An Improved Differential Evolution Algorithm Based on Statistical Log-linear Model

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1 Sesors & Trasducers, Vol. 59, Issue, November, pp Sesors & Trasducers by IFSA A Improved Dfferetal Evoluto Algorthm Based o Statstcal Log-lear Model Zhehuag Huag School of Mathematcs Sceces, Huaqao Uversty, Quazhou, 6, Cha Fuja Key Laboratory of the Bra-lke Itellget Systems, Xame, 65 E-mal: hzh98@hqu.edu.c Receved: 8 September /Accepted: 5 October /Publshed: November Abstract: Dfferetal evoluto () algorthm s a good optmzato techque based o populato whch has bee successfully appled may research ad applcato areas. Log-lear model s a statstcal model whch ca easly bled multple features, a varety of kowledge sources ca be added to the model the form of feature fuctos. Tradtoal dfferetal evoluto algorthm s easy to fall to local optmum value ad the covergece rate s slow. To solve these problems, a mproved dfferetal evoluto algorthm based o loglear model s proposed ad mplemeted ths paper. There are two maly works ths paper. Frstly, we troduce log-lear model to dfferetal evoluto algorthm whch ca ehace decso makg ablty. Secodly, some operatos are preseted to mprove global optmzato capablty. Expermets showed that the mproved algorthm has more powerful global explorato ablty ad faster covergece speed. Copyrght IFSA. Keywords: Dfferetal evoluto, Log-lear model, Statstcal model, Fucto optmzato.. Itroducto Dfferetal evoluto algorthm [, ] () s a ovel evolutoary algorthm proposed by Raer Stor ad Keeth Prce. Compared to tradtoal evolutoary algorthm, algorthm has better optmzato performace ad ca reduce the geetc complexty. At presets, has bee appled successfully to all kds of optmzato problems such as eural etworks [], Image Classfcato [], voltage cotrol [5, 6], ad mages segmetato [7], etc. However, for some complex optmzato problems, the covergece rate of algorthm s slow ad easy to fall to local optmal soluto. I order to mprove the optmzato performace of dfferetal evoluto algorthm, may researchers have proposed some mproved algorthm [8, 9], but t s stll dffcult to get good results. At the selecto of probablty model, we sged by the log-lear model of thkg []. I statstcs, lear model s used dfferet ways accordg to the cotext. Log-lear model s a mathematcal model that takes the form of a fucto whose logarthm s a polyomal fucto of the parameters of the model. Log-lear model s wdely used part-of-speech taggg [], statstcal mache traslato [] ad other felds. I ths paper, we proposed a mproved algorthm based o log-lear model. The advatage of loglear model s the ablty to easly bled multple features whch ca ehace decso makg ablty of the model. Ths paper s orgazed as follows. I Secto the backgroud of s preseted. The mproved algorthm s preseted Secto. I Secto some expermetal tests, results ad coclusos are gve. Secto 5 cocludes the paper. Artcle umber P_57 77

