Hidden Markov Models with Kernel Density Estimation of Emission Probabilities and their Use in Activity Recognition

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1 Hdden Markov Models wh Kernel Densy Esmaon of Emsson Probables and her Use n Acvy Recognon Massmo Pccard Faculy of Informaon echnology Unversy of echnology, Sydney massmo@.us.edu.au Absrac In hs aer, we resen a modfed hdden Markov model wh emsson robables modelled by kernel densy esmaon and s use for acvy recognon n vdeos. In he roosed aroach, kernel densy esmaon of he emsson robables s oeraed smulaneously wh ha of all he oher model arameers by an adaed Baum-Welch algorhm. hs allows us o rean maxmum-lkelhood esmaon whle overcomng he known lmaons of mxure of Gaussans n modellng ceran robably dsrbuons. Exermens on acvy recognon have been erformed on groundruhed daa from he CAVIAR vdeo survellance daabase and reored n he aer. he error on he ranng and valdaon ses wh kernel densy esmaon remans around 4-6% whle for he convenonal Gaussan mxure aroach vares beween 5 and 4%, srongly deendng on he nal values chosen for he arameers. Overall, kernel densy esmaon roves caable of rovdng more flexble modellng of he emsson robables and, unlke Gaussan mxures, does no suffer from beng hghly aramerc and of dffcul nalsaon.. Inroducon Auomac recognon of acves n vdeos s aramoun o many alcaons such as mulmeda annoaon and vsual survellance. As a consequence, has been wdely nvesgaed o dae (see [-5] and several ohers). Acves are ofen modelled as he saes of enes, eher a gven mes or along me nervals. For nsance, he sae of a erson a a gven frame may be recognsed as eher nacve or acve. By consderng sequences of sae values, we may wan o recognse more comlex aerns such as rasng an arm or collecng an obec. In many cases, he values of he sae varables canno be drecly measured, due o nose and oher nondeales such as occlusons and llumnaon changes, and have o be esmaed from he avalable observaons. Óscar Pérez Comuer Scence Dearmen - GIAA Unversdad Carlos III de Madrd, Colmenareo, San oscar.erez.concha@uc3m.es Hdden Markov models and her several varaons have been used exensvely for acvy recognon snce hey rovde a smoohed esmae of he sae values [6,7,3]. I may be argued ha smoohed sae esmaors nroduce a delay beween an observaon and he corresondng sae esmae: however, such a delay s ofen neglgble o he urose of he alcaon. In arcular, hs s he common case for vdeo analyss where observaons occur a vdeo rae and even a delay of a few hundred observaons ranslaes no a relavely shor me delay. A hdden Markov model (HMM) s fully descrbed by hree ses of quanes: he sae ranson robables, A; he emsson (or observaon) robables, B; and he robables of he nal saes, π. he saes are only allowed o assume dscree values; le us say,. herefore, A can be reresened by an x marx and π by an - dmensonal vecor. Insead, observaons are ofen drawn from connuous varables and, as such, emsson robables need o be modelled by robably densy funcons. he mos common aroach for modellng emsson robables s by use of mxures of Gaussans [6]. In such a case, B s fully descrbed by he weghs, means and covarances of all he Gaussan comonens. Once gven one or more sequences of observaons, he Baum-Welch algorhm can be used for learnng a corresondng HMM wh maxmum lkelhood. hs algorhm s an execaon-maxmzaon algorhm ha learns A, B and π n a smulaneous manner and s guaraneed o converge o a local omum n he arameer sace. Alernaves o he use of mxures of Gaussans (GMs) for modellng he emsson robables have been roosed n he leraure. Bourlard and Morgan n [8] roosed o relace he Gaussan mxures by Arfcal eural eworks n a hybrd A/HMM model. A number of varaons on hybrd A/HMM models s resened by renn n [9]. In a recen work [0], Krüger e al. roosed o relace he Gaussan mxures wh mxures of Suor Vecor Machnes. However, hese aroaches ycally ran he emsson robables n a suervsed manner, requrng knowledge of he groundruh values of he hdden sae varable. Even hough hey /07/$ IEEE

