Bounds on the Probability of Misclassification among Hidden Markov Models

Size: px

Start display at page:

Download "Bounds on the Probability of Misclassification among Hidden Markov Models"

Mildred Ball
6 years ago
Views:

1 20 50th IEEE Confrnc on Dcision and Contro and Europan Contro Confrnc (CDC-ECC) Orando FL USA Dcmr Bounds on th Proaiity of Miscassification among Hiddn Markov Mods Christoforos Krogou and Christoforos N Hadjicostis Astract Givn a squnc of osrvations cassification among two known hiddn Markov mods (HMMs) can accompishd with a cassifir that minimizs th proaiity of rror (i th proaiity of miscassification) y nforcing th maximum a postriori proaiity (MAP) ru For this MAP cassifir w ar intrstd in assssing th a priori proaiity of rror (for any osrvations ar mad) somthing that can otaind (as a function of th ngth of th squnc of osrvations) y summing up th proaiity of rror ovr a possi osrvation squncs of th givn ngth To avoid th high compxity of computing th xact proaiity of rror w dvis tchniqus for mrging diffrnt osrvation squncs and otain corrsponding uppr ounds y summing up th proaiitis of rror ovr th mrgd squncs W show that if on mpoys a dtrministic finit automaton (DFA) to captur th mrging of diffrnt squncs of osrvations (of th sam ngth) thn Markov chain thory can usd to fficinty dtrmin a corrsponding uppr ound on th proaiity of miscassification Th rsut is a cass of uppr ounds that can computd with poynomia compxity in th siz of th two HMMs and th siz of th DFA Indx Trms hiddn Markov mod proaiity of rror cassification proaiistic diagnosis stochastic diagnosr I INTRODUCTION W considr cassification among systms that can modd as hiddn Markov mods (HMMs) Givn a squnc of osrvations that is gnratd y undrying (unknown) activity in on of two known HMMs w anayz th prformanc of th MAP cassifir which minimizs th proaiity of miscassification [] y charactrizing th a priori proaiity of rror i th proaiity of rror for any osrvations ar mad Th prcis cacuation of th proaiity of rror (for squncs of osrvations of a givn finit ngth) is a cominatoria task of high compxity (typicay xponntia in th ngth of th squncs) In this papr w circumvnt this prom y focusing on otaining uppr ounds on th proaiity of miscassification In particuar w mpoy finit automata to mrg squncs of osrvations of th sam ngth in diffrnt ways; cacuating in ach cas an uppr ound on th proaiity of miscassification y summing up th individua proaiitis of miscassification ovr th mrgd squncs This matria is asd upon work supportd in part y th Europan Community (EC) 7 th Framwork Programm (FP7/ ) undr grants INFSO-ICT and PIRG02-GA Any opinions findings and concusions or rcommndations xprssd in this puication ar thos of th authors and do not ncssariy rfct th viws of EC Th authors ar with th Dpartmnt of Ectrica and Computr Enginring Univrsity of Cyprus Nicosia Cyprus E-mais: {krogouchristoforos chadjic}@ucyaccy Our anaysis and ounds can find appication in many aras whr HMMs ar usd incuding spch rcognition [2] [3] [4] pattrn rcognition [5] ioinformatics [6] [7] and faiur diagnosis in discrt vnt systms [] [8] [9] Our work aso rats to approachs daing with th distanc or dissimiarity twn two HMMs [0] [] [2] and th construction w dvis to otain our ounds ncompasss th concpt of a stochastic diagnosr [9] Dircty ratd prvious work can found in [] which introducs an uppr ound on th proaiity of miscassification appica to th cas whn th two HMMs hav diffrnt anguags Mor spcificay givn two mods S () and S (2) with anguags L(S () ) and L(S (2) ) rspctivy [] otains an uppr ound on th proaiity of miscassification y focusing on th proaiity of strings in L(S () ) L(S (2) ) or L(S (2) ) L(S () ) Undr crtain conditions (which rquir among othr things that L(S () ) L(S (2) )) this ound tnds to zro xponntiay with th numr of osrvation stps Th contriution of this papr is th charactrization of a cass of uppr ounds on th a priori proaiity of rror whn cassifying among two known HMMs that may not ncssariy hav diffrnt anguags By introducing an appropriat dtrministic finit automaton (DFA) w systmaticay mrg diffrnt squncs of th sam ngth in a way that aows asy computation of an uppr ound on th proaiity of miscassification In particuar for squncs of osrvations of a givn ngth n our ounds can otaind with inar compxity in n which shoud contrastd against th gnray xponntia compxity in n for otaining th xact proaiity of rror Our approach aso aows us to us Markov chain thory to otain an uppr ound for asymptoticay arg n (in a cass th approach has compxity poynomia in th siz of th two givn HMMs and th siz of th DFA that is usd) II NOTATION AND BACKGROUND An HMM is dscrid y a fiv-tup (Q E Λ π 0 ) whr Q {q q 2 q Q } is th finit st of stats; E { 2 E } is th finit st of outputs; : Q Q [0 ] capturs th stat transition proaiitis; Λ : Q E Q [0 ] capturs th output proaiitis associatd with transitions; π 0 is th initia stat proaiity distriution vctor For q q Q and σ E th stat Th anguag of an HMM consists of th st of a finit ngth squncs of outputs (osrvations) that can gnratd y th HMM starting from a vaid initia stat //$ IEEE 385

