A Dissimilarity Measure Based on Singular Value and Its Application in Incremental Discounting

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1 A Dssmarty Measure Based on Snguar Vaue and Its Appcaton n Incrementa Dscountng KE Xaou Department of Automaton, Unversty of Scence and Technoogy of Chna, Hefe, Chna Ema: kxu@ma.ustc.edu.cn MA Lyao Department of Automaton, Unversty of Scence and Technoogy of Chna, Hefe, Chna Ema: yma@ma.ustc.edu.cn WANG Yong Department of Automaton, Unversty of Scence and Technoogy of Chna, Hefe, Chna Ema: yongwang@ustc.edu.cn Abstract A new dssmarty measure between two basc probabty assgnments (BPA) s proposed. Wth ths method, two BPA s are represented by a two-row matrx caed BPA matrx. Aong wth a smarty matrx of foca eements to mantan the orgna reatons between foca eements, the smaest snguar vaue of the modfed BPA matrx s taken as the dssmarty measure between two BPA s. It satsfes severa basc propertes and has good behavors n ots of stuatons. Therefore ths new dssmarty measure s used n pace of the confct coeffcent n an ncrementa dscountng method so as to mprove ts performance. The ncrementa dscountng method aows us to dscount sources to dfferent eves accordng to predefned overa confcts. And a drect dscountng method can be seen as a speca case of the ncrementa one. Numerca exampes show effectveness of ths new proposed method. Keywords evdence theory; dssmarty measure; dscountng; snguar vaue; BPA matrx I. INTRODUCTION Evdence theory [1-3], beng effcent to mode and process uncertan and mprecse data, s wdey used n the fed of nformaton fuson for decson makng. In the frame of ths theory, nformaton from each source s represented as a body of evdence denoted by a beef functon or other reated functons. To aggregate severa bodes of evdence from dfferent sources, the combnaton rue pays a cruca roe. Dempster s rue of combnaton, beng assocatve and commutatve, s the most wdey used one n appcatons. However, when appyng ths rue, every body of evdence s treated euay wth the same reabty, whch s often not the case n rea appcatons. Besdes, a counterntutve resut may be generated when there s a hgh confct between two bodes of evdence as ponted out by Zadeh [4]. In the past years, ots of works on dscountng methods [5-9] have been done so as to sove the above probems. Most of these methods assume that the majorty opnon s the actua one f no pror knowedge s avaabe, and that a source wth a arge dssmarty wth others has a ow reabty and shoud be dscounted heavy. Dscountng factors are often defned as a functon of the dssmarty between bodes of evdence. Gven a set of BPA s and a certan dssmarty measure, then a set of dscountng factors coud be obtaned correspondngy. However, we can t adjust the dscountng eves accordng to dfferent reurements once the method s determned. The seuenta ncrementa dscountng method based on fasty proposed by Schubert et a [8] provdes us an aternatve. Each predefned overa confct woud ead to a set of dscountng factors. But the fasty s defned by confct coeffcent, causng t ess robust than a dstance based method [7]. From the above anayss we can see that the dssmarty measure between two bodes of evdence s cruca to a dscountng method, and t has attracted much attenton n recent years [1-12]. A comprehensve survey on dssmarty measures defned so far can be found n [13] by Jousseme et a. Athough mportant research efforts have been done to characterze the dssmarty between two BPA's, they each capture ony one aspect of the dssmarty and are sutabe for specfc appcatons. In ths paper, we propose a dssmarty measure from a new aspect. The BPA matrx s ntroduced to represent two bodes of evdence and ts smaest snguar vaue s taken as the measure of ther dssmarty, consderng