Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng entropy s a novel and effcent uncertanty measure to deal wth mprecse phenomenon, whch s an extenson of Shannon entropy. In ths paper, power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy are presented, whch partally uncover the nherent physcal meanngs of Deng entropy from the perspectve of statstcs. Ths ndcated some work related to power law or scale-free can be analyzed usng Deng entropy. The results of some numercal smulatons are used to support the new vews. Keywords: Deng entropy, Power law, Maxmum Deng entropy, Dmenson 1. Introducton Deng entropy has been proposed by Prof. Deng to manage the uncertan nformaton n the frame of Dempster-Shafer evdence theory (DST)[1], whch has acheved plenty of attenton recent years[2, 3, 4, 5, 6]. Deng entropy can be consdered as an extenson of Shannon entropy to deal wth Correspondng author: Bngy Kang, College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Emal address: bngy.kang@nwsuaf.edu.cn; bngy.kang@hotmal.com. Preprnt submtted to December 17, 2018
uncertan phenomenon n the probablty feld. In addton, Deng entropy can be appled to absorb the complex mprecse (or unknown) phenomenon n the belef fled (frame of DST) effcently. In ths paper, the work focuses two nvestgatons based on the maxmum values of the belef dstrbuton va the max Deng entropy wth dfferent scales of frame of dscernment (FOD). One s the relaton between the maxmum value of belef dstrbuton subjectng to the max Deng entropy and the scale of Deng nformaton correspondngly. The other s dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy. Some numercal smulatons have been made to acheve the two dscoverngs,.e., approxmate power law and approxmate constant dmenson. The paper s organzed as follows. The prelmnares brefly ntroduce some concepts about Dempster-Shafer evdence theory, Deng entropy and max Deng entropy n Secton 2. In Secton 3, the new vews about max Deng entropy are presented. One s the relaton between the maxmum value of belef dstrbuton subjectng to the max Deng entropy and the scale of Deng nformaton correspondngly. The other s dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy. Fnally, ths paper s concluded n Secton 4. 2. Prelmnares In ths secton, some prelmnares are brefly ntroduced. 2
2.1. Frame of Dempster-Shafer evdence theory Let X be a set of mutually exclusve and collectvely exhaustve events, ndcated by X = {θ 1, θ 2,, θ,, θ X } (1) where set X s called a frame of dscernment (FOD). The power set of X s ndcated by 2 X, namely 2 X = {, {θ 1 },, {θ X }, {θ 1, θ 2 },, {θ 1, θ 2,, θ },, X} (2) For a frame of dscernment X = {θ 1, θ 2,, θ X }, a mass functon s a mappng m from 2 X to [0, 1], formally defned by: m : 2 X [0, 1] (3) whch satsfes the followng condton: m( ) = 0 and A 2 X m(a) = 1 (4) where A s a focal element f m(a) s not 0. 2.2. Deng entropy Wth the range of uncertanty mentoned above, Deng entropy [1] can be presented as follows E d = m(f ) log m(f ) 2 F 1 (5) 3
where, F s a proposton n mass functon m, and F s the cardnalty of F. As shown n the above defnton, Deng entropy, formally, s smlar wth the classcal Shannon entropy, but the belef for each proposton F s dvded by a term (2 F 1) whch represents the potental number of states n F (of course, the empty set s not ncluded). Specally, Deng entropy can defntely degenerate to the Shannon entropy f the belef s only assgned to sngle elements. Namely, E d = m(θ ) log m(θ ) 2 θ 1 = m(θ ) log m(θ ) Next, the condton of the maxmum Deng entropy s dscussed[7]. 