Utilization of the Growing Hyperspheres Neural Network (GHS) for a Detection of Peaks in a Time Sequence of Total Factor Productivity

Transcription

1 Uilizaio of he Gowig Hypesphees Neual Newok (GHS) fo a Deecio of Peaks i a Time Sequece of Toal Faco Poduciviy Macel Ji ia, Tomáš Cahlík bsac: This aicle discusses a deecio of peaks (seep ime chages) i a ime sequece of oal faco poduciviy a esidual faco i he poducio fucio. The peaks ca be iepeed a leas i he Real Busiess Cycles (RBC) Theoy as shocks caused by sudde echological iovaios. Oe of he difficulies wih he RBC heoy is he deecio of sufficiely lage echological shocks i he eal ecoomy. The aim of his aicle is o es he gowig hypesphees (GHS) eual ewok fo such deecio. The effo is o coecly deec peaks so he o-peaks mus be fileed ou coecly. I is bee o o deec a peak ahe ha o poclaim a o-peak o be a peak. Two measues he sigal efficiecy ad backgoud eo ae ioduced o expess he success of peak deecio. The es of deecio is based o expeimeal daa obaied by simulaio o a geeal RBC model. Fou ohe ecoomic ime seies physical capial, cosumpio, labou ad icome ae used as ipus. Keywods: Peak Deecio, Neual Newok Classifie, Real Busiess Cycles Theoy. Ioducio I his pape, he use of some sophisicaed mehods of daa aalysis - icludig eual es - i ecoomics is demosaed. We ouch oe of he basic quesios of he Real Busiess Cycles (RBC) heoy: How ca we deec echology shocks? Basic poblem fo applicaios of eual es i ecoomics is he boleeck of daa. Tha is he easo why hese applicaios ae usually foud i he aalysis of fiacial makes, whee he elaively log daily ime seies ca be used. We ied o solve he daa poblem by geeaig sufficiely log ime seies abou e housads quaes o he basis of a geeal RBC model. We used he specificaio of he model ad he paamees coued by Campbell accodig o hei descipio i [Rom96]. The pocess of daa geeaio is descibed i sho i he secod pa of his aicle. I he hid pa of his aicle, a eual ewok based o gowig hypesphees (GHS) is descibed. The eual ewok has bee oigially ioduced i [Ji00a] ad is use ad some sligh modificaios i [Ji00b]. I he fouh pa, hee diffee appoaches o compilig of aiig daa, hei pocessig ad achieved esuls ae descibed. I he fis appoach we y o eveal he peaks i oal faco poduciviy empiically, i he secod appoach we use pue eual ewok appoach o peak deecio ad i he hid appoach we combie he pecedig mehods. Macel Ji ia, Depame of Cybeeics, Faculy of Elecical Egieeig, Czech Techical Uivesiy Pague, Techická 66 7 Pague 6 - Dejvice, Czech Republic, jiia@labe.felk.cvu.cz Tomáš Cahlík,, Isiue of Ecoomic Sudies, Chales Uivesiy Pague Faculy of Social Scieces, Oplealova 6, 0 00 Paha, Czech Republic, cahlik@mbox.fsv.cui.cz