that the snguar vaues of a matrx can refect ts structure property. The smaer the smaest snguar vaue, the coser two bodes of evdence are; and vce verse. A smarty matrx refectng the reatons among foca eements (as defned n defnton 1) s aso used to modfy the BPA matrx n order to get a reasonabe measure resut. Later on ths dssmarty measure s used to defne a new ncrementa dscountng approach. Smar to the seuenta dscountng method proposed by Schubert et a., wth ths ncrementa approach we coud obtan a smooth dscountng process and we are abe to reach any eve of predefned confct accordng to actua reurements. And the new method s more robust to outers (bodes of evdence wth dfferent opnons from the majorty opnon) than the seuenta dscountng method based on fasty. Note that the new dssmarty measure s extended from our eary work[14] wth organzatona modfcaton for better readabty and addtona exampes, whe the new ncrementa dscountng approach s frst proposed here. The rest of the paper s organzed as foows. In secton 2, some basc concepts n evdence theory are revewed, and some reated works are aso ntroduced. In secton 3, we present the new dssmarty measure n deta. The BPA matrx s defned and the smaest snguar vaue of ts

2 modfed form s proposed to be the dssmarty measure. A new ncrementa dscountng method based on ths dssmarty measure s put forward and some comparsons are made n secton 4; and a concuson s drawn n secton 5. II. BACKGROUND A. Fundamenta concepts Evdence theory was frst put forward by Dempster and then extended by Shafer, thus t s aso known as Dempster- Shafer theory. Here we st severa basc defntons that w be used n the foowng sectons. Defnton 1[1]. Let Θ, a fnte set of N mutuay excusve and exhaustve hypotheses, be the frame of dscernment. The set consstng of a ts subsets s caed the power set of Θ, denoted by Ω. A basc probabty assgnment (BPA) s defned as m: Ω [,1], such that ma ( ) A Ω, m( φ) =, ma ( ) = 1 A Θ m s aso caed the mass functon, a subset A s caed a foca eement f m(a)>. The vaue of m(a) s a measure of the beef attrbuted exacty to A, and to none of the subsets of A. One can aso defne severa other functons ke beef functon, pausbty functon and commonaty functon to descrbe a body of evdence. Snce these functons are n oneto-one correspondence, the BPA w be used n ths paper to measure the dssmarty between two bodes of evdence. Defnton 2[1]. Let m 1 and m 2 be two BPA s defned over frame Θ, then they can be combned by Dempster s rue of combnaton: X = φ m 1 m2( X) = (2) m1( A) m2( B) A, B Θ, A B= X X φ 1 k where k = m1 m2( φ ) = m1( A) m2( B) (3) A B= φ and k s caed the confctng coeffcent, whch descrbes the degree of evdence confct. Defnton 3[1]. Let m be a BPA defned over frame Θ, α [, 1] be a dscountng factor, then we can get the BPA dscounted by α as α α ma ( ), A Θ, A Θ m ( A) = (4) 1 α(1 ma ( )), A =Θ The dscountng factor of a body of evdence refects ts reabty, and the more reabe the source s, the bgger α s. Otherwse, f a source s not reabe, t w be dscounted to a bg extent wth a sma α. (1) B. Reated works As a way to manage the confct when usng Dempster s rue, many dscountng methods have been proposed. Dfferent from drect dscountng methods by others, Schubert et a[8,15] deveoped a seuenta dscountng method based on fasty. Wth ths method, dscountng s performed n a seuence of ncrementa steps unt the overa confct s brought down to a predefned acceptabe eve. The fasty of evdence e s defned as m where ( e ) = Δχ χ c c 1 c n n c m φ c m = 1 = 1, = ( ), = ( φ). Based on fasty, dscountng factors at step d was updated by ( d,) ( d, ) ( d, ) c c αε = 1 ε ( d, ) 1 c where ε s a gan factor and ε 1, and c ( d, ) s the vaue of c at step d. If ony one source s n hgh confct wth other