A Robust LS-SVM Regression

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1 PROCEEDIGS OF WORLD ACADEMY OF SCIECE, EGIEERIG AD ECHOLOGY VOLUME 7 AUGUS 5 ISS 37- A Robust LS-SVM Regresson József Valyon, and Gábor Horváth Abstract In comparson to the orgnal SVM, whch nvolves a quadratc programmng task; LS SVM smplfes the requred computaton, but unfortunately the sparseness of standard SVM s lost. Another problem s that LS-SVM s only optmal f the tranng samples are corrupted by Gaussan nose. In Least Squares SVM (LS SVM), the nonlnear soluton s obtaned, by frst mappng the nput vector to a hgh dmensonal kernel space n a nonlnear fashon, where the soluton s calculated from a lnear equaton set. In ths paper a geometrc vew of the kernel space s ntroduced, whch enables us to develop a new formulaton to acheve a sparse and robust estmate. Keywords Support Vector Machnes, Least Squares Support Vector Machnes, Regresson, Sparse approxmaton. I. IRODUCIO HIS paper focuses on the least squares verson of SVM [], the LS SVM [], whose man advantage s that t s computatonally more effcent than the standard SVM method. In ths case tranng requres the soluton of a lnear equaton set nstead of the long and computatonally hard quadratc programmng problem nvolved by the standard SVM. he method effectvely reduces the algorthmc complexty, however for really large problems, comprsng a very large number of tranng samples, even ths least-squares soluton can become hghly memory and tme consumng. Whereas the least squares verson ncorporates all tranng data n the network to produce the result, the tradtonal SVM selects some of them (the support vectors) that are mportant n the regresson. he sparseness of tradtonal SVM can also be reached wth LS SVM by applyng a prunng method [3][]. Unfortunately f the tradtonal LS SVM prunng method s appled, the performance declnes proportonally to the elmnated tranng samples, snce the nformaton (nputoutput relaton) they descrbed s lost. Another problem s that ths teratve method multples the algorthmc complexty. he tranng data s often corrupted by nose, whch f not handled properly msleads the tranng. Another modfcaton of the method, called weghted LS SVM [][5], s amed at reducng the effects of non-gaussan nose (e.g. outlers). he bggest problem s that prunng and weghtng hs work was partly sponsored by atonal Fund for Scentfc Research (OKA) under contract 77. J. Valyon s wth Budapest Unversty of echnology and Economcs- Department of Measurement and Informaton Systems, Budapest, Hungary, H- 5, pf. 9. (phone ; fax +3 3-; e-mal: valyon@ mt.bme.hu). G. Horváth (e-mal: horvath@mt.bme.hu). although ther goals do not rule out each other cannot be used at the same tme, because they work n opposton. he generalzed approach presented n ths paper enables us to accomplsh both goals by allowng a more unversal constructon and soluton of the LS SVM equaton set. hs paper proposes a geometrc vew of the kernel space and the lnear soluton that s based on the mapped tranng samples. In the LS-SVM soluton, the tranng samples are mapped to a kernel space, where a hyperplane s ftted on these ponts. In ths case all the tranng samples are used to acheve a result, whch consequently sn t sparse. o trade off between tranng error and a smooth soluton a regularzaton parameter s used, whch s the same for all samples, and can be consdered as a predefned, ntentonal error term n the kernel space fttng. Our proposton s to use a kernel space of smaller dmensonalty, whch means that by mappng all tranng samples, the hyperplane can be ftted many ways, snce there are more equatons than unknown. hs means that the poston of the hyperplane and consequently the tranng errors the dstances from ths plane can be automatcally determned accordng to the dstrbuton of many mapped ponts. he LS SVM method s capable of solvng both classfcaton and regresson problems. he classfcaton approach s easer to understand and more hstorc. he present study concerns regresson, but t must be emphaszed that all presented methods can be appled to classfcaton as well. hs paper s organzed as follows. Before gong nto the detals LS SVM, and ts extensons