Parametric Estimation in MMPP(2) using Time Discretization. Cláudia Nunes, António Pacheco

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1 Paramerc Esmaon n MMPP(2) usng Tme Dscrezaon Cláuda Nunes, Anóno Pacheco Deparameno de Maemáca and Cenro de Maemáca Aplcada 1 Insuo Superor Técnco, Av. Rovsco Pas, 1096 Lsboa Codex, PORTUGAL In: J. Janssen e N. Lmnos, ed., Proceedngs of he 2ndInernaonal Symposum on Sem-Marov Models: Theory and Applcaons -Compégne, 9-11 Dec. 1998, In secon ``Sascal Esmaon II'' (6 pp.), 1998 Absrac: Marov Modulaed Posson Processes (MMPP, for shor) are a specal class of Marov renewal processes. Informally, a MMPP s a doubly sochasc Posson process n whch he arrval raes vares accordng o an rreducble Marov chan. The MMPP models specally well bursy pon processes appearng n elecommuncaons. Snce ypcally he ransons on he Marov chan are no observable, one of he mos mporan problems relaed o he MMPP concerns he esmaon of he parameers of he process: he ranson raes of he Marov chan and he arrval raes. In hs wor we propose a new esmaon mehod for he parameers of he MMPP and compare numercally wh oher mehods. Key Words: MMPP, parameer esmaon, EM esmaors, me-dscrezaon. 1. Marov Modulaed Posson Processes The Marov modulaed Posson process (MMPP) s a generalzaon of he Posson process, n whch he arrval rae s deermned by an M -sae rreducble, homogeneous Marov chan (also nown as he envronmen) and he arrvals come accordng o a Posson process of rae λ, whenever he sae of he chan s ( 0 M 1). The process s fully characerzed by he nfnesmal generaor of he envronmmen σ 0 σ σ 0, M 1 σ 01 σ 1... σ 1, M 1 Q = (1) σ 0, M 1... σ M 2, M 1 σ M 1 (where σ = j σ j ) and he dagonal marx Λ, wh Λ jj = λ j ( 0 j M 1). If we denoe he counng arrval process by X and he envronmen by J, hen we refer o ( X, J ) as an MMPP (M) wh ( Q, Λ) - source (Saraman (1991). The MMPP s a versale model, whch has been specally used o model bursy pon processes arsng n elecommuncaons; n parcular, has been used o model he arrval process n he so-called ATM newors (Deng and Mar (1993)). Classcally, MMPP s have been suded n he conex of doubly sochasc processes. Bu hey can be suded n dfferen and rcher framewors, namely: Marov addve processes MAP (Pacheco and Prabhu (1995)): n a MAP he marx of arrval raes Λ s no necessarly dagonal; we can have hree ypes of evens (ype 1- arrvals whou envronmen ransons, ype 2 envronmen ransons whou arrvals and ype 3 - smulaneous arrvals and envronmen ransons), whle n a MMPP he hrd ype of evens s no allowed and so Λ s necessarly dagonal. Marov renewal processes MRP (Çnlar (1969)) : f we defne he sequence of arrval mes T = { T } and he embedded Marov chan (defned a hese mes) J, hen he process ( T, J ) s a MRP, whose densy ranson marx s compleely deermned afer he arrvals and ranson raes Q, Λ of he orgnal process. 1 Ths research was also suppored by he projec Praxs/PCEX/P/Ma/41/96.

