Research Article Quantitative Analyses and Development of a q-incrementation Algorithm for FCM with Tsallis Entropy Maximization

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1 Fuzzy Systems Volume 215, Artile ID 4451, 7 pages Researh Artile Quatitative Aalyses ad Developmet of a q-iremetatio Algorithm for FCM with Tsallis Etropy Maximizatio Makoto Yasuda Departmet of Eletrial ad Computer Egieerig, Gifu Natioal College of Tehology, Kamimakuwa, Motosu-shi, Gifu , Japa Correspodee should be addressed to Makoto Yasuda; yasuda@gifu-t.a.jp Reeived 4 Marh 215; Revised 16 July 215; Aepted 2 August 215 Aademi Editor: Nig Xiog Copyright 215 Makoto Yasuda. This is a ope aess artile distributed uder the Creative Commos Attributio Liese, whih permits urestrited use, distributio, ad reprodutio i ay medium, provided the origial work is properly ited. Tsallis etropy is a q-parameter extesio of Shao etropy. By extremizig the Tsallis etropy withi the framework of fuzzy -meas lusterig (FCM), a membership futio similar to the statistial mehaial distributio futio is obtaied. The Tsallis etropy-based DA-FCM algorithm was developed by ombiig it with the determiisti aealig (DA) method. Oe of the halleges of this method is to determie a appropriate iitial aealig temperature ad a q value, aordig to the data distributio. This is omplex, beause the membership futio hages its shape by dereasig the temperature or by ireasig q. Quatitative relatioships betwee the temperature ad q areexamied,adtheresultsshowthat,iordertohageu q ik equally, iverse hages must be made to the temperature ad q. Aordigly, i this paper, we propose ad ivestigate two kids of ombiatorial methods for q-iremetatio ad the redutio of temperature for use i the Tsallis etropy-based FCM. I the proposed methods, q is defied as a futio of the temperature. Experimets are performed usig Fisher s iris dataset, ad the proposed methods are ofirmed to determie a appropriate q value i may ases. 1. Itrodutio Statistial mehais ivestigates the marosopi properties of a physial system osistig of multiple elemets. I reet years, a popular area of researh has bee the appliatio of statistial mehaial models or tools to iformatio siee. There exists a strog relatioship betwee the membership futios of fuzzy -meas lusterig (FCM) [1] ad the maximum etropy or etropy regularizatio methods [2, 3] ad the statistial mehaial distributio futios. Iotherwords,FCM,wheregularizedormaximizedwith a Shao-like etropy, yields a membership futio that is similar to the Boltzma (or Gibbs) distributio futio [2, 4], ad whe regularized or maximized with fuzzy-like etropy [5], FCM yields a membership futio similar to the Fermi-Dira distributio futio [6]. These membership futios are suitable for aealig methods, beause they otai a parameter orrespodig to a system temperature. The advatage of usig etropy maximizatio methods is that fuzzy lusterig a be iterpreted ad aalyzed from both statistial physial ad iformatio-proessig poits of view. Tsallis [7] ahieved a oextesive extesio of Boltzma-Gibbs statistis by postulatig a geeralized form of etropywithageeralizatioparameterq, whih,ithe limit as q goes to 1, approahes the Shao etropy. Tsallis etropy is appliable to umerous fields, iludig physis, hemistry, biosiee, etworks, ad omputer siee, ad it has proved to be useful [8 1]. For example, Tsallis etropy a be appliable for attribute seletio i etwork itrusio detetio [11]. It also a be utilized as a optimizatio futio of thresholdig image segmetatio [12]. I [13, 14], Meard et al. disussed fuzzy lusterig i the framework of oextesive thermostatistis. By takig the possibilisti ostrait ito aout, the possibilisti membership futio was derived, ad its properties were osidered from various viewpoits. O the other had, based o the Tsallis etropy, aother form of etropy (or a measure of fuzziess) for a membership futio a be defied. A form of the membership futio

