Internatonal Journal of Mathematcal Archve-3(3), 2012, Page: 1136-1140 Avalable onlne through www.ma.nfo ISSN 2229 5046 ARITHMETIC OPERATIONS OF FOCAL ELEMENTS AND THEIR CORRESPONDING BASIC PROBABILITY ASSIGNMENTS IN EVIDENCE THEORY PALASH DUTTA * & TAZID ALI Dept. of Mathematcs, Dbrugarh Unversty, Dbrugarh,-786004, Inda E-mal: palash.dtt@gmal.com, tazdal@yahoo.com (Receved on: 19-02-12; Accepted on: 06-03-12) ABSTRACT Dempster-Shafer theory of evdence s a very mportant tool wdely used n many felds such as Informaton Fuson and decson-makng. In evdence theory when focal elements and ther correspondng basc probablty assgnments (bpa) of varables are gven then focal elements and ther correspondng probablty assgnment (bpa) can be combned under arthmetc operaton. In [5] authors proposed methods to combne fuzzy focal elements and ther basc probablty assgnments of two varables. Here, we make further nvestgaton for nterval focal elements. Keywords: Evdence Theory, Basc Probablty Assgnment, Focal elements 1. INTRODUCTION Probablty theory s ntended only for aleatory uncertanty (.e., uncertanty arses from heterogenety or the random character of natural processes) and t s napproprate to represent epstemc uncertanty (.e., uncertanty arses from the partal character of our knowledge of the natural world). To overcome the lmtaton of probablstc method, Dempster put forward a theory and now t s known as evdence theory or Dempster- Shefer theory (1976). Ths theory s now a days wdely used for the epstemc and aleatory uncertanty analyss. The use of Dempster-Shefer theory n rsk analyss has many advantages over the conventonal probablstc approach. It provdes convenent and comprehensve way to handle engneerng problems ncludng, mprecsely specfed dstrbutons, poorly known and unknown correlaton between dfferent varables, modelng uncertanty, small sample sze, and measurement uncertanty. R. R. Yager (1986) had shown that when focal elements and correspondng basc probablty assgnments of varables are gven they can be combned under arthmetc operatons such as addton. He consdered all the arthmetc operatons between focal elements but the operaton for the correspondng basc probablty assgnment (bpa) of the resultng focal elements was consdered as multplcaton (product). In [5], authors have also consdered all the arthmetc operatons between fuzzy focal elements by takng the operaton for the correspondng basc probablty assgnment (bpa) of the resultng focal elements based on the operaton between the focal elements. In ths paper, we study the proposed methods for nterval focal elements. 2. BASIC CONCEPT OF DEMPSTER-SHAFER THEORY OF EVIDENCE [7] A frame of dscernment (or smply a frame), usually denoted as Θ s a set of mutually exclusve and exhaustve propostonal hypotheses, one and only one of whch s true. Evdence theory s based on two dual non-addtve measure,.e. belef measure and plausblty measure. There s one mportant functon n Dempster-Shefer theory to defne belef measure and plausblty measure whch s known as basc probablty assgnment (bpa). A functon m : 2 Θ [0,1] s called basc probablty assgnment (bpa) on the set Θ f t satsfes the followng two condtons: m( φ ) 0 A Θ ma ( ) 1 Where φ s an empty set and A s any subset of Θ. *Correspondng author: PALASH DUTTA *,*E-mal: palash.dtt@gmal.com Internatonal Journal of Mathematcal Archve- 3 (3), Mar. 2012 1136
The Basc Probablty Assgnment functon (or mass functon) s a prmtve functon. Gven a frame, Θ, for each source of evdence, a mass functon assgns a mass to every subset of Θ, whch represents the degree of belef that one of the hypotheses n the subset s true, gven the source of evdence. The subset A of frame Θ s called the focal element of m, f m (A) > 0. Usng the basc probablty assgnment (bpa), belef measure and plausblty measure are respectvely determned as Bel( A) m( B), A Θ and Pl( A) m( B) B A B A Φ Here m(b) s the degree of evdence n the set B alone, whereas Bel(A) s the total evdence n set A and all subset B of A and the plausblty of an event A s the total evdence n set A, plus the evdence n all sets of the unverse that ntersect wth A. Where Bel(A) and Pl(A) represent the lower bound and upper bound