of prunng and weghtng are summarzed n Secton II. Secton III provdes a geometrc nterpretaton of the kernel space and summarzes the man dea behnd the propostons. Secton IV contans the detals of the soluton: It shows how partal reducton s used to acheve an overdetermned equaton set, and proposes robust solutons for ths. Secton V. contans some expermental results, whle n secton VI. the conclusons are drawn. II. A BRIEF IRODUCIO O HE LS-SVM MEHOD Gven the { } p x,d = tranng data set, where x R represents a p dmensonal nput vector and d = y + z, d R s a scalar measured output, whch represents the y system output corrupted by some z nose. Our goal s to PWASE VOLUME 7 AUGUS 5 ISS 37-5 WASE.ORG

2 PROCEEDIGS OF WORLD ACADEMY OF SCIECE, EGIEERIG AD ECHOLOGY VOLUME 7 AUGUS 5 ISS 37- construct an f ( x) y = functon, whch represents the dependence of the output y on the nput x. Let s defne the form of ths functon as formulated below: h y = w ϕ ( x) + b = w ϕ( x) + b, () = w = [ w w,..., ], = [ ϕ ϕ,..., ], w h p h ϕ., he ϕ (.) : R R s a mostly non-lnear functon, whch maps the data nto a hgher possbly nfnte dmensonal feature space. he man dfference from the standard SVM s that LS-SVM nvolves equalty constrants nstead of nequalty ones and works wth a least squares cost functon []. he optmzaton problem and the nequalty constrants are defned by the followng equatons ( =,..., ): mn J p ( w, e) = w w + C e () w, b, e = d = w ϕ x + b + e. ϕ h wth constrants: ( ) + he C R s the trade off parameter between a smoother soluton, and tranng errors. From ths, a Lagrangan s formed L( w, b, e; α ) = J ( w, e) α w ϕ( x ) + b + e d. (3) p k = { } he soluton concludes n a constraned optmzaton, where the condtons for optmalty lead to the followng overall soluton: r b r =, + Ω C I α d d d d,..., α = α α,..,, (5) = [, d ]., [, α ] = [,...,] = ( x, x ) = ϕ ( r, Ω, j K j x ) ϕ( x j ), where K( x ) j x, s the kernel functon, and Ω s the kernel matrx. he result s: ( ) y = = αk x, x + b () A detaled descrpton of LS-SVM can be found n refs. []- [5]. LS-SVM prunng - One of the man drawbacks of the least squares soluton s that, t s not sparse, because unlke the orgnal SVM t ncorporates all tranng vectors n the result. In order to get a sparse soluton, a prunng method must be used. Snce the α support values are proportonal to the errors at data ponts: α = Ce, (7) the rrelevant ponts are left out, by teratvely leavng out the least sgnfcant vectors. hese are the ones correspondng to the smallest α values. In the case of the classcal SVM sparseness s acheved by the use of such loss functons, where errors smaller than ε are gnored (e.g. ε-nsenstve loss functon). hs method reduces the dfference between SVM and LS SVM, because the omsson of some data ponts mplctly corresponds to creatng an ε-nsenstve zone []. he descrbed method leads to a sparse model, but some questons arse: How many neurons are needed n the fnal model? How many teratons t should take to reach the fnal model? Another problem s that a usually large lnear system must be solved n each teraton. Prunng s especally mportant f the number of tranng vectors s large. In ths case however, the teratve method s not very effectve. Weghted LS-SVM - hs method addresses the problem of nosy data lke outlers n a dataset, by usng a weghtng factor n the calculaton based on the error varables determned from a prevous frst an unweghted soluton. he method uses a bottom-up approach by startng from a standard soluton, and calculatng one or more weghted LS SVM based on the prevous result. he weghted LS SVM s formed as: r b r = () Ω + VC α d where V C = dag,...,. (9) Cv Cv he v weghtng s desgned such, that the results mprove n vew of robust statstcs. Large e -s mean a small weght and vce versa. A common property of the descrbed methods s that they are all teratve, where every step s based on the result of an LS-SVM learnng. hs means, that the entre large problem must be solved at least once, and a relatvely large one n every further teraton step. Another drawback s that prunng and weghtng cannot be easly combned, because the methods favor contradctory types of ponts. Whle prunng drops the tranng