SOMETHING ELSE ABOUT GAUSSIAN HIDDEN MARKOV MODELS AND AIR POLLUTION DATA

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1 UNIVERSIÀ CAOLICA DEL SACRO CUORE ISIUO DI SAISICA Robera AROLI e Luigi SEZIA SOMEHING ELSE ABOU GAUSSIAN HIDDEN MARKOV MODELS AND AIR OLLUION DAA Serie E N 96 - Marzo 2000

2 SOMEHING ELSE ABOU GAUSSIAN HIDDEN MARKOV MODELS AND AIR OLLUION DAA Robera aroli Isiuo di Saisica Universià Caolica SC di Milano Luigi Spezia Diparimeno di Ingegneria Universià degli Sudi di Bergamo Absrac New versions of he explici formulas of he esimaors of he parameers of gaussian hidden Markov models will be inroduced and compared o hose obained in a previous work (aroli and Spezia, 1999) hese new expressions are based on he join probabiliy densiy funcions of he observaions and he unobserved saes of he Markov chain he esimaors obained in his paper can be used also when he sequence of observaions conains unrecorded daa An applicaion abou a ime series of a polluan wih unrecorded daa will be shown Keywords: discree ime sochasic processes, Markov chains, maximum likelihood esimaors, unrecorded daa Inroducion 1 In a previous work (aroli and Spezia (1999), henceforh denoed S99) we have sudied gaussian hidden Markov models (GHMMs) and begun o ackle he esimaion problems, performing he EM algorihm he main resuls of S99 are roposiions 1, 2, 3: 1 Work parially suppored by MURS 1998 gran 1

3 roposiion 1 - he funcion Q ³φ; φ obained a he E sep of he (k+1) h ieraion of he EM algorihm is Q ³φ; φ = E φ (ln L c (φ) y) = α 1 (i) β = i S X l S X α 1 (i) (l) β +i S X j SX =1 1 =1 α +i S X l S X α (l) ln δ i+ α (i) β (i) (l) β (i) γ i,j l S X α (l) ln f(y i), f (y +1 j) β +1 (j) (l) β (l) ln γ i,j + where f(y i) = " 1 exp 1 2πσi 2 µ # 2 y µ i σ i roposiion 2 - Given a generic HMM {Y ; X }, he expression of he unknown ransiion probabiliy γ i,j of he Markov chain {X },obainedahe (k +1) h ieraion of he EM algorihm, is, for large, γ (k+1) i,j = 1 =1 α (i) γ i,j f (y +1 j) β +1(j), =1 1 α (i) β (i) for any sae i and j (i 6= j) ofhemarkovchain{x } roposiion 3 - Given a GHMM {Y ; X }, he expressions of he unknown parameers µ i and σ 2 i of he gaussian random variable Y (i), for any, obained a he (k +1) h ieraion of he EM algorihm, are µ (k+1) i = =1 σ 2(k+1) i = =1 α =1 α α (i) β (i) y (i) β (i) ³ (i) β (i) =1 α for any sae i ofhemarkovchain{x } y µ (k+1) i (i) β (i) In he case of convergence of he algorihm a he (k +1) h ieraion, given ha he four regulariy condiions of Wu (Wu, 1983, condiions (5), (6), (7), (10), pp 96, 98) are saisfied, ³φ (k+1) ;lnl ³φ (k+1) is a saionary poin of 2, 2