2 may oenally acheve hgher accuracy han maxmumlkelhood mehods, hey canno be aled n he general case where sae ground-ruh s no avalable. Conversely, maxmum-lkelhood HMMs can be raned us wh a sequence of observaons ha are, by defnon, avalable and herefore we focus our aenon on hem n he followng. In our work, we wan o rean he maxmumlkelhood smulaneous esmaon of all arameers offered by he Baum-Welch algorhm, whle overcomng he known lmaons of mxure of Gaussans. Gaussan mxures suffer from lmaons n modellng df s n a leas wo well-known crcumsances: a) when he number of modes n he df s greaer han ha of he Gaussan comonens n he mxure, b) when he df has unform regons. Kernel densy esmaon (KDE) has generally roven sueror o GMs n hese cases. On he oher, hand he esmaon of he omal kernel bandwdh n KDE s exremely crcal for s erformance. Several crera for omaly and corresondng mehods have been roosed o hs am []. In hs aer, we resen a model for he emsson robables based on KDE ha can be drecly lugged n he Baum-Welch algorhm (referred o as KDE/HMM n he followng). For esmaon of he kernel bandwdh, we roose an execaon-maxmzaon algorhm under a maxmum seudo-lkelhood creron. Exermens are erformed on vdeos from he CAVIAR daabase and accuracy s measured agans he sae ground ruh rovded by exer annoaon []. he exermenal resuls show he mroved erformance of KDE/HMM over convenonal HMMs based on GMs. Whle hs aer lms s analyss o vdeo daa, he roosed KDE/HMM lends self o general alcaon and can mrove erformance n oher domans. Exermenal resuls on synhec daa omed from hs aer due o sace lmaons reassure n hs drecon. he res of he aer s organsed as follows: Secon descrbes kernel densy esmaon by execaonmaxmzaon and comares modellng of df s wh ha rovded by a mxure of Gaussans wh a fxed number of comonens. Secon 3 exends he kernel densy esmaon o he modellng of emsson robables n HMMs. Secon 4 resens he exermens erformed and dscusses resuls. he conclusons summarse he man conrbuons of hs work.. Kernel densy esmaon wh an execaon-maxmzaon algorhm Gaussan mxures can be used o model he robably densy funcon of a random varable, (x), as: M ( x) = G( x; µ, σ ) α l l l () l = wh G( ) he Gaussan funcon and M he number of Gaussan comonens. We consder here he unvarae case for he sake of smlcy of noaon, bu we wll evenually exend resuls o he mulvarae case. he arameerzaon of such a GM requres he esmae of he wegh, α, mean, µ, and varance, σ, for each of he M Gaussan comonens (he sum of weghs has o be unary). hs esmae s ycally erformed so as o maxmze he lkelhood, L, over a se of samles, x, =,...,, ndeendenly drawn from hs dsrbuon: ( ) L ( x,..,x ) = x. () = Oeraonally, he lkelhood s convenenly comued n log form wh he man advanage of avodng rad underflow. Maxmzaon of () can be obaned by esmang he GM arameers hrough an execaonmaxmzaon (EM) algorhm. EM s an erave algorhm ha mroves he esmae of he arameers a each eraon and s guaraneed o converge o a local maxmum of he lkelhood (or a saddle on n he mulvarae case). he equaons used o esmae he arameers a each eraons are: new αl = ( l x, Θ ) (3) = x = ( l x, Θ ) new = µ l = (4) ( l x, Θ ) new ( x µ l ) ( l x, Θ ) new = σ l = (5) ( l x, Θ ) = where Θ reresens he curren se of arameers and (l x,θ) s smly he robably of he l-h Gaussan comonen a samle x : αl G ( ) ( x ; µ l, σ l ) l x, Θ M αk G( x ; µ k, σ k ) = (6) k= Equaon (6) s ofen referred o as a membersh funcon, exressng he membersh of x n each of he Gaussan comonens, and s comuaon s he execaon se of

