the convenience, each class is labeledby its corresponding focal element.2) Combine the bbas in the same group. Denote the com-bined bba in group A k by SSF ˆ m k = ( A k ) ˆ w k , k = 1 , , · · · , c. Let the number of bbas in group A k is s k . If theconjunctive rule is adopted, we have ˆ m k = ∩ (cid:13) j =1 , ··· ,s k m j = ( A k ) s k (cid:89) j =1 w j . (30)3) Reliability-based discounting. Suppose the fused bba ofall the mass functions in A k is ˆ m k . At this time, eachgroup can be regarded as a source, and there are c sources in total. The reliability of one source can beestimated as compared to a group of sources. In ouropinion, the reliability of source A k is related to theproportion of bbas in this group. The larger the numberof bbas in group A k is, the more reliable A k is. Thenthe reliability discounting factor of ˆ m k can be definedas: α k = s kc (cid:88) i =1 s i . (31)In order to keep the mass function representing totalignorance as a neutral element of the rule, in Eq. (31) welet a k = 0 for the group with A k = Θ . Another versionof the discounting can be given by a factor taking intoaccount the precision of the group by: α k = β ηk s kc (cid:88) i =1 β ηi s i , (32)where β k = | Θ || A k | . (33)Parameter η can be used to adjust the precision of thecombination results. The larger the value of η is, theless imprecise the resulting bba is. The discounted bbaof ˆ m k can be denoted by SSF ˆ m (cid:48) k = ( A k ) ˆ w (cid:48) k with ˆ w (cid:48) k =1 − α k + α k ˆ w k . As we can see, when the number of bbasin one group is larger, α is closer to 1. That is to say,the fused mass in this group is more reliable.4) Global combine the fused bbas in different groups usingthe conjunctive rule: m LNS-CR = ∩ (cid:13) k =1 , ··· ,c ˆ m (cid:48) k = ∩ (cid:13) k =1 , ··· ,c ( A k ) ˆ w (cid:48) k . (34) Remarks: • The reliability estimation method proposed here is verysimple compared with the previous mentioned methodsin Section II-C, where usually the distance between bbasshould be calculated or a special learning process isrequired. In the LNS-CR rule, to evaluate the reliabilitydiscounting factor, we only need to count the number ofSSFs in each group. Note that other reliability estimationmethods can also be used here. • In the last step of combination, as the number of massfunctions that take part in the global combination issmall (at most n ), other combination rules such as DP rule and PCR rules are also possible in practice insteadof Eq. (34). B. LNSa-CR rule for the approximated combination
If there is a large number of mass functions in each group,an approximation method is suggested here to calculate thecombined mass in the given group. Suppose the mass functionsin group with focal element A k ( k = 1 , , · · · , c ) are: m j ( A ) = − w j A = A k ,w j A = Θ , otherwise , ≤ w j < , j = 1 , , · · · , s k . (35)The combination of the masses in this group using the con-junctive rule is ˆ m k ( A ) = − s k (cid:81) j =1 w j A = A k , s k (cid:81) j =1 w j A = Θ , otherwise . (36)It is easy to get lim s k →∞ ˆ m k ( A ) = A = A k , A = Θ , otherwise . (37)This is an illustration of the conjunctive property. After thediscounting with factor α k , the fused bba using for the globalcombination is lim n k →∞ ˆ m (cid:48) k ( A ) = α k A = A k , − α k A = Θ , otherwise . (38)It can be represented by SSF ˆ m (cid:48) k = ( A k ) − α k , (39)where α k is shown in Eq. (31) or (32). If the conjunctive ruleis adopted for the global combination at step 4, the final bbawe get is m LNSa-CR = ∩ (cid:13) ( A k ) − α k . (40)In this approximate rule for the large number of sources,the initial mass functions is no longer considered, and thecombination process of the bbas inside each group is notrequired any more. This can accelerate the algorithm to a large extent. The LNS-CR and LNSa-CR rule provide differentresults when the number of sources is small. However, whenthe number of sources is large enough, they can be regardedas equivalent. C. Properties
The proposed rule is commutative, but not associative. Therule is not idempotent, but there is no absorbing element. Thevacuous mass function is a neutral element of the LNS-CRrule.There are four steps when applying LNS-CR rule † : decom-position (not necessary for simple support mass functions),inner-group combination, discounting and global combination.The LNS-CR rule has the same memory complexity as someother rules such as conjunctive, Dempster and cautious rulesif all the rules are combined globally using FMT method. Only DP and PCR6 rules have higher memory complexity becauseof the partial conflict to manage. Suppose the number of massfunctions to combine is S , and the number of elements in theframe of discernment is n . The complexity for decomposing ‡ mass functions to SSFs is O ( Sn n ) . For combining themass functions in each group, due to the structure of thesimple support mass functions, we only need to calculate theproduct of the masses on only one focal element Θ . Thusthe complexity is O ( S ) . The complexity of the discountingis O (2 n ) . In the process of global combination, the bbasare all SSFs. If we use the Fast M¨obius Transform method,the complexity is O ( n n ) . And there are at most n massfunctions participating the following discounting and globalconjunctive combination processes. Since in most applicationcases with a large number of mass functions, we have n (cid:28) S ,the last two steps are not very time-consuming. The totalcomplexity of LNS-CR is O ( Sn n + S + 2 n + n n ) and sois approximately equivalent to O ( Sn n ) .For the approximate method, we can also save the time forinner combination and the discounting. The fused mass in eachgroup is calculated by the proportions, and the complexity isalso O ( S ) . Although the approximate method does not reducethe complexity, in the experimental part, we will show that itwill save some running time in applications when S is quitelarge.We remark here that one of the assumptions of LNS-CRrule is that the majority of sources are reliable. However, thiscondition is not always satisfied in every applicative context.Consider here an example with two sensor technologies: TAand TB. The system has two TA-sensors ( S and S ), andone TB-sensor S . Suppose also a parasite signal causes TAsensors to malfunction. In this situation, the majority of sen-sors are unreliable. And we could not get a good result if theLNS-CR rule is used directly as LNS-CR ( S , S , S ) at thistime. Actually there is an underlying hierarchy in the sourcesof information, LNS-CR rule could be evoked according tothe hierarchy, such as LNS-CR ( LNS-CR ( S , S ) , S ) . We willstudy that more in the future work. † The source code for LNS-CR rule can be found in R package ibelief [53]. ‡ In the decomposing process, the Fast M¨obius Transform method is used.