2 Hdden Marov models - HMM (MacDonald and Zucchn (1997)): f, assocaed o he MRP, J Y = (where Y =T +1 -T, 0 ), hen he process (, J ) T, we defne he sequence of nerarrval mes { } Y s a HMM n dscree me. fne mxure models (Terngon e al. (1985)) : n hs case, he densy of he ner-arrval mes s M 1 1 ( 1) a fne mxure of he ype f y = π f y, where π s he probably ha he j= 0 j j envronmen s n sae j by he me of he ( 1) h arrval and Y j ( y) f j s he densy of he h nerarrval nerval, f a he begnnng of he nerval he envronmen s n sae j. Typcally, he mos mporan quesons concernng MMPP s are: reconsrucon of he underlyng Marov chan; he forward-bacward algorhm conaned n he erave lelhood-maxmzaon algorhm of Baum e al. (1970) may be used for hs purpose. esmaon of he parameers ( Q, Λ) and sudy of her properes (namely, conssency and assmpoc dsrbuon); alhough he maxmum-lelhood esmaon mehod has nce properes, due o several numercal problems new esmaon mehods have been recenly proposed. denfcably of he parameers of he Marov chan; for nsance, he ndces of he saes of he chan can be permued whou changng he law of he process (Leroux (1992). enropy for MMPP; hs queson s relevan no only from he applcaons pon of vew, bu also allows he sudy of he generalzed Kullbac-Lebler dvergence and he defnon of equvalence classes of parameer pons (Leroux (1992)). In hs paper we wll sudy he esmaon problem for he MMPP wh a wo-saes Marov chan (MMPP(2)). In secon 2 we presen brefly hree esmaon ypes: momen-machng, lelhood and recursve; n parcular, we refer o a echnque proposed by Deng and Mar (1993), based on he me dscrezaon of he axs, whch, by smulaon resuls, proved o be beer han any of he prevous classes of esmaors. In secon 3 we propose a new esmaor, based also on me dscrezaon. Fnally, n secon 4 we compare he numercal resuls of he hs new esmaor and he one proposed by Deng and Mar (1993). 2. Paramerc Esmaon n MMPP(2) The esmaon mehods can be roughly classfed n hree caegores. The frs one s based on momen machng: we begn by obanng he probably generang funcon of he MMPP(M) and compung s momens; he esmaors are hen chosen so as o ensure ha he low-order momens become numercally dencal o he emprcal momens of he raffc daa. Ths esmaon mehod s based on a poor parameer esmaon creron (Deng and Mar (1993)). For M larger han 2 hs mehod may no be feasble o apply because nvolves he resoluon of a se of M 2 equaons, wh M 2 varables, subjec o feasble consrans. The second caegory ncludes he maxmum lelhood esmaors, whch n general exhb nce properes (namely conssency and assmpoc normaly). For he case of mssng observaons, as s he case (we don observe he ransons on he envronmen), he maxmum lelhood esmaors are numercally found usng he Expecaon-Maxmzaon (EM) algorhm (for furher deals on hs algorhm, see McLachlan and Krshnan (1997); for mplemenaon deals of he EM-algorhm for MMPP s see Rydén (1996)). In general, he applcaon of hs algorhm s raher nvolved for MMPP (specally when M s large and some of he parameers are zero). To crcunven hs problem, some auhors have proposed he reparamerzaon of he MMPP and he use of heurscs; for nsance, Meer (1984) has devsed an heursc algorhm, based on a recursve esmaon procedure. Recursve esmaon echnques n mxure models wh ndependen observaons have been suded snce he decade of 70 s, by Kazaos (1977). Afer he wor of Meer (1984), many ohers auhors have consdered recursve esmaon for he MMPP; he dfference beween her approaches essenally concerns he scalng marces ha are used n he recursve procedure. Alhough here are many wors on he numercal behavour of hese esmaors, no resuls on he asympoc properes are proved, wh he excepon of he esmaor proposed by Rydén (1997).

3 Deng and Mar (1993) developed alernave mehods for he MMPP parameer esmaon whch are based on he maxmum lelhood prncple bu, nsead of a reparamerzaon and he use of recursve equaons, hey employed a compleely dfferen dea: dscrezaon of he me axs. Suppose ha he MMPP ( J ) ] 0, x] n subnervals of he form ]( 1) h, h] (wh =,2,..., T = x h process, ( X, J ) = {( X, J ) = ( X, J )( h), 0} Prabhu (1995)): X, s observed ll me x. Le h be a un me lengh and dvde he nerval 1 ). Consder now he embedded, whose ranson probables are of he form (Pacheco and pj ( n) = P[ ( X +, J + 1 ) = ( m + n, j) ( X, J ) = ( m, ) ] = pjb ( n) ( P = ( p j ) s one-sep ranson probably marx of J and { ( n), n 0} 1 (2) b s he vecor of he arrval probables when he envronmen s n sae ). Specal cases of hese processes are: MMBP (Marov Modulaed Bernoull Process): ( n) = b I { n= 1 } + ( 1 b ) I { n=0} λ Dscree MMPP: b ( n) e ( λ ) n! n =. b ; The dea behng he procedure proposed by Deng and Mar (1993) s he followng: suppose ha we choose h small enough enough so ha on each subnerval here s a mos one arrval. The embedded process, ( X, J ) = {( X, J ) = ( X, J )( h), 0}, s hen approxmaed by a MMBP, wh parameers p = σ h and b = λ h. The man advanage of hs procedure s ha now boh phases of he EM j j erave algorhm (phases E and M) are que easy o perform. A eraon 1, he esmaors are: ( ) ( ) ( ) = 0 ξ, j pˆ j = ( ) ; ˆ ( ) = O ( = ) + 1 λ = (3) T 1 = O = O1, O2,..., O T (wh O = X X 1) and where ξ (, j) = P( J =, J + = j O, Ψ) ; = P( J = O, Ψ) 1 γ (, j E ) (4) are probables ha can be effcenly calculaed usng he forward-bacward algorhm (Rabner (1989). 