Sequential Dynamic Classification Using Latent Variable Models

Transcription

1 The Auhor 2. Publshed by Oxford Unversy Press on behalf of The Brsh Compuer Socey. All rghs reserved. For Permssons, please emal: Advance Access publcaon on January 27, 2 do:.93/comjnl/bxp27 Sequenal Dynamc Classfcaon Usng Laen Varable Models Seung Mn Lee and Sephen J. Robers Paern Analyss & Machne Learnng Research Group, Deparmen of Engneerng Scence, Unversy of Oxford, Oxford, UK Correspondng auhor: sjrob@robos.ox.ac.uk Adapve classfcaon s an mporan onlne problem n daa analyss. The nonlnear and nonsaonary naure of much daa makes sandard sac approaches unsuable. In hs paper, we propose a se of sequenal dynamc classfcaon algorhms based on exenson of nonlnear varans of Bayesan Kalman processes and dynamc generalzed lnear models. The approaches are shown o work well no only n her ably o rack changes n he underlyng decson surfaces bu also n her ably o handle n a prncpled manner mssng daa. We nvesgae boh suaons n whch arge labels are unobserved and also where ncomng sensor daa are unavalable. We exend he models o allow for acve label requesng for use n suaons n whch here s a cos assocaed wh such nformaon and hence a fully labelled arge se s prohbve. Keywords: dynamc classfcaon; sequenal Bayesan nference; parally observed daa; nonsaonary decson processes. INTRODUCTION Many daa analyss problems requre onlne learnng. In hs paper we consder he problem of sequenal classfcaon n envronmens n whch he daa are by naure nosy, nonlnear and nonsaonary. Furhermore, we consder scenaros n whch componens of he daa are delayed and/or mssng. We use he framework of dynamc models o ackle hs problem doman, n parcular we use generc sae-space models. For a bnary classfcaon problem, a logsc regresson model s raned o produce poseror class probables. By modellng he parameers weghs) of he logsc regresson classfer as me-varyng parameers, he nonsaonary naure of he problem can be capured. In [, 2] he exended Kalman fler EKF) s employed o solve he nonlnear dynamc classfcaon model. In hs paper we exend hs approach and demonsrae how he unscened Kalman fler UKF) [3, 4] can be appled o he adapve classfcaon problem. Alernavely, hs problem can be regarded as a specal case of dynamc generalzed lnear models [5]. Bnary oucomes are known o have a bnomal dsrbuon, whch s hen modelled by a generalzed lnear model. By allowng he paramerc weghs of he model o evolve over me, Receved Augus 29; revsed 3 November 29 Handlng edor: Nck Jennngs we can exend canoncal generalzed lnear models o he dynamc classfcaon problem. The goal of dynamc sreamng classfcaon s nrnscally smlar o ha of classfcaon n he presence of concep drfs, normally handled usng ensembles of weak classfers ha are adapvely combned so as o form robus decsons n he presence of drfs [6 8]. Ths paper depars somewha from hs approach n ha we am o adap he base learnng model self. Furhermore, we am o do hs n an opmal Bayesan) framework. A combned approach n whch no only he base learners, bu also he ensemble mxure are adapve s explored n [8]. In Secon 2 we brefly revew he logsc regresson model and s nonsaonary verson usng he framework of he sandard EKF. In Secon 3 we exend he algorhm for parameer nference n he nonsaonary logsc regresson model by use of he UKF. Secon 4 revews he dynamc generalzed lnear model and we examne how dynamc classfcaon can be deal wh n ha conex. In Secon 5 we apply he proposed models o synhec and real daa and carry ou expermens o evaluae he performance of hese models n cases of ncompleely observed nformaon. Secon 6 concludes he paper wh dscussons abou fuure work. Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

2 46 S. M. Lee and S. J. Robers 2. DYNAMIC LOGISTIC REGRESSION 2.. Logsc regresson We consder a se of sequenal sensor observaons whch we denoe by he vecor h. Assocaed wh each of hese observaons or feaure) vecors s a arge classfcaon label whch s denoed by he varable y. For he bnary decson problem whch we concenrae on n hs paper), hese arge labels are represened by y and y. Our objecve s o nfer he class probably defned by πh) Pry h), ) where h s he gven npu vecor of, for example, sensor observaons. By defnon, Pry h) πh). Logsc regresson models he class probably as follows: πh) lh T w), 2) where w s a vecor of weghs and l ) s a logsc funcon defned by lh T w) expht w) + exph T w). 3) Insead of usng he npu vecor per se, we can consder an exended form as follows: h ϕh) φh), 4) where φ represens a se of nonlnear kernel funcons such as Gaussan radal bass funcons. We may hence see hs model as a dynamc Radal Bass Funcon classfer. The addon of he uny componen n he above equaon allows for a me varyng offse decson bas) o be modelled [2]. Throughou hs repor we consder, unless oherwse saed, a logsc funcon wh acvaon a ϕh) T w, no a h T w. Wh hs kernel npu vecor s possble o fnd a nonlnear boundary beween wo classes Bayesan approach Suppose ha he parameer or wegh vecor w has a probably dsrbuon pw). The class probably gven n Equaon 2) gnores any uncerany assocaed wh he wegh vecor, beng n effec condoned on a mos lkely pon value. A Bayesan approach akes no accoun he full dsrbuon, and accordngly n cases where uncerany s nrnscally hgh, moderaes he class probably such ha less confden decsons are made. If we ake he dsrbuon of w s a mulvarae Gaussan wh mean ŵ and covarance P w, hen he dsrbuon of acvaon a s also a Gaussan wh mean â and varance c 2 defned by â ϕh) T ŵ, 5) c 2 ϕ T h)p w ϕh). 6) By negrang a ou, he poseror class probably π s moderaed; ha s, π la)pa)da. 7) Snce s mpossble o compue he probably analycally, s approxmaed by [9]: where π lκc 2 )ā), 8) ) /2 κc 2 ) + πc2. 9) 8 Here π represens he sandard mahemacal consan approxmaely equal o If he varance of acvaon s close o zero, he exen of moderaon s neglgble. However, as he varance becomes very large, he poseror class probably for bnary sysems) s moderaed o le closer o he pror a Nonsaonary logsc regresson In he saonary logsc regresson model gven n Equaon 2), he wegh vecor w s assumed o be sac. In [] a dynamc logsc regresson model s proposed n whch he wegh vecor evolves accordng o a Gaussan random walk whch has he followng form: π lϕh ) T w ), ) w w + v, ) where v s a dffuson varable assumed o be a Gaussan wh mean and covarance q I. For noaonal convenence we wll use he noaon ϕh ) o ϕ. Also we noe ha he wegh vecor of he above equaon can be seen as he sae varable of a dynamcal sysems model. In he dynamc logsc regresson he nonsaonary s capured by he me-evolvng sae varable w, and he bnary decson s made accordng o he logsc funcon. To perform adapve classfcaon, we have o esmae he sae varable w n an onlne manner, and hs nference problem s deal wh n he followng subsecons Dynamc classfcaon usng EKF The EKF s he mos commonly used fler for handlng nonlnear sysems []. I s based on lnearzng a nonlnear funcon; he nonlnear funcon s approxmaed by he frs wo Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