2 . Geeaig he Daa Fo he geeaio of daa, we use he followig pocedue: Fom [Rom96], we ake ove he specificaio of a geeal busiess cycle model. We ca summaize his specificaio as follows. The ecoomy cosiss of a lage umbe of ideical ad pice akig households ad fims. The households ae ifiiely lived. The poducio fucio is Cobb-Douglas wih capial, labo ad echology as ipus. Oupu is divided amog cosumpio, ivesme ad goveme puchases. Depeciaio ae of capial is δ. The goveme puchases ae fiaced by lump-sum axes. Labo ad capial ae paid accodig o hei magial poducs. The epeseaive household maximizes is expeced uiliy. Thee ae wo divig vaiables, echology ad goveme puchases G. The geeal busiess cycle model, specified accodig o [Rom96], cao be solved aalyically. Campbell showed, ha he model ca be log-lieaized aoud he balaced gowh pah. The, we ca apply he followig ules fo cosumpio C, labo L, capial K ad icome Y: C = ack K + ac + acgg () L = alk K + al + algg () K + = bkk K + bk + bkg G (3) Y = [ α + ( α ) a ] K + ( α)( + a ) + a G (4) LK L LG whee all vaiables deoe he diffeeces bewee he log of he acual value ad he log of he balaced gowh pah. Techology ad goveme puchases G follow he fis-ode auoegessive pocess: ρ + ε, G ρ GG + ε G, = (5) =, (6) whee ε a ε G, ae adom vaiables., Paamees α, ρ ad ρ G ae amog he baselie paamees of he geeal busiess cycle model. Campbell cous fom he baselie paamees he ohe paamees a ad b i equaios ()-(4). Each peiod coespods o a quae. Fo he geeaio of daa, we use equaios ()-(6) wih he values of paamees fom Campbell. I he peiod zeo, all he vaiables ae se o zeo. I each ohe peiod, we geeae ε ad ε G as fis. ε is o-zeo wih pobabiliy 0.4. I he case of o-zeo value, ε is equally disibued i ieval <0;>. ε G is o-zeo wih pobabiliy 0.. I he case of o-zeo value, ε G is equally disibued i ieval <-0.5;0.5>. The we cou ad G. Theeafe we cou K ad as he las vaiables we ca cou C, L ad Y. We have expeimeed wih vaious possibiliies fo geeaig ε ad ε G. The way we fially use gives afe abou wo huded quaes, whe he pocess sabilizes, he gowh of he acual value of icome abou % pe yea, which is close o he eal

3 ecoomy. We geeae daa fo peiods, 5000 fo aiig ad 5000 fo esig. 3. GHS Neual Newok The GHS eual ewok is a epeseaive of a goup of classifies, i.e. is aim is o classify he daa io seveal classes. Geeally, classificaio is a pa of he ecogiio ask ad ca be divided io feaue mehods ad sucual mehods. I his case we ae ieesed i he feaue mehods oly. The sucual mehods ae descibed e.g. i [Ko93]. The feaue mehods wok wih quaiaive valued feaues of he objecs obseved. The umbes expessig hese feaues ad havig sigificace of feaue measues ae called feaues. ll feaues descibig a objec ae called feaue veco ad he space of all feaue vecos is called feaue space. The feaue vecos ca be obaied by pocessig he measued daa fom he objecs. I he simples case he feaue vecos ae epeseed by he measued daa. Such feaue vecos ae called (aiig) paes i he field of eual ewoks. The classifie is a machie ha pefoms classificaio. Classificaio is a pocess ha assigs a symbol epeseig a class o each feaue veco (pae). The classifie ca be se i wo ways: By aalysig a poblem ad defiig he decisio cieio befoe he classificaio. By ceaig a decisio cieio usig objecs whose classificaio is kow befoehad (seig a classifie by leaig). The classifies use wo cieios fo classificaio: oe of miimal eo ad he ohe of miimal disace. epeseaive of classifies of miimal eo is he well-kow Bayes s classifie [Ko93], [Roj96]. I is based o he pobabilisic appoach ad uses Bayes s fomula. The secod goup of classifies usig he cieio of miimal disace is widely epeseed. Coay o ohe classifies usig complex hypesufaces fo sepaaio of classes, he peseed classifie uses hypesphees. This piciple gives a ahe simple, udesadable ad quaifiable isigh io classificaio. 3.. Classificaio usig Sepaaio by meas of Hypesphees The mai idea of he suggesed GHS classifie is based o a exacio of leaig paes of he same class wihi a sphee. The cee of he sphee ad is adius deemie a aea which coais leaig paes of he same class. disjucio of such sphees demacaes he aea of oe class ha coais all paes of his class ad sepaaes he leaig paes fom hose of diffee classes. Fo a -dimesioal space -dimesioal sphees (hypesphees) will be used. I a wo-dimesioal space he hypesphees fom simple cicles. Fo a classificaio i k classes, k goups of hypesphees ca be used. Howeve, fo k classes oly k- goups of hypesphees ae sufficie. The las, say he k-h, class is jus he oe ha does o belog o ay ohe, i.e. he pevious k-, classes. So defied a classificaio classifies i k classes exacly. O he coay, all ukow paes ha makedly diffe fom hose colleced i he leaig se, will be classified jus i he k-h class. Such paes ca