2 transition proaiitis ar dfind as (q q ) P (q[n + ] q q[n] q) and th output proaiitis associatd with transitions ar givn y Λ(q σ q ) P (q[n + ] q E[n + ] σ q[n] q) whr q[n] (E[n]) is th stat (output/osrvation) of th HMM at tim stp n Th output function Λ dscris th conditiona proaiity of osrving th output σ associatd with th transition to stat q from stat q Th stat transition function nds to satisfy Q (q q ) σ E Λ(q σ q ) () and aso (q q i ) q Q i W dfin th Q Q matrix A σ associatd with output σ E of th HMM as foows: th (k j) th ntry of A σ capturs th proaiity of a transition from stat q j to stat q k that producs output σ i A σ (k j) Λ(q j σ q k ) Not that A σ E A σ is a coumn stochastic matrix whos (k j) th ntry dnots th proaiity of taking a transition from stat q j to stat q k without rgard to th output producd i A(k j) (q j q k ) Suppos that w ar givn two HMMs capturd y S () (Q () E () () Λ () π () 0 ) and S(2) (Q (2) E (2) (2) Λ (2) π (2) 0 ) with prior proaiitis for ach mod givn y P and P 2 P rspctivy Givn E (j) { (j) (j) 2 (j) } j { 2} for th E (j) two HMMs w dfin E E () E (2) with E { 2 E } and t i th transition matrix for S (j) j { 2} undr th output symo i E W st i to zro if i E E (j) If w osrv a squnc of n outputs Y n y[] y[2] y[n] y[i] E that is gnratd y on of th two undrying HMMs th cassifir that minimizs th proaiity of rror nds to impmnt th maximum a postriori proaiity (MAP) ru Spcificay th MAP cassifir compars P (S () Y n ) > < P (S (2) Y n ) P (Y n S () ) P (Y n S(2) ) > < P 2 P and dcids in favor of S () (S (2) ) if th ft (right) quantity is argr It is ovious that whn w dcid in favor of on or th othr mod thn w hav proaiity of rror proportiona to th proaiity of th mod that was not sctd With som agra it can shown that P (rror Y n ) min{p P (Y n S () ) P 2 P (Y n S (2) )} Cary if E () E (2) and at ast on symo y[i] is uniqu to S () (i y[i] E E (2) ) or to S (2) (i y[i] E E () ) thn w wi choos th mod with nonzro proaiity of rror (assuming th squnc of osrvations was indd gnratd y on of th two mods) and wi mak an rror with zro proaiity III PROBABILITY OF MISCLASSIFICATION Stp Proaiity of Miscassification To cacuat th a priori proaiity of rror for th squnc of osrvations of ngth n is osrvd w nd to considr a possi osrvation squncs of ngth n so that P (rror at n) P (rror Y n ) (2) Y n En whr E n is th st of a squncs of ngth n with outputs from E (som of ths squncs may hav zro proaiity undr on of th two mods or vn oth mods) W aritrariy indx ach of th d n (d E ) squncs of osrvations via Y (i) i { 2 d n } and us P (j) i to dnot P (j) i P (Y (i) S (j) ) Th proaiity of miscassification twn th two systms aftr n stps can thn xprssd as P (rror at n) d n i d n i P (rror Y (i)) min{p P () i P 2 P (2) i } (3) W can cacuat P (j) i P (Y (i) S (j) ) with an itrativ agorithm a dscription of which can found in [] For squnc Y n y[] y[2] y[n] w cacuat ρ (j) n y[n] A(j) y[n ] A(j) y[] π(j) 0 which is ssntiay a vctor whos k th ntry capturs th proaiity of raching stat q k Q (j) whi gnrating th squnc of outputs Y n (i ρ (j) n (k) w P (q[n] q k Y n )) If w sum up th ntris of ρ (j) n otain P (j) Y P (Y n n S (j) ) Q (j) k ρ(j) n (k) W can otain th proaiity of rror ovr a squncs of n osrvations y cacuating and comparing th ρ (j) n j 2 for a possi squncs of osrvations of ngth n W can arrang th computations in trms of two d-ary trs of dpth n as shown in Fig Each nod at v L rprsnts ρ (j) L j 2 aftr a spcific squnc (of xacty L) osrvations has n sn For ach nod at v L w crat d chid-nods and w rpat this procdur unti having n-vs in th tr Onc w xpand ths trs to n-vs ach of th d n af nods corrsponds to a uniqu squnc of ngth n which in th worst cas scnario can producd y oth HMM mods W assign to ach af-nod a proaiity of occurring P (j) i P (Y n Y (i) S (j) ) whr j { 2} rprsnts th mod and i { 2 d n } corrsponds to th indx of ach squnc of n osrvations Examp : Suppos w ar givn th two HMMs shown in Fig 2 with E () E (2) E { } π () 0 π (2) 0 [ 0 ] T and P P 2 05 Th corrsponding A () A () as foows:»» A () A () A(2) A (2) ar 386