of belef n A. Hence, nterval [Bel(A), Pl(A)] s the range of belef n A. Gven two mass functons m 1 and m 2, Dempster-Shafer theory also provdes Dempster's combnaton rule for combnng them, whch s defned as follows: 3. INTERVAL ARITHMETIC m1( Am ) 2( B) A B C mc ( ) 1 m( Am ) ( B) A B φ 1 2 For the ntervals A [a 1, a 2 ] and B [b 1, b 2 ] the arthmetc operatons are defned as below: 3.1 Addton of ntervals A+ B [a 1, a 2 ] + [b 1, b 2 ] [a 1 + b 1, a 2 + b 2 ] 3.2 Subtracton of ntervals A- B [a 1, a 2 ] - [b 1, b 2 ] [a 1 b 2, a 2 b 1 ] 3.3 Multplcaton of ntervals A. B [a 1, a 2 ]. [b 1, b 2 ] [mn (a 1 b 1,a 1 b 2,a 2 b 1,a 2 b 2 ), max(a 1 b 1,a 1 b 2,a 2 b 1,a 2 b 2 )] where mn (.) and max (.). produce the smallest and the largest number n the brackets correspondngly. 3.4 Inverse of an nterval A -1 [a 1, a 2 ] -1 [1/a 2, 1/a 1 ], a1 a2 0 [, ] 3.5 Dvson of ntervals A/B [a 1, a 2 ] / [b 1, b 2 ] [a 1, a 2 ] / [1/b 2, 1/b 1 ], 0 [ b1, b2] [mn (a 1 /b 1, a 1 /b 2, a 2 /b 1, a 2 /b 2 ), max (a 1 /b 1, a 1 /b 2, a 2 /b 1, a 2 /b 2 )] 4. PROPOSED COMBINATION OF FOCAL ELEMENTS [5] Let X 1 and X 2 be two varables whose values are represented by Dempser-Shafer structure wth focal elements A 1,A 2,A 3,,A n and B 1,B 2,B 3,,B m whch are consdered as ntervals and ther correspondng basc probablty assgnment (bpa) are as follows m (A ) a and m(b ) b, 1,2,3,...,n; 1,2,3,,m respectvely. n m Where a 1 and b 1 1 1 2012, IJMA. All Rghts Reserved 1137
Intally we combne all the focal elements usng nterval arthmetc whch wll produce nm number of focal elements and thereafter the correspondng basc probablty assgnment of resultng focal elements wll be calculated as follows: 4.1 Addton of Focal Elements: 4.2 Subtracton of Focal Elements: 4.3 Multplcaton of Focal Elements: 4.4 Dvson of Focal Elements: mc ( ) ma ( + B) mc ( ) ma ( B) m( c ) m( AB ) ma ( ) + mb ( ) ( ma ( ) + mb ( )) ma ( )(1 mb ( )) ( ma ( )(1 mb ( ))) ( ma ( ) mb ( )) ( ma ( ) mb ( )) (4.1) (4.2) (4.3) mc ( ) ma ( / B) ( ma ( ) / mb ( )) ( ma ( ) / mb ( )) (4.4) Fnally, we arrange all the focal elements n ncreasng order of the left end pont. 5. NUMERICAL EXAMPLE: Suppose basc probablty assgnment (bpa) of two parameters s assgned by an expert and whch are gven n the followng tables: Focal elements Bpa Focal elements Bpa [5,9] 0.15 [7,15] 0.20 [16,20] 0.35 [18,22] 0.30 [25,28] 0.05 [29,32] 0.12 [30,35] 0.43 [34,42] 0.25 [42,47] 0.15 Table 1: Bpa of the frst parameter Table 2: Bpa of the second parameter 5.1 Addton of Focal Elements: Number of focal elements of the frst parameter s 4 and second parameter s 5 respectvely. After summng up all the focal elements usng nterval arthmetc we get 20 numbers of focal elements. Now, the correspondng basc probablty assgnments of resultng focal elements are calculated usng (4.1) and arrangng all the focal elements n ncreasng order of the left end pont are gven n the followng table 3. Table3: Basc probablty assgnment of resultng focal elements usng algebrac addton. [30, 37] 0.022 [32, 43] 0.027 [34, 41] 0.03 [35, 44] 0.06 [36, 47] 0.035 [37, 50] 0.07 [39, 51] 0.044 [41, 48] 0.044 [41, 57] 0.05 [43, 50] 0.039 [45, 52] 0.052 [46, 55] 0.087 [47, 54] 0.046 [47, 56] 0.033 [48, 57] 0.081 [49, 62] 0.038 [50, 62] 0.066 [52, 64] 0.061 [58, 67] 0.055 [60, 69] 0.06 2012, IJMA. All Rghts Reserved 1138
5.2 Subtracton of focal elements: Subtractng all the focal elements usng nterval arthmetc we get 20 numbers of focal elements (ntervals). Now, the correspondng basc probablty assgnments of resultng focal elements are calculated usng (4.2) and arrangng all the focal elements n ncreasng order of the left end pont are gven n the followng table 4. Table4: Basc probablty assgnment of resultng focal elements usng algebrac Subtracton. [-42,-33] 0.032 [-40,-27] 0.042 [-37,-25] 0.028 [-35,-19] 0.038 [-31,-22] 0.074 [-30,-21] 0.021 [-29,-20] 0.064 [-28,-15] 0.028 [-27,-20] 0.033 [-26,-14] 0.066 [-25,-14] 0.044 [-24,-12] 0.056 [-23,-16] 0.036 [-21,-10] 0.048 [-19,-10] 0.05 [-17,-8] 0.043 [-16,-9] 0.066 [-14,-7] 0.077 [-12,-5] 0.083 [-10.