A New Evidence Combination Method based on Consistent Strength

Transcription

1 TELKOMNIKA, Vol.11, No.11, November 013, pp. 697~6977 e-issn: X 697 A New Evdence Combnaton Method based on Consstent Strength Changmng Qao, Shul Sun* Insttute of Electronc Engneerng, He Longang Unversty (XU), Harbn, , Chna, Ph.: *Correspondng author, e-mal: hlu501@16.com Abstract A lot of practcal applcatons show that DS (Dempster-Shafer) theory of evdence cannot deal wth hghly conflcted evdence. For ths problem, Murphy proposed a method to solve t by modfyng source model of evdence. But ths method ust calculated the mean of the conflcted evdence and ddn t pay much attenton on ther correlaton. Based on the combnaton method of Murphy, a new method to calculate the weghted average of the conflcted evdence usng consstent strength s proposed by takng the correlaton nto account. The proposed combnaton method can combne hghly conflcted evdences to be more satsfactory. A numercal example shows the effectveness of the proposed method. Keywords: evdence theory, combnaton rule, evdence conflct, consstent strength Copyrght 013 Unverstas Ahmad Dahlan. All rghts reserved. 1. Introducton As an effectve method of ntellgent reasonng, the DS theory of evdence [1-] has got more and more attenton for many years and has acheved good results n many areas [3-4]. However, t s gradually dscovered that there are some defects n the DS theory of evdence n practcal applcatons [5-7]. On the one hand, a hgh conflct exstng among evdences sometmes results n the falure of combnaton. On the other hand, t can cause an exploson of the number of focal elements and the computatonal cost s hgh. These two defects affect the applcaton of the DS theory of evdence [8-10]. Consderng these shortcomngs, there have been two aspects to perfect the evdence theory. One s to modfy the combnaton rule; the other s to modfy the model of evdences at the source. The supporters of the former beleve that the key to mprove the evdence combnaton s the management of conflcts. The new combnaton rule needs to solve the conflct redstrbuton problem. So many methods have been proposed [11-13]. These rules can be seen as the specal cases of reference [14]. One only needs to allocate dfferent conflct n each subset to get dfferent effects and have t work as a dfferent combnaton rule. But the scholars who support the latter hold the opnon that the DS combnaton rule tself does not have errors, and the conflct of evdence should be preprocessed frstly when the degree of conflct s hgh, and then combned. In 000, Murphy proposed a method to modfy the source model of evdence [15]. Ths method calculates the mean of the basc probablty assgnment (BPA) of the evdence drectly; and then combnes teratvely wth the Dempster combnaton rule. Compared wth other methods, ths combnaton rule can deal wth conflcted evdences and converge quckly. But the shortcomng of the Murphy method s that the mean of the conflcted evdence s only calculated and the correlaton among varous evdences s not taken nto account. So the conflct problem n some cases cannot be solved effectvely. In reference [16], Deng Yong proposes an mproved method whch s manly focused on the process of calculatng the mean of evdence. Ths method frstly consders the correlaton degree among varous evdences; and then calculates the credblty of the evdence accordng to the dstance among varous evdences; fnally, the credblty s used to calculate the weghted average of the evdence. Its expermental result shows that ths method s more reasonable and more effcent. However, t needs to calculate a knd of dstance among varous evdences when handng the masses. Moreover, the calculaton range ncludes the whole framework, so the computatonal complexty s relatvely hgh. In reference [17], Zhang Jun put forward another method to reduce Receved June 4, 013; Revsed August, 013; Accepted August 14, 013