4 aise, fo example, due o measueme misakes ec. Tha is why a classificaio i k classes is ecommeded. No-epeseaive paes will o he be foud wihi he hypesphees ad hus classified i he viual do kow k+ class. advaage of such a appoach is ha hee is a simple epeseaio of hypesphees. Each ode coais ceal coodiaes of a hypesphee ad is adius. sucue of he classifie is hus a simple liked lis of odes. The umbe of coodiaes of he cee is he same as he space dimesio. Theefoe he classifie has ipus. The adius is epeseed by oly oe eal umbe ad is idepede o he space dimesio. Moeove, due o seveal classes we eed o add o each ode he ifomaio abou he class which will epese he ode. fuhe eally impoa feaue is ha he hypesphees ca easily cu ou paes ha ae eclosed iside he paes of diffee classes ad fa fom he mai goups of paes. Oce we fid he hypesphees i he opimisaio pocess we ca simply classify ew ukow paes i idividual classes. We u hough all soed odes ad fid oe whose hypesphee coais his pae. appopiae class assiged o his hypesphee is jus he class o which he pae belogs. The mos difficul poblem is o fid a suiable algoihm ha fids hese hypesphees o he basis of a se of leaig paes. suggesed algoihm is pecisely descibed i he ex secio. I is obvious ha we wa o epese all classes by meas of a miimal umbe of hypesphees ad hus odes. The mai idea of he leaig pocess cosiss i movig he hypesphees i aeas of highe coceaio of paes i a defied diecio ad coicideal iceasig of hei diamees. hypesphee ca icease is diamee oly if a umbe of ieal paes gows o keeps cosa. 3.. Neual ppoach o Daa Classificaio The leaig algoihm woks exacly his way. Le be a space dimesio, k a umbe of k classes, N,, N k umbes of paes i idividual classes ad N = N i= i he oal umbe of paes i he aiig se. The leaig pocess will go ove all k (o k-, see Secio.) classes ad he same leaig subpocedue will u fo each class. Leaig will fiish afe pocessig all classes. Noe ha i is impoa i which ode we pocess he idividual classes. popely chose ode of classes could sigificaly educe he oal umbe of odes used fo he leaig se epeseaio ad hus he ewok size. Ufouaely, i is difficul o esimae his ode. Wihou a loss of geealiy we will suppose ha we pocess he classes fom class o k. subpocedue woks his way. Fis we adomly choose a pae fom a se coaiig paes belogig o he fis class. This pae will epese cee w = ( w,, w ) of a hypesphee ow. I is ecessay o fid adius of he hypesphee. I boh [Ji00a] ad [Ji00] we have ioduced he followig algoihm fo fidig he adius. Fis, we fid he eaes pae h = ( h,, h ) o ay diffee class. The disace bewee he hypesphee cee ẅ ad he foud pae h is he pimay adius. Such seleced adius does o seem o be opimal because he bouday bewee he classes deemied by he hypesphee suface is oo close o he diffee class (classes). possible soluio is o ace he bouday

5 bewee he classes appoximaely i a half of he fee space occuig bewee he classes. This fac ca be achieved his way. Usig he jus cosuced hypesphee we fid a pae g = ( g,, g ) wihi he hypesphee ha is eaes o pae h. Le us deoe he coespodig adius. The fial adius is he ecalculaed accodig o fomula = + = i= + hi g wi i. () This algoihm woks sufficiely well bu a simple halvig he fee space bewee diffee classes pefes appae educio of he hypesphee wih iceasig space dimesio. Theefoe we sugges he followig soluio. Isead of akig he mea value of he adii ad, we calculae he fial adius fom he value of mea volumes of he hypesphees deemied by he adii ad. Moe pecisely, he pimay adius deemies he hypesphee volume V ad he adius he hypesphee volume V. The mea volume is he V ( ) + V ( ) V ( ) =. The fial adius is he ecalculaed accodig o fomula + =. (b) Noe ha he fial adius appoaches he adius wih iceasig space dimesio. Le ad. The lim = lim = lim + + = = lim + =. = lim + = lim + = lim + = The umbe of paes wihi he hypesphee wih cee w ad adius is coued ad all hese paamees ae icluded i a ew ode. I is pobable ha he hypesphee is o placed popely. Theefoe a shif of he hypesphee o a ew bee posiio ha would maximise he umbe of ieal paes is ecessay. The * * * w = w,, w of he shif is deived fom hypesphee adius. The ew cee ( ) hypesphee is w * i = wi + d si, i =,, ()