IV. E
XPERIMENTS
In this section, several experiments will be conductedto illustrate the behavior of the proposed combination ruleLNS-CR and to compare with other classical rules. Somedifferent types of randomly generated mass functions will beused. The function
RandomMass in R package ibelief [53] isadopted to generate random mass functions [54].
Experiment 1 (Elicitation of the majority opinion). In someapplications, the elicitation of the majority opinion is veryimportant. In this experiment, it is assumed that reliablesources can provide some imprecise and uncertain information,which is assumed to be in the form of the mass functions m j ( j = 1 , , · · · , over the same discernment frame Θ = { θ , θ , θ } : m : m ( { θ } ) = 0 . , m (Θ) = 0 . ,m : m ( { θ } ) = 0 . , m (Θ) = 0 . ,m : m ( { θ } ) = 0 . , m (Θ) = 0 . ,m : m ( { θ } ) = 0 . , m (Θ) = 0 . ,m : m ( { θ } ) = 0 . , m (Θ) = 0 . ,m : m ( { θ } ) = 0 . , m (Θ) = 0 . . As can be seen, the first five sources share similar belief(supporting { θ } ) whereas the sixth one delivers a mass func-tion strongly committed to another solution (supporting { θ } ).These six mass functions cannot be regarded as conflicting,because the majority of evidence shows the preference of { θ } .Here, source 6, is assumed not reliable since it contradicts withall the other sources.The combination results by conjunctive rule, Dempster rule, disjunctive rule, DP rule, PCR6 rule, cautious rule,average rule and the proposed LNS-CR rule § are depictedin Table I. As can be observed, the conjunctive rule assignsmost of the belief to the empty set, regarding the sourcesas highly conflictual. Dempster rule, DP rule, PCR6 ruleand average rule redistribute all the global conflict to otherfocal elements. The disjunctive rule gives the total ignorancemass functions. The cautious rule and the proposed LNS-CRrule keep some of the conflict and redistribute the remaining.But the belief given to { θ } is more than that to { θ } whenusing Dempster , DP , PCR6 , cautious and the average rules,which indicates that these rules are not robust to the unreliableevidence. The obtained fused bba by the proposed rule assignsthe largest mass to focal element { θ } , which is consistent withthe intuition. It keeps a certain level of global conflict, and atthe same time reflects the superiority of { θ } compared with { θ } . From the results we can see that only the LNS-CR rulecan correctly elicit the major opinion.The LNS-CR rule is a conjunctive based combination rulefor mass functions with different reliability degrees. As men-tioned before, the principle of the LNS-CR rule is similarthat of Schubert’s method [32]. Table II lists the results bySchubert’s combination method with different values of k . Ascan be seen, the result by the use of the LNS-CR rule issimilar to that by Schubert’s method with a small value of § As the focal elements are singletons except Θ , parameter η has no effectson the final results when using LNS-CR rule. TABLE IT
HE COMBINATION OF SIX MASSES . F
OR THE NAMES OF COLUMNS , θ ij IS USED TO DENOTE { θ i , θ j } . Conjunctive
Dempster
Disjunctive DP PCR6 Cautious Average LNS-CR ∅ { θ } { θ } { θ , θ } { θ } { θ , θ } { θ , θ } Θ threshold k . When k is set small, the discounting process inSchubert’s method needs more steps. And in each step, theconjunctive rule should be evoked to calculate the falsity. It ismore complex compared with the reliability estimation processof the LNS-CR rule in that sense. TABLE IIT
HE COMBINATION OF SIX MASSES BY S CHUBERT ’ S METHOD WITHDIFFERENT VALUES OF k . k ∅ { θ } { θ } { θ , θ } { θ } { θ , θ } { θ , θ } Θ We also compare with another reliability discounting basedcombination method proposed by Martin et al. [10]. Sameas Schubert’s method, after the reliability degree of eachsource is estimated, the bbas are discounted following witha conjunctive combination. There is a parameter λ in themethod to adjust the discounting factor. The results varyingwith different values of λ are shown in Table III. We can seethis rule is similar to LNS-CR rule when λ is set to be around1. When λ is not well set, the results are not good. Moreover,in this method, the distance between bbas should be calculatedfirst. Consequently, it increases the complexity and makes themethod not feasible for combining a large number of sources. TABLE IIIT
HE COMBINATION OF SIX MASSES BY M ARTIN ’ S METHOD WITHDIFFERENT VALUES OF λ . λ ∅ { θ } { θ } { θ , θ } { θ } { θ , θ } { θ , θ } Θ Experiment 2 (The discounting mechanism). In this experi-ment, we will discuss the reliability discounting mechanismof the LNS-CR rule. Two reliability discounting methodsproposed by Schubert [32] and Martin et al. [10] will be used to compare. Same as the LNS-CR rule, after the discountingprocess by these two methods, the conjunctive rule is adoptedto combine the new mass functions. For simplicity, here wecall the combination rule, where the Schubert’s discountingmethod (or Martin’s discounting method) is first evoked andthen the conjunctive combination rule is used, “Schubert’smethod” (Martin’s method, correspondingly). A set of ∗ x bbas on a frame of discernment Θ = { θ , θ } are generated, x of them are unreliable while ∗ x are reliable. The reliablesources assign a large mass to the singleton { θ } . The unre-liable