An EM algorithm for maximum likelihood estimation given corrupted observations. E. E. Holmes, National Marine Fisheries Service

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1 An M algorihm maimum likelihood esimaion given corruped observaions... Holmes Naional Marine Fisheries Service Inroducion M algorihms e likelihood esimaion o cases wih hidden saes such as when observaions are corruped and he rue populaion size is unobserved. he following M algorihm is based on he summar b Ghahramani and Hinon (996 of he algorihm originall b Shumwa and Soffer (98 esimaing he parameers of linear dnamical ssems from corruped observaions. he algorihm consiss of an esimaion sep ( sep which esimaes he rue sae using a Kalman-Rauch filer combined wih a measuremen sep ( M sep which gives he maimum likelihood esimaes of he parameers given he daa and he esimae of he rue sae. M algorihms and he Kalman filer are well-known and heavil used in engineering and compuer science applicaions. For some general background on M algorihms he reader is referred o McLachlan (996 and o Harve (99 M algorihms ime series daa. here are a muliude of books on he Kalman filer. One of he more penerable inroducions is chaper of Mabeck (979. his presenaion of an M algorihm closel follows Ghahramani and Hinon s noaion and derivaion bu adaps and es i o he sochasic populaion model case. he sae-space model he diffusion approimaion a sochasic eponenial growh model can be wrien as a linear sae space model (wrien in he noaion familiar in he engineering lieraure: w where w ~ Normal(0 [ v where v ~ f (0 R [ where log N is he rue log populaion size and log O is he log observaions of he populaion size. is µ he mean populaion growh rae. is he σ oherwise known as he process error or environmenal variabili. R is he variabili associaed wih sampling error or oher non-process error. Onl is observed; he underling parameers and R and he underling rue populaion size is hidden. If we make he assumpion ha v is normall disribued hen he model is a linear Gaussian sae-space model and we can use he algorihm b Shumwa and Soffer (98 esimaing he parameers of he sae-space model from alone. From qns. and and he assumpion ha f( is normal we can wrie he following condiional probabiliies: ( ( / ep ( R [3 R

2 ( ( / ( ep ( [4 Given qn. and ha we are esseniall observing an ongoing sochasic process which we happen o begin observing a hen is iself a random variable ha is normall disribued wih some mean and variance and hus ( ( / ep ( [5 Using he Markov proper implici in he model he join probabili of he observed ime series and he rue sae is he join log likelihood of is: ( ( ( (. [6 log ( ( ( log R R ( ( log log log [7 he goal is o find he esimaes of R and ha maimize log (. he following M algorihm does his. he M algorihm his M algorihm an eension of he Shumwa and Soffer (98 algorihm has four basic seps: 0 Compue some iniial parameer esimaes R from which o sar he algorihm. Generae an esimae of b esimaing ( using he Kalman-Raush recursion which gives he maimum likelihood esimaes of ( given R and. he maimum likelihood esimae of ( is denoed.

3 Updae he R given he b finding he esimaes which maimize he updaed epeced log likelihood (using he updaed : (log (. 3 Check o see if has converged and no longer increases. If no converged reurn o sep. Wriing ou and describing he algorihm is raher edious and long however he acual code is quie rivial and encompasses abou a page of e minus he commens. Malab code is given a he of his wrie-up. Sep 0. Compue iniial parameer esimaes o ge good final esimaes one needs o sar he algorihm wih reasonable iniial parameer esimaes. I use he following iniial esimaes which are based on he parameer esimaes presened in Holmes and Fagan (00: ~ 3 i ~ ( ~ 3 4 ar( ~ 4 3 R i ~ ( ( ar( ar( ~ he esimae of R is based on he esimae of he non-process error presened briefl in he appi of Holmes and Fagan (00. Iniial esimaes of and are also needed. I used and 0.. ~ Sep. he Kalman-Raush recursion he firs par uses he Kalman recursion o esimae (. his is a ward recursion since we work ward o generae i. he second par uses he Raush recursion o work backwards and compue from. Firs some noaion: ( (

4 [ [ Cov[ ar[ [ [ K he ulimae goal of hese recursions is o compue and - which will be needed sep of he algorihm. he Kalman recursion o compue and sar a and sep ward o. A each sep compue: ( ( > > K K R K his is he well-known Kalman filer bu i looks a lile differen han wha ou ll see in engineering es. Firs generall i is assumed ha is a series of measuremens from muliple insrumens hus he Kalman filer is alwas wrien in mari m. Here since is one measuremen i can be wrien in scalar m. Second he Kalman filer is usuall presened he model w u A v C. In his applicaion A C and u so he filer is simplified quie a bi. he Rauch recursion Ne we work backwards from back o o compue and. his recursion requires he and ha were generaed during he Kalman recursion.

