ENSEMBLE OF NEURAL NETWORKS FOR IMPROVED RECOGNITION AND CLASSIFICATION OF ARRHYTHMIA

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1 XVIII IMEKO WORLD COGRESS Metrology for a Sustanable Development September, 7 22, 2006, Ro de Janero, Brazl ESEMBLE OF EURAL ETWORKS FOR IMPROVED RECOGITIO AD CLASSIFICATIO OF ARRHYTHMIA S. Osowsk,2, T. Markewcz, L. Tran Hoa 3 Warsaw Unversty of Technology,Warsaw, Poland, sto@em.pw.edu.pl 2 Mltary Unversty of Technology,Warsaw, Poland, 3 Hano Unversty, Vetnam Abstract: The paper presents dfferent methods of combnng many neural classfers nto one ensemble system for recognton and classfcaton of arrhythma. Maorty and weghted votng, Kullback-Lebler dvergence and modfed Bayes methods wll be presented and compared. The numercal experments wll be performed for the problems concernng the recognton of dfferent types of arrhythma on the bass of ECG waveforms of MIT BIH AD. Keywords: neural classfers, ensemble of classfers, methods of ntegratons, arrhythma recognton.. ITRODUCTIO The paper deals wth the problem of combnng many neural classfers nto one commttee machne performng the task of the recognton of heart rhythms. It s known fact that each classfer consders the recognton problem from dfferent pont of vew (dfference n data preprocessng, recognton algorthm and methodology). Usually for a specfc applcaton problem each classfer, relyng on dfferent feature sets, may attan dfferent degree of success. one of them s perfect or as good as expected. The dea s to combne dfferent solutons of classfers so that a better result could be obtaned. Combnng the traned networks, nstead of dscardng them, helps to ntegrate the knowledge acqured by the component classfers and n ths way to mprove the accuracy of the fnal classfcaton. The paper wll present and compare few dfferent ways of combng neural classfers nto one ensemble system. Smple maorty votng, weghted votng, Kullback-Lebler dvergence as well as the nave modfed Bayes combnaton wll be nvestgated and checked on the examples of the real lfe problem of arrhythma recognton by ECG waveform analyss. The consdered task of the arrhythma recognton s an mportant problem n automated pattern recognton n medcne [,4,7]. The ndvdual classfers consdered for ntegraton are bult on the bass of dfferent classfer platforms and data preprocessng methods. The consdered classfers nclude: the neuro-fuzzy networks of the modfed Takag-Sugeno- Kang structure, the hybrd network and the Support Vector Machne. The recognton of arrhythma s proceeded on the bass of the regstered ECG waveform (QRS segment) for patents sufferng from dfferent knds of rregulartes of the heart beats. Two preprocessng technques are employed for the dagnostc features generaton: the hgher order statstcs (HOS) characterzaton of the QRS complex and expanson of the QRS complex nto Hermte bass functons (HER). The results of numercal experments concernng the recognton of 6 types of arrhythma and the normal snus rhythm wll be presented and dscussed. 2. THE ITEGRATIO METHODS Fg. presents the general scheme of ntegraton of many classfers nto one ensemble system [7] Fg. : The general scheme of classfcaton usng many classfers The measured sgnals of the process form the vector x n, subect to the preprocessng n the preprocessng blocks P (=, 2,, M). The preprocessors may be of varous knds, stressng dfferent aspects of sgnal. The features generated by the preprocessors form the vectors x appled to the neural classfers C. These vectors may vary n many aspects, ncludng even the sze (the number of dagnostc features). Each classfer has outputs ( classes) and the output sgnals of each classfer are arranged n the form of vectors y for =, 2,, M, where M s the number of classfcaton channels. These vectors are combned n the ntegratng unt to form one fnal output vector z of the classfer ( z R ). The hghest value of elements of z ndcates the membershp to the approprate class.