sources assign a large mass to the singleton { θ } . Thegain factor for sequential discounting in Schubert’s method isset to be 0.1 here. Schubert and Martin’s methods are evokedwith different values of k and λ respectively. Let x = 10 , thefused bbas by the use of different rules are listed in Table IV.From the table we can see, the behavior of Martin’s dis-counting method is similar to that of LNS-CR rule when λ is set around 0.4. The conjunctive combination based onSchubert’s discounting does not give any belief to { θ } and Θ = { θ , θ } at all although there are / of sourcessupporting { θ } . Moreover, when k is larger, most of themass is assigned to the empty set in this rule. From theseresults we can see that only LNS-CR rule can give more beliefon { θ } which can be regarded as the major opinion. Thetime elapsed for Schubert’s method with different values ofthreshold k is listed in Table V. The smaller the value of k is, the more discounting steps are required in Schubert’smethod. Consequently, the time consumption becomes larger.The running time for both LNS-CR rule and Martin’s methodis less than one second. Schubert’s method is much more time-consuming.We have also tested the combination methods based on thediscounting factors proposed by Schubert [32] and Martin et al.[10] on some simple support mass functions with arbitraryfocal elements. The results are not shown here as we can getsimilar conclusions from the results: The reliability estimationprocess of these methods takes more time compared withthat of LNS-CR rule. The behavior of these two methods issimilar to that of LNS-CR rule when the parameter k or λ is set to be in a fixed range. But they are much more time-consuming compared with LNS-CR rule. This confirms thatthe reliability discounting method in LNS-CR rule is effectivefor the following conjunctive combination. Experiment 3 (The influence of parameter η ). We test here the TABLE IVT
HE COMBINATION RESULTS BY DIFFERENT RULES . Schubert’s method Martin’s method LNS-CR k = 0 . k = 0 . k = 0 . k = 0 . λ = 0 . λ = 0 . λ = 0 . λ = 1 ∅ { θ } { θ } Θ TABLE VT
IME ELAPSED FOR S CHUBERT ’ S METHOD WITH DIFFERENT VALUES OF k .1 2 3 4 5 6 7 8 9 k influence of parameter η in the LNS-CR rule. Simple supportmass functions are utilized in this experiment. Suppose that thediscernment frame under consideration is Θ = { θ , θ , θ } .Three types of SSFs are adopted. First s = 60 and s = 50 SSFs with focal elements { θ } and { θ } respectively (the otherfocal element is Θ ) are uniformly generated, and then s = 50 SSFs with focal element θ (cid:44) { θ , θ } are generated. Thevalue of masses are randomly generated. Different values of η (see Eq. (32)) ranging from 0 to 6 are used to test. Themass values in the fused bba by LNS-CR varying with η are displayed in Figure 1.a, and the corresponding pignisticprobabilities are shown in Figure 1.b.From these figures, we can see that η can have someeffects on the final decision. Figure 1.a shows that with theincreasing of η , the mass assigned to the singleton focalelements increases. On the contrary, the mass given to the focalelement whose cardinality is bigger than one decreases. Infact parameter η in LNS-CR aims at weakening the impreciseevidence which gives only positive mass to focal elementswith high cardinality, and the exponent η allows to controlthe degree of discounting. If η is larger, more weight is givento the sources of evidence whose focal elements are morespecific, and more discount will be committed to the impreciseevidence. As a result, in the experiment when η is larger than1.2, BetP( θ ) > BetP( θ ) (Figure 1.b). At this time the massfunctions with focal element { θ , θ } make little contributionto the fusion process, while the final decision mainly dependson the other two types of simple support mass functions withsingletons as focal elements.In real applications, η could be determined based on specificrequirement. This work is not specially focusing on how todetermine η , thus in the following experiment we will set η = 1 as default. Experiment 4 (The principle for the global conflict). Thegoal of this experiment is to show how Dempster’s degreeof conflict is dealt with by most of rules when combining alarge number of conflicting sources.In this experiment, the frame of discernment is set to
Θ = { θ , θ } . Assume that there are only 2 focal elementson each bba. One is the whole frame Θ , and the other is anyof the singletons ( { θ } or { θ } ). The number of bbas which . . . . . . . η M a ss ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● m ( θ ) m ( θ ) m ( ∅ ) m ( θ ) Θ a. bba . . . . . . . η be t P ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● betP ( θ ) betP ( θ ) betP ( θ ) b. Pignistic probability Fig. 1. Combination results for three types of SSFs using LNS-CR rule.The mass functions are generated randomly, and LNS-CR rule is evoked withdifferent values of η ranging from 0 to 6. have the focal element { θ } is denoted by s , while that with { θ } is s . We first fix the value of s , and let s = t ∗ s ,with t a positive integer. We generate S = s + s such kindof bbas randomly, but only withholding the bbas for which themass value assigned to { θ } or { θ } is greater than 0.5.Four values of t are considered here: t = 1 , , , . If t = 1 , s = s = S/ . If t = 2 , the number of mass functionssupporting { θ } is two times of that supporting { θ } , and soon. The global conflict (mass given to the empty set) after thecombination with different values of s for the four cases isdisplayed in Figures 2– 5 respectively. The mass assigned tothe focal element { θ } with different combination approachesis shown in Figures 6 – 9. . . . . . . G l oba l C on f li c t ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 2. The global conflict after the combination with s ranging from [0,100]and s = s . . . . . . . G l oba l C on f li c t ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 3. The global conflict after the combination with s ranging from [0,100]and s = 2 ∗ s . . . . . . . G l oba l C on f li c t ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 4. The global conflict after the combination with s ranging from [0,100]and s = 3 ∗ s . . . . . . . G l oba l C on f li c t ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 5. The global conflict after the combination with s ranging from [0,100]and s = 4 ∗ s . . . . . . . m ( θ ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 6. The mass on { θ } after the combination with s ranging from [0,100]and s = s . . . . . . . m ( θ ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 7. The mass on { θ } after the combination with s ranging from [0,100]and s = 2 ∗ s . . . . . . . m ( θ ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 8. The mass on { θ } after the combination with s ranging from [0,100]and s = 3 ∗ s . It is intuitive that when t becomes larger, the global conflictshould be smaller and we should give more belief to the focalelement { θ } . From Figures 2 – 9 we can see that only theresults by LNS-CR rule are in accordance with this commonsense. The simple average rule assigns larger bba to { θ } ,but it does not keep any conflict. In Figures 6 – 9, the massgiven to { θ } by Dempster rule cannot be displayed when S is large (and also for some small S ), because in these casesthe global conflict is 1 and the normalization could not beprocessed. As we can see, Dempster rule could not work atall when s is larger than 20. Although the conjunctive rule andcautious rule could work when combining a larger number ofmass functions, the obtained fused mass function is m ( ∅ ) ≈ ,which is useless for decision in practical situations. . . . . . . m ( θ ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ConjunctiveCautiousAverage s DempsterLNS-CRLNSa-CR
Fig. 9. The mass on { θ } after the combination with s ranging from [0,100]and s = 4 ∗ s . The results also confirm the equivalent of the LNS-CRrule and LNSa-CR rule when the number of sources is large,although the results provided by the two rules are not the samewhen there are not many mass functions to combine. FromFigures 2 – 5 we can see a kind of limit of the global conflictfor the LNS-CR rule. In fact, the mass on the empty set forthis rule depends on the size of the frame of discernment andmore directly on the number of groups created in the first stepof the rule. The limit value of the global conflict will tend to 1with the increase of the size of discernment when consideringonly categorical bbas on different singletons.
Experiment 5 (The complexity). In this experiment, thecomplexity of LNS-CR rule will be compared with othercombination rules in terms of time consumption. Simplesupport mass functions defined on a frame of discernment witheight elements are considered first. The focal elements of eachbba are set to be a random subset of Θ and Θ itself. The timeelapsed (and also the log value of the time elapsed) with thenumber of sources S varying from 10,000 to 100,000 is shownin Figure 10 ¶ . We can see that the running time of LNS-CR ismuch smaller than that of the conjunctive rule. LNSa-CR ruletakes almost the same time as cautious rule. Average rule isthe best among the five rules. As S increases, the applicationof LNSa-CR rule can save more time compared with the useof LNS-CR rule. The increment of time consumption withrespect to S is moderate. This tends to show that LNS-CR ruleis suitable for combining a large number of SSFs. Remark thatthe decomposition process is not required when the cautiousrule or LNS-CR(a) rule is adopted for combining SSFs.As mentioned before, for the combination of general separa-ble mass functions (not SSFs), LNS-CR needs four steps: de-composition, inner-group combination, discounting and globalcombination. The difference between the combination of anykind of separable bbas and of SSFs is the decompositionprocess, which is not necessary for the latter. We have designedanother experiment on consonant bbas (cid:107) over a frame of ¶ The result of Dempster rule is the same as that of conjunctive rule. (cid:107)
All consonant bbas are separable. S Log v a l ue o f T i m e E l ap s ed ( s ) l l l l l l l l l l l l l l l l l l l ll ConjunctiveCautiousAverageLNS−CRLNSa−CR a. Time lapse by five different rules
LNSa CR2e+04 4e+04 6e+04 8e+04 1e+05 − − S Log v a l ue o f T i m e E l ap s ed ( s ) l l l l l l l l l l l l l l l l l l l ll ConjunctiveCautiousAverageLNS CR b. The log value of Time lapse by five different rules