2 Sesors & Trasducers, Vol. 59, Issue, November, pp Itroducto to A Suppose the dvdual of geerato G s D represeted as XG = ( xg, xg, xg), =,. Basc operatos such as mutato, selecto ad crossover s the bass of the dfferece algorthm. ) Mutato operato. A dvdual ca be geerated by the followg formula: XG, + = X r, G+ + F *( X r, G Xr, G), where r, r, r are the radom umbers geerated, varato factor F s a real umber. ) Crossover operato. I dfferece algorthm, the cross operate s troduced to the dversty of the ew populato. New dvduals ca be represeted as follow: X = ( x, x x ), =,. G, +, G+ G, + DG, + Where V j, G +, f ( radb( j) CR) or ( j = mbr( )) x j, G + = V j, G +, f ( radb( j) > CR) ad ( j mbr( )) ( j =, D), ad radb( j ) s uformly dstrbuted the terval [, ], CR s crossover probablty. mbr() meas a radom umber. ) Selecto operato. Selecto operato s a greedy strategy, the caddate dvdual geerated from mutato ad crossover operato competto wth target dvdual. x G, + UG,, f ( f ( UG, ) > f ( xg, + )) =, xg, +, f ( f ( UG, ) f ( xg, + )) where f s the ftess fucto. The basc dfferetal evoluto () algorthm s show as Algorthm. Algorthm. The dfferetal evoluto algorthm. ) Italze the umber of populato NP, the maxmum umber of evoluto Max t er, the scale factor ad cross-factor; ) Italze the populato pop ; ) Follow the / rad // b polcy eforcemet optos, ad produce a ew geerato of dvdual: a) Mutato operato; b) Crossover operato; c) Selecto operato. ) Utl the termato crtero s met.. A Improved Dfferetal Evoluto Algorthm Based o Log-lear Model I ths secto, we mplemet a mproved algorthm based o log-lear model. Frstly, loglear model s used to mplemet a mult-probablty adaptve model. Secodly, release operato ad reproductve operato are proposed to mprove the performace for the algorthm... Log-lear Model Supposed the state vector of dvduals s X = ( x, x x ), where x, x x s status of the dvdual. The basc form of the decso makg probablty model based o the log-lear model ca be defed as equato (). P( X ) = exp[ αmfm( X)] m=, M () exp[ α f ( X )] X M m= m m where f m( X ), ( m =, M ) are the feature fucto, ad α m( m=, M) s the weght of feature fucto. The settg of feature fucto has a mportat role for the etre performace of the log-lear model. I order to realze the sgfcace of the model, we must select some effectve feature fuctos. I the expermet, we ca choose dversty fucto ad average dstace metrcs fucto as feature fucto... New Operato ) Release operato If P( X ) below threshold T, the the space of the dvdual s released. f (P( X ) > T ) Perform release operato. ) Reproductve operato The global explorato ablty of tradtoal algorthm s poor. So we propose a mproved algorthm based o reproductve behavor. f (P( X ) < T ) Perform reproductve operato. The reproductve operato ca be defed as equato (). X = X + radom( + * step, * step), () where s the umber of ew dvduals, step s step sze. 78

3 Sesors & Trasducers, Vol. 59, Issue, November, pp The Improved Algorthm Based o Log-lear Model (L) The mproved algorthm s show as follows. Algorthm. The mproved algorthm based o log-lear model. ) Italze the umber of populato NP, the maxmum umber of evoluto t er max, the scale factor ad cross-factor; ) Perform the followg behavor: Mutato, Crossover ad Selecto, ad produce a ew geerato of dvduals; ) Costruct log-lear probablty model by equato (); ) f (P( X ) > T ), the perform release operato by equato; 5) f (P( X ) < T ), the perform reproductve operato by equato (); 6) Utl the termato crtero s met.. Expermet A set of ucostraed real-valued bechmark fuctos was used to vestgate the effect of the mproved algorthm are show Table. Table. Fuctos used to test the effects of dfferet algorthm. Fucto Fucto expresso Optmal value Sphere fucto ( ) Rastrgr fucto f ( x) x ( π x ) Ackey fucto Shaffer's fucto f ( x) f x = x = ( cos + ) t t t xt cos t. f ( x) = + e e e (s x + x ).5 5 =.5 + ( x x ) ( +. + ) ( π x ) Sphere fucto s a sgle-peak fucto, we ca fd the optmal value s through the aalyss for fucto expresso, the fucto mage s show Fg.. Fg.. The mage of Rastrgr fucto. Fg.. The mage of Sphere fucto. Rastrgr fucto s a mult-peak fucto, we ca fd the optmal value s through the aalyss for fucto expresso, the fucto mage s show Fg.. Ackey fucto s a mult-peak fucto, we ca fd the optmal value s through the aalyss for fucto expresso, the fucto mage s show Fg.. Shaffer fucto s a mult-peak fucto, we ca fd the optmal value s through the aalyss for fucto expresso, the fucto mage s show Fg.. The results are show Table. Each pot s made from average values of over repettos. 79