4 ³ ln L (φ), andhereforeφ (k+1) = γ (k+1) 1,2,γ (k+1) 1,3,,γ m,m 1,µ (k+1) (k+1) 1,,µ m (k+1), σ 2(k+1) 1,,σ 2(k+1) m is he maximum likelihood esimaor of he unknown parameer φ In S99, an applicaion of GHMMs o real daa has been sudied: a ime series abou he daily mean concenraion of sulphur dioxide (SO 2 ) in he second semeser of 1998, recorded by he air polluion esing saion placed in via Goisis in Bergamo, has been analyzed Now we are ineresed in he dynamics of hourly mean concenraions of SO 2 during a week In he case of daily daa (S99, Secion 4 and Appendix A) we have a complee series of 184 observaions, while in he case of hourly daa (see Appendix), wihin a series of lengh =168, we have 12 means ha have no been recorded or validaed he aim of his paper is o sudy he esimaion problems of GHMMs wih unrecorded daa he problem is based on he fac ha he explici formulas obained in roposiions 2 and 3 can no be applied when he sequence of he observaions conains unrecorded daa, because he recursive formulas of he forward and he backward pdf s can no be compued So, firs, in Secion 1, we reinroduce he basic GHMM, as in S99; hen, in Secion 2, we consider some join pdf s of he process {Y ; X } ha will be used in Secion 3 o obain new versions of he explici formulas of he esimaors of he unknown parameers of GHMMs hese new expressions may be used when he sequence of observaions conains unrecorded daa, oo; so, finally, in Secion 4, a ime series wih unrecorded daa will be examined 1 he basic Gaussian Hidden Markov Model Le {X } (1,, ) N be a discree, homogeneous, aperiodic, irreducible Markov chainonafinie sae-space S X = {1, 2,,m} he ransiion probabiliy from sae i, a ime 1, o sae j, a ime, isdenoedbyγ i,j, for any sae i, j and for any ime : γ i,j = (X = j X 1 = i) =(X 2 = j X 1 = i) he ransiion probabiliies (m m) marix is Γ =[γ i,j ],wihj S X γi,j =1,foranyi S X he iniial disribuion is he vecor δ = (δ 1,δ 2,,δ m ) 0, where δ i = (X 1 = i), foranyi =1, 2,,m,wihi S X δi =1 Since {X } is a homogeneus, irreducible Markov chain, definedonafinie sae-space, i has an iniial disribuion δ which is saionary, ha is δ i = (X = i), foranysaei =1, 2,,m and for any ime Since δ is a saionary disribuion, he equaliy δ 0 = δ 0 Γ holds: δ ishelefeigenvecorof he marix Γ, associaed wih he eigenvalue one, which always exiss, because Γ is a sochasic marix Finally, he hypoesis characerizing HHMs is ha he Markov chain {X } is unobservable 3

5 Le {Y } (1,, ) N be some discree sochasic process, on a coninuous sae-space S Y R he process {Y } mus saisfy wo condiions: (1) condiional independence condiion - he random variables (Y 1,,Y ), given he variables (X 1,,X ), are condiionally independen; (2) conemporary dependence condiion - he disribuion of any Y,given he variables (X 1,,X ), depends only on he conemporary variable X By hese wo condiions, given a sequence of lengh of observaions, y 1,y 2,,y and a sequence of lengh of saes of he unobserved Markov chain i 1,i 2,,i, i resuls f(y 1,y 2,,y i 1,i 2,,i )= = =1 Q f(y i 1,i 2,,i )= =1 Q f(y i ), where he generic f(y i) is he pdf of he gaussian random variable Y,when X = i, henceforh denoed Y (i),forany1 : Y (i) N µ i ; σ 2 i,foranyi =1,,m he so-defined model {Y ; X } (1,, ) N is said gaussian hidden Markov model and i is characerized by he saionary iniial disribuion δ, by he ransiion probabiliies marix Γ and by he sae-dependen pdf s f(y i) he model is called Markov because he unobserved sequence of saes is a Markov chain; he model is called hidden because he sequence of lengh of realizaions of he sochasic process {Y } is observed, bu no he sequence of lengh of saes of he Markov chain, which is hidden in he observaions; he model is called gaussian because f(y i) is a gaussian pdf he GHMM can equivalenly be wrien as Y (i) = µ i + E (i), where E (i) denoes he gaussian random variables E, when X = i, wih zero mean and variance σ 2 i E(i) N (0; σ 2 i ), for any i S X, wih he discree process {E },given{x },saisfies he condiional independence and he conemporary dependence condiions In his paper, new versions of he formulas of he maximum-likelihood esimaors φ (k+1), obained by he EM algorihm, will be sudied 2 Some join probabiliy densiy funcions of he process Our iniial sep is he examinaion of some join pdf s of he process {Y ; X } Firsweshallobainhejoinpdf s of he observed variables (Y 1,,Y ), boh for a complee sequence of daa and for a sequence wih unrecorded daa; 4