sources, then ts remova w yed a steep reducton of the confct degree, and the correspondng dscountng factor w be sma,.e. t w be dscounted heavy. However, when there are more than one sources n confct wth the majorty opnon, removng one of the confctng sources may ony ead to a sma change of the confct degree. Thus, a dstancebased method woud be more robust as ponted out by Ken et a [7], who proposed an teratve method to compute dscountng rates usng Jousseme s evdenta dstance: 1 1 α j j ( j = α dbpa m, mmean) (7) 1 where ( 1, 2) ( 1 2) ( 1 2) 2 m m T d D m m BPA m m = (8) s the evdenta dstance[1] and m mean s the average N N evdence. D s a 2 2 smarty matrx of foca eements. These two works nspred the new ncrementa dscountng method presented n the foowng sectons. The proposed approach s smar to Schubert s method n form whe a new dssmarty measure s used n pace of the confct coeffcent. III. DISSIMILARITY MEASURE BASED ON SINGULAR VALUE The dssmarty between two bodes of evdence conssts of severa aspects, makng t dffcut to uantfy. The confct coeffcent, defned by euaton (3), s commony used to uantfy the degree of confct between two bodes of evdence. However, by ths method there may be a bg confct between two dentca bodes of evdence whe two dfferent bodes of evdence may have no confct at a, unsutabe for (5) (6)

3 many appcatons. Thus many other measures have been put forward, among whch, the evdenta dstance (euaton (8)) proposed by Jousseme et a s an nterestng one. But ts computaton burden w become an mportant ssue as the frame of dscernment ncreases, and t s not good enough to capture the dfference between BPA s n some cases as shown n [5]. In ths secton, we ntroduce a new dssmarty measure between two BPA s from a dfferent aspect. Before that, et s frst reformuate the propertes of dssmarty measure n terms of basc probabty assgnments. Property 1. Gven any two BPA s m 1 and m 2, then ds(m 1, m 2 ) ; Property 2. Gven any two BPA s m 1 and m 2, then ds(m 1,m 2 )= ds(m 2, m 1 ); Property 3. Gven any two BPA s m 1 and m 2, then ds(m 1, m 2 )= ff m 1 = m 2. A. The new dssmarty measure Matrx s an usefu mathematca too for smpfyng notatons and t s effectve for provng theorems as we. It was therefore used for severa probems reated to beef functons by Smets[16]. To the authors knowedge, few reatons have been drecty estabshed between the propertes of a matrx and beef functons so far. In ths secton, we try to estabsh a smpe reaton. Snguar vaues of a matrx refect ts structure propertes. Generay, they are not ony decded by the scaes of ts eements, but aso reated to the dfferences between them. However, when t comes to a speca knd of matrx wth one more constrant that the sum of each row s fxed,.e. the nfuence of eements scaes s amost removed, the snguar vaues w many depend on dstrbuton features,.e. the dfferences between eements. Before ntroducng the new approach, et s see a smpe but enghtenng exampe. x 1 x Exampe 1. Set matrx A = 1, x 1 x B =, 1 x x then when x ncreases contnuousy from to 1, we can get a seres of snguar vaues of A and B as Fg. 1 shows. When x ncreases n nterva [,.5], the two rows n matrx A separate from each other whe the two rows n B become coser and coser. From the fgure, we can fnd that the smaest snguar vaue of A ncreases whe that of B decreases. Smary, when x ncreases n nterva [.5, 1], the two rows separate from each other n both A and B. Correspondngy, ther smaest snguar vaues ncrease as the ncrement of x. Thus, n the entre nterva, the coser the two rows of a matrx are, the smaer ts smaest snguar vaue s, and vce verse. As to the bggest snguar vaue, t decreases n entre nterva n the eft fgure and keeps the same vaue n the rght fgure, thus t shows no necessary connecton wth the