2 The ntegraton of many classfers nto one ensemble of networks may be done usng dfferent methodologes. We wll apply here four dfferent approaches. They nclude: the smple maorty votng, the weghted votng, Kullback- Lebler dvergence method and the modfed Bayes combnaton. 2.. The maorty votng Suppose we have M neural network classfers, whch were traned on the same data. The commttee of these classfers assgns the pattern to the class that obtans the maorty of the votes. Each classfer has the same nfluence on the fnal score. The maorty votng s effectve when the probablty pr for each classfer to gve the correct class label s equal for all nput vectors x and at the same tme the classfer outputs are ndependent. However even n ths case we can expect mprovement over the ndvdual accuracy pr only when pr s hgher than 0.5 [6]. In the other case the maorty votng ntegraton does not brng any mprovement over the ndvdual classfer results. 2.2 The weghted votng If the classfers n the ensemble system are not of the same accuracy then t s reasonable to gve more competent classfers more power for the fnal decson. The weghted maorty votng combnes the results of M classfers wth the weghts accordng to the accuracy of each classfer obtaned for the learnng data. Ths s done through the ntegratng matrx W to form one response of the classfyng system []. Let us denote by y the vector of the classfcaton results of th classfer and by z the output vector of the ensemble system. The number of learnng data pars s denoted by p. The result of ntegraton of all classfers at the presentaton of one partcular nput vector x n can be expressed now by the relaton z = Wy () y = y y,..., y and, 2 T T T where [ ] T M W R M. The poston of the hghest value element of z ndcates the membershp to the approprate class. In adustng the values of elements of the weght matrx W we have appled the mnmzaton of the sum of squared error of the whole ensemble of the classfers, measured on the learnng data set []. Ths mnmzaton leads to the soluton expressed through the + Moore-Penrose pseudonverse n the form W = DY, where Y s the M p matrx composed of p vectors y correspondng to p results of ndvdual M classfcatons for learnng data and D s the approprate p matrx formed by the destnaton vectors assocated wth each learnng par of data. 2.3 Kullback-Lebler dvergence method Kullback-Lebler dvergence measures the dstance between the pror dstrbuton and a posteror dstrbuton. It s nterpreted as the amount of nformaton needed to change the pror probablty dstrbuton nto the posteror one. In Kullback-Lebler dvergence method [6] we calculate the ensemble probablty µ supportng the th class gven the actual nput vector x n, as the normalzed arthmetc mean M µ = (2) d M = where d means the probablty of ndcatng th class by th classfer for the data of ths class. Ths probablty s determned n the testng mode for each multple output classfer on the bass of the sgnal values on each output. In the case of one output classfer (for example SVM) we apply the one aganst one approach and the probablty of each class s equal to the rato of the number of vctores of th class to all possble ndcatons. Observe that at two classes and 0- membershp value to the partcular class the Kullback-Lebler method s equvalent to the smple maorty votng. 2.4 The modfed nave Bayes combnaton Ths method assumes that the classfers are mutually ndependent gven a class label. We apply here the modfcaton of the nave Bayes combnaton [6] snce t gves more relable results at zero estmated probablty of any classfer. Accordng to ths modfcaton the ensemble probablty µ supportng the th class s determned on the bass of the known results of testng the networks on the learnng data and s gven n the form ( ) M cm + / s µ = (3) = n + where n s the number of elements n tranng set for class () and cm s the element of the confuson matrx generated s for learnng data of th classfer. The (,s)th entry of the confuson matrx s the number of elements of the data set whose true class label was and were assgned by th classfer to sth class. 3. THE EURAL CLASSIFIERS Dfferent classfer solutons can be appled n practce. In ths paper we wll consder only the neural classfers of dfferent types. The consdered classfers nclude: the neuro-fuzzy networks of the modfed Takag-Sugeno-Kang (TSK) structure, the hybrd network and Support Vector Machne (SVM). 3. Hybrd fuzzy network The hybrd fuzzy network [0] s the combnaton of the fuzzy self-organzng layer and the multlayer perceptron (MLP) connected n cascade (generalzaton of the so called Hecht-elsen counter-propagaton network). The fuzzy self-organzng layer s responsble for the fuzzy clusterzaton of the nput data, n whch the vector x s preclassfed to all clusters wth dfferent membershp grades. The partcular membershp value of some data vector x to the cluster of the center c s defned by the equaton u (x ) (4) = c k = / ( d d ) k