3 an EM eraon. he comuaon of udae equaons (3-5) s s maxmzaon se. I s moran o noe ha each of (3-5) rovdes an omum for he resecve arameer ndeendenly of he oher wo. Kernel densy esmaon models a df as: KDE ( x) = h = x x K h where K( ) s a funcon wh arcular roeres, called kernel, and h s he kernel bandwdh (or smoohng facor). By usng he Gaussan kernel and nong he kernel bandwdh wh σ, (7) becomes: ( x) = G( x;, ) (7) KDE x σ. (8) = Alhough (8) reduces KDE o anoher Gaussan mxure, he smlary beween () and (8) s manly aaren: frs, n () he number of Gaussan comonens, M, s ycally very low comared o he number of samles,. Moreover, he wegh, oson and wdh (α, µ, σ) of each comonen are deermned as a rade-off over he samle se. In (8), nsead, each Gaussan comonen s frmly locaed on a samle. he only arameer o be esmaed s he varance, σ (or, equvalenly, he sandard devaon, σ), common o all he Gaussan comonens. Esmae of he omal varance, σ, can be erformed accordng o a number of dfferen crera (see [] for a comrehensve revew). I s neresng o noe ha maxmzng he lkelhood for he KDE case leads o an obvous bu mraccal soluon: LKDE ( x,.., x ) = KDE ( x ) = = (9) = ( ;, ) G x x σ as σ 0 = = Whle several crera could be chosen o deermne an omal value for σ, here we are neresed n reanng he maxmum lkelhood framework so ha our resuls can be more easly ransferred o he esmae of arameers of a hdden Markov model. hus, we use he seudo-lkelhood defned as []: PL ( ; x ) KDE ( x,.., x ) = G x, σ. (0) = =, x Essenally, he robably of each x samle s comued by excludng he Gaussan comonen cenred on x self. Maxmzaon of (0) can be erformed n varous ways. Dun n [3] suggesed comung he frs dervave of (0) wh resec o σ and eravely calculang s zero crossngs. He reored ha, by usng a secally adaed verson of he regula fals algorhm, 5-0 eraons were needed o reach an accuracy of 0-3 n he value of σ over a se of exermens. Here, we use he execaon-maxmzaon algorhm by adang udae equaon (5) o he case of a common value for σ for all he Gaussan comonens. From [7], can be easly roven ha: M new ( x µ l ) ( l x, Θ ) new l= = σ = () M ( l x ), Θ l= = rovdes he omal value for σ when such a value s consraned o be he same for all he M Gaussan comonens. Agan, f we use () drecly for esmang σ n he KDE case, we wll converge o he undesred value σ = 0. hus, for KDE we adus () o reflec he defnon of seudo-lkelhood gven n (0) as: ( x x ) ( x, Θ ) new = =, x x σ =. () = =, x x ( x, Θ ) Equaon () can be derved as follows: for GM, he dervaon of (3-5) s ossble under he smlfyng assumon ha he generave model of each samle x s no he whole GM, bu only he bes Gaussan comonen ndcaed by an unobserved ndcaor varable. For KDE, under he furher assumon of seudo maxmum lkelhood, he robably a (6) s assumed null when x = x ; hus, () derves from (). For esng, we have run a se of exermens on a range of smulaed daa: convergence of σ was always obaned, and always o a local maxmum of he seudo-lkelhood... Comarng KDE and GM df esmaon GMs show lmaons n modellng ceran dsrbuons. One lmaon s n he modellng of dsrbuons whch show more modes han he Gaussan comonens. In hs case, one sngle Gaussan comonen has o be f over mulle modes, hus leadng o oor modellng based on eye udgmen and relavely low lkelhood. Alhough esmang he rgh number of modes s ossble hrough rocedures such as he mean-shf vecor, s ofen unfeasbly me consumng. he same oor modellng occurs when he dsrbuon shows unform regons whch