3 Sequenal Dynamc Classfcaon Usng Laen Varable Models 47 erms of s Taylor seres expanson. Suppose ha he poseror mean and covarance of sae varable w are represened by Ew D ) ŵ, 2) Covw D ) P, 3) where D {y,y 2,...,y } s a se of observed labels up o me. Snce he sae process s lnear, he wo momens of he pror dsrbuon of he sae varable a me can be easly compued as follows: ŵ ŵ, P P + q I. 4) As he logsc observaon funcon s nonlnear n w, we approxmae usng where π lϕ T w ) lϕ T ŵ ) + A T w ŵ ), 5) A lw) w ŵ lϕ T ŵ ) lϕ T ŵ ) ) ϕ. 6) The one-sep ahead predcon for he class probably s compued accordng o and s varance s ˆπ lϕ T ŵ ), 7) Varπ D ) A T P A + r, 8) where r s he pror observaon nose varance for he observed label y.as he bnary label varable has a Bernoull dsrbuon, we can esmae he observaon nose varance usng r ˆπ ˆπ ). 9) Afer observng a new label y, we updae he pror dsrbuon of he sae varable. The poseror mean and covarance of he sae varable are compued accordng o he Kalman updae equaons, namely ) ŵ ŵ + K y ˆπ, 2) ) P P K A T P A + r K T. 2) The Kalman gan K s compued as follows []: K P A A T P A + r ). 22) In he predcon conex Jazwnsk [] proposed an dea for adapvely esmang nose varances by maxmzng he evdence of observaons. In [, 2] hs concep s modfed o make suable for he classfcaon conex by maxmzng he evdence of model predcons nsead. For example, Lowne e al.[2] suggess updang sae nose varance q accordng o where q max { u u, }, 23) u ˆπ ˆπ ) 24) s he uncerany varance) n he predcve Bernoull dsrbuon. 3. DYNAMIC CLASSIFICATION USING THE UKF 3.. The UKF The UKF s proposed n [3, 4]. The UKF s based on he unscened ransformaon, whch s a mehod for calculang he sascs of a random varable ha undergoes a nonlnear ransformaon and s founded on he prncple ha s easer o approxmae a probably dsrbuon han an arbrary nonlnear funcon [3]. The unscened ransformaon resembles Mone Carlo samplng mehods n ha a number of pons are seleced and he pons are propagaed hrough a nonlnear funcon. However, he unscened ransformaon selecs a se of pons referred o as sgma pons) no randomly bu deermnscally, so ha hey preserve he sascs e.g. mean and varance) of her underlyng dsrbuon. Suppose ha we have wo random varables a and c and a nonlnear funcon b ). We wsh o nfer he probably densy funcon for c ba); we denoe he means and covarances of a and c by ā, c, P a and P c, respecvely. We frs selec 2L + L s he dmenson of a) sgma vecors A n he followng deermnsc way: A ā, ) A ā + L + λ)pa ) A ā L + λ)pa,,,...,l, L +,...,2L, 25) where λ α 2 L + κ) L and L + λ)p a ) represens he h column of he marx square roo. The varables α and κ are parameers of he mehod: he former s a scalng parameer ha deermnes he spread of he sgma pons around ā and s normally se o a small posve value, and he laer s a secondary scalng parameer and s normally se o []. To compue he sascs of he ransformed vecor c, he sgma vecors A are propagaed hrough he nonlnear funcon o provde anoher se of sgma vecors: C ba ),,,...,2L. 26) The unscened ransformaon approxmaes he mean and covarance of c as weghed sums of he propagaed sgma Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