ponts belongng to small α -s, the weghted LS SVM ncreases the effects of these ponts. III. HE MAI IDEA When an LS-SVM s constructed from tranng samples:. he samples are mapped to an + dmensonal kernel space, where dmensons are defned by the kernel functons and one s the desred output.. A hyperplane s ftted on these mapped samples. he hyperplane s determned by mapped ponts, and one addtonal constrant ( α = ). = he approxmated answer for a new sample results from mappng t nto dmensons, and calculatng the correspondng pont on the ftted hyperplane (dmenson +). herefore n case of ths soluton, we expect that when a new sample s transformed to ths kernel space, the desred output wll be close to ths hyperplane. For the sake of generalzaton and to avod overfttng the accuracy of the ft can be adjusted through regularzaton (the C hyperparameter). Some questons stll need answers: How many and whch PWASE VOLUME 7 AUGUS 5 ISS WASE.ORG

3 PROCEEDIGS OF WORLD ACADEMY OF SCIECE, EGIEERIG AD ECHOLOGY VOLUME 7 AUGUS 5 ISS 37- dmensons are needed n the kernel space? How good s my approxmaton, for a specfc mappng? What s a good value of C, or more generally, how should the hyperplane be placed n the kernel space? Our proposton s to take control of the problem n the kernel space by: controllng the dmensonalty of ths space (see IV. A.), fndng a better lnear hyperplane n the kernel space (see IV. B.), choosng an approprate kernel space (IV. C.). Havng fewer dmensons n the kernel space, results n a sparse soluton, whle at the same tme t ncreases the number of mapped ponts that can be used to determne the lnear ft. Havng more ponts than dmensons n the kernel space allows us to optmze the lnear ft. he dmensonalty of the kernel space hgh s enough, f samples (not used n determnng ths space) fall close to ths plane after mappng (see Fg..). a.) b.) Fg. he mage of tranng samples n a kernel space of dfferent dmensons. Usng all three samples as support vectors (kernel centers), a three dmensonal kernel can space guarantee exact ft for the samples. he dashed lnes represent a zone n whch errors can be accepted (correspondng to the ε-nsenstvty of SVM) IV. HE PROPOSED MEHODS hs secton proposes some modfcatons and extensons to the standard LS SVM. her purpose s to gan control over network sze, to reduce complexty and to mprove the qualty of the results. A. Usng an Overdetermned Equaton Set If the tranng set conssts of samples, then our orgnal lnear equaton set (see eq. 5) wll have ( +) unknowns, the α -s and b, ( +) equatons and ( +) multplers. hese factors are manly the values of the K ( x, x j ) kernel functon calculated for every combnaton of the tranng nput pars. he cardnalty of the tranng set therefore determnes the sze of the kernel matrx, whch plays a major part n the soluton, as algorthmc complexty; the complexty of the result etc. depends on ths. o reduce the equaton set, columns and/or rows may be omtted. If the k-th column s left out, then the correspondng α k s also deleted, therefore the resultng model wll be = smaller. he α = condton automatcally adapts, snce the remanng α -s wll stll add up to zero. If the j-th row s deleted, then the condton defned by x, tranng sample s lost, because the j-th the ( ) j d j equaton s removed. he most mportant component of the man matrx s the Ω kernel matrx; ts elements are the results of the kernel functon for pars of tranng nputs: Ω, j = K( x, x j ) () o reduce the sze of Ω some tranng samples should be omtted. Each column of the kernel matrx represents an addtve term n the fnal soluton, wth a kernel functon centered on the correspondng x nput. he rows however, represent the nput output relatons, descrbed by the tranng ponts. It can be seen that n order to reach sparseness the number of columns must be reduced. he followng reducton technques can be used on the kernel matrx (the names of these technques are ntroduced here for easer dscusson): radtonal full reducton A tranng sample ( x k, d k ) s fully omtted, therefore both the column and the row correspondng to ths sample are elmnated. In ths case however reducton also means that the knowledge represented by the numerous