2.3. The maxmum Deng entropy Assume F s the focal element and m(f ) s the basc probablty assgnment for F, then the maxmum Deng entropy for a belef functon happens when the basc probablty assgnment satsfy the condton m (F ) = 2 F 1 2 F 1, where = 1, 2,..., 2 X 1, and X s the scale of the frame of dscernment. Theorem 1 (The maxmum Deng entropy). The maxmum Deng entropy: E d = m(f ) log m(f ) f and only f m (F 2 F 1 ) = 2 F 1 2 F 1 More nformaton refers to Appendx A. As shown n Fg. 1, belef dstrbutons wth the maxmum Deng entropy are changng wth the scale of FOD, X =1,..8. The pont n ths paper les n the maxmum value of each belef dstrbuton. 4
Belef dstrbuton changng wth scale of FOD,va Max Deng Entropy 1 Belef Dstrbuton (X=1) 0.9 0.8 0.7 Mass functon 0.6 0.5 0.4 Belef Dstrbuton (X=2) Belef Dstrbuton (X=3) 0.3 0.2 Belef Dstrbuton (X=4) Belef Dstrbuton (X=5) 0.1 Belef Dstrbuton (X=6) Belef Dstrbuton (X=7) Belef Dstrbuton (X=8) 0 0 50 100 150 200 250 300 Fgure 1: Belef dstrbuton wth the maxmum Deng entropy changng wth the scale of FOD, X =1,..8. 5
Next, Two focuses are presented. One s the relaton between the maxmum value of belef dstrbuton subjectng to the max Deng entropy and the scale of Deng nformaton correspondngly. The other s dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy. 3. Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy In statstc, a power law s a relatonshp n whch a relatve change n one quantty gves rse to a proportonal relatve change n the other quantty, ndependent of the ntal sze of those quanttes. Power law s a pervasve phenomenon n many felds, such as complex network (scale-free network) [8], Fractals[9]. Frstly, the power law functon s establshed between the maxmum value for belef dstrbuton va max Deng entropy and the maxmum Deng nformaton scale correspondngly. 3.1. Power law of the maxmum value for belef dstrbuton wth the max Deng entropy Suppose the maxmum value for belef dstrbuton va max Deng entropy max [m (F )] relates to a functon P (r). In addton, assume the correspondng maxmum Deng nformaton scale ( 2 F 1 ) relates to the varable r. ( ( A pow law functon P 2 F 1 )) wth a scale nvarance (d 0.59) s establshed usng Eq. (6). P (r) = r d (6) 6
where r = ( 2 F 1 ), P (r) = max [m (F )], m (F ) = 2 F 1 ),d 0.59. (2 F 1 As shown n Fg. 2, when the scales of FOD ( X ) change from 1 to 10, all the few hgh belef (the maxmum values of the belef dstrbuton va max Deng entropy, max [m (F )]) are contaned n the front of the plane, most of the other low belef (the maxmum values of the belef dstrbuton va max Deng entropy, max [m (F )]) are dstrbuted n the followng wde plane. ( Ths s an approxmate power law dstrbuton. A pow law functon ( P 2 F 1 )) wth a scale nvarance (d) s easly observed by Fg. 2. What s more, the scale nvarance d 0.59, whch wll be dscussed n the next secton. 1 Max value of belef dstrbuton changng wth scale of FOD,va Max Deng Entropy 0.9 0.8 max value of Mass functon 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 200 400 600 800 1000 1200 Sum(2 F -1), X =1,...,10 Fgure 2: The maxmum value of belef dstrbuton wth the maxmum Deng entropy changng wth the maxmum scale of Deng nformaton, the scale of FOD, X =1,..10. Ths s an approxmate power law dstrbuton. 7