4 48 S. M. Lee and S. J. Robers vecors accordng o c P c 2L 2L The weghs ω are gven by ω m) λ L + λ, ω c) λ L + λ + α2 + β, ω m) ω c) ω m) C, 27) ω c) C c)c c) T. 28) 2L + λ),,...,2l, 29) where β s anoher parameer ha ncorporaes pror knowledge of he dsrbuon of a. For Gaussan dsrbuons and arbrary smooh nonlneares, β 2 s opmal []. In he UKF, he pror and poseror means and covarances of he sae varable of a dynamc nonlnear model are approxmaed by he unscened ransformaon. There are wo major advanages of he UKF over he EKF. Frs, whereas he lnearzaon mehod of he EKF approxmaes he rue mean and covarance up o he frs order, he mean and covarance obaned by he UKF, for Gaussans, are accurae up o hrd order for all smooh monoone nonlneares, and for non-gaussans up o a leas he second order s guaraneed [2]. Secondly, despe he mprovemen n approxmaon accuracy, he compuaonal cos requred by he unscened ransformaon s equal o he lnearzaon mehod of he EKF. A comparson beween he lnearzaon approach, he unscened ransformaon and a samplng mehod s excellenly llusraed n [] Dynamc logsc regresson usng he UKF We repea he dynamc logsc regresson model of he prevous secon, namely π lϕh ) T w ), 3) w w + v, 3) where v s a Gaussan sae nose wh mean and covarance q I. In Secon 2.2. we derved an algorhm for solvng hs nonlnear sysem usng he EKF. We here presen an algorhm n whch he UKF s appled o he dynamc classfcaon problem. Assumng ha he poseror mean and covarance of he sae varable w are ŵ and P, respecvely, he wo momens of he pror sae dsrbuon a me are ŵ ŵ, 32) P P + q I, 33) n whch q s nferred usng Equaons 23) and 24). To compue he mean and varance of he one-sep ahead predcve dsrbuon, he unscened ransformaon mehod s employed. We frs draw 2L + ) sgma vecors for he dsrbuon of w from s esmaed mean and covarance, ŵ and P : W [ W ],...,W,...,W2L [ ) L + λ)p ŵ, ŵ + ) ŵ L + λ)p, ], 34) where L s he dmenson ) of w, λ α 2 L + κ) L and L + λ)p s he h column of he marx square roo. The sgma vecors are propagaed hrough he logsc funcon: l ϕ T )) W,,,...,2L. 35) Usng hese propagaed sgma vecors we can compue a onesep ahead predced class probably and s varance, whch are gven by ˆπ P yy 2L 2L C m), 36) C c) ˆπ ) 2 + r. 37) The pror observaon nose varance r can be eher fxed o a sac small value or esmaed accordng o r ˆπ ˆπ ). In addon, we compue he covarance beween he sae varable and he class probably accordng o P wy 2L The weghs C m) C c) W ŵ ) ˆπ ) T. 38) and C c) C m) λ L + λ, are gven by C c) λ L + λ + α2 + β, C m) C c) 2L + λ),,...,2l. 39) Afer observng a new label y, he sae varable dsrbuon s updaed. The poseror mean and covarance of w are compued accordng o where he Kalman gan s ) ŵ ŵ + K y ˆπ, 4) P P K P yy K T, 4) K P wy P yy. 42) In Secon 2.. we nroduced a Bayesan approach o he logsc regresson model n order o ake no accoun he Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

5 Sequenal Dynamc Classfcaon Usng Laen Varable Models 49 uncerany assocaed wh he sae varable w. We noe ha he UKF represens he dsrbuon of w by selecng sgma pons from he dsrbuon. Once he probably densy funcon over w has been nferred usng he unscened fler, he dynamc classfcaon scheme proceeds dencally as for he EKF. 4. DYNAMIC GENERALIZED LINEAR MODELS Generalzed lnear models are a powerful mehod for analysng dsrbuons n he exponenal famly. They can be generalzed o capure he nonsaonary naure of daa such as we consder n hs paper by allowng he model parameers o evolve over me, gvng rse o a famly of approaches whch we refer o as dynamc generalzed lnear models. Snce a bnary label sequence follows a Bernoull dsrbuon, whch s a specal case of a bnomal dsrbuon and n he exponenal famly, he dynamc classfcaon problem can be solved n he framework of dynamc generalzed lnear models. In hs secon we revew he generalzed lnear model self and exend o allow for dynamcs usng a Bayesan perspecve. 4.. Generalzed lnear models In an ordnary lnear regresson model an observaon random varable y s modelled by y μ + n, 43) where μ h T w. Noe ha, as n prevous secons, h represens a known npu varable and w s a model parameer vecor. In addon, n s a Gaussan nose varable wh mean and varance σ 2. The observaon varable s also aken o have a mulvarae Gaussan dsrbuon wh he followng mean and varance: EY ) μ, VarY ) σ 2. 44) Noe ha he mean s a lnear funcon of he predcor varable, h. In a generalzed lnear model wo generalzaons are made from he sandard lnear model above: ) he probably dsrbuon of he observaon varable s no resrced o be a Gaussan bu s allowed o have any dsrbuon n he exponenal famly; ) s no he mean of he varable bu a funcon of he mean ha s lnearly relaed o he predcor varable. An observaon random varable, y, ha has an assocaed dsrbuon n he exponenal famly akes he form, ) dy)θ bθ) py θ,φ) exp + cy, φ), 45) aφ) where θ s referred o as he canoncal parameer, φ he dsperson parameer and a, b, c, d are all known funcons for a gven paramerc dsrbuon. The mean and varance of he varable are relaed o he canoncal and dsperson parameers as follows: Edy) θ,φ) μ b θ), 46) Vardy) θ,φ) b θ)aφ). 47) Members of he exponenal famly nclude Gaussan, bnomal, Posson, exponenal, gamma, nverse Gaussan dsrbuons ec. In he ordnary Gaussan lnear model he mean s lnearly relaed o he predcor varable, h; ha s, μ h T w. 48) The rgh-hand sde of he equaon s denoed by η h T w, and η s referred o as he lnear predcor. Noe ha he lnear predcor and he mean of a Gaussan varable can ake any real values. However, hs may or may no be rue for oher dsrbuons n he exponenal famly; for nsance, he mean of a Posson dsrbuon mus be non-negave. A lnk funcon g.) s used o relae he lnear predcor o he mean, ha s gμ) η h T w. 49) The mean can herefore be compued smply by nverng he lnk funcon: μ g η) g h T w). 5) Comparng Equaon 48) wh Equaon 49), we can see ha he lnk funcon for he sandard Gaussan lnear model s an deny funcon. A parcular form of he lnk funcon s of mporance. A lnk funcon ha makes he lnear predcor η equal o he canoncal parameer θ s referred o as canoncal lnk: gμ) η θ. 5) The canoncal lnk s mporan due o he exsence of mnmal suffcen sascs for he model parameers [3], hereby beng used n mos cases Exponenal famly dynamc models Suppose ha an observaon varable y, ha has a dsrbuon n he exponenal famly, s observed over me. The observaon model of a dynamc generalzed lnear model s of he form ) dy )θ bθ ) py θ,φ ) exp + cy,φ ), 52) aφ ) wh he followng lnk equaon gμ ) η h T w, 53) where μ Ey ).Ify has a Gaussan dsrbuon, hen he observaon model and lnk equaon can be merged no a sngle observaon equaon as follows: y h T w + n, 54) where n N,r ) and μ h T w. Ths s equvalen o he observaon equaon of a sandard dynamc lnear model. Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