3 L 0 L L n S d vnts S 2 a) ) d vnts P P 2 P d n P d n 2 Fig Computation on two d-ary trs of dpth n on for S () and on for S (2)» A (2) A (2)» Fig 2 S () (ft) and S (2) (right) in Examp If th squnc Y () aa is osrvd w hav P () Q () k ρ () 4 (k) 005 whr ρ() 4 A () a A () A () a A () π () 0 and P (2) A (2) a A (2) Q (2) k A (2) a A (2) π (2) 0 ρ (2) 4 (k) 095 whr ρ(2) 4 Thus th proaiity of rror twn th two mods if this spcific squnc is osrvd is P (rror Y ()) 0025 Stp 2 Uppr Bound for Proaiity of Error If w hav two squncs Y () and Y (2) of ngth n w can otain an uppr ound on th proaiity of rror for ths squncs as foows: P (rror {Y () Y (2)}) 2 i min{p min{p P () i P 2 P (2) i } 2 i P () i P 2 2 i P (2) i } (4) Th aov can shown asiy y considring th diffrnt cass and osrving that min{a a 2 } + min{ 2 } min{a + a } W can asiy gnraiz th aov discussion to any numr of mrgd squncs of th sam ngth Th nxt stp is to find an uppr ound for th proaiity of rror at n stps In particuar if w tak any partition of th indx st I { 2 d n } into susts D D 2 D m (such that D i D j for i j and m i D i I) thn w hav P (rror at n) d n P (rror Y ()) m min{p P () P 2 P (2) } k D k m min{ P P () P 2 P (2) } k D k D k (5) Stp 3 Cacuation of Uppr Bound via a DFA W now discuss how w can otain a partition of th indx st I via a dtrministic finit automaton (DFA) H with anguag E Th rason w considr this particuar partitioning of I wi com carr atr whn w discuss fficint ways of cacuating th quantitis P P (j) D k j 2 A DFA H is dscrid y a four-tup (X E δ x 0 ) whr X {x x 2 x } is th finit st of stats; E { 2 E } is th finit st of inputs (aphat); δ : X E X is th transition function; and x 0 X is th initia stat For a squnc of vnts s s[n]s[n ]s[] s[i] E i 2 n w dfin δ(q s) δ(δ(δ(q s[]) s[2]) s[n]) A sufficint condition for th rquirmnt that th anguag of H is E is that δ is dfind for a pairs of stats x X and outputs E Considr th foowing susts of squncs of osrvations of ngth n: D k {s E n δ(x 0 s) x k } k 2 It is not hard to argu that D k whr k 2 form a partition of E n For ach E w can construct th inary transition matrix T of H foowing th ru that if δ(x i ) x i thn T (i i) othrwis T (i i) 0 This matrix capturs a possi transitions from a stat to anothr undr vnt ; sinc H is dtrministic T for E is a inary matrix with xacty a sing in ach coumn W can aso dfin th inary coumn vctor π 0 to hav a sing nonzro mnt with vau at its i th ocation if x 0 x i (in othr words π 0 is an indicator vctor for th initia stat of H) With this notation at hand δ(x 0 s) x k for s s[n]s[n ]s[] is quivant to π n T s[n] T s[n ] T s[] }{{} π 0 ing a vctor T s with a zro ntris xcpt a sing at th k th ocation This is asy to staish y induction Mor gnray th ntris of th matrix T s T s[n] T s[n ] T s[] ar such that T s (k i) {0 } and T s (k i) if and ony if δ(x i s) x k If w t th 387