6 hen we shall obain he join pdf s of he observed variables and one or wo consecuive saes of he Markov chain hese pdf s will be used in Secion 3 o obain new versions of he explici formulas of he esimaors of he parameers γ i,j,µ i,σ 2 i, no based on he backward and he forward pdf s 21 he join pdf of (Y 1,,Y ) Given a sequence of observaions y 1,y 2,,y and a sequence of saes of he Markov chain i 1,i 2,,i from a HMM {Y ; X }, we may obain he join pdf f(y 1,y 2,,y,i 1,i 2,,i )= = δ i1 γ i1,i 2 γ i 1,i f(y 1 i 1 )f(y 2 i 2 ) f(y i )= (1) = δ i1 f(y 1 i 1 ) =2 Q γi 1,i f(y i ), applying he condiional independence, he conemporary dependence and he Markov dependence condiions Summing over i 1,i 2,,i he equaliy (??), we have he join pdf f(y 1,y 2,,y )= = i 1 S X i2 S X i S X δi1 γ i1,i 2 γ i 1,i f(y 1 i 1 )f(y 2 i 2 ) f(y i )= Q = i 1 S X i2 S X i S X δi1 f(y 1 i 1 ) =2 γi 1,i f(y i ) (2) Seing F = diag(f(y 1),f(y 2),,f(y m)), for any =1,,, we obain f(y 1,y 2,,y )=δ 0 F 1 ΓF 2 ΓF, (3) where is he m-dimensional vecor of ones Replacing δ 0 wih δ 0 Γ,given ha he iniial disribuion δ is saionary, and seing ΓF = G, we have! Y f(y 1,y 2,,y )=δ Ã 0 =1 G We can noice ha he join pdf f(y 1,,y ) may be compued even if some daa are no available If, for example, a subsequence of w 1 observaions, y v+1,,y v+w 1, is no available wihin a sequence y 1,,y,wih 5

7 1 <v+1 v + w 1 <,hepdf (??) becomes f(y 1,,y v,y v+w,,y )= = i 1 S X iv S X iv+w S X i S X δi1 γ i1,i 2 γ iv 1,i v γ iv,i v+w (w)γ iv+w,i v+w+1 γ i 1,i f(y 1 i 1 ) f(y v i v )f(y v+w i v+w ) f(y i )= = δf 1 ΓF v Γ w F v+w ΓF 1 0 (m) = = δ 0 µ =1 v Q G Γ w 1 µ = v + w Q G, (4) where γ i,j (w) is he w-seps ransiion probabiliy, γ i,j (w) =(X v+w = j X v = i) =(X 1+w = j X 1 = i); he w-seps ransiion probabiliies marix is Γ(w) = γ i,j (w) and, by Chapman-Kolmogorov equaion, wehaveγ(w) =Γ w he difference beween Formulas (??) and(??) lies in replacing he marix F, for any = v +1,,v+ w 1, wih he ideniy marix 22 he join pdf of he observaions and one sae of he Markov chain Now we wan o obain he join pdf s of he observaions y 1,,y and he sae i a ime of he Markov chain, ie f(y 1,,y,X = i), for any =1,, We separaely analyze he following hree siuaions: =1, 1 <<,= Henceforh we shall denoe he i h row of Γ wih Γ i and he i h column of Γ wih Γ i (a) =1: f(y 1,,y, X 1 = i) =δ i f(y 1 i)γ i F 2 ΓF 3 ΓF In fac: summing f(y 1,,y,X 1 = i) over i 2,,i and applying he condiional independence, he conemporary dependence and he Markov dependence condiions, we obain f(y 1,,y,X 1 = i) = = i 2 S X i S X δi γ i,i2 γ i 1,i f(y 1 i)f(y 2 i 2 ) f(y i )= = δ i f(y 1 i)γ i F 2 ΓF 3 ΓF (b) 1 <<: f(y 1,,y, X = i) =δ 0 F 1 ΓF 2 ΓF 1 Γ i f(y i)γ i F +1 ΓF 6

8 In fac: summing f(y 1,,y,X = i) over i 1,,i 1,i +1,,i and applying he condiional independence, he conemporary dependence and he Markov dependence condiions, we obain f(y 1,,y, X = i) = = i 1 S X i 1 S X i+1 S X i S X δi1 γ i1,i 2 γ i 1,iγ i,i+1 γ i 1,i f(y 1 i 1 ) f(y 1 i 1 )f(y i)f(y +1 i +1 ) f(y i )= = δ 0 F 1 ΓF 2 ΓF 1 Γ i f(y i)γ i F +1 ΓF (c) = : f(y 1,,y, X = i) =δ 0 F 1 ΓF 2 ΓF 1 Γ i f(y i) In fac: summing f(y 1,,y,X = i) over i 1,,i 1 and applying he condiional independence, he conemporary dependence and he Markov dependence condiions, we obain f(y 1,,y,X = i) = = i 1 S X i 1 S X δi1 γ i1,i 2 γ i 1,if(y 1 i 1 ) f(y 1 i 1 )f(y i) = = δ 0 F 1 ΓF 2 ΓF 1 Γ i f(y i) 23 he join pdf of he observaions and wo consecuive saes of he Markov chain Finally we wan o obain he join pdf sofheobservaionsy 1,,y and he consecuive saes i, j a imes, +1 of he Markov chain, ie f(y 1,,y,X = i, X +1 = j), for any =1,, 1 We separaely analyze he following hree siuaions: =1, 1 << 1, = 1 (a) =1: f(y 1,,y, X 1 = i, X 2 = j) =δ i f(y 1 i)γ i,j f(y 2 j)γ j F 3 ΓF 4 ΓF In fac: summing f(y 1,,y,X 1 = i, X 2 = j) over i 3,,i and applying he condiional independence, he conemporary dependence and he Markov dependence condiions, we obain f(y 1,,y,X 1 = i, X 2 = j) = 7