3 where c s the number of clusters and d = x c. The poston of the center of each cluster s adusted n the learnng procedure over all learnng vectors x. In our work we have appled the c-means algorthm [5]. The sgnals of the self-organzng neurons (the membershp grades) form the nput vector to the second subnetwork of MLP. MLP conssts of many smple neuronlke processng unts of sgmodal actvaton functon, grouped together n layers. Informaton s processed locally n each unt by computng the dot product between the correspondng nput vector and the weght vector of the neuron. Tradtonally tranng the network to produce a desred output vector when presented wth an nput vector nvolves systematcally changng the weghts of all neurons untl the network produces the desred output wthn a gven tolerance (error). The MLP part of the hybrd network s responsble for the assocaton of the nput vector wth the approprate class (the fnal classfcaton). It s traned after the frst selforganzng layer has been establshed. The tranng algorthm s dentcal to that used n tranng MLP alone [2]. 3.2 TSK neuro-fuzzy network Another neuro-fuzzy network nvolved n comparson s the modfed Takag-Sugeno-Kang (TSK) network [2]. It s mplemented n the neuro-lke structure realzng the fuzzy nference rules wth the crsp TSK concluson, descrbed by the lnear functon. The TSK network can be assocated wth the approxmaton functon y(x) K y( x ) = µ ( x ) p0 + pk xk (5) = k = where µ (x) s descrbed by (4) and p k are the coeffcents f k of the lnear TSK functons (x) = p0 + = pk xk. The parameters of the premse part of the nference rules (the membershp values µ (x )) are selected very precsely usng Gustafson-Kessel self-organzaton algorthm [2]. After then they are frozen and don t take part n further adaptaton. It means that at applcaton of the nput vector x ( =, 2,..., p) to the network, the membershp values µ (x ) are constant. The remanng parameters p of the lnear TSK functons can be then easly obtaned by solvng the set of lnear equatons followng from equatng the actual values of y(x ) and the destnaton values d for =, 2,..., p. The determnaton of these varables can be done n one step by usng the sngular value decomposton (SVD) algorthm and the pseudo-nverse technque. 3.3 SVM classfer The last classfer nvolved n the ensemble s the Support Vector Machne network [3,4]. It s known as the effcent tool for the classfcaton problems, of a very good generalzaton ablty. The SVM s a lnear machne workng n the hgh dmensonal feature space formed by the nonlnear mappng of the n-dmensonal nput vector x nto a K-dmensonal feature space (K>n) through the use of the nonlnear functon ϕ (x). The equaton of the hyperplane separatng two classes s defned n terms of these functons y( ) = w ( x) + b = 0 K x ϕ, where b s the = bas, and w the synaptc weght of the network. The parameters of ths separatng hyperplane are adusted n a way to maxmze the dstance between the closest representatves of both classes. In practce the learnng problem of SVM s solved n two stages nvolvng the soluton of the prmary and dual problems [3,4]. The most dstnctve fact about SVM s that the learnng task s smplfed to the quadratc programmng by ntroducng the Lagrange multplers α. All operatons n learnng and testng modes are done n SVM usng kernel functons K( x, x ), satsfyng the Mercer condtons [3,4]. The most known kernels are Gaussan, polynomal, lnear or splne functons. The output sgnal y(x) of the SVM network s fnally determned as p = y( x ) α d K( x, x) b (6) = + where d = ± s the bnary destnaton value assocated wth the nput vector x. The postve value of the output sgnal means membershp of the vector x to the partcular class, whle the negatve one to the opposte one. Although the SVM separates the data nto two classes only, the recognton of more classes s straghtforward by applyng ether one aganst one or one aganst all methods [3]. The more powerful s one aganst one approach, n whch many SVM networks are traned to recognze between all combnatons of two classes of data. For classes we have to tran (-)/2 ndvdual SVM networks. In the retreval mode the vector x belongs to the class of the hghest number of wnnngs n all combnatons of classes. 