other samples are lost. hs s exactly the case n tradtonal LS SVM prunng snce prunng teratvely omts some tranng ponts. he nformaton emboded n these ponts s entrely lost. o avod ths nformaton loss, one may use the technque referred here as partal reducton. he proposed partal reducton In partal reducton, the omsson of a tranng sample ( x k, d k ) means that only the correspondng column s elmnated, whle the row whch defnes an nput-output relaton s kept. Elmnatng the k-th column reduces the model complexty, whle keepng the k-th row means that the weghted sum of that row should stll meat the d k regresson goal (as closely as possble). By selectng some (e.g. M, M < ) vectors as support PWASE VOLUME 7 AUGUS 5 ISS WASE.ORG

4 PROCEEDIGS OF WORLD ACADEMY OF SCIECE, EGIEERIG AD ECHOLOGY VOLUME 7 AUGUS 5 ISS 37- vectors, the number of α varables are also reduced, resultng n more equatons than unknowns. he effect of partal reducton s shown on the next equaton, where the removed elements are colored grey. () hs proposton resembles to the bass of the Reduced Support Vector Machnes (RSVM) ntroduced for standard SVM classfcaton n []. For further dscussons, let s smplfy the notatons of our man equaton as follows: r b A = r, u = Ω +, v = C I α. () d he omsson of columns wth keepng the rows means that the network sze s reduced; stll all the known constrants are taken nto consderaton. hs s the key concept of keepng the qualty, whle the equaton set s smplfed. It s mportant to menton that the hyperparameter C s not necessarly needed n case of partal reducton. As t wll be seen later, the overdermned system means that errors are nherently expected at the samples. C s used to show how our proposton reduces the orgnal formulaton, but t can be left out from the formulas entrely. he dmensonalty of the x nput vectors only affects the calculaton of K ( x, x j ), but nothng n the rest of the method, therefore the descrbed process works rrespectvely of the nput dmensonalty. It s also ndependent from the kernel functon, snce after calculatng the kernel matrx, the proposed methods can be appled wthout any change. he deleted columns can be selected many ways e.g. randomly, or by usng the method proposed n the sequel. B. Solvng the Overdetermned System It s easy to see that partal reducton leads to a sparse soluton, but havng an overdetermned equaton set has several other advantages. By havng more equatons than unknowns we have means to analyze ths nformaton set. he soluton of ths equaton set corresponds to a lnear fttng problem, where we have to ft an M+-dmensonal hyperplane on the ponts defned by the rows of the matrx. Snce >>M+, ths can be done several ways. he resdual for the -th data pont corresponds to the e error, whch s defned as the dfference between the observed desred response value d and the ftted response value y. e = d y (3) he solutons dffer n the way they calculate the accumulated error (resduals), whch s then mnmzed. he optmal soluton depends on the statstcal propertes of the dataset. (he term statstcal here does not necessarly mean a large number of samples, but t means more than one whch s the case n the orgnal formulatons.) Some possble solutons: Lnear least squares (for Gaussan nose LS SVM) Weghted lnear least squares Custom weghtng Robust bsquare weghts method It s mportant to emphasze, that the proposed partal reducton s essental, snce t allows us to have more samples than dmensons n the kernel space, whch allows optmzng further n ths space. ) Lnear least squares Usually there are two mportant assumptons that are made about the nose ( z ): he error exsts only on the output. he errors are random and follow a normal (Gaussan) dstrbuton wth zero mean and constant varance σ. In ths case we mnmze the summed square of the resduals: = S = e = ( d y ) () = he soluton of equaton () can be formulated as A Au = A v. (5) he modfed matrx A has ( +) rows and ( M +) columns. After the matrx multplcatons the results are obtaned from a reduced equaton set, ncorporatng A A, whch s of sze ( M + ) ( M + ) only. Our proposton, to use partal reducton along wth the lnear least squares soluton have already been presented