4 are naccuraely modelled by means of only a few Gaussans. KDE can overcome boh hese lmaons. We argue ha n some cases feaure values obaned from human acves n vdeos such as seed and osons exhb such unform regons. Moreover, we argue ha KDE could also lead o mroved hdden Markov models of such acves. Fgures and show an examle of densy esmaon wh a GM wh wo comonens based on (3-6) and wh KDE based on our esmaor for σ. For he laer case, he fng of he dsrbuon over he samles seems generally very good. As an obvous consequence, he lkelhood obaned for KDE has always been greaer or equal han ha for GM n all our exermens. Obvously, hs comes a an ncreased comuaonal cos for he evaluaon of (8) wh resec o (). Fgure : GM densy esmaon of a seemngly unform dsrbuon. Esmaon seems generally naccurae (nal arameers: α = α = 0.5; µ = 85, µ = 70; σ = σ = 6). Fgure : KDE from he same se of samles as Fgure wh he roosed esmaor for σ. Esmaon seems generally accurae (nal arameer: σ = 6). 3. Hdden Markov models wh kernel densy esmaon of emsson robables A hdden Markov model (HMM) offers a mean o esmae he on robably of a sequence of medscree observaons O, =.., and corresondng hdden saes, X {..} [7]. he model s fully descrbed by he se of arameers λ = {A, B, π}: { a } = ( X = X = ), { b ( o )} = ( O = o X = ) o A = (3) B = (4), π { π } = ( X = ) = (5) A are called he sae ranson robables and exress he Markovan hyohess ha he value,, of he curren sae, X, s only deenden on he value,, of he revous sae, X -. B are called he emsson (or observaon) robables, quanfyng he robably of observng value o when he curren sae s. Evenually, π are called he nal sae robables and quanfy he robables of values for he nal sae. When observaon values are connuous, HMMs ycally use GMs o model her emsson robables. Each sae s value, =.., has a corresondng GM. hus, he observaon robably b (o ) s gven by: M b ( o ) = l G( o ; µ l, σ l ) α. (6) l= he Baum-Welch algorhm rovdes udae equaons for he erave omsaon of he model s arameers. Herewh, we concenrae on B. Frs, smlarly o (6), we ose: αl G ( ) ( o ; µ l, σ l ) l o, Θ =. (7) M αk G( o ; µ k, σ k ) k= o exress he robably of he l-h comonen n he GM of sae. hen, he weghs, means and varances of he emsson robables are obaned n a way smlar o (3-5) over he se of observed values, o, =... he basc dfference s ha he erms n he numeraors and denomnaors are mulled by he robably of beng n sae a me, γ (): ( l o, Θ ) γ ( ) new = α l = (8) γ ( ) = o ( l o, Θ ) γ ( ) new = µ l = (9) ( l o, Θ ) γ ( ) = new ( o µ l ) ( l o, Θ ) γ ( ) new = σ l = (0) ( l o, Θ ) γ ( ) = In urn, γ () can be exressed from he curren esmaes of A and B (see [7] for deals).

5 From (8-0) and he consderaons addressed n Secon, we can fnally derve he udae equaons for he KDE case: new l =, o ol ( l o, Θ) γ = ( ) ( ) α = () γ new µ l = ol () ( o o ) ( l o, Θ ) γ ( ) l new = l=, ol o σ = (3) ( l o, Θ ) γ ( ) = l=, o o In (), he cenres of he Gaussan comonens are no subec o udae and s, as usual, on he samles. (3) s he re-wrng of () negraed by γ (). Agan, we exclude he Gaussan comonen cenred on he samle self o reven convergence o σ = 0. We convenenly oban hs by seng (l o l, Θ) = 0 a he begnnng of he eraon. Wegh adusmen s needed also n he KDE case snce observaons need o be dsached o he saes n any case. o hs am, () s dencal o (8) and us follows he way EM udaes he GM weghs. he only dfference n () s ha (l o l, Θ) s, agan, se equal o 0. In hs way, he weghs are essenally defned by he neghbourng kernels, no he one cenred on he on self, lke n udae equaon (3). Alernaves for wegh assgnmen are ossble, such as a smle α l = γ (l), bu hey have no been exermened n real daa, us n he synhec daa showed n he secon 4.. Overall, equaons a (-3) defne he KDE/HMM roosed n hs aer. Merely o rove ha hese resuls obvously exend o he mulvarae case, we conclude hs secon by showng (3) for he case of mulvarae observaons: Σ new = = l=, o o 4. Exermens l ( o o )( o o ) ( l o, Θ) γ ( ) = l=, o o l l ( l o, Θ) γ ( ) (4) In hs secon, we reor resuls dvded n wo ses of exermens. Frs of all, n order o show he erformance of he KDE/HMM, we resen some resuls wh synhec daa and he wegh assgnmen as smle as smle α l = γ (l). Subsequenly, we carred ou some ess for he human acvy classfcaon wh he well known daabase of CAVIAR []. 