6 42 S. M. Lee and S. J. Robers The sae varable or model parameer vecor) w s assumed no o be sac bu o evolve over me va he sae process as w F w + v, 55) where v, Q ). Noe ha he noaon a b, c) represens ha he mean and varance of a are b and c, respecvely, whou specfyng he probably densy funcon of he varable a. We frs consder he sequenal nference mehod for he sae varable n he Bayesan seng [5]. I s an approxmae mehod n he sense ha dsrbuons are only specfed n erms of her frs and second momens whou full dsrbuonal nformaon. Suppose ha, gven D, sae varable w has mean ŵ and covarance P ; ha s, w D ŵ, P ). 56) Snce he sae process of Equaon 55) s lnear, he pror dsrbuon of he sae varable a me can be readly compued as where w D ŵ, P ), 57) ŵ F ŵ, 58) P F P F T + Q. 59) As he lnear predcor.e. η gμ ) h Tw ) s lnearly relaed o he sae varable, we can readly compue he pror dsrbuon of η, gven D. Therefore, may evaluae he jon dsrbuon of η and w s where [ η w ] D [ ] ˆη, ŵ [ S h TP ]) P T h, 6) P ˆη h T ŵ, 6) S h T P h. 62) The one-sep ahead forecas dsrbuon.e. he dsrbuon of y gven D ) can be obaned by margnalzng over he canoncal parameer assumng ha he dsperson parameer s known: py D ) py θ )pθ D ) dθ. 63) To solve he equaon we need o specfy he pror dsrbuon of he canoncal parameer, pθ D ).Aconjugae pror s wdely used because allows us o derve he forecas dsrbuon analycally. The conjugae pror dsrbuon s of form pθ D ) ωk,m ) expk θ m bθ )), 64) where ω s a known funcon provdng he normalzng consan, and k and m are hyperparameers of he conjugae pror. The complee form of he one-sep ahead forecas dsrbuon may hence be obaned n he followng way: py D ) pθ D )py θ ) dθ ωk,m ) expk θ m bθ )) ) dy )θ bθ ) exp + cy,φ ) dθ aφ ) ωk,m ) expcy,φ )) exp k + dy ) ) θ aφ ) m + ) ) bθ ) dθ aφ ) ωk,m ) ω k + dy )/aφ ), m + /aφ )) exp cy,φ )). 65) To employ he one-sep ahead forecas dsrbuon we need o deermne he values of he hyperparameers of he conjugae pror, k and m. If a canoncal lnk s used, whch ensures θ η, hen we know ha Eθ D ) Eη D ) ˆη, 66) Varθ D ) Varη D ) S. 67) From he form of he conjugae pror n Equaon 64), we can oban Eθ D ) and Varθ D ) n erms of he hyperparameers. In addon, we know he values of ˆη and S, whch are gven n Equaon 62). We can, herefore, calculae he values of he hyperparameers. Va Bayes heorem, he poseror dsrbuon of he canoncal parameer θ can be represened by pθ D ) pθ D )py θ ), 68) whch we may evaluae as pθ D ) ω k + dy ) aφ ),m + ) aφ ) exp k + dy ) ) θ aφ ) m + aφ ) ) ) bθ ). 69) Snce usng he canoncal lnk makes he poseror dsrbuon of η equvalen o ha of θ, we oban he poseror dsrbuon of η as η D ˆη, S ), 7) where ˆη Eη D ) Eθ D ), 7) S Varη D ) Varθ D ). 72) We noe ha Eθ D ) and Varθ D ) can be compued from Equaon 69). Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

7 Sequenal Dynamc Classfcaon Usng Laen Varable Models 42 The poseror dsrbuon of he sae varable may herefore be rewren as pw D ) pη, w D ) dη pη, w D )py η ) dη pw η, D )pη D )py η ) dη pw η, D )pη D ) dη. 73) The frs wo momens of pη D ) are gven n Equaon 72). The mean and covarance of pw η, D ) are esmaed usng lnear Bayesan esmaon [5], and are gven by Êw η, D ) ŵ + PT h S η η ), 74) Varw ˆ η, D ) P PT h h T S P. 75) Usng hese frs wo momens of pw η, D ) and pη D ), we can fnally compue he poseror mean and covarance of w as follows: and ŵ Ew D ) E[Ew η, D ) D ] [ E ŵ + PT h ] η ˆη ) S D P Varw D ) ŵ + PT h S Eη D ) ˆη ) ŵ + PT h S ˆη ˆη ) 76) Var[Ew η, D ) D ]+E[Varw η, D ) D ] [ Var ŵ + PT h ] η ˆη ) S D [ + E P PT h ] h T S P D PT h S ) 2 ht P Varη D ) + P PT h h T S P P PT h h T S P S ). 77) S 4.3. Dynamc classfcaon usng he dynamc bnomal model The oal number of successes y n n ndependen expermens follows a bnomal dsrbuon wh probably of success π. The bnomal dsrbuon s a member of he exponenal famly defned by py n, π) n y) π y π) n y, 78) where π, n {,, 2,...} and y {,, 2,...,n}. We can rearrange he above densy funcon n he followng way: [ y py n, π) exp n n log + log ) )] π log π π n y)). 79) Comparng hs wh he general form of dsrbuons n he exponenal famly, ) dy)θ bθ) py θ,φ) exp + cy, φ), 8) aφ) we can readly fnd ha θ logπ/ π)), aφ) /n, bθ) log + expθ)), cy, φ) log y n ) and dy) y/n. The mean and varance of he bnomal varable are y ) E n, π μ π, 8) n Var y ) n, π n π π). 82) n For he bnomal dsrbuon, he canoncal lnk, whch makes he lnear predcor η equal o he canoncal parameer, θ,she log funcon he nverse of he logsc funcon) defned by ) p logp) log. 83) p To solve he dynamc bnary classfcaon problem we consder he dynamc bnomal model n whch boh y and π n Equaon 78) are funcons of me. The lnk equaon s defned by gμ ) η h T w. 84) In addon, we assume ha he sae varable w evolves accordng o a random walk w w + v, 85) where v,q I). Suppose ha we have he poseror mean and covarance of w, gven a se of observaons D {y,...,y }, Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