2 2 Fuzzy Systems a the be derived by extremizig (maximizig) this etropy withi the framework of FCM [15]. I ompariso with the ovetioal etropy maximizatio methods [2, 3], this method yields superior results [15]. Determiisti aealig (DA) [4] is a determiisti variat of simulated aealig, ad it a be applied to lusterig [16]. By applyig DA to FCM usig Tsallis etropy, a DA- FCM algorithm usig Tsallis etropy has bee developed [15]. As for aother appliatio example of DA, i [17], the q-parameterized DA expetatio maximizatio algorithm is proposed. Oe of the importat harateristis of the membership futio of this method is that eters of lusters are give as a weighted futio of the membership futio to the power of q(u q ik ). We also ote that it hages its shape i a similar way by dereasig the system temperature (or aealig) or by ireasig q. However, it remais ukow how appropriate q value ad iitial aealig temperature T high should be determied aordig to the data distributio. The purpose of the preset study is to overome the above problem, whih ivolves quatitative aalyses of the relatioships betwee the temperature ad q,adtodevelop q-iremetatio algorithms by itegratig q ad the temperature. The aalyses show that the temperature ad q affet u q ik almost iversely. Based o these results, we developed two kids of q-iremetatio algorithms for Tsallis etropybasedfcm,iwhihqis defied as a futio of the temperature. These algorithms are ompared with the ovetioal Tsallis etropy-based DA-FCM method. I the first algorithm, q is ireased so as to maitai similar shapes of u q ik with the ovetioal T-redutio method. I the seod algorithm, q is defied as a iverse of a dereasig pseudo-temperature. Experimets are performed usig Fisher s iris dataset [18], ad it was ofirmed that, i may ases, appropriate q value is determied automatially from the temperature. Furthermore, the proposed methods improve the auray of lassifiatio ad are superior to the ovetioal method. However, it was also foud that the umber of omputatio iteratios depeds o T high, ad sometimes it beomes greater tha that of the ovetioal method; this suggests that T high should be optimized to some extet. 2. Etropy Maximizatio Method Let X={x 1,...,x } (x k =(x 1 k,...,xp k ) Rp )beadataset i p-dimesioal real spae, whih is to be divided ito lusters. I additio, let V={k 1,...,k } (k i =(V 1 i,...,vp i ))be the eters of the lusters, ad let u ik [, 1] (i=1,...,;k= 1,...,) be the membership futios. Furthermore, let J= u m ik d ik (d ik = x k k i 2 ; m R; m > 1) (1) k=1 i=1 be the FCM objetive futio that is to be miimized Etropy Maximizatio for FCM. The Tsallis etropy is defied as S q = 1 W q 1 ( i=1 p q W i 1) ( i=1 p i =1; q R), (2) where W is the total umber of mirosopi possibilities of the system. Basedo(2),theetropy(orameasureofthefuzziess) of a membership futio is defied as follows: S= 1 q 1 ( u q ik k=1 i=1 The objetive futio a be writte as 1). (3) J= u q ik d ik. (4) k=1 i=1 Uder the ormalizatio ostrait of i=1 the Tsallis etropy futioal is give by δs k=1 α k δ ( i=1 u ik =1 ( k), (5) u ik 1) βδ (u q ik d ik), (6) k=1 i=1 where α k ad β are the Lagrage multipliers ad α k must be determied so as to satisfy (5). By extremizig (6) with respet to u ik,thestatioary oditio yields the followig membership futio: where u ik = {1 β(1 q)d ik} 1/(1 q), (7) Z Z= j=1 {1 β (1 q) d jk } 1/(1 q). (8) Ithesameway,theeterofthelusterisgiveby J ad S satisfy whih leads to k i = k=1 uq ik x k. (9) k=1 uq ik S βj= k=1 Z 1 q 1 1 q, (1) S =β. (11) J By aalogy with statistial mehais, this relatioship makes it possible to regard J as the iteral eergy ad β 1 as a artifiial system temperature T [19].