4.. Exermens wh synhec daa he frs se of exermens consss of a synhec daa dsrbued n hree clusers of wo-dmensonal daa: Class : A wo-dmensonal unform funcon beween x, y beween 0 and 8. Class : Four wo-dmensonal ndeenden Gaussans funcons beween x, y = and 6 and σ=. Class 3: Anoher wo-dmensonal unform beween x, y beween 30 and 60 Fgure 3. Dsrbuon of he ranng daa. he nalzaon of he arameers for he KDE and GMs are se as follows: Means (only for he GMs case): µ = [0 3 50] Covarance: Σ =, Σ =, Σ 3 =, Σ =, 4 Σ =, Σ 6 = 0 0 Wegh for each of he Gaussan comonens: random n he case of GMs, and for KDE. umber of Gaussans er sae (only GMs case): M =. Sae ranson marx (3 x 3): A = [ ; ; ]. Inal robables: Π = [ 0 0] Maxmum number of eraons for he EM algorhm: max-er=50 he exermens are carred ou by changng he value of he nal covarance and usng -fold cross valdaon. We ook 50 ons for he frs and hrd class and 60 for he second one.

6 able. oal classfcaon error for he ranng and valdaon ses of synhec daa. Error(%) KDE GMs () Σ ranng Valdaon () Σ ranng Valdaon (3) Σ 3 ranng Valdaon (4) Σ 4 ranng Valdaon (5) Σ 5 ranng Valdaon (6) Σ 6 ranng Valdaon used he CAVIAR vdeo daase [] and seleced he wo vdeos named Fgh_RunAway.mg and Fgh_OneManDown.mg. Among all he acves showed n hese vdeos, we focused on hree: {Inacve ( n ), Walkng ( wk ) and Runnng ( r )}. Boh vdeos come accomaned by he ground ruh rovded by he daase s auhors. Each erson n each frame s labelled wh an acvy value. hs ground ruh was deermned by hand-labellng and we mus ake no accoun he subecvy when classfyng, esecally beween classes walkng and runnng. he ool used for he exermens was he Kevn Murhy s Malab oolbox for HMM [4]. We laer modfed hs oolbox o add he mlemenaon of he KDE model. he feaures ha we seleced n order o classfy he acves are he magnudes of he subec s seed measured over dfferen me nervals. In arcular, we comued he seed a 5 and 5 frame nervals as follows: seed f = ( x x f ) + ( y y f ), (5) f where f s se o 5 and 5, resecvely, and (x, y ) and (x -f, y -f ) are he subec s osons n he mage lane. Fgure 3 shows hsograms of he wo veloces for he hree acves and how challengng he searaon of he acves romses o be based on such feaures. Fgure 4. Pcure of he classfcaon and confuson marx afer he ranng for exermens from 3 o 6 wh GMs. We can check ha he KDE ouerforms n all he exermens he GMs mehod by modelng erfecly he wo unforms and he grou of Gaussans. On he oher hand, he GMs mroves s erformance wh covarance values hgher han, whereas he classfcaon s very oor below hs hreshold. able. Confuson marx for he classfcaon wh he GMs (exermens from 3 o 6) GMs Acual Predced Exermens for he human acvy recognon Fnally, we reor resuls on he alcaon of KDE/HMM o he classfcaon of human acves. We Fgure 5: Hsograms of feaures seed 5 (lef column) and seed 5 (rgh column) for saes n (nacve), wk (walkng) and r (runnng). he exermens conssed of learnng he HMM arameers for boh models, and her subsequen use for acvy classfcaon,.e. Verb sae decodng. he ror robably was fxed o {sae = ; sae = 0; sae 3 = 0} as we assume ha he sae of an ndvdual frs aearng n he scene s always nacve. hs assgnmen resuls very useful when decodng he sequence, as we do no know a ror he corresondence beween he saes n he Verb ouu and hose n he ground ruh. By fxng hs robably we assure ha he frs code of he Verb ouu wll mach he nacve sae.