4 Sesors & Trasducers, Vol. 59, Issue, November, pp Performace omparso of dfferet algorthms L 5 Ftess value 6 8 Iteratos Fg. 5. Sphere fucto. Fg.. The mage of Ackey fucto Performace omparso of dfferet algorthms L 5 Ftess value Iteratos Fg. 6. Rastrgr fucto. Fg.. The mage of Shaffer fucto..5.5 Performace omparso of dfferet algorthms L Table. The performaces of dfferet algorthm. Algorthm L Optmal Tme (s) Optmal Tme (s) Sphere Rastrgr Ackley Shaffer...7 Compared wth the stadard algorthm, L algorthm ca effectvely mprove the accuracy such that the optmal value obtaed s much closer to the theoretcal oe for all the four fuctos. O the covergece tme, o algorthm has obvous advatages tha other algorthms. For Sphere fucto, Rastrgr fucto ad Shaffer fucto, L s better tha algorthm. For Ackley fucto, L s ot as good as the stadard algorthm, but the dfferece s small ad acceptable. The comparso of two methods wth coverget curves s show Fg. 5 to Fg. 8. These expermetal results show that mprovg algorthm ca effectvely mprove the covergece speed wth excellet covergece effect. Ftess value Ftess value Iteratos Fg. 7. Grewak fucto. Performace omparso of dfferet algorthms 6 8 Iteratos Fg. 8. Shaffer fucto. L 8

5 Sesors & Trasducers, Vol. 59, Issue, November, pp Coclusos I the paper, we proposed a mproved dfferetal evoluto algorthm base o log-lear model. Expermets show the mproved algorthm has more powerful global explorato ablty wth faster covergece speed. We ca get the followg coclusos. ) The log-lear model ca ehace decso makg ablty of the algorthm. ) The operato of dvduals s essetal for better performace. I the future, we wll troduce some ew operatos ad apply ths dea to other optmzato tasks. 6. Ackowledgmets Ths work was supported by the Natoal Natural Scece Foudato of Cha (Grat No. 655), the scece ad techology project of Quazhou (Grat No. Z9), the Fudametal Research Fuds for the Cetral Uverstes (Grat No. 68) ad the Natural Scece Foudato of Fuja Provce of Cha (Grat No. J69). Refereces []. Stor R., Prce K., Dfferetal evoluto-a smple ad effcet heurstc for global optmzato over cotuous spaces, Joural of Global Optmzato,, 997, pp []. Stor R., Prce K., Dfferetal evoluto for multobjectve optmzato, Evolutoary Computato,,, pp. 8-. []. Dhahr H., Alm A. M., The modfed Dfferetal Evoluto ad the RBF (M-RBF) Neural Network for Tme Seres Predcto, Proceedgs of the Iteratoal Jot Coferece o Neural Networks, Vacouver, USA, 6, pp []. Omra M. G. G., Egelbrecht A. P., Self-Adaptve Dfferetal Evoluto Methods for Usupervsed Image Classfcato, Proceedgs of the IEEE Coferece o Cyberetcs ad Itellget Systems, Bagkok, Thalad, 6, pp. -6. [5]. Yag Shwe, Ga Y. B., Qg Ayog, Sdebad Suppresso Tme-Modulated Lear Array by the Dfferetal Evoluto Algorthm, IEEE Tras o Ateas ad Wreless Propagato Letters,,,, pp [6]. Massa A., Pastoro M., Radazzo A., Optmzato of the Drectvty of a Moopoles Atea wth a Subarray Weghtg by a Hybrd Dfferetal Evoluto Method, IEEE Tras o Ateas ad Wreless Propagato Letters, 5,, 6, pp [7]. Aslatas V., Tuckaat M., Dfferetal evoluto algorthm for segmetato of woud mages, Proceedgs of the IEEE Iteratoal Symposum o Itellget Sgal Processg, WISP, 7. [8]. Wu Laghog, Wag Yaoa, Yua Xaofag, et al., Dfferetal Evoluto Algorthm wth Adaptve Secod Mutato, Chese Joural of Cotrol ad Decso,, 8, 6, pp [9]. Wu Zh-Feg, Huag Hou-Kua, Yag Be, et al., A modfed dfferetal evoluto algorthm wth selfadaptve cotrol parameters, Proceedgs of the rd Iteratoal Coferece o Itellget System ad Kowledge Egeerg, 8, pp []. Ae ath G., The multomal dversty model: lkg Shao dversty to multple predctors, Ecology, 9,,, pp []. Fraz Josef Och, Mmum Error Rate Trag for Statstcal Mache Traslato, Proceedgs of the st Aual Meetg of the Assocato for Computatoal Lgustcs (ACL' ), Japa, Sapporo, July. []. Fraz Josef Och ad Herma Ney, The algmet template approach to statstcal mache traslato, Computatoal Lgustcs,,,, pp Copyrght, Iteratoal Frequecy Sesor Assocato (IFSA). All rghts reserved. ( 8