Fig. 10. Time lapse for combining SSFs. discernment with eight elements, and the number of focalelements is set to 5. The focal elements are randomly set tofive nested subsets of Θ , and the mass values are generateduniformly. The average running time (and the log value of therunning time) of 10 trials by the use of different combinationrules with different number of sources S is displayed in Figure11.a (and Figure 11.b) ∗∗ . In order to show the complexity ofLNS-CR rule more clearly, the elapsed time in each of thefour steps is shown in Figure 12.As we can see from these figures, the time consumptionof LNS-CR is significantly smaller than the cautious rule,but a little worse than the conjunctive rule and the averagerule. Although the complexity of cautious rule is the sameas LNS-CR rule and both of them require a decompositionprocess, it takes more running time than LNS-CR rule. Thereason may be the different combination approach for themass functions in the same group. The complexity of thatprocess by cautious rule is O ( S n ) (The calculation is to find ∗∗ The result of cautious rule is not displayed for large S , as it has beenalready shown that cautious rule is significantly worse than the other rules interms of time consumption when S is small. S Log v a l ue o f T i m e E l ap s ed ( s ) l l l l l l l l l l l l l l l l l l l ll ConjunctiveCautiousAverageLNS−CRLNSa−CR a. Time lapse by five different rules − − S Log v a l ue o f T i m e E l ap s ed ( s ) l l l l l l l l l l l l l l l l l l l ll ConjunctiveCautiousAverageLNS−CRLNSa−CR b. The log value of Time lapse by five different rules
Fig. 11. Time lapse for combining consonant bbas. S T i m e E l ap s ed ( s ) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● DecompositionInner−group combinationDiscountGlobal combinaiton
Fig. 12. Time lapse of each step using LNS combination rule with S varyingfrom 10,000 to 100,000. the minimum of each row in a S × n matrix), while forLNS-CR is O ( S ) . LNSa-CR is faster than LNS-CR when S is large. Figure 12 shows that the most time-consuming step in LNS-CR rule is the decomposition. Moreover as S increases,the increase of time lapse for the inner-group combination,discount, and global combination is limited. This is compliantwith the complexity analysis of each step for LNS-CR rulein Section III-C. In many applications the mass functions aredirectly SSFs in which case there is no need to perform thedecomposition, and LNS-CR is the best choice to fuse a largenumber of bbas.V. P ERSPECTIVE ON APPLICATIONS
Pattern recognition is a class of problems where the theoryof belief functions has proved to allow increased performances[2]. In such problems we can be facing many bbas to combine.Denœux [2] proposed Evidential KNN method (EKNN) as anextension of KNN in the framework of the theory of belieffunctions to better model the uncertainty in neighbor pointinteractions. The
Dempster rule is adopted to combine themass evidence from K neighbors in EKNN.The problem considered here is to classify an input pattern x into n categories or classes, denoted by Θ = { θ , θ , · · · , θ n } .The available information is assumed to consist of a trainingset L = (cid:8) ( x (1) , θ (1) ) , ( x (2) , θ (2) ) , · · · , ( x ( N ) , θ ( N ) ) (cid:9) of N patterns x ( i ) i = 1 , , · · · , N with known class labels θ ( i ) ∈ Θ . To classify pattern x , each pair ( x ( i ) , θ ( i ) ) constitutes adistinct item of evidence regarding the class membership of x .If the K nearest neighbors according to the distance measureare considered, K items of evidence can be obtained. Thesebbas can be constructed according to a relevant metric betweenpattern x and its j th neighbor x ( i ) m i ( { θ q } ) = αφ ( d ( i ) ) ,m i (Θ) = 1 − αφ ( d ( i ) ) ,m i ( A ) = 0 ∀ A ∈ Θ \ {{ θ q } , Θ } , (41)where d ( i ) is the (Euclidean) distance between x and its j th neighbor x ( i ) with class label θ ( i ) = θ q , α is a discountingparameter and φ ( · ) is a decreasing function on R + defined as φ ( d ( i ) ) = exp (cid:18) − γ q (cid:16) d ( i ) (cid:17) (cid:19) (42)with γ q being a positive parameter associated to class θ q . Itcan be heuristically set to the inverse of the mean Euclideandistance between training data belonging to class θ q . In EKNN,the K bbas for each neighbor are aggregated using the Dempster rule to form a resulting bba. A decision has to bemade regarding the assignment of sample x to one individualclass. The maximum of pignistic probability can be used fordecision-making. A. A small data set with noisy training sample
Figure 13 illustrates a simple two-class (red circle and greentriangle) data set, where there are seven objects in each class.The pattern x marked by blue star is the sample data to beclassified. The K bbas using the distance to its neighbor couldbe constructed by Eq. (41), and the five nearest neighbors aredenoted by N i orderly in the figure. Set α = 0 . and γ i isthe inverse of the average distance between the points in class θ i , i = 1 , . The fused mass function by different combinationrules with K = 4 and K = 5 are listed in Table VI and VIIrespectively. −2 −1 0 1 2 3 4 − − ● ● ● ●●●● x N N N N N ● Test data θ θ KNN of x
Fig. 13. A small data set. . . . . . . K be t P ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● DS −θ DS −θ Conjunctive −θ Conjunctive −θ Cautious −θ Cautious −θ Average −θ Average −θ −θ LNS-CR −θ LNS-CR
Fig. 14. Pignistic probability.