n [7] and [], where we named ths method LS SVM, snce t gves the least squares soluton of a least squares SVM method. ) Weghted methods If the assumpton that the random errors have constant varance does not hold, weghted least squares regresson may be used. Instead of levelng the errors statstcally, t s assumed that the weghts used n the fttng represent the dfferng qualty of data samples. he error term s: = S = w e = = w ( d y ) () he weghted soluton can be formulated as: A WAu = A Wv. (7) where the W weght matrx s: W = dag { w,..., w }. () he weghts are used to adjust the amount of nfluence each data pont has on the estmated lnear ft to an approprate level. hs formulaton s exactly the same that was reached by Suykens n the Weghted LS SVM [] but the way t s derved dffers greatly. Suykens ntroduces dfferent regularzaton parameters (Cv -s) for the samples, whle n the proposed method the weghts are ntroduced n the lnear fttng method. he most mportant dfference, however s that the use of partal reducton leads to an overdetermned system, PWASE VOLUME 7 AUGUS 5 ISS WASE.ORG

5 PROCEEDIGS OF WORLD ACADEMY OF SCIECE, EGIEERIG AD ECHOLOGY VOLUME 7 AUGUS 5 ISS 37- so the weghts can be calculated from the statstcal propertes of the ponts (dstrbuton of many ponts) n the kernel space. Another mportant dfference s that the proposed weghted soluton s also sparse. CUSOM WEIGHIG - ths method can be used f one had a pror knowledge about the qualty of the samples. If so, weghts can be defned to determne how much each learnng sample nfluences the ft. Samples known to have less nose are expected to ft more, than low-qualty ones. he weghts should transform the response varances to a constant value. If the varances of the data are known, the weghts are gven by: w σ =. (9) BISQUARE WEIGHS a method that mnmzes a weghted sum of squares, where the weght of each data pont depends on ts dstance from the ftted lne. he farther away s the pont, the less weght t gets. hs method fts the hyperplane to the bulk of the data wth the least squares approach, whle t mnmzes the effect of outlers (Fg..). More detals on robust regresson can be found n [9][]. 5-5 data samples least squares ft roboust ft - Fg. he least squares and the roboust (bsquare) fttng n two dmensons C. Selectng Support Vectors Standard SVM automatcally selects the support vectors. o acheve sparseness by partal reducton, the lnear equaton set has to be reduced n such a way, that the soluton of ths reduced (overdetermned) problem s the closest to what the orgnal soluton would be. As the matrx s formed from columns we can select a lnearly ndependent subset of column vectors and omt all others, whch can be formed as lnear combnatons of the selected ones. hs can be done by fndng a bass (the quote ndcates, that ths bass s only true under certan condtons defned later) of the coeffcent matrx, because the bass s by defnton the smallest set of vectors that can solve the problem. he lnear dependence dscussed here, does not mean exact lnear dependence, because the method uses an adjustable tolerance value when determnng the resemblance (parallelsm) of the column vectors. he use of ths tolerance value s essental, because none of the columns of the coeffcent matrx wll lkely be exactly dependent (parallel). he reducton s acheved as a part of transformng the A matrx nto reduced row echelon form, usng a slght modfcaton of Gauss-Jordan elmnaton wth partal pvotng []. hs method returns a lst of the column vectors whch are lnearly ndependent form the others consderng a tolerance ε. he tolerance (ε ) can be related to the ε parameter of the standard SVM, because t has smlar effects. he larger the tolerance, the fewer vectors the algorthm wll select. If the tolerance s chosen too small, than a lot of vectors wll seem to be ndependent, resultng n a larger network. As stated earler the standard SVM s sparseness s due to the ε-nsenstve zone, whch allows the samples fallng nsde ths boundary to be neglected. Accordng to ths, t may not be very surprsng to fnd that an addtonal parameter s needed to acheve sparseness n LS SVM, and ths parameter corresponds to the one, whch was orgnally