7 he nalzaon of he varables for he EM algorhm was carred ou as follows: Means (only for he GMs case) : µ = [ ] and µ = [0. 4] Covarance: we sared wh hgh and low values of covarance as we do no know a ror o wha values he algorhm s execed o converge. Moreover, we chose a dagonal covarance marx and wo osve semdefne marx and no dagonal Σ =, Σ = 4 5 and Σ3 = 8 9 Wegh for each of he Gaussan comonens: random n he case of GMs, and for KDE. umber of Gaussans er sae (only GMs case): M =. Sae ranson marx (3 x 3): A = [ ; ; ]. he runnng acves are very few and her duraon s very shor. ha s he reason why he a 3, s such a hgh value whereas a 3,3 s low. Maxmum number of eraons for he EM algorhm: max-er=50 he daa are dvded no wo ses of sequences for ranng and esng: one of 7 sequences of 975 daa n oal and anoher of 6 sequences and 605 daa. he frs se s used for ranng whle boh are searaely used for valdaon. able 3 shows he oal classfcaon error on he ranng and valdaon ses for he GMs and he KDE. able 3. oal classfcaon error for he ranng and valdaon ses. Error (%) ranng Valdaon () GMs (µ, Σ ) 3,59 7,3 () GMs (µ, Σ ) 6, (3) KDE (Σ ) 4,48 6,45 (4) GMs (µ, Σ ) 8,38 5,07 (5) GMs (µ, Σ ) 7,37 5,3 (6) KDE (Σ ) 4,7 6,0 (7) GMs (µ, Σ 3) 3,59 7,3 (8) GMs (µ, Σ 3) 3,59 7,3 (9) KDE (Σ 3) ,6 covarance arameers. hs shows he lmaon of he GM model as a hghly aramerc echnque of dffcul nalzaon. he error on he ranng and valdaon ses for he KDE model remans around 4-6% whle for he GMs model vares beween 5 and 4% deendng on he dfferen combnaons of nal covarances and means. o rovde furher deal no hese resuls, able 4 shows he confuson marx for he GMs and he KDE cases for exermens, 3, 5 and 6 n able 3. Each column of he marx reresens he nsances n a redced class, whle each row reresens he nsances n an acual class (ground ruh). able 4 shows ha he beer overall resuls of KDE also corresond o mroved ner-class errors wh resec o he GMs model. able 4. Confuson marx for he classfcaon wh he GMs (exermen ) and KDE (exermens 3 and 6) models for he valdaon daa. GMs () Acual KDE (3) Acual KDE (6) Acual Predced n Wk r n wk r Predced n Wk r n wk r Predced n Wk r n wk r Fnally, Fgure 6 shows he df s of he emsson robables for he GMs and KDE for exermens 5 and 6 for each of he saes. he df s show he nuvely dfferen modellng of GMs and KDE. In arcular, he KDE emsson robables are no requred o be comac and sonaneously adus o model non-clusered daa and wh daa wh unform regons. he exermens show he sable resuls of KDE/HMM ndeendenly of he nalsaon of s covarance arameer. I aears ha he arameer sace s very smle o search and he learnng converges o he same value of Σ rresecvely of very dfferen nal values. Conversely, he GMs HMM obans sgnfcanly dfferen error raes deendng on he nal values of s means and (a) (d)