dfference of the two rows. Fg. 1. Snguar vaues when x vares Ths exampe gves us enghtenment that t s a good choce to take the smaest snguar vaue as the dssmarty measure between the two rows of a matrx, and f we coud represent two bodes of evdence by a two-row matrx, then the smaest snguar vaue may aso be used to measure the dssmarty between them. For the case where a foca eements are sngetons and excusve, we can mmedatey construct such a matrx, e.g., f there are two BPA s m 1 (ω 1 )=x, m 1 (ω 2 )=1-x; m 2 (ω 1 )= 1-x, m 2 (ω 2 )= x; then we coud use matrx B n exampe 1 to represent them. Generay two BPA s often have dfferent foca eements and some foca eements are not sngetons. As to ths case, the above method cannot be apped drecty and some modfcatons shoud be done. Defnton 4. Let F 1, F 2 be the foca sets of two BPA s defned over Θ, and F=F 1 F 2 ={C 1,C 2,,C n }, then a matrx can be constructed as foows: m1( C1),, m1( Cn ) M = m2( C1),, m2( Cn ) mj( C), C Fj where mj( C) = j = 1, 2; = 1, 2, n., C Fj Through defnton 4, we coud represent two BPA s by a matrx M whch w be caed BPA matrx hereafter, and then the smaest snguar vaue of M coud be used to measure the dssmarty. Nevertheess, evdence theory s dfferent from Bayesan theory that a subsets of the frame coud be foca eements. As a conseuence, the foca eements of a BPA may often have ntersectons and are not ndependent n many cases. But once two BPA s are represented as a BPA matrx, ony ndependent vaues are eft n ths matrx and a the orgna reatons are ost. Thus, n order to mantan the reatons between foca eements and to refect the dssmarty reasonaby, some modfcatons shoud be done to the BPA matrx. So far many works have been done on the measure of smarty between foca eements. In [13], Jousseme et a. made a comprehensve concuson about t. Here we are not gong to st them a snce t s not the man topc of ths paper,

4 and we drecty take A B / A B as the smarty measure between foca eements A and B. Accordng to ths defnton, a smarty matrx w be formed to modfy the BPA matrx. Defnton 5. Let m 1, m 2 be two BPA s defned over frame Θ, M be the BPA matrx, and F={C 1,..., C n }, then the dssmarty between m 1 and m 2 coud be defned as foows: dsv( m1, m2) = mn( σ( MD )). (9) Where σ(m D) denotes snguar vaues of the matrx M D. D s the smarty matrx of foca eements to modfy the BPA matrx and D s an n n matrx wth eements denoted by: j Dj = C C, C, Cj F C Cj (1) The smaest snguar vaue, rather than the bggest one, s seected, because the smaest snguar vaue refects the dssmarty between two BPA s more propery, whe the bggest snguar vaue to a great extent refects the focused degree of each BPA as exampe 1 and many other numerca exampes show. Accordng to the basc propertes of snguar vaue, t s trva to prove that the dssmarty measure defned by euaton (9) s rreevant to the order of coumns of the BPA matrx. Thus, the coumns can be arranged n any order when construct a BPA matrx. However, arrangng them by cardnaty s a good suggeston for the sake of convenence. Agan, from the propertes of snguar vaue, property 1 and property 2 formuated at the begnnng of ths secton can be verfed mmedatey. Property 3 s not so obvous, but t s aso easy to prove and we omt t here. B. Exampes and anayss In ths subsecton, to further verfy the effectveness of the proposed approach, two exampes are ustrated. The frst one s taken from [1], whch s often used to check the behavors of a dssmarty measure. Exampe 2. Let Θ be the frame of dscernment wth 2 eements, and for convenence, 1, 2, etc. w be used to denote eement 1, eement 2, etc. over ths frame. The frst BPA s defned as: m 1 ({2,3,4})=.5, m 1 ({7})=.5, m 1 (A)=.8, m 1 (Θ)=.1; where A s a subset of Θ. The second BPA s defned as: m 2 ({1,2,3,4,5})=1. There are 2 cases where subset A ncreases one more eement at a tme, startng from Case 1 wth A = {1} and endng wth Case 2 wth A = Θ. The vaues of dsv n each case are graphcay ustrated n Fg. 2 wth the bue curve. And the red curve represents Jousseme s evdenta dstance. From the fgure, we can see that the two curves have the same trend. In the begnnng, they decrease as the eements of A ncrease. And they reach the mnmum vaue at the pont where A s {1,2,3,4,5}. After that, they keep ncreasng as the ncrement of A. Ths trend s consstent wth the fact that the Fg. 2. Dssmarty measures as the ncrement of A two BPA s become coser and coser at frst and they separate from each other when A keeps ncreasng afterwards. Exampe 3. Let the frame of dscernment be Θ={ω 1,ω 2,ω 3 }, and three BPA s are defned over ths frame: m 1 ({ω 1 })=1/3, m 1 ({ω 2 })=1/3, m 1 ({ω 3 })=1/3; m 2 (Θ)=1; m 3 ({ω 1 })=1. Athough both m 1 and m 2 are hepess for makng a decson, they are not dentca. They represent two extreme cases,.e. m 1 has a fu randomness whe m 2 s totay gnorant. Athough ther ntrnsc nature of uncertanty s dfferent, from a decson makng pont of vew, they shoud be reatvey coser and ther dssmarty shoud be sma. Snce m 3 supports ω 1 absoutey, dssmar to m 1 and m 2, t shoud have bg dssmarty measures wth both m 1 and m 2. But m 1 supports ω 1 to a certan extent whe m 2 gves no drect support to ω 1, so m 3 and m 1 are coser than m 3 and m 2. From the above anayss, a concuson coud be made that the dssmarty among these three BPA s shoud obey the neuaty: ds(m 1,m 2 )<ds(m 1,m 3 ) <ds(m 2,m 3 ). Accordng to euaton (8), we can compute the evdenta dstance: d BPA (m 1,m 2 )=d BPA (m 1,m 3 )=.5774 and d BPA (m 2,m 3 )=.8165, whch s nconsstent wth our expectaton. However, usng the dssmarty measure based on smaest snguar vaue, we have dsv(m 1,m 2 )=.2961, dsv(m 1,m 3 )=.4419, dsv(m 2,m 3 )=.6667, whch satsfes the order we want. IV. INCREMENTAL DISCOUNTING METHOD WITH THE NEW DISSIMILARITY MEASURE In the above secton we proposed a new dssmarty measure.e. the smaest snguar vaue of the modfed BPA matrx. Inspred by Schubert s seuenta ncrementa dscountng strategy, n ths secton we propose a new ncrementa dscountng approach usng ths new dssmarty measure. An experment s conducted and some comparsons are made between the new proposed ncrementa approach and the orgna one.

5 A. Incrementa dscountng method Suppose we have a set of BPA s m 1, m 2,, m n, then accordng to euaton (9), the average dssmarty between m and the other BPA s can be defned as: 1 d = dsv( m, mj), = 1,2,, n (11) n 1 j A bg d means that m s dssmar to other BPA s, and that t s not supported by other BPA s to a certan extent. Thus, the bgger d s, the more ncredbe m s, and we defne the reatve ncredbty of a BPA as foows d RI =, = 1,2,, n (12) dmax where max 1 2 d = max{ d, d,, d n } The ncredbty of a BPA rests wth not ony the reatve ncredbty, but aso the overa consstency among BPA s. If a set of BPA s are consstent, then a BPA n ths set shoud have a ow ncredbty even wth a bg reatve ncredbty. Generay, the average dssmarty can refect the degree of consstency, and we defne the absoute ncredbty as d AI = RI d = d (13) mean mean d max where dmean = ( d1 + d2 + + dn ) / n AI coud be drecty taken as the dscountng rate. But n order to contro the overa confct accordng to dfferent actua reurements, smar to [8], we take an ncrementa step by assgnng d α = 1 ε AI = 1 ε d mean (14) d max at step, where ε s a gan factor, and ε<1. Agorthm 1. Incrementa dscountng factors computaton for =1 to n, do Compute d usng euaton (11); Compute α usng euaton (14); end for 1 whe K>K, do for =1 to n, do Compute dscounted BPA m usng euaton (15); Compute d usng euaton (11); Compute