3. Proposed Esmaor Deng and Mar (1993) presen a smulaon o compare her esmaor wh classcal esmaors (namely recursve and EM esmaors), and conclude ha her esmaor leads o smaller esmaon errors han he ohers. In he mehod proposed by Deng and Mar, he choce of he me lengh h plays an mporan role. Frs of all, h should be chosen as o ensure ha no more han one even occurs whn each me nerval. Bu he choce of very small h can also lead o serous mplemenaon problems, as he bnary sequence O = can become oo large o handle n a normal compuer! { } O Unforunaely, hs s no he only problem. From smulaons ha we have done, we have noced a hgh dependence beween he resuls and he choce of he un sep h. For nsance: Table 1: Esmaves (for dfferen h ) of he parameers of he MMPP(2): ( λ =, λ = 20, σ = 1, σ 1) = Esmaves h= h= h=0.001 h=0.005 h=0.01 h=0.03 h=0.045 h=0.05 ˆλ ˆλ ˆ σ ˆ10 σ As would be expeced, he arrval raes esmaon error ncrease wh h ; bu hs s no he case for he esmaon of he ranson raes! In fac, for he ranson raes he bes resul s obaned wh a larger h ( h =0.03) and he wors wh a smaller one ( h =0.0004).

4 Two obvous quesons are hen: Queson 1. How o handle mulple arrvals ( 2) n one subnerval? Queson 2. How o handle he fac ha usually he arrval raes are beer esmaed wh small h s and he ranson raes are beer esmaed wh larger h s? 3.1. Mulple Arrvals The esmaor proposed by Deng and Mar assume ha he number of arrvals n each subnerval s lmed by one and so O has a Bernoull dsrbuon,. In order o handle mulple arrvals, we assume nsead ha O ( ) has a Posson dsrbuon, wh parameer λ h, where J s he sae of he envronmen n he beggnng of he h subnerval; hus we approxmae he embedded process X menoned n secon 2 by a dscree MMPP wh p = σ h and λ = h. Applyng now he, J erave EM algorhm, we ge he followng esmaor a eraon : ( ) ( ) = 0 ξ, j pˆ j = ( ) ; ˆ ( ) = O ( = ) + 1 λ = (5) T 1 = Noce ha even f all O 1, esmaors (5) wll no be numercally equal o esmaors (3), because he j γ are calculaed usng dfferen dsrbuons (Bernoull for (3) and Posson probables ξ and for (5))., 3.2. Two-Sage Esmaon As he arrval raes are usually beer esmaed wh small h s whle he ranson raes are beer esmaed wh larger h s, we wll consder he followng wo-sage esmaon procedure: Sage 1: Esmae eravely he ranson raes (usng he pˆ j esmaor from (3) or (5)); Sage 2: Once he ranson raes are esmaed, use her values o esmae he arrval raes (usng he esmaor λˆ from (3) or (5) and eepng pˆ j fxed). 4. Numercal Resuls In order o compare he esmaon error assocaed wh he esmaors proposed by Deng and Mar (1993) and he ones ha we have proposed n he prevous secon, we made a smulaon sudy, for 3 dfferen cases (also consdered by Deng and Mar (1993)): case 1: λ 0 = 100 ; λ1 = 10 ; σ 01 = 10 ; σ 10 = 1 case 2: λ 0 = 100 ; λ1 = 20 ; σ 01 = 1 ; σ 10 = 1 case 3: λ 0 = 100 ; λ1 = 30 ; σ 01 = 3 ; σ 10 = 2 j j ( ) Four esmaon mehods were consdered, namely: B: esmaors (3) P: esmaors (5) B-B: wo-sage esmaon, usng esmaors (3) n boh sages : wo-sage esmaon, usng esmaors (3) n sage 1 and esmaors (5) n sage 2 Gven he smulaed MMPP daa whose rue underlyng model parameers were nown n advance, s possble o compare he parameer esmaes obaned from dfferen mehods. We have chosen several values for he me lengh h n order o see s effec on he esmaon resuls. The cases ha we have chosen are arranged n able 2 by ncreasng degree of dffculy n parameer esmaon (hs degree ncreases as he Posson nenses from he wo saes become closer o each oher). By a quc readng of able 2, we see ha: When usng mehod B, we observe a clear rend of degradaon n he esmaon accuracy of he arrval raes as h ncreases; he accuracy of he mehod P seems o depend less on he me lengh used, and so we may say ha s a more sable mehod o esmae he arrval raes. J 1 λ 1

5 Case 1 P Table 2. Esmaon Resuls of MMPP(2) 2 h B B-B B Case 2 P B-B B Case 3 P B-B In all cases, he wo-sage mehods (B-B and ) are noorously beer han he B and P mehods, and seem o combne he goods properes of he B and P mehods. From hese wo-sage mehods, he mehod shows, n general, beer accuracy and robusness. Remar as well ha: 2 The able should be read n he followng way: n each enrance, we have he esmaves for ˆ λ 0, ˆ λ1, ˆ σ 01, ˆ σ 10, by hs order means ha he algorhm dd no converge or converged o a non-feasble soluon, because durng he erave procedure a leas one esmae was such ha: esmae h >1.