4. PREPROCESSIG OF THE ECG SIGALS The mportant step n buldng the effcent classfer system s the generaton of the dagnostc features, on the bass of whch the classfer wll recognze the pattern. In our approach to the problem we have appled two preprocessng methods of the data. One apples the Hermte representaton of the QRS complex of the ECG and the second characterzes the QRS complex by the cumulants. 4. Hermte representaton of theecg In Hermte bass functon expanson method we represent the QRS complex by the seres of Hermte functons [7]. Denote the QRS complex of the ECG curve by x(t). Its expanson nto Hermte seres may be wrtten n the way x( t) = n= 0 c n φ ( t, σ ) (7) n where c n are the expanson coeffcents, φn ( t, σ ) - the Hermte bass functons of n th order and σ s the wdth parameter. The coeffcents c n of Hermte bass functons expanson may be treated as the features used n the recognton

4 process. They may be obtaned by mnmzng the sum E = x( t ) c n= 0 n n φ ( t, σ ) squared error, [ ] 2. Ths error functon represents the set of lnear equatons wth respect to the coeffcents c n. They can be easly solved by usng sngular value decomposton. In numercal calculatons, we have presented the QRS segment of the ECG sgnal by 9 data ponts around the R peak (45 ponts before and 45 ones after). At the data sample rate 360 Hz, ths gves a wndow of 250 ms, whch s long enough to cover most of QRS sgnals. The data has been also addtonally expanded by addng 45 zeros to each end of the QRS segment. The extra zeros are added to enforce that the beats are closed to zero outsde the QRS complex. The wdth σ was chosen proportonal to the wdth of the QRS complex. The modfed QRS complexes have been decomposed onto a lnear combnaton of 5 Hermte bass functons. These coeffcents together wth 2 classcal features: the nstantaneous RR nterval of the beat (the tme span between two consecutve R ponts) and the average RR nterval of 0 precedng beats, form the 7-element feature vector x appled to the nput of the classfer. 4.2 HOS characterzaton of the ECG Another approach to the feature generaton s the applcaton of the statstcal descrpton of the QRS curves. Three types of statstcs have been appled: the second-, thrd- and fourth-order cumulants [9]. Applcaton of the cumulant characterzaton of QRS complexes reduces the relatve spread of the ECG characterstcs belongng to the same type of heart rhythm and n ths way makes the classfcaton easer. As the features used n the heart rhythm recognton we have appled the values of the cumulants of the 2nd, 3rd and 4th orders at fve ponts dstrbuted evenly wthn the QRS length (for the 3rd and 4th order cumulants the dagonal slces have been calculated). For 9-element vector representaton of the QRS complex the cumulants correspondng to the tme lags of 5, 30, 45, 60 and 75 have been chosen. Addtonally we have added two temporal features: one correspondng to the nstantaneous RR nterval of the beat and the second representng the average RR nterval of 0 precedng beats. In ths way each beat has been represented here by the 7- element feature vector, wth the frst 5 elements correspondng to the hgher order statstcs of QRS complex (the second, thrd and fourth order cumulants, each represented by 5 values) and the last two - the temporal features of the actual QRS sgnal. 5. THE UMERICAL EXPERIMETS 5. The data base The numercal experments have been drected for the recognton of the heartbeat on the bass of the ECG waveform. The recognton of arrhythma s proceeded on the bass of the QRS segments of the regstered ECG waveforms of 7 patents. The data have been taken from the MIT BIH Arrhythma Database [8]. The mportant dffculty of the accurate recognton of the arrhythma type s the large varablty of the morphology of the EEG rhythms belongng to the same class [8]. Moreover the beats belongng to dfferent classes are also morphologcally alke to each other. Hence the confuson of dfferent classes s very lkely. In our numercal experments we have consdered sx types of arrhythma: left bundle branch block (L), rght bundle branch block (R), atral premature beat (A), ventrcular premature beat (V), ventrcular flutter wave (I), ventrcular escape beat (E), and the waveforms correspondng to the normal snus rhythm (). All these 7 rhythms have been dscovered at one patent. So ths knd of experment may be regarded as the ndvdual classfer specalzed for the sngle patent data pars have been generated for the purpose of learnng and 3068 were used for testng purposes. Table presents the number of representatve of the beat types used n testng only. Table The number of testng samples of each beat type Beat type L R A V I E o The lmted number of representatves of some beat types (for example E or I) s the result of the lmtaton of the MIT BIH database [8]. 