3 Fuzzy Systems u q (x).6.4 u q (x) T=1. T = x (a) q=2. T= q = 2. q =5. (b) T= x q = 1. Figure 1: Plots of u q (x) = {1 β(1 q)x 2 } q/(1 q), parameterized by (a) T ad (b) q. 3. Depedeies of u q ik o Temperature ad q I (9), u q ik worksasaweightvaluetoeahx k, ad it determies k i.ithispaper,forsimpliity,k i issettobe.thismakesthe deomiator of (7) beome the sum of the same forms of its umerator. I Figures 1(a) ad 1(b), the umerator of u q ik is plotted as a futio of x k, parameterized by T ad q,respetively. I these figures, i order to examie the shape of u q (x) (the subsript ik is omitted i this formula) as a futio of the distae betwee the eter of the luster ad various data poits, x k is osidered to be a otiuous variable x. The extet of u q ik beomes arrower with ireasig q ad as the temperature dereases, the distributio beomes arrower. This leads to q-iremetatio lusterig istead of aealig or T-redutio. 4. Quatitative Relatioship betwee Temperature ad q As stated i the previous setio, T ad q iversely affet the extet of u q ik, whih hages i a similar way with ireasig q or dereasig T. Aordigly, i order to examie the quatitative relatioship betwee T ad q, wehagethem idepedetly, as follows. First, we defie q/(1 q) u q (1 q) x2 (x,t,q)={1 }. (12) T The, u q (x,t,q)is alulated by fixig T ad q to some ostats T ad q. Next, by dereasig T, we determie the q values that miimize the sum of squares of the residuals of these two futios: k max u q (Δxk, T,q ) u q 2 (Δxk,T,q). (13) k= I these alulatios, the parameters are set as follows: T high (= T ) is set to 2.;thedomaiofx is set to x 1; theumberofsampligpoitsofthesumofresidualsis 11 (k max ad Δx i (13) are set to 1 ad.1,resp.). For q(= q ) values of 1.1, 2., 6., ad1. ad for T dereasig from T high,theq value that miimizes the sum of squares of the residuals (expressed by q mi )isshowi Figure 2(a). Figure 2(b), o the other had, shows the results of ases i whih q is set to 2. ad T is lowered from T high = 2.,2., 1., 2.. Approximate urves i Figures 2(a) ad 2(b) are obtaied by fittig the data to the followig formula: q mi =at b, (14) where a ad b are the fittig parameters. Optimal values for these parameters are summarized i Tables 1 ad 2. It was foud that b is early equal to 1., suggestigthatq is iversely proportioal to T. I additio, it a be see that though b does ot hage its value, a ireases with ireasig T. Aordigly, by usig the relatioship of T ad q mi as show i Tables 1 ad 2, q-iremetatio lusterig is possible. 5. q-iremetatio FCM Algorithm I this setio, we develop q-iremetatio FCM that uses Tsallis etropy istead of aealig. We begi by osiderig parameters a ad b i (14) for q = 2. ad T high = 2..Ithis ase, q is derived from T by the followig equatio: q = 4.67T (15) The temperature is held at T high durig the lusterig.

4 4 Fuzzy Systems q mi 4 q mi q = 1.1 q = 2. T q=6. q = T high = 2. T high = 2. T T high = 1. T high = 2. (a) T high =2. (b) q=2. Figure 2: Plots of q mi as a futio of T parameterized by (a) q ad (b) T high. Table 1: Parameters of approximate urves (T high = 2.). q a b Table 2: Parameters of approximate urves (q = 2.). T high a b The q-iremetatio FCM algorithm usig Tsallis etropy maximizatio is preseted as follows: (1) Set the umber of lusters, the highest temperature T high, the temperature redutio parameter, the thresholds of the overgee test δ 1 ad δ 2,adthe q-iremetatio parameter. (2) Geerate iitial lusters at radom positios ad assig eah data poit to the earest luster. Set the urret temperature to T high. (3) Calulate the membership futio u ik usig (7). (4) Calulate the luster eters k i usig (9). (5) Compare the differees betwee the urret eters ad the eters of the previous iteratio k i.ifthe overgee oditio max 1 i k i k i < δ 1 is satisfied, the go to Step (6). Otherwise, retur to Step (3). Table 3: Maximum, miimum, ad average umbers of mislassified data poits. Maximum Miimum Average T-redutio q-iremetatio (6) Compare the differee betwee the urret eters ad the eters obtaied i the previous iteratio k i. If the overgee oditio max 1 i k i k i <δ 2 is satisfied, the stop. Otherwise, update q usig (15) ad retur to Step (3). 6. Experimet 1 I Experimet 1, lassifiatio results of the ovetioal Tredutio method ad the q-iremetatio methods i the previous setio are ompared to examie if they give similar results. I the experimet, we used Fisher s iris dataset [18], osistig of 15 four-dimesioal vetors of iris flowers. The dataset otais three lusters of flowers: versiolor, virgiia, ad setosa. Eah luster osists of 5 vetors. The parameters were set as follows: =3, T high = 2., δ 1 =δ 2 =.1,theiitialvalueofq was set to 2.,adq=2.wasuseditheovetioalaealig method. For the oolig shedule of the aealig method, we used very fast aealig (VFA) [15, 2] with d = Classifiatio Results of the T-Redutio ad q-iremetatio Methods. The maximum, miimum, ad average umbers of mislassified data poits of the ovetioal Tredutio ad the q-iremetatio methods for 1 trials are summarized i Table 3.