6 whee d is a shif cosa ad ( ) calculaed by fomula s = s,, s a ui diecioal veco whose coodiaes ae s i = w h i i j= ( w j h j ), i =,,. (3) The shif cosa d deemies he shif of he hypesphee cee bu also he size of he ew hypesphee. Fom he same easo as meioed above fo he fial adius calculaio, he shif cosa should be depede o he space dimesio. I ohe wods, we should y o shif he hypesphee by a poio of is oigial volume. Hece, we sugges o use his shif cosa d d = v, whee v expesses he icease of hypesphee volume. I mus be lage ha o esue he icease of hypesphee volume. Fo example, if v =, i.e. he ew hypesphee will have double volume. his mome we have a ew cee!w * of he hypesphee bu we eed o ecalculae is adius. The calculaio of he ew adius us i he same way as meioed above fo he iiial hypesphee. Fo his ew hypesphee a umbe of ieal paes is compued. Now i is ecessay o decide if he ew shifed hypesphee is bee ha he oigial oe. The qualiy of he hypesphee is measued by a umbe of ieal paes. If he ew hypesphee has a geae umbe of paes, we coiue wih is expasio. If he umbe of paes wihi he ew hypesphee is smalle ha i he oigial oe o equal o i, he we do he followig seps. I his place i is ecessay o emphasise ha we should o simply sop he hypesphee shif. The easo is he followig. bouday of classes b " h * # h a Fig.. wo-dimesioal hypesphee expasio i a coe Fig. shows expasio of a wo-dimesioal hypesphee (cicle) i he diecio a afe seveal shifs. The las expasio of a hypesphee wih cee!w * iefees a diffee class ad hus he maximal pemied hypesphee is he oe wih cee $w. I is obvious ha such

7 a hypesphee is o placed opimally ad should expad i diecio b o iclude moe paes. Theefoe a moe sophisicaed algoihm is ecessay. The idea cosiss i allowig a hypesphee expasio eve if he ew shifed hypesphees ae wose (i he sese of he umbe of ieal paes) ha he up-o-ow bes placed hypesphee. I ohe wods, we allow he ew shifed hypesphee o be empoaily wose duig hese seveal seps. This allows a hypesphee o bouce fom a bouday wih a diffee class ad eu back o he space of he cue class. Fo example, i Fig. he hypesphee will bouce zigzag fom he boudaies ad hus move i diecio b. The umbe of empoay seps allowed is a paamee give o a ode. If his umbe oveseps he pescibed umbe of he seps (cool paamee of GHS e), we sop he seachig fo a bee hypesphee ad say ha he bes-foud hypesphee is he opimal oe. Geeally we ca say ha he highe he umbe of allowed empoay seps he bee fiig of a hypesphee i a space is doe bu also moe leaig ime is cosumed. Noe ha he qualiy of a hypesphee is measued by he umbe of paes i icludes ad o by he hypesphee diamee. Eve a small hypesphee ca iclude moe paes ha a lage oe. Oce we have foud he odes i he leaig pocess we ca classify he ew ukow paes i idividual classes. The acual pocess us his way. The ecallig algoihm simply goes ove all soed odes ad fids he fis ode whose hypesphee coais his pae (Noe ha hee ca be seveal odes whose hypesphees coai he pae bu all epese he same class). appopiae class assiged o his ode is jus he class o which he pae belogs. If o ode is foud, he pae belogs o he las k-h class i he case of k- goups of odes o does o belog o ay class i he case of k goups of odes. Noe ha afe fiishig he leaig pocess we have all he foud odes soed i a daa sucue like fo example a aay o a liked lis. If we have added idividual ew odes oe by oe o he sucue we obai a ime-odeed lis of he odes. If he ime eeded fo acual classificaio is o cucial we ca wok wih such a ode lis of odes. If he pocess of ecallig is epeaed may imes i he fuue, i is good o speed up he classificaio by soig he odes accodig o he umbe of paes ha he hypesphees of he odes ca coai i he leaig se. I his case we have o add a coue o each ode ad evey pae fom he leaig se es if he hypesphees coai his pae ad icease he coespodig coues. We have o bea i mid ha oe pae ca belog o seveal hypesphees as has aleady bee meioed. Noe ha his couig cao be doe duig leaig. fe he couig we so all odes i a deceasig ode idepedely o he class hey epese. We use such ceaed lis fo he classificaio he same way as is descibed above. The easo of his soig is ha we y o fid a ukow pae wihi mos pobable hypesphees. 4. Compilig ad Pocessig of Taiig Daa, Resuls Thee ae fou ipu daa sequeces: physical capial K, cosumpio C, labou L ad icomey ad oe oupu epeseig peak i he oal faco poduciviy. The ask is o deec peaks i o he basis of kow K, C, L ad Y.