4 two vctors c (j) P j [] of siz Q (j) for j 2 w can show that th proaiity of rror in Eq (5) is smar or qua to min{ k c () A () s π () 0 c (2) A (2) s π (2) whr for s s[n]s[n ]s[] w hav s s[n] A(j) 0 } (6) π (j) 0 s[n ] A(j) s[] π(j) 0 W now discuss how th aov ound can computd rathr fficinty W dfin th matrix T j E 2 whr T dnots th Kronckr product dfind as th ( Q (j) ) ( Q (j) ) matrix T ( ) T (2 ) T ( ) T ( 2)A (j) T ( ) T (2 2) T (2 ) T ( 2) T ( ) Not that ach T (i i) x i x i X is a matrix of siz ( Q (j) ) ( Q (j) ) W aso dfin th (i i) ock of as (B i B i ) ( (j) i : f (j) i (j) i : f (j) i ) i a ( Q j ) ( Q j ) sumatrix starting from row (j) i (i )Q (j) + to row f (j) i iq (j) and from coumn (j) i (i )Q (j) to coumn f (j) i i Q (j) Ltting p (j) 0 π 0 π (j) 0 w can writ2 (for s s[n]s[n ]s[] E n ) p (j) n ( ( ) n p (j) 0 ) n T (π 0 π (j) 0 ) E (T s[n] T s[] )π 0 ( s[n] A(j) s E n k k T s π 0 ρ (j) ns u k ρ (j) ns u k k ρ (j) ns s[] )π(j) 0 whr u k is a coumn vctor of siz with zros on a of its ntris xcpt a sing on at its k th ntry and ρ ns (j) is th vctor ρ (j) n for th squnc of osrvations s If w focus on th k th ock of p (j) n of siz Q (j) (i ntris (k )Q (j) + to kq (j) ) w s that p (j) n (B k ) ρ (j) ns s π (j) 0 Foowing Eqs (5) and th ound in (6) w can writ P (rror at n) min{c () p () n (B k ) c (2) p (2) n (B k )} (7) k 2 On of th proprtis of th Kronckr product is that (A B)(C D) (AC) (BD) for matrics A B C D of appropriat sizs [3] which can usd to comput an uppr ound on th proaiity of rror twn th two systms (S () and S (2) ) y taking advantag of how th DFA H crats th partitions D k k Fig 3 DFA H s for Examp 2 Examp 2: Considr th two HMMs in Fig 2 and th DFA H s in Fig 3 with X { 2 3} anguag E ( + ) and initia stat x 0 (which mans that π 0 [ 0 0] T ) Assum that th priors ar P 06 P 2 04 so that and aso that c () [ ] c (2) [ ] π () 0 [ 0 ] T π (2) 0 [ ] T W crat according to th prvious dfinitions th matrics T T for H s as T T and otain th matrics A () A (2) as A () A (2) Simiary w otain p (j) 0 π 0 π (j) 0 for j 2 as p () 0 [ ] T p (2) 0 [ ] T For a squnc of osrvations of ngth n w can writ P (rror at n) min{c () p () n (B i ) c (2) p (2) n (B i )} i {23} 388