left when changng from the SVM to the standard least squares soluton. he basc dea of dong a feature selecton n the kernel space s not new. he nonlnear prncpal component analyss technque, the Kernel PCA uses a smlar dea []. A bass selecton from the kernel matrx has been shown n [3]. V. EXPERIMES he next fgures show the results for a smple llustratve experment, the snc(x) regresson. he tranng set contans 5 data samples corrupted wth Gaussan nose. Fg. 3 shows the results of custom weghtng. We have samples wth addtve Gaussan nose, where the σ of the nose s known for all samples. It can be seen, that the effect of nose s reduced. he orgnal LS SVM s plotted, because the Weghted LS SVM would gve almost the same results as the partally reduced soluton, but n ths case we have a sparse soluton nput data ponts support vectors partal reducton LS-SVM SIC Fg. 3 Custom weghtng s appled wth partal reducton (he PWASE VOLUME 7 AUGUS 5 ISS WASE.ORG

6 PROCEEDIGS OF WORLD ACADEMY OF SCIECE, EGIEERIG AD ECHOLOGY VOLUME 7 AUGUS 5 ISS 37- LS SVM s not weghted.) he followng experment (Fg. ) shows the same problem as Fg. 3, but n ths case a few data ponts are corrupted to provde outlers nput data ponts support vectors roboust partal reducton LS-SVM [3] J. A. K. Suykens, L. Lukas, and J. Vandewalle, Sparse approxmaton usng least squares support vector machnes, IEEE Internatonal Symposum on Crcuts and Systems ISCAS', [] J. A. K. Suykens, L. Lukas, and J. Vandewalle, Sparse least squares support vector machne classfers, ESA' European Symposum on Artfcal eural etworks,, pp. 37. [5] J.A.K. Suykens, J. De Brabanter, L. Lukas, and J. Vandewalle, Weghted least squares support vector machnes: robustness and sparse approxmaton, eurocomputng,. pp. 5-5 [] Yuh Jye Lee and O. L. Mangasaran, RSVM: Reduced Support Vector Machnes, Proceedngs of the Frst SIAM Internatonal Conference on Data Mnng, Chcago,. Aprl 5 7. [7] J. Valyon and G. Horváth, A generalzed LS SVM, SYSID'3 Rotterdam, 3, pp [] J. Valyon and G. Horváth, A Sparse Least Squares Support Vector Machne Classfer, Proceedngs of the Internatonal Jont Conference on eural etworks IJC,, pp [9] Holland, P. W., and R. E. Welsch, "Robust Regresson Usng Iteratvely Reweghted Least-Squares," Communcatons n Statstcs: heory and Methods, A, 977, pp [] Huber, P. J., Robust Statstcs, Wley, 9. [] W. H. Press, S. A. eukolsky, W.. Wetterlng and B. P. Flannery, umercal Recpes n C, Cambrdge Unversty Press, Books On-Lne, Avalable: [] B. Schölkopf, S. Mka, C.J.C. Burges, P. Knrsch, K.-R. Müller, G. Rätsch, and A. Smola, Input space vs. feature space n kernel-based methods. IEEE ransactons on eural etworks, 999, (5), pp. 7. [3] G. Baudat and F. Anouar, Kernel-based methods and functon approxmaton. In Internatonal Jont Conference on eural etworks, pages 9, Washngton DC,. July 5 9. [] H. Golub and Charles F. Van Loan, Matrx Computatons, Gene Johns Hopkns Unversty Press, 99. Fg. he contnuous black lne plots the result for a partally reduced LS-SVM solved by the bsquare weghts method. he dashed lne s the orgnal LS SVM It can be seen that by usng a robust bsquare fttng, the effect of the outlers was successfully reduced. It s mportant to menton that the result of the LS SVM s sparse, consstng of only support vectors. If the number of tranng samples s very hgh for the problem complexty, than the gan n the network sze can be rather large. VI. COCLUSIO In ths paper a geometrc vew and a generalzed formulaton of the least squares support vector machne was presented. he basc dea s that by reducng the dmensonalty of the kernel space, the hyperplane ftted to the mapped tranng samples can be optmzed accordng to ther dstrbuton. hs s especally mportant, to deal wth non Gaussan nose. he descrbed soluton acheves two mportant results smultaneously: a sparse LS SVM soluton, the effect of nose s reduced. REFERECES [] V. Vapnk, "he ature of Statstcal Learnng heory", ew York: Sprnger Verlag., 995 [] J. A. K. Suykens, V.. Gestel, J. De Brabanter, B. De Moor, J. Vandewalle, Least Squares Support Vector Machnes, World Scentfc, PWASE VOLUME 7 AUGUS 5 ISS WASE.ORG

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