Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 Avalable at http://www.jofcs.com A New Grey Relatonal Fuson Algorthm Based on Approxmate Antropy Yun LIN, Jnfeng PANG, Ybng LI College of Informaton and Communcaton Engneerng, Harbn Engneerng Unversty, Harbn 150001, Chna Abstract The theory nstrument of grey relatonal analyss s grey relatonal grade between the sequences. The grey relatonal grade s a measure of the degree of assocaton between two factors, or between two systems. It descrbes the relatve changes among the varous factors n the development process of the system. In turn, we can calculate the grey relatonal grade, compare and judge the grey relatonal grade, and then judge the degree of assocaton between the compared factors. Ths paper based on the tradtonal gray relatonal decson fuson algorthm, proposed a new gray relatonal decson fuson algorthm based on the approxmate entropy. The smulaton shows that recognton rate mproved sgnfcantly compared wth the orgnal gray relatonal decson fuson algorthm. Keywords: Grey Relatonal; Decson Fuson; Shannon Entropy; Approxmate Entropy; Informaton Fuson 1 Introducton In the feld of decson fuson research, gray relatonal analyss s an mportant method of system analyss, t s a quanttatve descrpton method that compares the development trend of the system changes. The theory nstrument of grey relatonal analyss s grey relatonal grade between the sequences. The grey relatonal grade s a measure of the degree of assocaton between two factors, or between two systems. If the development trend between two factors s bascally the same, then the grey relatonal grade between the two factors s larger than others, and the degree of assocaton s hgher, n turn, the degree of assocaton s lower [1]. So we often compare and judge the grey relatonal grade, and then judge the degree of assocaton between the compared factors. The grey relatonal grade method transform fuzzy thoughts and deas nto a mathematcal model. And ts calculatons are smple, ts prncples are easy to understood, moreover there has Ths wor s supported by the Naton Nature Scence Foundaton of Chna No. 61201237, the Fundamental Research Funds for the Central Unverstes No. HEUCFZ1129, No. HEUCF130810 and No. HEUCF130817. Correspondng author. Emal address: lybng0920@sna.cn (Ybng LI). 1553 9105 / Copyrght 2013 Bnary Informaton Press DOI: 10.12733/jcs6759 October 15, 2013
8046 Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 no specal requrements n the type of data dstrbuton and the analyss result of varables. Ths maes the gray relatonal analyss have great practcal value. On the other hand, the results of the grey relatonal analyss are ntutve sort of sze, and the result s clear. Compared wth classcal mathematc method, grey relatonal analyss brea through the tradtonal precse mathematcal whch are not tolerate ambguty constrants, these advantages mae the the applcaton of ths method greatly expand. Ths paper proposed a new grey relatonal grade algorthm, compared wth the orgnal grey relatonal grade algorthm and mproved algorthm, ths method has a hgher recognton rate and actual applcaton value. 2 The Tradtonal Grey Relatonal Fuson Algorthm 2.1 Correlaton coeffcent and correlaton degree Defne two sequences, X 0 = {X 0 () = 1, 2,, n} and X = {X () = 1, 2,, n}, the correlaton coeffcent ξ () n the pont between these two sequences s ξ () = mn mn x 0 () x () + ρ max x 0 () x () + ρ max max max x 0 () x () x 0 () x () (1) Where X 0 = {X 0 () = 1, 2,, n} and X = {X () = 1, 2,, n}, = 1, 2,, m are the ntalzaton sequences or equalzaton sequences of X 0 and X ndvdually. ρ (0, 1]. Defne the correlaton degree γ between X and X 0 s γ = 1 n n ξ () (2) =1 2.2 Steps (1) Calculate the ntalzaton sequences or equalzaton sequencesthat s X = (2) Calculate deference sequence X x (1) = (x (1), x (2),, x (n)), = 0, 1, 2,, m (3) () = x 0() x (), = ( (1), (2),, (n)), = 1, 2,, m (4) (3) Fnd maxmum devaton and mnmum devaton M = max m = mn max () () mn (5)
Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 8047 (4) Calculate the correlaton coeffcent γ 0 () = m+ξm ()+ξm, ξ (0, 1), = 1, 2,, n; = 1, 2,, m (6) (5) Calculate grey relatonal grade γ 0 = 1 n n γ 0 (), = 1, 2,, m (7) =1 3 Grey Relatonal Fuson Algorthm Based on Shannon Entropy From the calculaton formula of grey relatonal grade we can see that, n the calculaton process, each features are gven the same weght, whle n actual case, because of every features have dfferent contrbuton, the tradtonal grey relatonal grade cant use well n n actual case. We now that, n the nformaton applcaton research feld, entropy s a uncertanty descrpton of the system, and the greater of the entropy the greater uncertanty of the system, n turn the uncertanty of the system s lower, so we can ntroduce entropy nto the grey relatonal grade algorthm [2]. Introduce entropy not only can get rd of the subjectve judgment from the experts, but also mprove the practcal applcaton value. After dfferent features gven dfferent weghts, we defne weghted grey relatonal grade γ = N ξ (j)α (j), = 1, 2,, M j=1 N α (j) = 1, α (j) 0 j=1 And the steps of ths algorthm are shown as follows, (1) Frst of all, accordng to the absolute dfference of the sequence and the j sequence, we establsh matrx : = ( j ) M N = [ 1, 2,, N ] (9) Where = 1, 2,, M, j = 1, 2,, N. (2) Calculate p j = j / M =1 j. (3) Calculate the redundancy of the j feature. The greater devaton of the feature has a better reflecton of the dfference between the types relatvely, so we suppose that the feature wth a greater devaton has a greater nfluence to the system, then we defne the redundancy D j of the j feature s D j = 1 e j (10) Where E j = M =1 p j ln p j, E max = InM, e j = E j /E max. (4) Calculate the weght a (j) of the j feature. a (j) = D j / (8) N D j (11) j=1
8048 Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 4 Grey Relatonal Fuson Algorthm Based on Approxmate Entropy It s dffculty to solve the entropy n chaotc phenomena, so Pncus proposed a new concept approxmate entropy. Approxmate entropy s a non-negatve quanttatve descrpton of the complexty of nonlnear tme seres [3]. Approxmate entropy has the followng characterstcs: (1) It just need very short data sequence to estmate the relatvely stable statstcal value; (2) It accordng to the complexty of nonlnear tme seres to measure the overall characterstcs of the sgnal; (3) It can apply to a determnaton sgnal or a random sgnal; (4) It s good at ant-nterference and nose mmunty. In order to get approxmate entropy, we need two fxed-parameters m, r, where m s the length of the compared sequences, e. wndow length, also called pattern dmenson. r s a effectve threshold, called smlarty tolerance. These two parameters n the entre calculaton process s fxed. Gven the value of N ponts u(1), u(2),, u(n), for the fxed-parameter m, r, we defne another two parameter, one s lmt value ApEn(m, r), another s statstcal estmaton ApEn(m, r, N) of these N ponts [2, 7]. The concrete steps are as follows: (1) Form a m-dmenson vector X() through arrange X sequence n order, that s accordng to consecutve numbers mae up a m-dmenson vector: X() = [x(), x( + 1),, x( + m 1)] = 1 N m + 1 That s start from the pont and have m contnued x value. (2) We defne the maxmum devaton d[x(), X(j)] of the correspondng element vector X() and vector X(j): d[x(), X(j)] = dstance(x()x(j)) (13) (3) For the gven threshold r, calculate the number of d[x(), X(j)] less than r for every N m + 1, and then calculate the rato of ths value and N m, we wrte as C m (r): (12) C m (r) = count{d[x(), X(j)) < r}/(n m) = 1 N m + 1 (14) (4) Calculate the logarthm of C m (r), and then calculate the average of all, wrte as Φ m (r): Φ m (r) = 1/(N m + 1) ln C m (r) = 1 N m + 1 (15) (5) Plus 1 to m, repeat 2 5 steps, then we can obtan C m+1 (r) and Φ m+1 (r). (6) The approxmate entropy value of the sequence theoretc s: ApEn(m, r) = lm[φ m (r) Φ m+1 (r)] N (16)
Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 8049 In all nstances, the above lmt value s exst n the case of probablty s 1, but n practcal wor, N s a lmted number, t cant be, so when N s a lmted number, accordng to the above steps, we can get the estmated value of ApEn when the length of sequence s N, wrte as: ApEn(m, r, N) = Φ m (r) Φ m+1 (r) (17) In our calculaton, we use ApEn(m, r, N) as the result of approxmate entropy value, by all appearances, ApEn has a certan relatonshp wth m and r. Accordng to practcal experence, we advce that m = 2 and r = 0.1 0.2ST D (ST D s the standard devaton of orgnal data). (7) Calculate the redundancy of the j feature: D j = 1 e j (18) Where E max = InM, e j = ApEn j /E max. The same to Shannon entropy, E max and e j represent the maxmum entropy and relatve entropy of the j feature. (8) Calculate the weght a (j) of the j feature: a (j) = D j / (9) Calculate grey relatonal grade accordng to the fuson formula. The calculaton of the approxmate entropy process s shown n Fg. 1. N D j (19) j=1 Fg. 1: The calculaton of the approxmate entropy process