5 whr p (j) n ( ) n π (j) 0 j 2 Th pot of th ound as a function of n is providd in Fig 4 As n coms infinit this ound staiizs at Fig 6 Actua proaiity of rror (continuous in) and uppr ound (dashd in) with th DFA H in Fig 5 Fig 4 Actua proaiity of rror (continuous in) and uppr ound (dashd in) with DFA H s in Fig Fig 5 DFA H in Examp Examp 3: W can xtnd th construction of th prvious xamp to th argr DFA H in Fig 5 with X { 2 5} anguag E ( + ) and initia stat x 0 (which mans that π 0 [ 0 0 0] T ) W omit th dtais of th construction du to spac considrations (ut th stps ar idntica to th stps in Examp 2) Th rsuting uppr ound on th proaiity of rror is pottd in Fig 6 as a function of th numr of osrvations As n w s that this uppr ound tnds to th constant vau 0066 Not that this ound can prhaps rducd y mpoying a DFA with mor stats and/or diffrnt transition functionaity (to try and achiv a ttr partitioning of th st of possi squncs) In this particuar xamp in ordr to find this H w trid a possi DFAs of 5 stats and prsntd th on that asymptoticay rsuts in th ast uppr ound IV CONNECTIONS TO A STOCHASTIC DIAGNOSER W can rduc th numr of stats or vn th siz of a transition sumatrics j 2 for ach mod (S () S (2) ) if w ar a to rmov a stats that ar not racha undr spcific conditions (g unrachaiity from a spcific starting stat) An xamp of such a dtrministic finit automaton was th stochastic diagnosr introducd in [9] for th purpos of faut diagnosis W dscri this connction via th foowing xamp whr w us an appropriat DFA to crat th stochastic diagnosr for th two mods shown in Fig 2 Examp 4 Suppos that th mods in Fig 2 captur th Norma (S ) and Fauty (S 2 ) haviour of a systm Aso w dfin Q () {N 2N} and Q (2) {F 2F } with priors P P 2 05 and initia stats q () 0 {N} q (2) 0 {F } W want to find a transition matrics for th stochastic diagnosr and rat thm to th prvious anaysis (th origina work in [9] uss th transpos of th matrics w us hr) W anayz th systm using th prvious mthod with th ony diffrnc ing that th construction of th matrics A E considrs th havior in ach systm simutanousy g A [ A () 0 0 A (2) ] N 2N F 2F N N F F If w ony kp mnts on nonzro rows and coumns w otain th rducd matrix N F A (s) 2N 0 2F 0 X 0 X 2 0 N 2N F 2F a X 3 N 2N F 2F a Fig 7 Stochastic Diagnosr for S () and S (2) 389