-3] 0.071 5.3 Multplcaton of focal elements: Multplyng all the focal elements usng nterval arthmetc we get 20 numbers of focal elements (ntervals). Now, the correspondng basc probablty assgnments of resultng focal elements are calculated usng (4.3) and arrangng all the focal elements n ncreasng order of the left end pont are gven n the followng table 5. Table5: Basc probablty assgnment of resultng focal elements usng algebrac multplcaton. [125,252] 0.0075 [145,288] 0.018 [150,315] 0.0645 [170,378] 0.0375 [175,420] 0.01 [203,480] 0.024 [210,423] 0.0225 [210,525] 0.086 [238,630] 0.05 [294,705] 0.03 [400,560] 0.0175 [450,616] 0.015 [464,640] 0.042 [480,700] 0.1505 [522,704] 0.036 [540,770] 0.129 [544,840] 0.0875 [612,924] 0.075 [672,940] 0.0525 [756,1034] 0.045 5.4 Dvson of focal elements: Dvdng all the focal elements usng nterval arthmetc we get 20 numbers of focal elements (ntervals). Now, the correspondng basc probablty assgnments of resultng focal elements are calculated usng (4.4) and arrangng all the focal elements n ncreasng order of the left end pont are gven n the followng table 6. Table6: Basc probablty assgnment of resultng focal elements usng algebrac Dvson. [0.10638,0.21429] 0.0242 [0.11904,[0.26471] 0.0145 [0.14285,0.3] 0.0084 [0.14893,0.35715] 0.0322 [0.15625,0.31035] 0.0302 [0.1666,0.44118] 0.0194 [0,17857,0.36] 0.0726 [0.2,0.5] 0.0113 [0.21875,0.51725] 0.0403 [0.25,0.6] 0.0968 [0.34042,0.4762] 0.0565 [0.38095,0.58824] 0.0339 [0.38297,0.52381] 0.0484 [0.42857,0.64706] 0.0290 [0.45714,0.6667] 0.0.0197 [0.5,0.68966] 0.0706 [0.51428,0.7334] 0.0169 [0.5625,0.75863] 0.0605 [0.57142,0.8] 0.1694 [0.64285,0.88] 0.1452 2012, IJMA. All Rghts Reserved 1139
CONCLUSION Evdence theory based uncertanty quantfcaton s a recent trend, as t can possess the computaton wth mprecse nformaton. Probablstc methods can handle only aleatory uncertanty. Evdence theory can handle both aleatory and epstemc uncertanty. Three mportant functons n evdence theory: the basc probablty assgnment functon (bpa), Belef functon (Bel), and Plausblty functon (Pl) are used to quantfy the gven varable. One of the advantages of evdence theory s that focal elements and ther correspondng basc probablty assgnments of varables can be combned. Here, we found that that methods proposed by authors n [5] are also sutable to combne nterval focal elements and ther basc probablty assgnments of two varables. ACKNOWLEDGEMENT The work done n ths paper s under a research proect funded by Board of Research n Nuclear Scences, Department of Atomc Energy, Govt. of Inda. REFERENCES [1] Deng, Y., Sh, W. K., Zhu Z. F., & Lu, Q. (2004). Combnng belef functons based on dstance of evdence, Decson support systems, Vol. 38, No. 1, pp.-489-493. [2] Deqang, H., Chongzhao, H. & Y, Y. (2008). A modfed evdence combnaton approach based on ambguty measure, Informaton fuson, 11 th Internatonal Conference. [3] Dubos, D. & Prade, H. (1988). Default reasonng and possblty theory, Artfcal Intellgence, Vol. 35No. 2, pp. 243-257. [4] Dubos, D. & Prade, H. (1988). Representaton and combnaton of uncertanty wth belef functons and possblty measures, Computatonal Intellgence, Vol. 4, pp. 244-264. [5] Dutta P., Al T., Arthmetc Operatons of Fuzzy Focal Elements n Evdence Theory. Internatonal Journal of Latest Trends n Computng. Volume 2 No. 4 (2011), pp.: 528-534. [6] Murphy, C. K. (2000). Combnng belef functon when evdence conflcts, Decson support systems, Vol. 29, No. 1, pp.-1-9. [7] Shafer, G. (1967). A mathematcal Theory of Evdence, Prnceton Unversty Press, Prnceton. [8] Sun, Q., Ye, X. Q. & Gu, W. K. (2000) A new combnaton rules of evdence theory, Acta Electronca Snca, Vol. 28, No. 8, pp. 117-119. [9] Voorbraak, F. (1991). On the ustfcaton Dempster s rule of combnaton, Artfcal Intellgence, vol. 48, pp. 171-197. [10] Xn, G., Xao, Y., & You, H. (2005). An Improved Dempster-Shafer Algorthm for Resolvng the Conflctng Evdences, Internatonal Journal of Informaton Technology, Vol. 11, No. 12. [11] Yager R. R. (1986). Arthmetc and other operatons on Dempster-Shafer Structures, nternatonal ournal of manmachne Studes, Vol. 25, pp. 357-366. [12] Yager, R. R. (1987). On the Dempster- Shafer Framework and new combnaton rules, nformaton Scence, Vol. 41, No. 2, pp. 93-137. ******************** 2012, IJMA. All Rghts Reserved 1140