As we can see from Figure 13, pattern x is closer to class θ .Among pattern x ’s five nearest neighbor N j , j = 1 , , · · · , ,four belong to class θ while only 1 to class θ . The realclass of object N is θ , but it is located in the boundaryof the class and far from the other data points in the class.It may be a noisy item of θ . The standard KNN rule cancorrectly classify object x to θ when K > . However, if theevidential KNN model is applied, due to the existence of asuch neighbor, the behavior of the combination rules has beenaffected. From Table VI we can see, when K = 4 , the fusedbbas by all combination rules all assign more mass to θ thanto θ . Consequently, pattern x will be classified into class θ if the pignistic probability is considered for making decision.The same phenomenon also occurs when K is smaller than 4 (see Figure 14). When K = 5 (Table VII), only the LNS-CRrule could partition pattern x into class θ , which seemsmore reasonable. The pignistic probabilities (Figure 14) by the Dempster , conjunctive, cautious and average rules for class θ are significantly higher than those for class θ , even when K is large. These rules are not robust to the noisy trainingdata. Pattern x could be correctly classified to θ by LNS-CRrule when K is between 5 and 10.It is indicated that when there are some noisy data in thetraining data set, the performance of the combination rulemay become worse with small K . We should increase K moderately to improve the performance of the classifier. Butas we analyzed before, the existing combination rules do notwork well for aggregating a large number of mass functions.This is a limit of the use of evidential classifier. TABLE VIT
HE FUSED BBA BY DIFFERENT COMBINATION RULES ( K = 4) .Conjunctive Dempster
Cautious Average LNS-CR ∅ { θ } { θ } Θ HE FUSED BBA BY DIFFERENT COMBINATION RULES ( K = 5) .Conjunctive Dempster
Cautious Average LNS-CR ∅ { θ } { θ } Θ B. Real data sets
In this section, we consider some well known real datasets from the UCI repository †† summarized in Table VIII.The classification rates by using different combination rulesin evidential KNN model are displayed in Figure 15. Notethat the “leave-one-out” method is adopted here to test theclassifier. TABLE VIIIA
SUMMARY OF
UCI
DATA SETS .Data set No. of objects No. of cluster No. of attributesIris 150 3 4Yeast 1484 10 8Digits 5620 10 64
As we can see from Figure 15, for all the three data sets,the performance is almost the same for the two combinationrules, LNS-CR and DS, in terms of classification rates. Butthere is a little improvement by the use of LNS-CR rule when K is large. To make it clear, we specially depict the results onDigits data set in Figure 16. It is shown that when K > ,the classification rates by the use LNS-CR rule are a littlelarger than those through DS rule. We show the mass givento the empty set (global conflict) after the combination using †† http://archive.ics.uci.edu/ml/datasets.html conjunctive rule and LNS-CR rule with different values of K in Figure 17. The y -axis is the maximal assignment to ∅ among all the mass functions for the test data. As we can see,the global conflict tends to 1 quickly as K increases, whileLNS-CR rule keeps a moderate degree of global conflict. AsDS rule is a normalized conjunctive rule, there is not sense tonormalize a mass assignment with high global conflict. . . . . . . . K C l a ss i f i c a t i on R a t e Iris−DSIris−LNSYeast−DSYeast−LNSDigits−DSDigits−LNSIris−DSIris−LNSYeast−DSYeast−LNSDigits−DSDigits−LNS
Fig. 15. Classification results with different values of K on UCI data set.In the figure, the legend “Iris-DS” means it is the classification rates on Irisdata set using DS combination rule. Same as the other legends. . . . . . . K C l a ss i f i c a t i on R a t e Digits−DS
Digits-LNS-CR
Fig. 16. Classification rates on Digits data set.
C. Perspective
The above two examples are just two perspectives on theapplication of LNS-CR rule. In the first example, there aresome special noisy data in the training data set. At this time,the sources should not be considered with equal reliability. . . . . . . KIris−DSIris−LNSYeast−DSYeast−LNSDigits−DSDigits−LNS G l oba l C on f li c t Fig. 17. Global conflict using conjunctive rule and LNS-CR rule varyingwith different values of K . In the figure, the legend “Iris-DS” means it isthe conflict on Iris data set using DS combination rule. Same as the otherlegends. In this situation, using the DS rule or the conjunctive rulein EKNN model could not get good results. In the secondexample, it is shown that the global conflict may tend to onequickly as K increases. Sometimes we even could not do thenormalization process for DS rule because of the machineprecision.In real world social networks, the available information canbe uncertain, or even noisy. At this time, if we want to do aclassification task such as for recommendation, the conjunctiverule could not be applied as the sources are not all reliable.Even if the sources are reliable, the global conflict may tend to1 quickly if the bbas are not consistent. At this time, LNS-CRrule can be an alternative choice. In the future work, we willstudy how Dempster’s degree of conflict is distributed in thefeature space, and to study what special information containedin the moderate degree of global conflict kept by LNS-CR rule.VI. C ONCLUSION