α usng euaton (14); end for +1; end whe In ths dscountng process we may derve a set of ncrementay dscounted BPA s as 1 1 ( ),, α m A A Θ A Θ m ( A) = (15) α + α m ( Θ ), A =Θ The computaton procedure of ths method s summarzed by Agorthm 1 as Fg. 3 shows. B. Experments and anayss In ths secton we conduct two experments wth sx BPA s as sted n tabeⅠ. Assumng that the majorty opnon s the actua one when no further nformaton s avaabe, then m 2 and m 5 coud be seen as outers and the aggregaton resut shoud gve {A} the bggest support. TABLE I. SIX BPA S OVER THE FRAME Θ={A, B, C} φ A B A, B C A, C B, C Θ m m m m m m As the frst exampe, we consder the case where a the sx BPA s w be aggregated to get a fna decson. Set ε=.1 and K =.3, we coud get the dscountng factors at each step (Fg. 4(a)) and the correspondng fuson resuts by Dempster s rue of combnaton (Fg. 5). For comparson, Schubert s method [8] s aso conducted wth the same parameters (Fg. 4(b)). A steady decne n the overa confct coud be found n Fg. 5 wth the back curve. As a conseuence, the dscountng factors n Fg. 4(a) ncrease sowy as the dscountng steps contnue. What s more, dscountng factors of m 2 and m 5 are aways smaer than other BPA s and there s an obvous gap snce they are outers and are reatvey ncredbe. However, t s not the case n Fg. 4(b) where a dscountng factors tend to be eua after forty steps, whch seems unreasonabe. Fg. 3. Agorthm for ncrementa dscountng (a). Agorthm 1

6 (b). Seuenta dscountng based on fasty Fg. 4. Dscountng factors of dfferent methods Fg. 6. (a) Seuenta dscountng based on fasty Fg. 5. Fuson resuts and overa confct of Agorthm 1 The aggregaton resut of agorthm 1 s m(a)=.4527, m(b)=.1113, m(c)=.1618, m(θ)=.2742; and the resut of Schubert s method s m(a)=.3937, m(b)=.13, m(c)=.1717, m(θ)=.346. Wth the same predefned overa confct eve K, our method has a ower uncertanty, whch s good for decson makng. A hgher K w ead to a more certan resut, f necessary. It coud aso be seen that our method needs more steps to reach the predefned overa confct n contrast to Schubert s method. However, ths coud be overcome by a bgger gan factor ε. As the second exampe, we move away m 2 and m 5. ε=.1 and K =.3 are st used for Schubert s method, and the correspondng resuts are shown n Fg. 6. Athough t s ceary that a four BPA s support proposton A and they are consstent wth each other, a BPA s are heavy dscounted just as n the frst exampe and the fna aggregaton resut s: m(a)=.61, m(b)=.855, m(c)=.1165, m(θ)= Dfferent from the frst exampe, the maxmum dsv measure between any two BPA s s aso used n pace of the Fg. 6. (b) Fuson resuts and overa confct of seuenta dscountng overa confct n agorthm 1 to decde whether the teratvey dscountng process shoud stop. If we st set dsv =.3, then no dscountng s needed at a and the fna resut s the same as drecty combnng them usng Dempster s rue : m(a)=.9923, m(b)=.24, m(c)=.53, m(θ)=. It has a much ower uncertanty compared to Schubert s method, and we can thnk t more reasonabe wth the fact that a bodes of evdence have the same decson. Ths exampe shows the advantage of dsv over the confct coeffcent when usng as the stop condton and that t may be a good choce to use the new dssmarty both n the dscountng process and as the stop condton. Note that drect dscountng coud be seen as a speca case of the ncrementa dscountng process. If we take euaton (13) as the dscountng rate and use (1-AI ) as the dscountng factor of the -th BPA, then the aggregaton resut by drect dscountng s: m(a)=.8351, m(b)=.534, m(c)=.173, m(θ)=.42, wth the overa confct after dscountng be Whe f we set K =.9347 and ε=.1, a smar resut coud be obtaned by the ncrementa dscountng method: m(a)=.8121, m(b)=.627, m(c)=.1195, m(θ)=.56.