5 Fuzzy Systems 5 Table 4: Maximum, miimum, ad average umbers of omputatio iteratios required. Maximum Miimum Average T-redutio q-iremetatio u q (x, T, q) T = 2., q = 2. T = 2., q = x T =.1, q = 2. T = 2., q = Figure 3: Plot of u q (x,t,q), parameterized by T ad q; q is determied from T by (15). The maximum, miimum, ad average umbers of omputatio iteratios required for the ovetioal T-redutio ad the q-iremetatio methods for 1 trials are summarizeditable4. Tables 3 ad 4 show that the q-iremetatio method redues mislassifiatios ad requires fewer omputatio iteratios tha does the T-redutio method; this is true eve though q isireasediordertomiimizethesumofsquares of the residuals of u q (x, T = ost., q) ad u q (x,t,q = ost.). This suggests that there exists a sigifiat differee betwee the shapes of u q ik i T-redutio ad those i qiremetatio. I Figure 3, hages i u q (x,t,q)as a futio of T are plotted. As show i this figure, q = 2.23 ad were obtaied from T = 2. ad.1,respetively,byusig(15). By omparig the plots of u q (x, 2., 2.) ad u q (x, 2., 2.23) ad those of u q (x,.1, 2.) ad u q (x,.1, 54.36), it a be see that whe T = 2., bothu q s have asimilarshape.however,whet =.1 ad x is large, u q (x,.1, 54.36) has a steeper slope tha does u q (x,.1, 2.); this results i a lak of agreemet of the lusterig results. 7. Modified q-iremetatio FCM Algorithm I the previous setio, it was ofirmed that q-iremetatio lusterig is available for FCM usig Tsallis etropy maximizatio. I this setio, we osider a very simple ad geeral algorithm, i whih T is fixed to T high ad q is defied as the iverse of T,whereT is a pseudo-temperature that is dereased usig the DA method. That is, q is give as q= T high +e T, (16) where e is ay small ostat (the small ostat e is added i order to prevet q from reahig 1. whe T =T high ;this eeds to be avoided beause, i the limit of q = 1., the Tsallis etropy equals the Shao etropy). Steps (1), (2), ad (6) ithealgorithmpresetedisetio5shouldbehagedas follows: (1) Set the umber of lusters, the highest temperature T high, the temperature redutio parameter, the thresholds of the overgee test δ 1 ad δ 2,adthe iitial q value. (2) Geerate iitial lusters at radom positios ad assig eah data poit to the earest luster. Set the urret temperature T to T high. (6) Compare the differee betwee the urret eters ad the eters obtaied i the previous iteratio k i. If the overgee oditio max 1 i k i k i <δ 2 is satisfied, the stop. Otherwise, derease T,update q usig(16),adreturtostep(3). 8. Experimet 2 I Experimet 2, we ompare the lassifiatio results of the T-redutio method ad those of the modified qiremetatio method (hereafter the proposed method) that was preseted i Setio 7. Fisher s iris dataset was lustered usig the same parameters as were used i Experimet 1, with the exeptio that q was haged from 1.1 to 1. for the ovetioal method with T high = 2.,adT high was haged from.1 to 1. for the proposed method. VFA was used as the oolig shedule for both methods, ad e i (16) was set to Classifiatio Results of the T-Redutio ad Modified q-iremetatio Methods. The maximum, miimum, ad average umbers of mislassified data poits ad required umber of omputatio iteratios for 1 trials eah of the ovetioal T-redutio ad the proposed method are summarized i Tables 5 ad 6, respetively. I both ases, T high was set to 2.. Tables 5 ad 6 show that the ovetioal method mislassifies fewer data poits as q ireases. The umber of omputatio iteratios required ireases with ireasig q, reahig a maximum at q=5.. The umbers of mislassified data poits ad the umber of omputatio iteratios required for the proposed method with T high = 2. arelosetothemiimumsofthoseumbers for the ovetioal method with q = 1. ad q = 1.1, respetively. I the ovetioal method, as q ireases, the umber of mislassified data poits dereases; this ours beause u q ik is arrowly distributed whe q is large, ad thus lusterig is