6 - he esmaed values of he ranson raes when usng mehod B are relavely sable for a broad range of h values and, even more mporan, hs sably regon s que clear (meanng ha here s no a progressve rend of degradaon bu raher a wo-value suaon: n one regon he esmaed values are smlar, no maer whch value of h s used, and a he rgh of hs regon hey vary a lo). For nsance, n case 1, for h 0. 01, ˆ σ 01 s, roughly speang, beween 8.0 and 10.5; bu for h = 0. 02, ˆ σ 01 s (less han en mes smaller han he real value). - hs sable behavour of he esmaed values for he ranson raes s also shared by he esmaes of he arrval raes when performng sage 2 (assumng ha we pc up values n he sably regon for he esmaed ranson raes). For nsance, for case 2, usng h =0.003 n he second sage: Frs Sage Second Sage Table 3. (Non)Influence of he esmaed ranson raes n he esmaon of he arrval raes h (1 s sage) ranson raes arrval B-B raes Fnal Remars In general, wh he excepon of maxmum lelhood esmaors and he Rydén recursve esmaor, no much has been sad abou he heorecal properes of he esmaors of he parameers of he MMPP; hs s rue n parcular for he mehods based on me dscrezaon. The sudy of he properes of he esmaors based on me dscrezaon, wh poseror use of he EM algorhm, consue an mporan research problem ha needs o be adressed. We beleve ha f esmaors (3) and (5) are conssen, hen we can expec conssency as well n he wosage esmaors. For nsance, n a raher dfferen conex, Par (1993) proved he equvalence of maxmun lelhood esmaors and erave wo-sage esmaors (for he SUR models Seemngly Unrelaed Regresson models). Bblography Baum, L.E., Pere, T.A., Wess, N. (1970). A maxmzaon echnque occurng n he sascal analyss of probablsc funcons of Marov chans, Ann. Mah. Sas., 41, p Çnlar, E. (1969). Marov renewal heory, Adv. Appl. Prob. 1, p Deng, L. and Mar, J.W. (1993). Parameer esmaon for Marov modulaed Posson processes va he EM algorhm wh me dscrezaon, Tel. Sysems 1, p Kazaos, D. (1977). Recursve esmaon of pror probables usng mxure. IEEE Trans. Inf. Theo., vol IT-23. p Leroux, B.G. (1992). Max-lelhood esmaon for HMM. Sochasc. Process. Appl., 40, p MacDonald, I.L. e Zucchn, W. (1997). Hdden Marov and Oher Models for Dscree-valued Tme Seres, Chapman & Hall. McLachlan, G.J. and Krshnan, T. (1997). The EM and Exensons, Wley. Meer, K.S. (1984). A Sascal Procedure for Fng Marov Modulaed Posson Processes, Ph.D. Dsseraon, Unv. Delaware. Pacheco, A. e Prabhu, N.U. (1995). Marov addve processes of arrvals, Advances n Queueng: Theory, Mehods and Open Problems, J.H. Dshalow ed., CRC Press, p Par, T. (1993). Equvalence of maxmum lelhood esmaon and erave wo-sage esmaon for seemngly unrelaed regresson models. Commun. Sas.- Theory, vol. 22, nº8, p Rabner, L.R. (1989). A uoral on hdden Marov models and seleced applcaons n speech recognon. Proc. of he IEEE, vol. 77, nº2, p Rydén, T. (1997). On recursve esmaon of hdden Marov models. Sochas. Process. Appl., 66, p Saraman, H. (1991). Approxmaons of some Marov modulaed Posson processes. ORSA J. Comp., vol. 3, nº1, p Terngon, D.M., Smh, A.F.M. and Maov, U. E. (1985). Sascal Analyss of Fne Mxure Dsrbuons, Wley.