5 ( ( ( One more recursion Using from he Rauch recursion wih K and from he Kalman recursion we do anoher backwards recursion o compue. Saring from work backwards o and a each sep compue ( < K - ( uing i all ogeher Using he hree recursions we can hen compue he following which are needed sep of he algorihm. Sep Generae parameer esimaes he epeced log likelihood funcion is given b qn. 7 wih he esimae : ( (log using. o compue he parameer esimaes we find he R ha maimize he. o do his we ake he parial derivaive of wih respec o each parameer se he derivaive o zero and solve he maimizing parameer: C R R R 0

6 ( 0 ( ( ( ( ( given ha ( ( ( 0 ( ( 0 Sep 3 Check convergence A simple wa o do his is o compare he o he previousl esimaed and check if he difference is less han some hreshold. Once he log likelihood converges ou re done. References Ghahramani Z. and G.. Hinon arameer esimaion linear dnamical ssems. echnical repor CRG-R-96-. Universi of orono Deparmen of Compuer Science orono Canada. Harve A. C. 99. Forecasing srucural ime series models and he Kalman filer. Cambridge Universi ress Cambridge UK. Holmes.. and W. F. Fagan. 00. alidaing populaion viabili analsis corruped daa ses. colog 83: McLachlan G. M he M algorihm and eensions. Wile USA. Mabeck. S Sochasic models esimaion and conrol. olume. Academic ress New York USA.

7 Shumwa R. H. and Soffer D. S. 98. An approach o ime series smoohing and ecasing using he M algorihm. ournal of ime Series Analsis 3(4: Malab code funcion [ R ini LL AKalman(logdaa %AKalman Find he ML parameers of a sochasic corruped eponenial growh %ime series using M % % [ R ini ini LL AKalman(logdaa % fis he parameers which are defined as follows % ( ( w( w ~ N(0 (0 ~ N(ini ini % ( ( v( v ~ N(0R % logdaa( is a (: vecor of he logged observaions; no missing ears % LL is he vecor of he log likelihood values a each ieraion; he idea is % o maimize his. % % he algorihm used here is an eension of he mehod described in % Shumwa R. H. and Soffer D.S. 98. An approach o ime series smoohing and %ecasing using he M algorihm. ournal of ime Series Analsis 3(4: % which is described in Ghahramani Z. and Hinon G arameer simaion % LDS. U. orono echnical repor CRG-R-96-. % he noaion used here follows Ghahramani and Hinon ecep ha I drop he %"" subscrips. %S 0 se he iniial esimaes %Iniialize b geing D-H esimaes lengh(logdaa; daaep(logdaa; runsum daa(:(-3daa(:(-daa(3:(-daa(4:; mean(log(runsum(:./runsum(:(-; ovar var(logdaa(:-logdaa(:(-; (/3*(var(log(runsum(4:./runsum(:( var(log(runsum(:./runsum(:(-; if( < ; R (ovar - /; if(r < 0 R 0.000; ini logdaa(; %pi is Ghahramani and Hinon's noaion bu pi reserved 0.; %variance of ini logdaa; LL[; converged 0; previous_loglik -Inf; ma_ier 500; num_ier 0; %run unil he ma log likelihood is found while ~converged & (num_ier < ma_ier %S using hese iniial esimaes generae an esimae of ( %using a ward and backward pass of he Kalman filer. his gives %ou he ML esimae of ( given (: and he parameer esimaes. %iniialize zeros(; zeros(; zeros(; zeros(; zeros(; zeros(; zeros(; zeros(; %ward pass ges ou [( given (: 0ini;

8 0; (: if( ( ini; %denoes _^0 ( ; %denoes _^0 else ( (- ; % denoes _^(- ( (- ; K (/((R; ( ( K*(( - (; ( (-K*(; K K; %backward pass ges ou [((: from [((: ( (; ( (; (:-: (- (-/(; (- (- (-*((-((-; (- (- (-*((-(*(-; ha ; %esimae of (.*; %(^ %run anoher backward recursion o ge [((-( ( ( - K*(-; %his is ar(((-( (:-:3 (- (-*(- (-*((-(-*(-; [NaN (:(:.*(:(-; %( NA since - 0 %Calculae negaive log likelihood his ha Riniini combo loglik - sum((-ha.^/(*r - *log(abs(r/... - sum((ha(:-(ha(:(-.^/(* - (-*log(abs(/... - (ha(-ini^/(* - log(abs(/ - *log(*pi; LL[LL loglik; %S Re-esimae Riniini via ML given ( esimae R (/*sum(.* - ha.*; (ha(-ha(/(-; sum((: - *(: (:(- - ^/(-; ini ha(; (-ha(*ha(; %S 3 check convergence num_ier num_ier; converged em_converged(loglik previous_loglik; %subfuncion below previous_loglik loglik; %while no converged funcion converged em_converged(loglik previous_loglik hreshold % M_CONRGD Has M converged? % [converged decrease em_converged(loglik previous_loglik hreshold % % We have converged if % f( - f(- / avg < hreshold % where avg (f( f(-/ and f is log lik. % hreshold defauls o e-4. % his sopping crierion is from Numerical Recipes in C p43

9 if nargin < 3 hreshold e-4; %log likelihood should increase if loglik - previous_loglik < -e-3 % allow a lile imprecision fprinf( '******likelihood decreased from %6.4f o %6.4f!\n' previous_loglik loglik; dela_loglik abs(loglik - previous_loglik; avg_loglik (abs(loglik abs(previous_loglik eps/; if (dela_loglik / avg_loglik < hreshold converged ; else converged 0;

Kalman filtering for maximum likelihood estimation given corrupted observations.

Kalman filtering for maximum likelihood estimation given corrupted observations. alman filering maimum likelihood esimaion given corruped observaions... Holmes Naional Marine isheries Service Inroducion he alman filer is used o eend likelihood esimaion o cases wih hidden saes such