6 6 Fuzzy Systems Table 5: Maximum, miimum, ad average umbers of mislassified data poits of the T-redutio method parameterized by q (T high = 2.). Method q Maximum Miimum Average T-redutio Proposed Table 7: Maximum, miimum, ad average umbers of mislassified data poits with the modified q-iremetatio method parameterized by T high. T high Maximum Miimum Average Table 6: Maximum, miimum, ad average umbers of omputatio iteratios required for the T-redutio method parameterized by q (T high = 2.). Method q Maximum Miimum Average T-redutio Proposed Table 8: Maximum, miimum, ad average umbers of omputatio iteratios required by the modified q-iremetatio method parameterized by T high. T high Maximum Miimum Average doe loally ad optimally. O the other had, the umber of required omputatio iteratios teds to derease with dereasig q, beause whe q is small, lusterig has bee doe widely ad effiietly. Itheproposedmethod,q is iitially give as q=(t high + e)/t high,whihisearlyequalto1.. Thus,ataearlystage i aealig, lusterig is automatially doe widely. This is beause the modified method does ot require as may iteratios. Isummary,theovetioalmethodhasaiosistey, i that the q value that miimizes the umber of mislassified data poits ireases the umber of omputatio iteratios that are required. However, by settig T high to be thesameasthevalueuseditheovetioalmethod,the proposed method is better able tha the ovetioal method to balae the umber of mislassified data poits with the umber of omputatio iteratios that are required Properties of the Modified q-iremetatio Method. I this subsetio, we examie the reaso why the proposed method has improved lusterig. Tables 7 ad 8 summarize, for T high hagig from 1. to.1, the maximum, miimum, ad average umbers of mislassified data poits ad the umber of omputatio iteratios required for 1 trials of the proposed method. I Table 8, the umber of omputatio iteratios required by the proposed method dereases with dereasig T high util T high = 2..Afterthatpoit,itbegistoirease,suggestig that there exist miima i T high.thereasosforthisproperty are osidered to be as follows. Whe T high is as high as 1., therelativesizeofqis too small to hage the shape of u q ik,ad this ireases the required umber of omputatio iteratios. However, whe T high =.1,thewidthofu q ik is very arrow, ad a log time is required for the eters of the lusters to overge. For these reasos, we assume there is at least oe miimum i T high. As stated i the previous subsetio, the proposed method a limit the umber of mislassified data poits to as few as 11 to 13 poits.thus,weoludethatt high does ot sigifiatly affet the umber of mislassifiatios. I summary, our experimets ofirmed that whe q ireases as the iverse of the dereasig pseudotemperature, the proposed method works at least as well as the ovetioal method. However, the umber of omputatio iteratios required by the proposed method apparetly depeds o the value of T high, ad it is ot yet kow how to determie a appropriate value of T high for a give dataset. 9. Colusio We formulated the membership futio of Tsallis etropybased FCM by maximizig the Tsallis etropy futioal. I this formulatio, the q-parameter of the Tsallis etropy strogly affets the auray of the lusterig. I order to determie a appropriate value of q for a give data distributio, it is first eessary to examie quatitatively the effet of q o the extet of u q ik. We determied that, i order to miimize the square of the residual of u q ik for the T-redutio ad q-iremetatio, q must be ireased as the iverse of the temperature.