8 422 S. M. Lee and S. J. Robers denoed by ŵ and P, respecvely. Owng o he lnear sae process, he pror dsrbuon of he sae varable s where w D ŵ, P ), 86) ŵ ŵ, 87) P P + q I, 88) where, as before, q s esmaed va Equaons 23) and 24). The mean and varance of he pror dsrbuon of he lnear predcor can be easly compued by ˆη h T ŵ, 89) r h T P h. 9) To derve he one-sep ahead forecas dsrbuon gven by py D ) py θ )pθ D ) dθ, 9) we need o specfy he conjugae pror, pθ D ). Usng he dervaon gven n Appendx A., we fnd ha he pror conjugae dsrbuon s Ɣm ) expk θ ) pθ D ) Ɣk )Ɣm k ) + expθ )), 92) m and s frs wo momens are approxmaely ) k Eθ D ) log, 93) m k Varθ D ) +. 94) k m k We know ha he mean and varance of he canoncal parameer are equvalen o ha of he lnear predcor because a canoncal lnk s used. Hence, we can calculae he values of he hyperparameers as follows: k r + exp ˆη )), 95) m r + exp ˆη )) + exp ˆη )). 96) Wh hese hyperparameer values, he one-sep ahead forecas dsrbuon can be analycally obaned as py D ) py θ )pθ D ) dθ, Ɣm ) Ɣk )Ɣm k ) Ɣk + y )Ɣm k + n y ) Ɣm + n) n. 97) y) A full dervaon s gven n Appendx A.2. Ths s a beabnomal dsrbuon wh parameers k and m k, and accordngly he mean and varance of he forecas dsrbuon are Ey D ) nk m, 98) Vary D ) nk m k )m + n) m 2. 99) + m ) Afer observng a new daum y, we updae he dsrbuons of θ and η. The poseror dsrbuon of θ s of he form pθ D ) pθ D )py D ) exp k + y )θ m + n) log + expθ ))). ) The approxmae values of he mean and varance of he poseror dsrbuon are ) k + y Eθ D ) log, ) m k + n y Varθ D ) +. 2) k + y m k + n y They are equvalen o he mean and varance of he poseror dsrbuon of η, whch are represened by ˆη and r, respecvely. Fnally, he poseror mean and covarance of w are ŵ ŵ + PT h ˆη ˆη ), r 3) P P PT h h T r P r ). r 4) The dynamc classfcaon problem can be solved by a specal case of he dynamc bnomal model n whch he number of rals, n, s se o and he number of success, y, s eher or. Hence, he expeced value of he observaon, Ey ),s equal o he success probably, π. In he alernae approach usng nonlnear Kalman flers, as dscussed n Secons 2.2. and 3.2, he class or success) probably π s drecly predced from npus va he logsc funcon. However, n he dynamc bnomal model approach s predced n a full Bayesan manner, raher han he requse lnk funcons beng approxmaed, as s he case wh he EKF and he UKF dealed earler. Ths allows us o work wh he explc dsrbuon form of he class varable and s conjugae pror dsrbuon. The drawback s however, ha we mpose a lnear model o descrbe he dynamcs of he suffcen sascs of he dsrbuons. We compare he performances of hese mehods laer n hs paper. 5. RESULTS In hs secon we nvesgae he performance of hree dynamc classfcaon models developed n hs paper: ) Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

9 Sequenal Dynamc Classfcaon Usng Laen Varable Models 423 dynamc logsc regresson usng he EKF, ) dynamc logsc regresson usng he UKF and ) he dynamc bnomal model. We refer o hese as DLR-E, DLR-U and DBM, respecvely. As we are ulmaely neresed n makng classfcaons sequenally, no only when npu and label nformaon s compleely gven, bu also when par of he nformaon s mssng or ncomplee, and expermens nvesgang he performance of he models o mssng labels and sensor daa are repored here. Fnally, we develop a mehod ha allows an adapve classfcaon model o acvely reques labels. Ths s of mporance n domans n whch obanng labels connuously may be oo cosly. 5.. Synhec daa As a smple example of he approaches, we consder wo overlappng Gaussan dsrbuons roang n a crcular fashon around a cenral pon a [, ], wh he wo dsrbuons ou of phase by π radans. Targe labels are nerleaved,.e. {,,,,...} Fg. ). Three daa ses are consruced wh dfferen Bayes errors Fg. 2). The daa are as presened n [2]. Owng o he nonsaonary naure of he daa any sac classfer fals o classfy he daa. In hese example expermens he kernel 2 Tme sep label: label: h 2 2 Tme sep 5 npu vecor s se o ϕh ) {, 5) h}.e. wh no nonlnear bass funcons, because we know ha he underlyng boundary s lnear. The vecor h represens an npu a me. The sae evoluon nose varance q s se o., and for DLR-E and DLR-U he observaon nose varance r s adapvely esmaed usng he scheme advocaed by [, 2, ]. We compued one-sep ahead predcons of he class probably.e. ˆπ ) wh all hree adapve classfers usng all observaons boh npu and label) up o me. The predced label class ŷ s deermned wh a fxed decson hreshold se o.5,.e. f ˆπ >.5, hen ŷ sor oherwse. Comparng he predced labels wh he rue labels, all hree dynamc classfers acheved performances near he Bayes errors of each daa se Table ). A sac classfer s unable o separae he daa and acheves a performance close o random a.5 as expeced Mssng labels In pracce we may only parally observe he class decson nformaon.e. some labels are mssng. We requre, label: label: h 2 2 Tme sep label: label: Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 h h h FIGURE. Example daa: wo Gaussan dsrbuons roae around a cenral poson over me. a) % Bayes error b) 4% Bayes error c) label: label: label: label: 22% Bayes error label: label: h 2 h h h h FIGURE 2. Example daa: roang Gaussan daa ses wh dfferng Bayes errors: a) %, b) 4% and c) 22%. The Compuer Journal, Vol. 53 No. 9, 2