Uncertainty in big data applications has attracted moreand more attention. The theory of belief functions is oneof the uncertainty theories allowing a model to deal withimprecise and uncertain information. This theory is also welldesigned for information fusion. However, despite that a lot ofcombination rules have been proposed in recent years in thisframework, they are not able to combine a large number ofsources because of the complexity or the absorbing element.In this paper, a new combination rule, named LNS-CR rule,preserving the principle of the conjunctive rule is proposed.This rule considers the mass functions given by the sourcesand groups them according to their set of focal elements(without auto-conflict). The mass functions of each group canbe summarized by one mass function after combination. Thereliability of the source is estimated by the proportion of bbasin one group. Therefore, after discounting the mass function ofeach group by the reliability factor, the final combination canbe proceeded by the conjunctive rule (or another rule according to the application). If the number of sources in each group ishigh enough, an approximation method is presented.The LNS-CR rule is able to combine a large number ofsources. The only existing method allowing to combine a largenumber of mass functions is the average rule. However, thatrule may give more importance to few sources with a highbelief (even if the source is not reliable) and cannot capturethe conflict between the sources. The proposed rule with areasonable complexity (lower than the DP and PCR6 rules)can provide good combination results.Overall, this work provides a perspective for the applica-tion of belief functions on big data. We will study how toapply LNS-CR rule on the problems of social network andcrowdsourcing in the future research work.A
CKNOWLEDGEMENTS
This work was supported by the National Natural ScienceFoundation of China (Nos.61701409, 61135001, 61403310,61672431), the Natural Science Basic Research Plan inShaanxi Province of China (No.2018JQ6005), and the Funda-mental Research Funds for the Central Universities of China(No.3102016QD088). R
EFERENCES [1] K. Zhou, A. Martin, and Q. Pan, “Evidence combinationfor a large number of sources,” in . IEEE, 2017, pp.1–8.[2] T. Denœux, “A k –nearest neighbor classification rulebased on dempster-shafer theory,” Systems, Man andCybernetics, IEEE Transactions on , vol. 25, no. 5, pp.804–813, 1995.[3] X. Deng, Q. Liu, Y. Deng, and S. Mahadevan, “An im-proved method to construct basic probability assignmentbased on the confusion matrix for classification problem,”
Information Sciences , vol. 340, pp. 250–261, 2016.[4] M.-H. Masson and T. Denœux, “ECM: An evidentialversion of the fuzzy c -means algorithm,” Pattern Recog-nition , vol. 41, no. 4, pp. 1384–1397, 2008.[5] K. Zhou, A. Martin, Q. Pan, and Z.-G. Liu, “Ecmdd: Ev-idential c -medoids clustering with multiple prototypes,” Pattern Recognition , vol. 60, pp. 239 – 257, 2016.[6] K. Zhou, A. Martin, Q. Pan, and Z.-g. Liu, “Medianevidential c -means algorithm and its application to com-munity detection,” Knowledge-Based Systems , vol. 74,pp. 69–88, 2015.[7] P. Smets, “Analyzing the combination of conflictingbelief functions,”
Information Fusion , vol. 8, pp. 387–412, 2007.[8] ——, “The combination of evidence in the transferablebelief model,”
Pattern Analysis and Machine Intelligence,IEEE Transactions on , vol. 12, no. 5, pp. 447–458, 1990.[9] P. Smets and R. Kennes, “The transferable belief model,”
Artificial intelligence , vol. 66, no. 2, pp. 191–234, 1994.[10] A. Martin, A.-L. Jousselme, and C. Osswald, “Conflictmeasure for the discounting operation on belief func-tions,” in
Information Fusion, 2008 11th InternationalConference on . IEEE, 2008, pp. 1–8. [11] W. Liu, “Analyzing the degree of conflict among belieffunctions,” Artificial Intelligence , vol. 170, no. 11, pp.909–924, 2006.[12] S. Destercke and T. Burger, “Toward an axiomatic defi-nition of conflict between belief functions,”
Cybernetics,IEEE Transactions on , vol. 43, no. 2, pp. 585–596, 2013.[13] A. Martin and C. Osswald, “Human experts fusion forimage classification,”
Information & Security: An In-ternational Journal, Special issue on Fusing Uncertain,Imprecise and Paradoxist Information (DSmT) , vol. 20,pp. 122–143, 2006.[14] E. Lef`evre and Z. Elouedi, “How to preserve the conflictas an alarm in the combination of belief functions,”
Decision Support Systems , vol. 56, pp. 326–333, 2013.[15] R. R. Yager, “On the dempster-shafer framework and newcombination rules,”
Information sciences , vol. 41, no. 2,pp. 93–137, 1987.[16] D. Dubois and H. Prade, “Representation and combina-tion of uncertainty with belief functions and possibilitymeasures,”
Computational Intelligence , vol. 4, no. 3, pp.244–264, 1988.[17] R. Ilin and E. Blasch, “Information fusion with belieffunctions: A comparison of proportional conflict redis-tribution PCR5 and PCR6 rules for networked sensors,”in .IEEE, 2015, pp. 2084–2091.[18] A. Martin, C. Osswald, J. Dezert, and F. Smarandache,“General combination rules for qualitative and quantita-tive beliefs,”
Journal of Advances in Information Fusion ,vol. 3, no. 2, pp. 67–89, 2008.[19] Y. Zhao, R. Jia, and P. Shi, “A novel combinationmethod for conflicting evidence based on inconsistentmeasurements,”
Information Sciences , vol. 367–368, pp.125–142, 2016.[20] P. Orponen, “Dempster’s rule of combination is P -complete,” Artificial Intelligence , vol. 44, pp. 245–253,1990.[21] W. T. Da Silva and R. L. Milidi´u, “Algorithms forcombining belief functions,”