5.2 The results of numercal experments In solvng the problem of arrhythma recognton we have reled on two sets of features. One set s related to the hgher order statstcs (HOS) and the second to the Hermte bass functon expanson (HER) of the QRS part of the ECG waveform. Three dfferent classfers have been appled: SVM, Hybrd and TSK. All of them have been traned separately on both sets of features (HOS and HER) and ther results have been combned together. In ths way the ensemble of 6 recognton systems have been created. The ntegraton of the results of all classfers has been done usng four presented above methods. We wll lmt the presentaton of the results to the testng mode only, the most mportant from the practcal pont of vew. The results are gven n the form of the relatve classfcaton error, calculated as the rato of all msclassfcaton cases to the number of samples used n testng. Table 2 presents the results of testng all ndvdual classfers and the ensemble system ntegrated accordng to dfferent methodologes. All classfer networks have been frst learned on the same learnng data set and then tested on another testng data set, the same n all cases. The best results of sngle classfers refer to the applcaton of SVM- HER methodology (Hermte expanson for generaton of features and SVM network classfer) and Hybrd-HOS (HOS representaton for generaton of features and hybrd network classfer). The worst results have been obtaned at the applcaton of TSK-HER soluton (TSK classfer n combnaton wth Hermte preprocessng of data). The relatve dfference between the accuracy of the best and worse classfer s very large (more than 60%). In spte of

5 large dfference of the qualty of the ndvdual recognton systems even the smple maorty votng was able to mprove results sgnfcantly. However the best results have been obtaned at the applcaton of the weghted maorty votng. The best ndvdual result of.96% of relatve msclassfcaton (SVM-HER) has been mproved to.37% (over 30% of relatve mprovement) n ths case. Observe that all ntegraton methods have mproved the fnal accuracy of recognton n comparson to the best ndvdual classfcaton system. Table 2 The average msclassfcaton rate for the famly of 7 beat types (the ndvdual classfers and ensemble of classfers) o Classfer system Testng error Hybrd-HER (H-HER) 2.93% 2 Hybrd-HOS (H-HOS) 2.35% 3 TSK-HER (T-HER) 3.26% 4 TSK-HOS (T-HOS) 2.7% 5 SVM-HER (S-HER).96% 6 SVM-HOS (S-HOS) 2.80% 7 Maorty votng (MV).63% 8 Weghted votng (WV).37% 9 Kullback-Lebler (KL).47% 0 Modfed Bayes (MB).56% Generally we may state that ntegraton of many classfers mproves the recognton results sgnfcantly. The mprovement rate depends on the appled ntegraton scheme and the qualty of the ndvdual classfers. Fg. 2 presents the relatve mprovement of the fnal classfcaton results of the ensemble obtaned thanks to the appled ntegraton method. Fg. 2a llustrates the mprovement wth regards to the best ndvdual classfer (SVM-HER) and Fg. 2b to the worst one (TSK-HER). Fg. 2 The relatve mprovement of dfferent ntegraton methods wth respect to a) the best, b) the worst ndvdual classfer The notatons used on the horzontal axs of the fgure mean the type of the recognton system, for example H-HER means Hybrd-HER system, etc. It s seen that the relatve mprovement of the best ntegraton scheme (weghted maorty votng) wth respect to the best ndvdual classfer (SVM-HER) s over 30% and wth respect to the worst one (TSK-HER)) almost 60%. The results prove that ntegratng the results of many classfers of even not equal qualty brngs the sgnfcant mprovement of the qualty of performance of the whole classfer system. The qualty of results can be assessed n detals on the bass of the error dstrbuton wthn dfferent beat types. Table 3 presents the dstrbuton of classfcaton errors for the testng data n the form of the confuson matrx dvded nto dfferent beat types. These results correspond to the best ntegraton scheme. The dagonal entres of ths matrx represent rght recognton of the beat type and the off dagonal the msclassfcatons. The column presents how the beats of partcular type have been classfed. The row ndcates whch beats have been classfed as the type mentoned n ths row. Thanks to the confuson matrx we can easly analyze whch classes have been confused by our classfyng system. Table 3 The confuson matrx of the ntegrated classfyng system for 7 types of rhythms of testng data L R A V I E L R A B I E The analyss of the error dstrbuton shows that some classes are confused more frequently than the others. It s evdent that most msclassfcatons have been commtted between two classes: and