8 he heavy low-level rocessng of foreground exracon and rackng. (c) (b) Fgure 6: GMs derved from he EM algorhm, exermen (5) (n a scale 0-7) for nacve (a), walkng (b) and runnng (c). KDE derved from he modfed EM algorhm, exermen (6) (n a scale 0-0.5) for nacve (d), walkng (e) and runnng (f). 5. Conclusons In hs aer, we have resened a modfed hdden Markov model wh KDE emsson robables (HMM/KDE) and s use for acvy recognon n vdeos. In he roosed aroach, kernel densy esmaon of he emsson robables occurs smulaneously wh ha of all he oher model arameers hanks o an adaed Baum- Welch algorhm. hs has allowed us o rean maxmumlkelhood esmaon whle overcomng he known lmaons of mxure of Gaussans n modellng ceran daa dsrbuons such as unform and non-clusered daa. Exermens on acvy recognon have been erformed on he CAVIAR vdeo survellance daabase and reored n he aer. Overall, he error on he ranng and valdaon ses wh kernel densy esmaon remans around 4-6% whle for he convenonal Gaussan mxure aroach vares beween 5 and 4%. he man advanage ha we denfy n he roosed KDE/HMM model s ha s accuracy seems subsanally ndeenden from he choce of he nal value of s only arameer, he covarance marx common o all s kernel comonens. On he conrary, he convenonal GMs modellng of emsson robables s a hghly aramerc echnque and roves of challengng nalsaon. Obvously n a way, he ncreased and more sable accuracy obaned by KDE comes a hgher comuaonal coss for boh model esmaon and evaluaon as he number of kernels n KDE s much greaer han ha ycal of GM comonens. However, hs does no seem o reresen a sgnfcan ssue n alcaons such as acvy recognon n vdeos as hey are however domnaed by (f) (e) References [] O. Masound and. Paankolooulos. Recognzng Human acves. In Proc. IEEE Conference on Advanced Vdeo and Sgnal Based Survellance, 57-6, 003. [] M. Brand and V. Kenaker. Dscovery and segmenaon of acves n vdeo. IEEE rans. Paern Analyss and Machne Inell., (8): , 000. [3].M. Olver, B. Rosaro, and A.P. Penland. A Bayesan comuer vson sysem for modelng human neracons. IEEE rans. on Paern Anal. and Machne Inell., (8): , 000. [4] J. Ben-Are, Z. Wang, P. Pand, and S. Raaram. Human Acvy Recognon Usng Muldmensonal Indexng. IEEE rans. on Paern Anal. and Machne Inell., 4(8): 09-04, 00. [5] Ju Han and B. Bhanu. Human Acvy Recognon n hermal Infrared Imagery. In Proc. IEEE CS Comuer Vson and Paern Recognon, 3:7-7, 005. [6] L. Rabner. A uoral on Hdden Markov Models and Seleced Alcaons n Seech Recognon. Proc. IEEE, 77:57-86, 989. [7] J. Blmes. A genle uoral on he EM algorhm and s alcaon o arameer esmaon for Gaussan mxure and Hdden Markov Models. ech. Re. ICSI-R-97-0, Unversy of Calforna Berkeley, 998. [8] H. Bourlard and. Morgan. Connecons Seech Recognon. A Hybrd Aroach. Kluwer Academc Publshers, 994. [9] E. renn. onaramerc Hdden Markov Models: Prncles and Alcaons o Seech Recognon. In Lecure oes n Comuer Scence 859:3-, 003. [0] S.E. Kruger, M. Schaffoner, M. Kaz, E. Andelc, and A. Wendemuh. Mxure of Suor Vecor Machnes for HMM based Seech Recognon. In Proc. 8h In. Conf. on Paern Recognon 4:36-39, 006. [] B. A. urlach. Bandwdh Selecon n Kernel Densy Esmaon: A Revew. echncal Reor Unversé Caholque de Louvan, Belgum, 993. [] CAVIAR: Conex Aware Vson usng Image-based Acve Recognon. h://homeages.nf.ed.ac.uk/rbf/caviar/ (las accessed: 30 ovember 006). [3] R. P. W. Dun. On he choce of smoohng arameers for Parzen esmaors of robably densy funcons. IEEE rans. on Comuers, 5():75-79, 976. [4] Hdden Markov Model (HMM) oolbox for Malab: h://