7 These exampes above show the performance of our proposed dssmarty measure and the modfed ncrementa dscountng method. In some cases they behave better than exstng methods. V. CONCLUSION As an effectve way to manage confct when combnng beef functons, dfferent dscountng methods have been proposed. Dfferent from others, the ncrementa dscountng method aows us to dscount sources to dfferent eves by settng varous predefned confcts. It provdes us wth more choces and s therefore more appcabe. To mprove the performance, we proposed a modfed ncrementa dscountng method usng a new dssmarty measure. In order to uantfy the dssmarty between two basc probabty assgnments, we use a two-row matrx to represent them. A drect reaton s estabshed between ths matrx and the dssmarty of two BPA s,.e. the smaest snguar vaue s taken as the dssmarty measure. It coud be an nterestng aternatve. Consderng that foca eements are not aways sngetons and excusve whe the BPA matrx aone cannot refect ther reatons, a smarty matrx of foca eements s used to modfy t. Ths new dssmarty measure has some basc propertes and shows ts effectveness n exampes. Based on ths new dssmarty measure, the reatve ncredbty and absoute ncredbty are defned and ncrementa dscountng factors are computed usng these concepts. Comparsons wth the orgna seuenta dscountng method show ts effectveness. We represent two BPA s as a BPA matrx and took the smaest snguar vaue as the dssmarty measure between them n ths paper. Athough t shows ts effectves n exampes, necessary theoretca verfcatons are not founded. Besdes, some other reatons may be estabshed between ths matrx and beef functons. Thus these aspects w be our future research work. REFERENCES [1] G. A. Shafer. A Mathematca Theory of Evdence. Prnceton: Prnceton Unversty Press, [2] P. Smets and R. Kennes, The Transferabe Beef Mode. Artfca Integence, vo. 66, pp , [3] R. R. Yager. On the Dempster-Shafer framework and new combnaton rues. Informaton Scences, vo. 41, pp , [4] L. A. Zadeh. On the vadty of Dempster's rue of combnaton of evdence. Berkey, USA: Eectroncs Research Laboratory, Unversty of Caforna, [5] Z. G. Lu, J. Dezert, Q. Pan, et a. Combnaton of sources of evdence wth dfferent dscountng factors based on a new dssmarty measure. Decson Support Systems, vo. 52, pp , 211. [6] M. C. Forea, A. L. Jousseme and E. Bosse. Dynamc Estmaton of Evdence Dscountng Rates Based on Informaton Credbty. Raro- Operatons Research, vo. 44, pp , 21. [7] J. Ken and O. Coot. Automatc dscountng rate computaton usng a dssent crteron. Proceedngs of the Workshop on the Theory of Beef Functons, Brest, France, pp. 1-6, 21. [8] J. Schubert. Confct management n Dempster-Shafer theory usng the degree of fasty. Internatona Journa of Approxmate Reasonng, vo. 52, pp , 211. [9] Y. Deng, W.K. Sh, Z.F. Zhu, et a. Combnng beef functons based on dstance of evdence. Decson Support Systems,vo.38,pp , 24. [1] A. L. Jousseme, D. Grener and É. Bossé. A new dstance between two bodes of evdence. Informaton Fuson, vo. 2, pp , 21. [11] W. R. Lu. Anayzng the degree of confct among beef functons. Artfca Integence, vo. 17, pp , 26. [12] A. Martn, A. L. Jousseme and C. Osswad. Confct measure for the dscountng operaton on beef functons. 11th Internatona Conference on Informaton Fuson, Coogne, Germany, 28. [13] A.L. Jousseme and P. Maupn. Dstances n evdence theory: Comprehensve survey and generazatons. Internatona Journa of Approxmate Reasonng, vo. 53, pp , 212. [14] X. L. Ke, L. Y. Ma, Y. Wang. A new method to measure evdence confct based on snguar vaue. Acta Eectronca Snca(to be pubshed). [15] J. Schubert. Confct Management n Dempster-Shafer Theory by Seuenta Dscountng Usng the Degree of Fasty. Proceedngs of IPMU, Torremonos, pp , 28. [16] P. Smets, The appcaton of the matrx cacuus to beef functons. Internatona Journa of Approxmate Reasonng, vo. 31, pp. 1-3, 22.