10 424 S. M. Lee and S. J. Robers TABLE. Proporon of correc classfcaons for he hree dynamc classfcaon approaches. Bayes error Classfer % 4% 22% DLR-E DLR-U DBM neverheless, o make a decson forecas. Bayesan dynamc models can cope easly wh mssng observaons. When class labels are no known, we may nfer he mssng label va he predcve dsrbuon of he dynamc model. In hs case we have ỹ Pry h ) ˆπ, 6) whch we may use as a quas-arge n place of a rue arge label y. Ths approach allows us no o dscard nformaon conaned whn he npu vecor h, such as slow drf, by dscardng he enre observaon even. However, as Lowne e al. [2] poned ou, reang he quas-arges as f hey were rue labels runs he rsk of a classfer becomng excessvely confden n s predcons. To compensae for hs llusory knowledge when exernal labelled feedback s unavalable, we may augmen he sae varable dffuson process usng he uncerany n he label. If he laer s fully observed, hen hs uncerany s zero. If we rely on mpung he mssng label, hen we have q q +ˆπ ˆπ ), 7) for he dynamc logsc regresson-based algorhms, and q q + Vary D ), 8) for he dynamc bnomal model. The quany q s he fxed value for he saonary sae nose varance and Vary D ) s he varance of he one-sep ahead forecasng dsrbuon gven n Equaon 99). We hereby ensure ha he classfers do no become overly confden oo quckly based on fcous feedback. We carred ou expermens n whch label nformaon s successvely removed, a random, from o % whn each daa sream. The performances of he classfers were evaluaed over runs Fg. 3). We can see ha he classfcaon performances of he models dd no worsen rapdly wh he fracon of unobserved labels. Performances wh only 5% labellng were degraded by.3.3% for he daa wh % Bayes error, by.99.7% for he daa wh 4% Bayes error and by.6 3.8% for he daa wh 22% Bayes error. Wh only 2% class labels avalable he classfers performances were degraded by only.5 5.2% n comparson wh he % labellng case. Ths resul ndcaes ha he adapve classfers can deal effecvely wh sparsely observed labels whle mananng good classfcaon performance Mounan fre scenaro We consder n hs secon a more realsc scenaro n whch here s a fores fre on a wooded mounan wh several vllages n he nearby regon, as schemacally depced n Fg. 4. To allocae lmed fre-fghng resources effecvely we mus classfy each of he vllages as beng n poenal danger or no based on local weaher condons measured by a se of weaher sensors. For example, consder he vllage n he boom lef of he fgure wh a rng). The sronger he wnd speed and he hgher he ar emperaure, he more lkely s for a vllage o be n danger. In addon, unless he wnd blows n a norherly drecon, he vllage s more vulnerable o he spread of fre. Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 a) % Bayes error b) 4% Bayes error c).8 22% Bayes error Fracon correc classfcaon DLR E DLR U DBM Probably of label observaon Fracon correc classfcaon DLR E DLR U DBM Probably of label observaon Fracon correc classfcaon.7.6 DLR E DLR U DBM Probably of label observaon FIGURE 3. Mssng label expermens: he fracons of observed labels ranged from close o o % for he daa ses wh a) %, b) 4% and c) 22% Bayes errors. The classfers mananed good classfcaon performances wh up o 7% of mssng labels. The Compuer Journal, Vol. 53 No. 9, 2

11 Sequenal Dynamc Classfcaon Usng Laen Varable Models 425 TABLE 2. Mounan fre scenaro: correc classfcaon proporons. All hree adapve models ouperformed a sac logsc regresson. Classfer Classfcaon accuracy Logsc regresson.739 DLR-E.9636 DLR-U.933 DBM.9623 FIGURE 4. Mounan fre scenaro: a fre sars n a wooded mounan wh fve vllages denoed by dos), whch are n poenal danger accordng o local weaher condons. Wh weaher daa colleced by a nework of weaher sensors [4], we creaed a realsc daa se conssng of hree npu varables wnd speed, wnd drecon and ar emperaure) and a bnary class varable danger or no danger ). The laer was obaned by runnng a smple dffusve fre spread model wh he ansoropc dffuson kernel beng a funcon of wnd drecon and wnd speed. The manner n whch npus and arge labels of he daa are dsrbued means ha a sac classfer s unable o separae he wo classes wh hgh accuracy see Fg. 5a). The rue labels alernae beween danger and no danger a rregular nervals as he weaher varables change Fg. 5b). We employed hree nonlnear Gaussan kernels o form he exended bass se ψ as n Equaon 4, locaed a random whn he daa space hs number was no opmzed and adapon of he locaon and number of bass funcons remans a curren research focus). For he sake of comparson, we ran a sandard logsc regresson model, wh npus beng he exended bass se, ψ; npu oupu bass arge label) pars were randomly sampled whou replacemen from he full daa se o use as a es se needed as hs baselne sac algorhm s non-sequenal). The performance of hs sac model was averaged over runs of dfferen ranng daa samples. The hree adapve classfers sgnfcanly ouperformed he sac classfer, achevng an mprovemen of beween 9.2 and 22.5% Table 2). Furhermore, when label nformaon s successvely randomly removed from he observed daa, he classfcaon performances of he models degraded slowly as he proporon of mssng labels ncreased Fg. 6) Acve label requesng In many scenaros we can conceve of suaons n whch rue class labels are cosly o oban. In hs secon we propose a mechansm for acvely requesng a label whereby he adapve classfer decdes wheher o reques a label or wheher o make mssng value nference. Ths problem s closely relaed o he problem of acve daa selecon [5, 6], as boh mehods ulmaely use uncerany as a gudelne o enable requess of nformaon. Consder a each me sep a classfer makng a one-sep ahead label predcon. If he uncerany assocaed Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 a) hgh Label: Label: b) Ar emperaure Labels low 36 Wnd drecon slow Wnd speed srong FIGURE 5. Mounan fre scenaro: a) a 3D plo of he hree npu varables and he class varable. b) A me plo of he class varable. The rue labels alernae beween danger and no danger a rregular nervals as he weaher varables change. Noe ha we deal wh a sngle vllage locaed a he lower lef n Fg. 4. The Compuer Journal, Vol. 53 No. 9, 2