International Journal ofApproximate Reasoning , vol. 7, no. 1-2, pp. 73 – 94,1992.[22] A. Martin and C. Osswald, “Toward a combination ruleto deal with partial conflict and specificity in belieffunctions theory,” in . IEEE, 2007, pp. 1–8.[23] P. Smets, “The α -junctions: the commutative and associa-tive non interactive combination operators applicable tobelief function,” in , 1997,pp. 131–153.[24] Y. Leung, N.-N. Ji, and J.-H. Ma, “An integrated infor-mation fusion approach based on the theory of evidenceand group decision-making,” Information Fusion , vol. 14,no. 4, pp. 410–422, 2013.[25] G. Shafer,
A mathematical theory of evidence . PrincetonUniversity Press, 1976.[26] P. Smets, “The application of the matrix calculus tobelief functions,”
International Journal of Approximate Reasoning , vol. 31, no. 1, pp. 1–30, 2002.[27] ——, “Decision making in the TBM: the necessity ofthe pignistic transformation,”
International Journal ofApproximate Reasoning , vol. 38, no. 2, pp. 133–147,2005.[28] K. Sentz and S. Ferson, “Combination of evidence indempster-shafer theory,” SAndia National Laboratorie,Tech. Rep., 2002.[29] P. Smets, “Belief Functions: the Disjunctive Rule ofCombination and the Generalized Bayesian Theorem,”
International Journal of Approximate Reasoning , vol. 9,pp. 1–35, 1993.[30] A. Martin, “Comparative study of information fusionmethods for sonar images classification,” in
InformationFusion, 2005 8th International Conference on , vol. 2.IEEE, 2005, pp. 7–pp.[31] A. Samet, E. Lefevre, and S. Ben Yahia, “Reliabilityestimation with extrinsic and intrinsic measure in belieffunction theory,” in . IEEE,2013, pp. 1–6.[32] J. Schubert, “Conflict management in dempster–shafertheory using the degree of falsity,”
International Journalof Approximate Reasoning , vol. 52, no. 3, pp. 449–460,2011.[33] Z. Elouedi, K. Mellouli, and P. Smets, “The evaluationof sensors’ reliability and their tuning for multisensordata fusion within the transferable belief model,” in
Symbolic and Quantitative Approaches to Reasoning withUncertainty . Springer, 2001, pp. 350–361.[34] A. Samet, E. Lef`evre, I. Hammami, and S. Ben Yahia,“Reliability estimation measure: Generic discounting ap-proach,”
International Journal of Pattern Recognitionand Artificial Intelligence , vol. 29, no. 07, p. 1559011,2015.[35] Y. Yang, D. Han, and C. Han, “Discounted combinationof unreliable evidence using degree of disagreement,”
International Journal of Approximate Reasoning , vol. 54,no. 8, pp. 1197–1216, 2013.[36] J. Klein and O. Colot, “Singular sources mining usingevidential conflict analysis,”
International Journal ofApproximate Reasoning , vol. 52, no. 9, pp. 1433–1451,2011.[37] P. Smets, “The canonical decomposition of a weightedbelief,” in , vol. 95, 1995, pp. 1896–1901.[38] T. Denœux, “Conjunctive and disjunctive combinationof belief functions induced by nondistinct bodies ofevidence,”
Artificial Intelligence , vol. 172, no. 2, pp. 234–264, 2008.[39] X. Ke, L. Ma, and Y. Wang, “Some notes on canonicaldecomposition and separability of a belief function,” in
Belief Functions: Theory and Applications , ser. LectureNotes in Computer Science, F. Cuzzolin, Ed., vol. 8764.Springer International Publishing, 2014, pp. 153–160.[40] R. Kennes, “Computational aspects of the m¨obius trans-formation of graphs,”
Systems, Man and Cybernetics,IEEE Transactions on , vol. 22, no. 2, pp. 201–223, 1992. [41] A. P. Dempster, “Upper and lower probabilities inducedby a multivalued mapping,” The annals of mathematicalstatistics , pp. 325–339, 1967.[42] C. Osswald and A. Martin, “Understanding the largefamily of dempster-shafer theory’s fusion operators-adecision-based measure,” in . IEEE, 2006, pp. 1–7.[43] P. Smets, “Belief functions: the disjunctive rule of com-bination and the generalized bayesian theorem,”
Interna-tional Journal of approximate reasoning , vol. 9, no. 1,pp. 1–35, 1993.[44] N. Wilson, “Algorithms for Dempster-Shafer theory,”in
Hanbook of defeasible reasoning and uncertaintymanagement , D. Gabbay and P. Smets, Eds. Boston:Kluwer Academic Publisher, 2000, vol. 5: Algorithmsfor uncertainty and Defeasible Reasoning, pp. 421–475.[45] T. Denœux and A. Ben Yaghlane, “Approximating thecombination of belief functions using the fast M¨obiustransform in a coarsened frame,”
International Journalof Approximate Reasoning , vol. 30, no. 1-2, pp. 77–101,2002.[46] A. Martin, “Implementing general belief function frame-work with a practical codification for low complexity,”in
Advances and Applications of DSmT for InformationFusion , F. Smarandache and J. Dezert, Eds. AmericanResearch Press Rehoboth, 2009, vol. 3, ch. 7, pp. 217–274.[47] T. Denœux, “The cautious rule of combination for belieffunctions and some extensions,” in . IEEE, 2006, pp.1–8.[48] K.-S. Chin and C. Fu, “Weighted cautious conjunctiverule for belief functions combination,”
Information Sci-ences , vol. 325, pp. 70–86, 2015.[49] C. K. Murphy, “Combining belief functions when evi-dence conflicts,”
Decision support systems , vol. 29, no. 1,pp. 1–9, 2000.[50] F. Smarandache and J. Dezert,
Advances and Appli-cations of DSmT for Information Fusion . AmericanResearch Press, Rehoboth, 2004–2009, vol. 1–3.[51] D. Dubois and H. Prade, “Consonant approximation ofbelief functions,”
International Journal of ApproximateReasoning , vol. 4, no. 5-6, pp. 419–449, 1990.[52] A. Aregui and T. Denœux, “Constructing consonantbelief functions from sample data using confidence setsof pignistic probabilities,”
International Journal of Ap-proximate Reasoning , vol. 49, no. 3, pp. 575 – 594, 2008.[53] K. Zhou and A. Martin, ibelief: Belief FunctionImplementation , 2015, r package version 1.2. [Online].Available: http://CRAN.R-project.org/package=ibelief[54] T. Burger and S. Destercke, “How to randomly generatemass functions,”