8050 Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 5 Smulaton Analyss Adopt the example n [4], the smulaton choose three characterstc parameters of sx nds of sgnals, the standard values of specfc characterstc parameters are shown n Table 1. Where Table 1: Three characterstc parameters of sx nds of sgnals P W (us) P RI(us) RF (GHz) Sgnal1 50 200 4 Sgnal2 35 240 3.4 Sgnal3 40 280 2.8 Sgnal4 60 340 2.3 Sgnal5 70 165 4.8 Sgnal6 85 140 5.8 P W represents sgnal pulse wdth, P RI s Pulse Repetton Interval and sgnal carrer frequency s RF. The smulaton adopts three algorthms to calculate grey relatonal grade, that s the tradtonal grey relatonal fuson algorthm, grey relatonal fuson algorthm based on Shannon entropy and grey relatonal fuson algorthm based on approxmate entropy. And the result are shown n Fg. 2 and Fg. 3. In the frst smulaton, the result show us that, the latter two algorthms have mprove the frst algorthm by change the weght, also the last algorthm s better than the second method, on the other hand, the calculaton process of grey relatonal fuson algorthm based on approxmate entropy s smple, moreover t doesnt need to pretreat the reference sequence. In the second smulaton, we suppose m = 2 whle we change the value of r, we can see that we get dfferent results, that s to say, effectve threshold can affect the recognton rate, so f we want to get a better result, we need to research how does effectve threshold nfluence the effectve threshold, ths s the next part I need to research. 6 Concluson Grey relatonal grade formula contans two-stage maxmum absolute dfference and two-stage least absolute dfference, f there have maxmum value or mnmum value n these data, then the grey relatonal grade of every pont wll be affected. For ths problem, some people put forward mproved grey relaton grade [5, 6] and absolute correlaton degree [7, 8], but these algorthms can t satsfy the Four axoms of grey related theoryand the mproved algorthm of the grey relatonal grade stll need further research. Because grey relatonal fuson algorthm based on Shannon entropy and grey relatonal fuson algorthm based on approxmate entropy put forward n ths paper gve every feature a correspondng weght accordng to ther contrbuton, so they get rd of the subjectve judgment of the experts, and mprove the recognton rate obvously, also ncrease the practcalty of the algorthm.
Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 8051 Fg. 2: The Smulaton Result of The Three Methods Fg. 3: The Smulaton Result of Dfferent Effectve Threshold Acnowledgment My deepest grattude goes frst and foremost to Professor Ybng L and my nstructor Doctor Yun Ln, and my frends, for ther constant encouragement and gudance. They have waled me through all the stages of the wrtng of ths thess. Wthout ther consstent and llumnatng nstructon, ths thess could not have reached ts present form. References [1] L Xuequan, L Songren,Han Xul, Research on the Computaton Model of Grey Interconnet Degree, Systems Engneerng theory And Practce, 1996, 11, pp. 91-95. [2] WANG Zheng-xn, DANG Yao-guo, CAO Mng-xa, Weghted degree of grey ncdence based on optmzed entropy, Systems Engneerng and Electroncs, 2010, 32(4), pp. 774-783. [3] XU Yong gang, LI Lng jun, HE Zheng ja, Approxmate entropy and ts applcatons n mechancal
8052 Y. Ln et al. /Journal of Computatonal Informaton Systems 9: 20 (2013) 8045 8052 fault dagnoss, Informaton and Control, 2002, 31(6), pp. 547-551. [4] Zhang Qshan, Guo Xjang, Deng Julong, Grey Relaton Entropy Method of Grey Relaton Analyss, Systems Engneerng-theory And Practce, 1996, 8, pp. 7-11. [5] WAN Shu-png, Method of mproved grey relatonal degree for mult-sensor target recognton, Computer Engneerng and Applcatons, 2009, 45(24), pp. 25-27. [6] ZHANG Le, CHANG Tan qng, WANG Qng-sheng, LI Yong, An Improvement Method of Grey Incdence Degree Mode, Fre Control and Command Control, 2012, 37(1). [7] Me Zhenguo, The Concept and Computaton Method of Grey Absolute Correlaton Degree, Systems engneerng, 1992, 10(5), pp. 43-44, 72. [8] Tang Wuxang, Some shortcomngs of Grey Ansolute Correlaton Degree, Systems engneerng, 1994, 12(5), pp. 59-62. [9] Zhao Yanln, We Shuyng, Me Zhanxn, A New Theoretcal Model for Analyss of Grey Relaton, Systems Engneerng and Electroncs, 1998, pp. 36-39. [10] CHEN Tao-we, JIN We-dong, CHEN Zhen-xng, Blnd classfcaton of radar emtter sgnals based on grey relatonal analyss, Computer Engneerng and Desg, 2009, 30(20), pp. 4686-4689. [11] YU Fanhua, LIU Renyun, ZHAO Hongwe, ZANG Xueba, Wavelet Neural Networ Based on Gray Relevancy Analyss Theory and Its Applcaton, Computer Engneerng, 2005, 31(22), pp. 21-28.