7 Fuzzy Systems 7 Based o this relatioship, we proposed two kids of qiremetatio methods ad ombied them with the DA method. I the first method, q is ireased aordig to the approximatio futio that miimizes the square of the residual of u q ik. The experimetal results show that, ompared to the ovetioal aealig method, the proposed method redues both the umber of mislassifiatios ad the umber of required omputatio iteratios. I the seod method, q is simply defied as the iverse of the dereasig pseudo-temperature. The experimetal results reveal that, i most ases, this method determies a appropriate q value,hasimprovedauray,adissuperior to the ovetioal method. However, the results also ofirm that the umber of omputatio iteratios depeds o T high,ad,isomeases, it a beome greater tha that of the ovetioal method. This should be avoidable by usig the value of T high that miimizes the umber of iteratios. I the future, first we ited to estimate the validity of our approximatiomethodusedisetios3ad4aurately. We the ited to explore ways to irease effiiey by performig overgee ad omputatio time tests usig various formulas of q ad T. Furthermore, we ited to develop a better shedule for aealig ad a method for optimizig T high. Coflit of Iterests The author delares that there is o oflit of iterests regardig the publiatio of this paper. Akowledgmet The preset study was supported by JSPS KAKENHI Grat o Referees [1] J. C. Bezdek, Patter Reogitio with Fuzzy Objetive Futio Algorithms, Preum Press, New York, NY, USA, [2] R.-P. Li ad M. Mukaidoo, A maximum etropy approah to fuzzy lusterig, i Proeedigs of the 4th IEEE Iteratioal Coferee o Fuzzy Systems (FUZZ-IEEE/IFES 95), pp , [3] S. Miyamoto ad M. Mukaidoo, Fuzzy C-meas as a regularizatio ad maximum etropy approah, i Proeedigs of the 7th Iteratioal Fuzzy Systems Assoiatio World Cogress (IFSA 97),vol.2,pp.86 92,Prague,CzehRepubli,Jue1997. [4] K. Rose, E. Gurewitz, ad G. Fox, A determiisti aealig approah to lusterig, Patter Reogitio Letters, vol. 11, o. 9, pp , 199. [5] A. De Lua ad S. Termii, A defiitio of a oprobabilisti etropy i the settig of fuzzy sets theory, Iformatio ad Cotrol,vol.2,o.4,pp ,1972. [6] M. Yasuda, T. Furuhashi, M. Matsuzaki, ad S. Okuma, Fuzzy lusterig usig determiisti aealig method ad its statistial mehaial harateristis, i Proeedigs of the 1th IEEE Iteratioal Coferee o Fuzzy Systems, vol.3,pp.797 8, Deember 21. [7] C. Tsallis, Possible geeralizatio of Boltzma-Gibbs statistis, Joural of Statistial Physis, vol.52,o.1-2,pp , [8] S.AbeadY.Okamoto,Eds.,Noextesive Statistial Mehais ad Its Appliatios, Spriger, 21. [9]M.Gell-MaadC.Tsallis,Eds.,Noextesive Etropy Iterdisipliary Appliatios, Oxford Uiversity Press, New York, NY, USA, 24. [1] C. Tsallis, Ed., Itrodutio to Noextesive Statistial Mehais, Spriger, 29. [11] C. F. L. Lima, F. M. Assis, ad C. P. de Souza, A omparative study of use of Shao, Réyi ad Tsallis etropy for attribute seletig i etwork itrusio detetio, i Proeedigs of the IEEE Iteratioal Workshop o Measuremets ad Networkig, pp , IEEE, Aaapri, Italy, Otober 211. [12] W. Wei, X. Li, ad G. Zhag, Fast image segmetatio based o two-dimesioal miimum Tsallis-ross etropy, i Proeedigs of the 2d Iteratioal Coferee o Image AalysisadSigalProessig(IASP 1), pp , April 21. [13] M. Meard, V. Courboulay, ad P.-A. Dardiga, Possibilisti ad probabilisti fuzzy lusterig: uifiatio withi the framework of the o-extesive thermostatistis, Patter Reogitio,vol.36,o.6,pp ,23. [14] M. Meard, P. Dardiga, ad C. C. Chibelushi, No-extesive thermostatistis ad extreme physial iformatio for fuzzy lusterig, Iteratioal Joural of Computatioal Cogitio, vol.2,o.4,pp.1 63,24. [15] M. Yasuda, Etropy maximizatio ad very fast determiisti aealig approah to fuzzy C-meas lusterig, i Proeedigs of the 5th Joit Iteratioal Coferee o Soft Computig ad 11th Iteratioal Symposium o Itelliget Systems,SU-B1-3, pp , 21. [16] F. Rossi ad N. Villa-Vialaeix, Optimizig a orgaized modularity measure for topographi graph lusterig: a determiisti aealig approah, Neuroomputig,vol.73,o.7 9, pp , 21. [17] W. Guo ad S. Cui, A q-parameterized determiisti aealig EM algorithm based o oextesive statistial mehais, IEEE Trasatios o Sigal Proessig,vol.56,o.7,pp , 28. [18] R. A. Fisher, The use of multiple measuremets i taxoomi problems, Aals of Eugeis, vol. 7, o. 2, pp , [19] L. E. Reihl, A Moder Course i Statistial Physis,JohWiley & Sos, New York, NY, USA, [2] B. E. Rose, Futio optimizatio based o advaed simulated aealig, i Proeedigs of the IEEE Workshop o Physis ad Computatio (PhysComp 92), pp , IEEE, Otober 1992.

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