6 Foowing this approach w can crat a possi diffrnt stats and appy th rducd transition matrics Th stochastic diagnosr for our xamp is shown in Fig 7 W can crat th S matrix which incuds a sumatrics according to ach stat {X X 2 X 3 } (g S( 7) capturs th transition proaiity from stat X N to X3 F ) If th stats ar ordrd as foows: stat X N stat 2 X F stat 3 X2 N stat 8 X3 2F th matrix S is givn y S Using S w can comput th uppr ound of th proaiity of rror as in th prvious xamp (using howvr ocks of diffrnt sizs du to th fact that ntris that ar zro in ach ock ar droppd) Atrnativy w can us th automaton shown in Fig 8 and foow th approach in th prvious sction to otain p (j) n ( ) n p (j) 0 Not that y construction a stochastic diagnosr chcks if an output symo is possi or not so that th undrind symos in Fig 8 do not appar in th stochastic diagnosr in Fig 7 For arg n w find th uppr ound to Fig 8 Equivant DFA to Stochastic Diagnosr in Examp 4 A proaiistic finit automaton that is AA-stochasticay diagnosa [9] is ssntiay an automaton for which th proaiity of miscassification 3 gos to zro as th numr of osrvations coms asymptoticay arg It is vidnt that our mthod can usd to staish whthr th proaiity of miscassification gos to zro (y dtrmining whthr its uppr ound gos to zro) using constructions quit distinct from a stochastic diagnosr Thus a sufficint condition for AA-stochastic diagnosaiity woud th xistnc of a DFA that ads to an uppr ound on th 3 Stricty spaking AA-stochastic diagnosaiity is ony concrnd with fauty havior that might considrd as non-fauty (and whthr its proaiity gos to zro as th numr of osrvations incrass); thus on shoud xcud th proaiity of miscassification that ariss from strings gnratd y th non-fauty systm that ar mor iky to hav n gnratd y th fauty systm proaiity of miscassification that gos to zro as th numr of osrvations incrass Rmark: Th compxity of computing th xact proaiity of rror is an xponntia function of n (it is of O(n d n ( Q () 2 + Q (2) 2 ))) In otaining th uppr ound w ony rquir compxity inar in n (th compxity is of O(n 2 ( Q () 2 + Q (2) 2 ))) In addition for an aritrariy arg numr of osrvations w can comput th asymptotic uppr ound with compxity of O( 3 ( Q () 3 + Q (2) 3 )) y mpoying ignvau dcomposition to otain th stady-stat of th Markov chains with transition matrics j 2 V CONCLUSIONS In this work w otain an uppr ound on th proaiity of rror whn cassifying among two HMMs asd on a squnc of osrvations of ngth n W us a spcific cass of DFAs to spit th squncs of osrvations into diffrnt partitions and appy Markov chain thory to fficinty comput an uppr ound on th a priori proaiity of miscassification among th two HMMs for squncs in ach partition Th choic of DFA affcts th partitioning which in turn affcts th tightnss of th uppr ound An opn prom is th choic of a spcific DFA (of a fixd numr of stats) that rsuts in th ast uppr ound REFERENCES [] E Athanasopouou and C N Hadjicostis Proaiity of rror ounds for faiur diagnosis and cassification in hiddn Markov mods in Procdings of th IEEE Confrnc on Dcision and Contro 2008 pp [2] L R Rainr Radings in spch rcognition A Wai and K-F L Eds Morgan Kaufmann Puishrs Inc 990 ch A tutoria on hiddn Markov mods and sctd appications in spch rcognition pp [3] F Jink Statistica mthods for spch rcognition MIT Prss 997 [4] L R Bah F Jink and R L Mrcr Radings in spch rcognition A Wai and K-F L Eds Morgan Kaufmann Puishrs Inc 990 ch A maximum ikihood approach to continuous spch rcognition pp [5] K S Fu Syntactic Pattrn Rcognition and Appications Prntic- Ha 982 [6] R Durin S R Eddy A Krogh and G Mitchison Bioogica Squnc Anaysis: Proaiistic Mods of Protins and Nucic Acids Camridg Univrsity Prss 998 [7] T Koski Hiddn Markov Mods of Bioinformatics Kuwr Acadmic Puishrs 200 [8] J Lunz and J Schrödr Stat osrvation and diagnosis of discrt vnt systms dscrid y stochastic automata Discrt Evnt Dynamic Systms: Thory and Appications vo no 4 pp [9] D Thorsy and D Tnktzis Diagnosaiity of stochastic discrt vnt systms IEEE Transactions on Automatic Contro vo 50 no 4 pp [0] B-H Juang and L Rainr A proaiistic distanc masur for hiddn Markov mods AT&T Tchnica Journa pp [] M Fakhausn H Riningr and D Wof Cacuation of distanc masurs twn hiddn Markov mods in Proc Eurospch 995 pp [2] S M E Sahraian and B Yoon A nov ow-compxity HMM simiarity masur IEEE Signa Procssing Lttrs vo 8 no 2 pp [3] P A Rgaia and S K Mitra Kronckr products unitary matrics and signa procssing appications Socity for Industria and Appid Mathmatics vo 3 no 4 pp Dcmr

Bayesian Decision Theory

Bayesian Decision Theory Baysian Dcision Thory Baysian Dcision Thory Know probabiity distribution of th catgoris Amost nvr th cas in ra if! Nvrthss usfu sinc othr cass can b rducd to this on aftr som work Do not vn nd training