12 426 S. M. Lee and S. J. Robers Fracon correc classfcaon.9.8 Mounan fre daa DLR E DLR U DBM Probably of label observaon FIGURE 6. Mounan fre scenaro: classfcaon performance of adapve models as a funcon of he fracon of observed labels. Noe ha he classfcaon performances of he models dd no degrade n proporon o he number of unobserved labels. wh hs predcon s hgher han a pre-defned hreshold, hen requess a label, oherwse, proceeds whou labellng. The hreshold s nmaely lnked o he cos assocaed wh obanng a label and he cos assocaed wh erang he model wh an mpued label and makng a wrong decson. If hese coss are known hen hs scheme allows for an adapve mnmum cos decson process. We presen proof of concep resuls n hs paper and more full exposon of hs approach s relegaed o fuure work. For dynamc logsc classfers, he predcon uncerany s he varance of he assocaed Bernoull dsrbuon, namely ˆπ ˆπ ). For he dynamc generalzed lnear bnomal model s he varance of a bea-bnomal predcve dsrbuon, Vary D ), as gven n Equaon 99). By varyng he uncerany hreshold whch he algorhms use o reques labels, we may oban performance measures for dfferen proporons of requesed labels. We carred ou such a se of expermens on he mounan fre daa se, leadng o dfferen proporons of labellng from o approxmaely 3% Table 3). Wh 5% requesed labellng, TABLE 3. Acve label requesng: proporon of correc classfcaons on he mounan fre daa when he classfers acvely reques labels. Labellng DLR-E DLR-U DBM % % % % % for example, classfcaon accuracy was degraded by.2 4.5% Fg. 7). In hs fgure he grey regons represen me seps when a classfer requesed a label. We can see ha when he models were confden enough abou her predcons hey dd no reques a label. Noe ha despe he relavely low overall classfcaon error, he dynamc bnomal model faled o deec a change of label on wo occasons, and he dynamc logsc regresson usng he EKF on one occason. We noe ha our models for acve selecon do no a presen nclude measures of wde-sense correlaon beween requesed samples, n ha hey look nsananeously a he expeced uncerany and use hs as a creron for acve requess. I s possble ha, for some applcaons, a Markov-syle look back or a mulsep reques look-forward may offer an mprovemen. Ths s an area of acve research Mssng npus We fnally consder he scenaro n whch no only labels bu also npu sensor daa can be mssng, for example, as a resul of a falure n he sensors or n ransmng subsequen nformaon. Our decson models are defned by for he DLR models, or π lϕh ) T w ), 9) y Bnomal wh η ϕh ) T w, ) for he DBM. Here l ) s he logsc funcon gven n Equaon 3) and η s he lnk equaon gven n Equaon 53). When he npu, h, s no avalable, we can mpue as ĥ.wh a se of observed npus up o me,.e. {h,...,h }, we can predc he mssng npu usng, for example, a Gaussan Process model [7] or a nonsaonary auoregressve model, such as he sequenal Bayesan mulvarae auoregressve model dealed n [8]. The daa sreams of he hree npus n he mounan fre scenaro.e. wnd speed, wnd drecon, ar emperaure) are llusraed n Fg. 8a. We removed a random 7% of he npu daa Fg. 8b) and predced he mssng npu values wh a sequenal dynamc mulvarae auoregressve model [8] of order 2 Fg. 8c). We can see ha he predced npus are very close o he rue ones. To compensae for usng hese predced npus, we fold he uncerany n he predced values of mssng npus denoe by u ) no he sae nose. We herefore updae q usng q q + u. ) The quany q s he consan value for he saonary nose varance as prevously defned. To evaluae how he adapve classfers perform wh mssng npus, we randomly removed npu nformaon from he mounan fre daa se. The performances of he hree adapve classfers were evaluaed over runs Fg. 9). We noe n parcular he performance of he DLR-E: wh only 4% of npus avalable sll classfed labels wh a more han 9% accuracy. Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

13 Sequenal Dynamc Classfcaon Usng Laen Varable Models 427 a) Dynamc logsc regresson usng EKF Class probably.5 b) Dynamc logsc regresson usng UKF Class probably.5 c) Dynamc bnomal model Class probably FIGURE 7. Acve label requesng on he mounan fre daa when 5% of oal labels s requesed by a) DLR-E, b) DLR-U and c) DBM. The one-sep ahead predcons for he class probably are represened by he nnermos of he hree races n each subplo, he label predcons by he mddle races and he observed labels by he ouermos races. The grey shaded regons represen me seps when he classfers requesed a label. Combned wh acve label requesng, s possble for a classfer o decde whch label o reques even when npu sensor nformaon s no avalable. We carred ou an expermen n whch 5% of npus are mssng and he classfers requess a) True npus b) Gven npus c) 3 25 Wnd Speed Wnd Drecon Ar Temperaure % of labellng. We compued he average proporon of correc classfcaon and he correspondng sandard devaon for he hree adapve classfers over runs:.8846/.229 DLR-E),.7344/.22 DLR-U) and.8274/.84 DBM). Wnd Speed Wnd Drecon Ar Temperaure 3 25 Predced npus Wnd Speed Wnd Drecon Ar Temperaure Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, FIGURE 8. Tme plo of a) rue npus, b) gven npus 7% mssng) and c) predced npus. A dynamc mulvarae auoregressve model of AR order 2 was used. The Compuer Journal, Vol. 53 No. 9, 2