8 ll expeimeal daa have bee geeaed by he simulaio o he RBC model. The geeal daa, sevig fo aiig ad esig, coai he fou chaaceisics K, C, L, Y ad coespodig peaks. The peaks ae ideified by ad o-peaks by daa ows have bee geeaed. The daa have bee divided io wo goups aiig ad esig each of 5000 values. Fo he opimisaio pocess ad esig of he classifie, simulaed daa ae used. Theefoe we kow exacly whee ae he peaks i ad hus we ca measue he success of he aiig pocess. I ay case, he daa used fo esig ae diffee fom he daa used fo he aiig. The deecio of peaks i he daa coespods o classificaio whee he classifie should deoe hose daa paes which belog o class (peaks, sigal). Of couse, he classificaio is o ideal ad so wo kids of eos aise. Fis, o all paes of class ae ecogised as class - he aio of popely ecogised paes of class o all paes of class is sigal efficiecy. Secod, some paes of class 0 (o-peaks, backgoud) ca be cosideed as paes of class. The aio of paes of class 0, which ae cosideed as paes of class o he oal umbe of paes of class 0, is backgoud eo. The elaio of sigal efficiecy vs. backgoud eo gives he chaaceisics of he classifie ad hus he success of he peak deecio. I followig subsecios, hee diffee appoaches o compilig ad pocessig of aiig daa ogehe wih achieved esuls ae descibed. 4.. Empiical Rules Fis, a empiical appoach has bee used o eveal he peaks i o he basis of K, C, L ad Y. The followig six empiical ules have bee used. X epeses ay of K, C, L o Y, is he discee ime ad d ( ) = X ( ) X ( ) ad d ( ) = X ( + ) X ( ).. if he d ( ) > 0 ad d ( ) < 0 he hee is a peak a ( ),. if he d ( ) < 0 ad d ( ) > 0 ad d( ) d ( ) he hee is a peak a ( ), 3. if he d ( ) > 0 ad d ( ) > 0 ad d( ) d ( ) he hee is a peak a ( ), 4. if he d ( ) > 0 ad d ( ) > 0 ad d( ) d ( ) he hee is a peak a ( ), 5. if he d ( ) < 0 ad d ( ) < 0 ad.5 d( ) d ( ) he hee is a peak a ( +) 6. if he d ( ) 0 ad d ( ) 0 ad ( ). d ( ) he hee is a peak a ( ). < < d O he basis of hese empiical ules we have obaied esuls show i Table., K C L Y Sigal Efficiecy Backgoud Eo Table. Sigal efficiecies ad backgoud eos fo he empiical ules Because he chaaceisics ae geeally diffee we have modified ad ued he ules o bee expess feaues of he idividual chaaceisics. Fo K we have modified he empiical ules so we have omied he ule 3 ad i he ule we have omied he las codiio:

9 . if he d ( ) 0 ad d ( ) 0 ad he hee is a peak a ( ) < >, ad i he ule 6 we have iceased he cosa. o.5 fo he L: 6. if he d ( ) 0 ad d ( ) 0 ad ( ). d ( ) he hee is a peak a ( ) < < d 5. Boh modificaios sigificaly impove he qualiy of peak deecio i K he sigal efficiecy iceases ad he backgoud eo deceases ad i L - he backgoud eo deceases, see Table K C L Y Sigal Efficiecy Backgoud Eo Table. Sigal efficiecies ad backgoud eos fo he modified empiical ules 4.. Pue Neual Newok ppoach o Peak Deecio The pue eual ewok appoach iself ca be divided io wo goups. The fis oe uses absolue values of he chaaceisics as aiig paes ad he secod oe uses elaive values obaied as fis diffeeces of eighboig absolue values of he chaaceisics as aiig paes. Moeove, we ca ai he eual ewok o each chaaceisics idividually o compile all fou chaaceisics ogehe o fom oe aiig pae. Fis, we have expeimeed wih,, 3 ad 4 absolue eighboig values of he chaaceisics ad he chaaceisics have bee pocessed idividually. The bes esuls have bee obaied fo 3 eighboig values. Resuls wih ohe umbes of eighboig values have bee expessively wose. So i seems ha he 3 eighboig values ae he opimum. The esuls ae show i Table 3. Noe ha hee ae o implici peiodicies o ohe simila depedecies i he chaaceisics, so i seems ha hee is o poi i icludig values ohe ha he eighboig oes i he aiig paes. K C L Y Sigal Efficiecy Backgoud Eo # Hypesphees Table 3. Resuls fo hee absolue eighboig values Fuhe, we have compiled he aiig pae fom all fou chaaceisics K, C, L ad Y ad fom each chaaceisics we have used hee absolue eighboig values. Each pae had 4 3 = elemes, ipus o he eual ewok. The sigal efficiecy was ad he backgoud eo was The GHS eual ewok used 39 hypesphees. Ohe umbes of values of he chaaceisics have bee esed bu he esuls wih 4 3 = elemes seem o be he bes. Secod, we have expeimeed wih elaive values of he chaaceisics, i.e. we have used wo fis diffeeces of he wo eighbouig values of each chaaceisic, X ( ) X ( ), X + X, whee X is oe of K, C, L ad Y. The aiig pae had so 4 = 8 ( ) ( )

10 elemes. Supisigly, we have obaied vey poo esuls. The sigal efficiecy was ad he backgoud eo was fo he K (fo ohe chaaceisics we have achieved simila bad esuls). The GHS eual ewok used 57 hypesphees. The esuls ae a big supise fo us, because o he same diffeeces hemselves ae based he empiical ules meioed i pevious subsecio Combiaio of Empiical Rules ad Neual ppoach The hid appoach o peak deecio is o use he empiical ules meioed i subsecio 4. ad impove hem by eual ewok. The chaace of he solved ask allows usig of his appoach. The mai idea cosiss i aiig he eual ewok o eos caused by he empiical ules. The equaliies o diffeeces of he deeced peaks by he empiical ules fom he coec kow peaks ae used as coec oupus fo he aiig se. The ipus o he eual ewok have bee jus he hee eighbouig absolue values of he idividual chaaceisics because of he bes esuls o such compiled aiig se, see pevious subsecio. The acual algoihm woks his way. Fis, empiical ules fo he ipu chaaceisic ae used. fe ha he veco of peaks deeced by he empiical ules is compaed wih coec peaks kow fom he aiig se. If he values i boh vecos ae he same a he same posiio, i.e. boh ae zeo o oe, he he eo veco has zeo a he coespodig place. If he values diffe he he eo veco has oe a he coespodig place. I his way we have compiled he veco of diffeeces bewee he esul of he empiical aalysis ad coec peak placemes. Now, he eual ewok is aied i he same way as descibed i subsecio 4. o he paes compiled fom he ipu chaaceisic wih coespodig oupus fom he eo veco. fe aiig, he ew pae is boh pocessed by he empiical ules ad applied o he ewok ipu. fe ha we coec he fial oupu his way. If he eual ewok oupu is zeo, he he fial oupu is jus he aswe o he empiical aalysis. I he opposie case, whe he ewok oupu is oe, i.e. he chage of he esul afe empiical aalysis is ecommeded, wo appoaches ca be ake io accou. The fis oe simply ives he aswe o he empiical aalysis. The secod oe chages oe o zeo bu does o chage zeo o oe. This secod appoach is safe fo ou ask because we wa o be almos sue o deec peaks coecly. Resuls of he fis appoach ae show i Table 4 ad he esuls of he secod appoach ae show i Table 5. 0 K C L Y Sigal Efficiecy Backgoud Eo # Hypesphees Table 4. Combied empiical-eual appoach wih coecio i boh diecios 0 K C L Y Sigal Efficiecy Backgoud Eo # Hypesphees Table 5. Combied empiical-eual appoach wih coecio i diecio 0