A (2 -rhythms have been classfed as A-rhythms and 7 A-rhythms have been recognzed as -rhythms). Ths confuson s the result of large smlarty of ECG waveforms for these two rhythms. The last but not least aspect of heart beat recognton s the analyss of how the abnormal rhythms have been separated from the normal one. In practce the most dangerous case s when the ll person s dagnosed as the healthy one (false negatve dagnosed patent). To deal wth such case we have ntroduced the qualty measure equal to the number of all false negatve dagnosed patents. Analyzng the obtaned results we have notced the evdent mprovement of ths qualty measure for the ntegraton schemes, both n learnng and n the testng mode. Table 4 presents the number of the false negatve dagnoses for the ndvdual classfers and for all ntegrated systems under nvestgaton. The results correspond to the testng data, not takng part n learnng. The best results n terms of the number of the false negatve dagnoses have been obtaned for most of the ntegraton methods (except Kullback- Lebler approach). Fg. 3 presents the dstrbuton of the

6 false negatve cases for all proposed solutons (the ndvdual classfers and all ntegraton schemes). Table 4 The comparson of the number of false negatve dagnoses for dfferent soluton of the classfyng systems o Classfer system o of false negatve cases Hybrd-HER 22 2 Hybrd-HOS 0 3 TSK-HER 23 4 TSK-HOS 36 5 SVM-HER 4 6 SVM-HOS 7 Maorty votng 9 8 Weghted 9 votng 9 Kullback- Lebler 0 Modfed Bayes 9 The nterestng s that most of the ntegraton schemes have produced the same number of false negatve dagnoses, much better than the average number obtaned at applcaton of ndvdual classfers. The Kullback-Lebler method has produced slghtly worse results. Fg. 3 The comparson of the number of the false negatve cases correspondng to ndvdual classfers and to all ensemble systems 6. COCLUSIOS The paper has presented and compared dfferent methods of ntegraton of the results of many ndvdual neural classfers combned nto one classfcaton system. The appled classfers nclude: hybrd neural network, neurofuzzy TSK network and support vector machne classfers. The ensemble system appled maorty and weghted votng, Kullback-Lebler dvergence and modfed Bayes methods. The experments performed for seven heart beat types taken from MIT BIH AD have shown that ntegraton of the results of many classfers mproves the qualty of the fnal classfcaton system. The mprovement s observed n terms of the accuracy of recognton as well as of the number of false negatve dagnoses. To the best ntegraton approaches belong the weghted maorty, modfed Bayes and Kullback-Lebler methods. They have resulted n the reducton of not only the total classfcaton errors and at the same tme also n the reducton of the most dangerous false negatve cases of dagnoss. The results presented n the paper confrm our conecture that a hghly relable classfer can be obtaned by combnng a number of classfers whch exhbt an average performance. REFERECES [] P. de Chazal P., M. O'Dwyer, R. B. Relly, Automatc classfcaton of heartbeats usng ECG morphology and heartbeat nterval features, IEEE Trans. on Bomed. Eng, 2004, vol. 5 pp [2] S. Haykn, eural networks, comprehensve foundaton, Prentce Hall, 999, ew Jersey [3] C. W. Hsu, C. J. Ln, A comparson methods for mult class support vector machnes, IEEE Trans. eural etworks Vol. 3, pp , 2002 [4] Y. H. Hu, S. Palreddy, W. Tompkns, A patent adaptable ECG beat classfer usng a mxture of experts approach, IEEE Trans. Bomed. Eng., 997, vol. 44, pp [5] L. Jang, C. T. Sun, E. Mzutan, euro-fuzzy and Soft Computng, Prentce Hall, ew Jersey, 997 [6] L. Kuncheva, Combnng pattern classfers: methods and algorthms, Wley,. J., 2004 [7] M. Lagerholm, C. Peterson, G. Braccn, L. Edenbrandt, L. Sornmo, Clusterng ECG complexes usng Hermte functons and self-organzng maps, IEEE Tr. Bomed. Eng., 2000, vol. 47, pp [8] R. Mark, G. Moody, MIT-BIH arrhythma database drectory, MIT [9] C. kas, A. Petropulu, Hgher order spectral analyss, Prentce Hall,. J., 993 [0] S. Osowsk, Tran Hoa Lnh, ECG beat recognton usng fuzzy hybrd neural network, IEEE Trans. on Bomedcal Engneerng, vol. 48, pp , 200 [] S. Osowsk, L. Tran Hoa, T. Markewcz, Support Vector Machne based expert system for relable heart beat recognton, IEEE Trans. on Bomedcal Engneerng, 2004, vol. 5, pp [2] S. Osowsk, L. Tran Hoa, On-lne heart beat recognton usng Hermte polynomals and neurofuzzy network, IEEE Trans. on Instrum. and Measur., 2003, vol. 52, pp [3] B. Schölkopf, A. Smola, Learnng wth Kernels, Cambrdge MA, MIT Press, 2002 [4] V. Vapnk, Statstcal learnng theory, Wley,.Y. 998