14 428 S. M. Lee and S. J. Robers Fracon correc classfcaon DLR E DLR U DBM Probably of Inpu observaon FIGURE 9. Mssng sensor daa expermen on he mounan fre daa. The performance of he DLR-E model, n parcular, degraded slowly as he fracon of npus mssng ncreased. In comparson o correspondng resuls when gven complee npu nformaon provded n Table 3, we can see ha for DLR-E classfcaon accuracy was degraded by only 5%. 6. CONCLUSIONS AND FUTURE WORK 6.. Conclusons We have demonsraed how a classfcaon decson can be made sequenally n hghly nonlnear, nonsaonary and ncomplee envronmens. We have suded wo dfferen approaches o he problem: ) dynamc logsc regresson usng nonlnear varans of he Kalman fler and ) a dynamc generalzed lnear model. The laer model provdes a mahemacally elegan Bayesan framework for dynamc models so long as he dsrbuon of an observaon varable s a member of he exponenal famly. Snce n bnary classfcaon an observaon varable has a smple Bernoull dsrbuon, a dynamc bnomal model has been developed as an adapve bnary classfer. All hree models acheved consderably beer performance compared wh sac fxed parameer) classfcaon for boh he smple roang daa example and he mounan fre scenaro. As he models are developed n a fully Bayesan framework, handlng mssng labels and npu sensor) daa can be managed. The performance of all algorhms dealed n hs paper dd no degrade rapdly as he amoun on unobservable daa ncreased. Labels, and ndeed sensor observaons, can be acvely requesed by all he algorhms f he uncerany and hence cos) of makng a predcve decson s hgh. We noe ha he approaches dscussed n hs paper may be naurally exended o mulclass decson makng and for mulsep decson forecasng. Along wh he co-ordnaon of sequenal adapve decsons from mulple auonomous agens, hese avenues of research consue acve ongong research. FUNDING Ths work was underaken as par of he ALADDIN Auonomous Learnng Agens for Decenralsed Daa and Informaon Sysems) projec and was jonly funded by a BAE Sysems and EPSRC Engneerng and Physcal Research Councl) sraegc parnershp. REFERENCES [] Penny, W.D. and Robers, S.J. 999) Dynamc Logsc Regresson. In. Jon Conf. Neural Neworks 999 IJCNN 99), Washngon, DC, USA. [2] Lowne, D.R., Robers, S.J. and Garne, R. 2) Sequenal nonsaonary dynamc classfcaon. Paern Recogn., 43, [3] Juler, S.J. and Uhlmann, J.K. 997) A New Exenson of he Kalman Fler o Nonlnear Sysems. Proc. AeroSense: h In. Symp. Aerosp./Defense Sensng, Smula. Conr., Orlando, FL, USA. [4] Juler, S.J. and Uhlmann, J.K. 24) Unscened flerng and nonlnear esmaon. Proc. IEEE, 92, [5] Wes, M. and Harrson, J. 997) Bayesan Forecasng and Dynamc Models. Sprnger. [6] Kuncheva, L. 24) Classfer Ensembles for Changng Envronmens. Lecure Noes n Compuer Scence, Vol. 377, pp. 5. Sprnger. [7] Koler, J.Z. and Maloof, M.A. 23) Dynamc Weghed Majory: A New Ensemble Mehod for Trackng Concep Drf. Proc. 3rd IEEE In. Conf. Daa Mnng ICDM 3), Washngon, DC, USA, p. 23. IEEE Compuer Socey. [8] Garne, R. and Robers, S. 28) Learnng from Daa Sreams wh Concep Drf. Techncal Repor, Paern Analyss Research Group, Unversy of Oxford. [9] MacKay, D.J.C. 992) A praccal Bayesan framework for backpropagaon neworks. Neural Compu., 4, [] Jazwnsk, A.H. 97) Sochasc Processes and Flerng Theory. Academc Press, New York. [] Wan, E.A. and van der Merwe, R. 2) The Unscened Kalman Fler. In Haykn, S. ed.), Kalman Flerng and Neural Neworks, pp Wley. [2] Juler, S.J. and Uhlmann, J.K. 996) A General Mehod for Approxmang Nonlnear Transformaons of Probably Dsrbuons. Techncal Repor, RRG, Deparmen of Engneerng Scence, Unversy of Oxford. [3] McCullagh, P. and Nelder, J.A. 989) Generalzed Lnear Models. Chapman & Hall/CRC. [4] Chan, T.Y.-K. 2) CHIMET: weaher repors from Chcheser bar. hp:// accessed November 29). [5] MacKay, D. 992) Informaon-based objecve funcons for acve daa selecon. Neural Compu., 4, [6] Osborne, M., Robers, S., Rogers, A., Ramchurn, S. and Jennngs, N. 28) Informaon Agens for Pervasve Sensor Neworks. Proc. PerSens-8, Hong Kong. [7] Rasmussen, C.E. and Wllams, C.K.I. 26) Gaussan Processes for Machne Learnng. Sprnger. Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2

15 Sequenal Dynamc Classfcaon Usng Laen Varable Models 429 [8] Lee, S.M. and Robers, S.J. 28) Tme Seres Forecasng Usng he EM Algorhm n Incomplee Envronmens. Techncal Repor, Paern Analyss Research Group, Unversy of Oxford. [9] Abramovz, M. and Segun, I.A. 965) Handbook of Mahemacal Funcons. Dover, New York. APPENDIX: DERIVATIONS A.. Conjugae pror dsrbuon for he bnomal dsrbuon The conjugae pror for he bnomal dsrbuon s of he form pθ D ) ωk,m ) expk θ m log + expθ ))). A.) Usng θ logπ /l π )), he probably densy funcon of π s of form ) pπ D ) p θ D π ) d π log dπ π ωk,m )π k π ) m k. A.2) Ths s a bea dsrbuon wh parameers k and m k, hereby we know ha ωk,m ) Ɣm )/Ɣk )Ɣm k ). The mean and varance of he conjugae pror are compued n he followng way. The momen generang funcon of θ s M θ z) Eexpzθ )) expzθ )pθ D ) dθ Ɣk + z)ɣm k z), A.3) Ɣk )Ɣm k ) and hus he cumulan generang funcon s C θ z) logm θ z)) log Ɣk + z) log Ɣm k z) Ɣk ) Ɣm k ). A.4) From he cumulan generang funcon he wo momens of he conjugae pror can be compued: Eθ D ) d dz C θ z) z ψk ) ψm k ) log k m k ), A.5) Varθ D ) d2 dz C 2 θ z) z ψ k ) + ψ m k ) k + m k. The funcon ψ ) denoes he dgamma funcon defned by ψx) d dx log Ɣx) Ɣ x) Ɣx). A.6) A.7) The dgamma funcon and s dervave are approxmaed by logx) and /x, respecvely, [9]. A.2. One-sep forecas dsrbuon for he bnomal dsrbuon When he observaon varable Y s a bnomal, he one-sep ahead forecas dsrbuon s compued as follows: py D ) py θ )pθ D ) dθ, [ y ] exp n n log + expθ )) )) n + log y Ɣm ) expk θ ) dθ Ɣk )Ɣm k ) + expθ )) m, Ɣm ) Ɣk )Ɣm k ) ) n y expk + y )θ ) + expθ )) dθ m +n Ɣm ) Ɣk )Ɣm k ) Ɣk + y )Ɣm k + n y ) Ɣm + n) n y ). A.8) Downloaded from comjnl.oxfordjournals.org a Oxford Unversy on June 9, 2 The Compuer Journal, Vol. 53 No. 9, 2