Next, dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy s presented. 3.2. Dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy The scale nvarance d n Eq.(6) s equal to the dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy. By the polyf t functon n the Matlab, the dmenson d 0.59 after nvestgatng the data of belef dstrbutons wth max Deng entropy (scale of FOD, X = 1, 2,..., 25). The result s shown n Fg. 3, whch ndcates the log values trendng between the maxmum value of belef dstrbuton wth the maxmum Deng entropy and the maxmum amount of Deng nformaton, the scale of FOD, X =1,..25. As shown n Fg. 3, an approxmate lnear relaton s obtaned, whch ndcate the scale-free and power law. d = lm ε 0 log N (ε) log (ε) s.t. m (F ) = 2 F 1 (2 F 1) log = lm 2 max [m (F )] log 2 (2 F 1) 0.59 (7) (8) 4. Concluson In ths paper, power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy are presented, whch partally uncover the nherent physcal meanngs of Deng entropy from the perspectve of statstcs. The results of some numercal smulatons are used to support 8
0 Trendng between Log2 of max value of Mass functon and Log2 Sum(2 F -1) Log2 of max value of Mass functon -5-10 d 0.59-15 0 5 10 15 20 25 Log2 Sum(2 F -1) Fgure 3: Dmenson of the maxmum value of belef dstrbuton wth the maxmum Deng entropy and the maxmum amount of Deng nformaton, the scale of FOD, X =1,..25. 9
the new vews. Ths ndcated some work related to power law or scale-free can be analyzed usng Deng entropy. Acknowledgments The author greatly apprecates Professor Yong Deng for hs encouragement to do ths research. The work s partally supported by Chna Scholarshp Councl, and ntal Doctoral Natural Scence Foundaton of Northwest A&F Unversty (Grant No. Z109021812). Appendx A. The maxmum Deng entropy Assume F s the focal element and m(f ) s the basc probablty assgnment for F, then the maxmum Deng entropy for a belef functon happens when the basc probablty assgnment satsfy the condton m (F ) = 2 F 1 2 F 1, where = 1, 2,..., 2 X 1, and X s the scale of the frame of dscernment. Theorem 2 (The maxmum Deng entropy). The maxmum Deng entropy: E d = Proof. Let m(f ) log m(f ) f and only f m (F 2 F 1 ) = 2 F 1 D = m (F ) log m (F ) 2 F 1 2 F 1 (A.1) m (F ) = 1 (A.2) 10
Then the Lagrange functon can be defned as D 0 = ( ) m (F ) log m (F ) 2 F 1 + λ m (F ) 1 (A.3) Now we can calculate the gradent, D 0 m (F ) = log m (F ) 2 F 1 m (F ) Then Eq. (A.4) can be smplfed as 1 m(f ) ln a 2 F 1 1 2 F 1 + λ = 0 (A.4) log m (F ) 2 F 1 1 ln a + λ = 0 (A.5) From Eq. (A.5), we can get m (F 1 ) 2 F 1 1 = m (F 2) 2 F 2 1 = = m (F n) 2 Fn 1 (A.6) Let m (F 1 ) 2 F 1 1 = m (F 2) 2 F 2 1 = = m (F n) 2 Fn 1 = k (A.7) Then m (F ) = k ( 2 F 1 ) (A.8) Accordng to Eq. (A.2), we can get k = 1 2 F 1 (A.9) 11
Accordng to Eq. (A.7), we can get m (F ) = 2 F 1 2 F 1 (A.10) Hence, the maxmum Deng entropy E d = m (F ) = 2 F 1 2 F 1 m(f ) log m(f ) 2 F 1 f and only f References [1] Y. Deng, Deng entropy, Chaos, Soltons & Fractals 91 (2016) 549 553. [2] J. Abellán, Analyzng propertes of deng entropy n the theory of evdence, Chaos, Soltons & Fractals 95 (2017) 195 199. [3] R. Jroušek, P. P. Shenoy, A new defnton of entropy of belef functons n the dempster shafer theory, Internatonal Journal of Approxmate Reasonng 92 (2018) 49 65. [4] H. Zheng, Y. Deng, Evaluaton method based on fuzzy relatons between dempster shafer belef structure, Internatonal Journal of Intellgent Systems 33 (7) (2018) 1343 1363. [5] Y. L, Y. Deng, Generalzed ordered propostons fuson based on belef entropy., Internatonal Journal of Computers, Communcatons & Control 13 (5) (2018) 792 807. [6] W. Jang, B. We, C. Xe, D. Zhou, An evdental sensor fuson method 12
n fault dagnoss, Advances n Mechancal Engneerng 8 (3) (2016) 1687814016641820. [7] B. Kang, Y. Deng, The maxmum deng entropy, vxra (2015) 1509.0119. URL http://vxra.org/abs/1509.0119 [8] C. Song, S. Havln, H. A. Makse, Self-smlarty of complex networks, Nature 433 (7024) (2005) 392. [9] D. Avnr, O. Bham, D. Ldar, O. Malca, Is the geometry of nature fractal?, Scence 279 (5347) (1998) 39 40. 13