2 TELKOMNIKA e-issn: X 6973 computatonal complexty when calculatng the masses. However, these two methods allocate the masses of evdences due to the proportons of the dstances of evdences and the sum of all dstances. The changng range of weght coeffcent s lmted and the computatonal complexty of the algorthms s relatvely hgh. Ths paper also consders the problem from the vew of modfyng the source model of evdence based on the Murphy combnaton method. The method of calculatng weghted average based on the consstent strength of the evdence s proposed. A new formula s appled to calculatng the consstent strength of the evdence frstly; and then the support and credblty are calculated; at last, the weghted average of the evdence s BPA s calculated accordng to the credblty. The expermental result shows that ths proposed method can deal wth the conflcted evdences effectvely and the combned result s further mproved and very sutable for practcal applcaton.. DS Theory of Evdence In 1967, Dempster frst ponted out the defntons of upper and lower probabltes whch do not satsfy the needs of addton. The opnon s more n lne wth the human habt of thnkng. It smulates uncertanty by usng a probablty range rather than a sngle probablty value. In 1968, Dempster dscussed the generalzaton of statstcal nference problems and presented the combned rule of two evdences. Based on the work of Dempster, Shafer proposed the evdence theory and extended t to the more general case. Ther research results form the DS theory of evdence..1. DS Combnaton Rule The DS theory of evdence s based on a nonempty set called the framework, and the elements n meet the condton of ncompatble. If m : [0,1] meets the followng condtons: A m ( A ) 1 (1) And ( ) 0 A, (A) m, where represents the empty set, m(a) m represents the exact belef degree of the proposton A ; f A, m(a) A and m ( A) 0 represents does not know how to allocate m ; f s called the BPA functon. If, A s called the focal element of m. DS theory of evdence gves a useful combnaton rule to combne evdences provded by many evdence sources. Assumng that m1 and m are the BPA correspondng to the two evdences n the same framework, the focal elements are A 1, A,, An and B 1, B,, Bn, respectvely. Then the functon m : [0,1] defned by the followng formula s the BPA after the unon of two evdences. A m( A) 0 1 B A m ( A ) m ( B ) 1 K A A () evdences. Where, K m1( A ) m ( B ), t reflects the degree of conflct among varous A B A New Evdence Combnaton Method based on Consstent Strength (Changmng Qao)

3 6974 e-issn: X.. Defects Analyss It can be known from formula () that the closer K approach to 1, the greater the conflct among the evdences. When K 1, the evdences are completely conflcted and cannot be combned. For example, the framework A, B, C, the two evdences are m 1 and m respectvely, m ( A) 0. 99, m ( B) 0. 01, m ( B) 0. 01, m ( C) From an ntutve 1 1 pont of vew, the degree of belef for focal element B s very low, so the result must not be the B, but the combned result of DS theory of evdence s K , m ( A) m( C) 0, m ( B) 1, whch s completely contrary to the result of human logc reasonng. Therefore, the DS theory of evdence cannot deal wth the evdences wth hgh conflct. 3. Improved Method Assumng that the framework s, the two evdences are E and F, and ther BPA are m and m, respectvely, A and B are focal elements, the consstent strength between E and F s defned by: m( A) m( A) m( B) m( B) (, ) ( ) ( ) ( ) ( ) ( ) 0.5 ( ) 0.5 E F m A m A m B m B m A or m B 1 m( A) m( A) m( B) m( B) m( A) 0.5 or m( B) 0.5 (3) Where, ( E, F) (0,1]. The descrbe degree of consstency between evdences. Frstly, the correlaton between evdences s consdered when calculates ; secondly, the case of,.e., the between one evdence and tself s analyzed n detal and t s the largest contrbuton of ths formula. Generally, we may ntutvely thnk that ( E, F) 1when, but ths way of thnkng s not comprehensve enough. In ths method, when, we thnk that s smlar to calculatng the between evdence E and anther evdence F, only the BPA of E and F are equvalent. In ths case, s a range of values rather than a constant, and ts range s 0 1. Therefore, s more obectve when descrbng the correlaton between all evdences. So t wll obtan more satsfactory results. The mplementaton steps of the method based on adopted n ths paper are as follows: Step 1: calculate the consstent strength between the evdences n accordance wth the formula (3); Step : all the consstent strengths are expressed as a matrx form, that s, CS 11 1 n1 1 n 1n n nn (4) Step 3: the support of m s the sum of each row n the consstent strength matrx: n 1 Sup ( m ), 1,,, n (5) TELKOMNIKA Vol. 11, No. 11, November 013:

4 TELKOMNIKA e-issn: X 6975 Step 4: obtan the credblty of the evdence from the normalzed support: Crd ( m ) Sup ( m ) n 1 Sup ( m ) (6) Step 5: calculate the weghted average of evdence s BPA accordng to the credblty; n BPA ( m ) Crd ( m ) m ( A) (7) 1 Step 6: combne the evdences after weghted average wth the DS combnaton rule. n. When the number of evdences s n, the tmes of combnaton are 1 4. Complexty Analyss There are total 7 methods to be analyzed n ths paper. The methods of Yager and Sun Quan belong to the way of modfyng the combnaton rule, and the methods of Murphy, Deng Yong, Zhang Jun and the method proposed n ths paper belong to the way of modfyng the modelng of evdences at the source. So we manly compare the complexty of the latter. Frstly, the method of Murphy only calculates the mean among varous evdences, so the complexty s the lowest. But t cannot gve the correct combnaton results when the number of evdences s few. Secondly, the method of Deng Yong consders the correlaton between varous evdences, so t needs to calculate the dstances among varous evdences. The dstance got form ths method s Jousselme dstance [18]. The nner product of m, the nner product of m and the nner product between m and m are needed to be calculated when calculatng the Jousselme dstance d, so the complexty s the hghest. The method of Zhang Jun also needs to calculate the dstance, but ths knd of dstance s the Eucldean dstance. Ths method frst calculates the mean of varous evdences, and then calculates the Eucldean dstance between each evdences and the mean. Although the complexty of ths method s lower than Deng Yong, the computatonal cost s stll relatvely hgh. Fnally, the method gven n ths paper also takes the correlaton between varous evdences nto account when calculatng the. Moreover, ths method does not need to calculate any knds of dstance, and the formula (3) clearly shows that t only needs to perform a smple calculaton. 5. Numercal Example The numercal example ncludes 5 evdences and 3 dentfy targets, as shown n Table 1. m (A), m (B) and m(c) represent the BPA of the target A, B andc. It can be seen from Table 1, for the target A, all evdences BPA are hgh except evdence m, whch s the typcal problem of hghly conflcted evdence. The combned result should be A accordng to the normal human thnkng, but the result of the classcal DS combnaton rule s that the belef of A s 0, whch s n contradcton wth the normal udgment. Table 1. BPA of Three Targets for Four Evdences m 1 m m 3 m 4 m 5 m(a) m(b) m(c) Usng the method of DS, Yager [11], Sun Quan [13], Murphy [15], Deng Yong [16], Zhang Jun [17] and the method proposed n ths paper to combne the data n Table 1. The way A New Evdence Combnaton Method based on Consstent Strength (Changmng Qao)

5 6976 e-issn: X to evaluate whch combnaton method s better s manly to see whch method can make the belef of A hgher as soon as possble. The results are shown n Table. It can be seen from Table that the method proposed n ths paper sgnfcantly mproves the belef of the focal element A, compared wth other methods. Specfcally, the DS method cannot get correct combnaton results. Yager combnaton rule gves the conflct to m (X ), so the belef of X s hgh, but others are very low. It wll not cause an error of ustce, but t also cannot help us to make the rght udgment. The method of Sun Quan s a lttle better than that of Yager. The belef of A s not equal to 0, but the belef of X s stll hgh. Moreover, wth the ncreasng of the number of the evdences, the belef of A ncreases very slowly, so the combned results are not so good. When the number of evdences s 3, the method of Murphy cannot also provde the correct combnaton results, and the methods of Deng Yong and Zhang Jun gve relatvely correct combnaton results, but the belef of focal element A s not hgh. When four evdences are gven, the methods of Murphy, Deng Yong and Zhang Jun gve the correct combnaton results, but the belefs of the focal element A are only 0.607, and Observng the new method proposed n ths paper, wth the ncreasng of the number of evdences, we can fnd that combnaton result gradually ncreased from to , and ; when all the 5 evdences are gven, the belef of focal element B s 0. Thus, the method proposed n ths paper s much better than the other sx methods when dealng wth the problem of evdences conflcted. The combnaton accuracy s greatly mproved, especally when the number of evdences s more. DS Yager Sun Quan Murphy Deng Yong Zhang Jun Ths paper Table. Results of Dfferent Combnaton Methods m 1,m m 1,m,m 3 m 1,m,m 3,m 4 m 1,m,m 3,m 4, m 5 m(a)=0 m(a)=0 m(a)=0 m(a)=0 m(b)= m(b)= m(b)=0.388 m(b)=0.188 m(c)=0.149 m(c)= m(c)=0.671 m(c)=0.877 m(a)=0 m(a)=0 m(a)=0 m(a)=0 m(b)=0.18 m(b)= m(b)= m(b)= m(c)=0.03 m(c)= m(c)= m(c)= m(x)=0.79 m(x)= m(x)= m(x)= m(a)=0.090 m(a)=0.160 m(a)=0.194 m(a)=0.11 m(b)=0.377 m(b)=0.01 m(b)=0.160 m(b)=0.138 m(c)=0.10 m(c)=0.15 m(c)=0.137 m(c)=0.144 m(x)=0.431 m(x)=0.486 m(x)=0.509 m(x)=0.507 m(a)= m(a)= m(a)=0.607 m(a)= m(b)= m(b)=0.54 m(b)=0.67 m(b)=0.093 m(c)= m(c)=0.176 m(c)= m(c)= m(a)= m(a)= m(a)= m(a)= m(b)= m(b)=0.439 m(b)=0.048 m(b)= m(c)= m(c)= m(c)=0.148 m(c)= m(a)= m(a)= m(a)=0.801 m(a)= m(b)= m(b)=0.5 m(b)= m(b)= m(c)= m(c)= m(c)= m(c)= m(a)= m(a)= m(a)=0.966 m(a)= m(b)= m(b)= m(b)=0.006 m(b)= m(c)= m(c)=0.136 m(c)= m(c)= Concluson The classcal DS theory of evdence cannot deal wth the problem of hghly conflcted evdences, especally when the degree of conflct s too hgh. The combned result s often contrary to the human reasonng. Ths paper proposes an mproved combnaton method by modfyng the source model of evdence. Based on the Murphy method, a new method of calculatng weghted average wth the consstent strength of the evdence s proposed. It has the advantage of smple calculaton and hgh accuracy. The example shows that the combned results of the conflctng evdences are mproved greatly compared wth some exstng methods. Moreover, the computaton of the dstances among varous evdences s avoded, so t s more sutable for practcal applcatons. TELKOMNIKA Vol. 11, No. 11, November 013:

6 TELKOMNIKA e-issn: X 6977 Acknowledgements Ths work was supported n part by Natural Scence Foundaton of Chna under Grant NSFC , Program for Hgh-qualfed Talents under Grant Hdtd010-03, and Electronc Engneerng Provnce Key Laboratory. References [1] Dempster AP. Upper and Lower Probabltes Induced by a Mult-valued Mappng. Annual Mach Statst. 1967; 38(4): [] Shafer G. A Mathematcal Theory of Evdence. Prnceton Unversty Press, Prnceton. 1976; [3] He Luo, Shan-ln Yang, Xao-an Hu, et, al. Agent orented ntellgent fault dagnoss system usng evdence theory. Expert Systems wth Applcatons. 01; 39(3): [4] Quan Wen, Wang Xaodan, Wang Jan, etc. New combnaton rule of DST based on local conflct dstrbuton strategy. Acta Electronc Snca. 01; 40(9): [5] Zadeh L. A smple vew of the Dempster-Shafer Theory of evdence and ts mplcaton for the rule of combnaton. Artfcal Intellgence. 1986; 7(): [6] Han Deqang, Han Congzhao, Deng Yong, etc. Weghted combnaton of conflctng evdence based on evdence varance. Acta Electronca Snca. 011; 39 (3A): [7] Nhan Nguyen-Thanh. Emprcal dstrbuton-based event detecton n wreless sensor networks: An approach based on evdence theory. IEEE Sensors Journal. 01; 1(16): -8. [8] Deqang Han, Yong Deng, Chongzhao Han. Weghted evdence combnaton based on dstance of evdence and uncertanty measure. Journal Infrared Mllm. Waves. 011; 30(5): [9] Klrg J, Lewsh W. Measurng Ambguty n the Evdence Theory. IEEE Transactons on Systems, Man and Cybernet, Part A: Systems and Humans. 008; 38(4): [10] Nhan Nguyen-Thanh, Insoo Koo. Evdence theory based cooperatve spectrum sensng wth effcent quantzaton method n cogntve rado. IEEE Transacton on Vehcular Technology. 010; 60(1): [11] Yager RR. On the Dempster-Shafer framework and new combnaton rules. Informaton Scence. 1989; 41(): [1] Smets P. The combnaton of evdence n the transferable belef model. IEEE Transacton on Pattern Analyss and Machne Intellgence. 1990; 1(5): [13] Quan Sun, Xuqng Ye, Wekang Gu. A new combnaton rules of evdence theory. Acta Electronca Snca. 000; 8(8): [14] Lefevree, Coloto, Vannoorenberghe P. Belef functon combnaton and conflct management. Informaton Fuson. 00; 3(3): [15] Murphy CK. Combnng belef functons when evdence conflcts. Deon Support Systems. 000; 9(1): 1-9. [16] Yong Deng, WenKang Sh, ZhenFu Zhu. Combnng belef functons based on dstance of evdence, Deon Support Systems. 004; 38(3): [17] Jun Zhang. Approaches to conflct evdence n D-S evdence theory and ts applcatons. Jang X, Nan Chang Unversty. Chna [18] Jousselme AL, Grener D, Bosse E. A new dstance between two bodes of evdence. Informaton fuson. 001; (1): A New Evdence Combnaton Method based on Consstent Strength (Changmng Qao)