11 5. Coclusios The use of a hypesphee as a sepaaio suface is pefecly aual because eal daa fom usually o-agula daa cluses i he pae space. Moeove, he daa wih commo feaues ad hus belogig o oe class ae coceaed ofe ea oe aohe aoud a cee ad fom ealy hypesphee s shape. class sepaaio hypesuface fo moe complicaed daa is cocaeaio of idividual hypesphees, i.e. simple disjucio of hese hypesphees. The mai advaage of he ioduced GHS eual classifie is is good abiliy o popely sepaae a abiay umbe of diffee classes i ay space dimesio. fuhe impoa advaage is a low memoy demad. The cee ad adius deemie each hypesphee. The umbe of cee coodiaes is equal o he umbe of elemes of he leaig paes; i.e. o he space dimesio, ad he adius is kep as oe eal umbe idepede of he space dimesio. We discussed he GHS classifie fo oly wo classes bu did o come o ay coclusios abou he class ode i geeal. We suppose ha he classes ae pocessed sep-by-sep i a iceasig ode such as hey have bee umbeed. Thee appoaches have bee used o deecio of peaks i he ime sequece of he oal faco poduciviy. Fis, a empiical aalysis has bee made. Is idea cosiss i compilaio of seveal ules o he basis of visual obsevig of he value chages of he chaaceisics. We have compiled six basic ad geeal empiical ules ha ae commo fo all chaaceisics, see Table. Fuhe, we have modified ad hus ued hese ules fo idividual chaaceisics o obai bee esuls, see Table. Secod, we have used he GHS eual ewok o ai he classifie o deec he peaks i. Thee ae seveal appoaches o do i. The fis oe uses absolue values of he chaaceisics as aiig paes ad he secod oe uses elaive values obaied as fis diffeeces of eighboig absolue values of he chaaceisics as aiig paes. Moeove, we ca ai he eual ewok o each chaaceisics idividually o compile all fou chaaceisics ogehe o fom oe aiig pae. The esuls of he GHS wih idividual absolue values of he chaaceisics as ipus ae cocluded i Table 3. I geeal, we ca say ha he achieved esuls ae compaable wih he esuls of he empiical aalysis. Boh sigal efficiecies ad backgoud eos ae shifed owads zeo. I ohe wods, he sigal efficiecy (abiliy o deec peaks) is wose bu he backgoud eo (pobabiliy of misake peak deecio) is bee. The GHS wih compiled chaaceisics epeseig oe aiig pae has achieved slighly bee boh sigal efficiecy (0.7599) ad backgoud eo (0.060) ha is he aveage of sigal efficiecies (0.745) especive backgoud eos (0.0794) of idividual chaaceisics fo he GHS. The expeimes wih elaive values of he chaaceisics oally failed. The hid appoach o deecio of peaks cosised i he combiaio of (modified) empiical ules ad he GHS eual ewok. Thee ae wo easoable possibiliies how o coec he fial esuls, 0 ad 0, see subsecio 4.3. The achieved esuls ae i boh cases slighly shifed owad oe. The oe-diecial (0 ) appoach is bee ha he bidiecioal (0 ) oe. Geeally, we ca coclude, ha we ae able o coecly deec he peaks i appoximaely

12 80 % ad ill-deec he peaks i appoximaely 5 %. I ou fahe eseach, we will es he developed GHS eual e o eal daa as well. We hope o eich he RBC heoy i his way. ckowledgeme The eseach has bee caied ou ude he suppo of he Czech Miisy of Educaio ga MSM VZ3, ad he Ga gecy of he Czech Republic ga 40/00/0999. Refeeces [Ji00a] Ji ia M., j., Ji ia M.: Neual ewok classifie based o gowig hypesphees, Neual Newok Wold Poceedig, 000 [Ji00b] Ji ia M., j., Ji ia M.: Neual Classifie Usig Hypesphees, ISCI 000 Poceedig, 000 [Ko93] Koek Z., Ma ík V., Hlavá% V., Psuka J., Zdáhal Z.: Meody ozpozáváí a jejich aplikace, cademia, Paha, 993 [Roj96] Rojas R. : Neual Newoks Sysemaic Ioducio, Beli, New Yok, 996 [Rom96] Rome D.: dvaced Macoecoomics, McGaw-Hill, 996.