9 = i 3 S X i S X δi γ i,j γ j,i3 γ i 1,i f(y 1 i)f(y 2 j)f(y 3 i 3 ) f(y i )= = δ i f(y 1 i)γ i,j f(y 2 j)γ j F 3 ΓF 4 ΓF (b) 1 << 1: f(y 1,,y, X = i, X +1 = j) = = δ 0 F 1 ΓF 2 ΓF 1 Γ i f(y i)γ i,j f(y +1 j)γ j F +2 ΓF In fac: summing f(y 1,,y,X = i, X +1 = j) over i 1,,i 1,i +2,,i and applying he condiional independence, he conemporary dependence and he Markov dependence condiions, we obain f(y 1,,y, X = i, X +1 = j) = = i 1 S X i 1 S X i+2 S X i S X δi1 γ i1,i 2 γ i 1,iγ i,j γ j,i+2 γ i 1,i f(y 1 i 1 ) f(y 1 i 1 )f(y i)f(y +1 j)f(y +2 i +2 ) f(y i )= = δ 0 F 1 ΓF 2 ΓF 1 Γ i f(y i)γ i,j f(y +1 j)γ j F +2 ΓF (c) = 1: f(y 1,,y, X 1 = i, X = j) =δ 0 F 1 ΓF 2 ΓF 2 Γ i f(y 1 i)γ i,j f(y j) In fac: summing f(y 1,,y,X 1 = i, X = j) over i 1,,i 2 and applying he condiional independence, he conemporary dependence and he Markov dependence condiions, we obain f(y 1,,y,X 1 = i, X = j) = = i 1 S X i 2 S X δi1 γ i1,i 2 γ i 2,iγ i,j f(y 1 i 1 ) f(y 2 i 2 )f(y 1 i)f(y j) = = δ 0 F 1 ΓF 2 ΓF 2 Γ i f(y 1 i)γ i,j f(y j) 3 arameers esimaion of GHMMs A he M sep of he (k+1) h ieraion, o obain φ (k+1), he funcion Q ³φ; φ a roposiion 1 mus be maximized wih respec o he m 2 m parameers γ i,j s, for any i; j S X, wih i 6= j, he m parameers µ i s and he m parameers σ 2 i s, for any i S X 8

10 roposiion 4 - Given a generic HMM {Y ; X }, he expression of he unknown ransiion probabiliy γ i,j of he Markov chain {X },obainedahe (k +1) h ieraion of he EM algorihm, is, for large, γ (k+1) i,j = 10 ( 1) A 1 0 ( 1) B, for any sae i and j (i 6= j) ofhemarkovchain{x },where δ i f (y 1 i)γ i,j f (y 2 j)γ j F 3 Γ F 4 Γ F A δ 0 F 1 Γ = F 2 Γ F 1Γ i f (y i)γ i,j f (y +1 j)γ j F +2 Γ F δ 0 F 1 Γ F 2 Γ F 2 Γ i f (y 1 i)γ i,j f (y j) and B = δ i f (y 1 i)γ i F 2 Γ F 3 Γ F δ 0 F 1 Γ F 2 Γ F 1Γ i f (y i) Γ i F +1 Γ F δ 0 F 1 Γ F 2 Γ F 2 Γ i f (y 1 i)γ i F roof - By roposiion 2, we have γ (k+1) i,j = 1 =1 α (i) γ i,j f (y +1 j) β +1(j), (5) =1 1 α (i) β (i) and, by definiion (S99, pp 5, 10, 11), we have α (i) =f (y 1,,y,X = i); β (i) =f (y +1,,y X = i) (6) Replacing (??) in(??), we obain γ (k+1) i,j = 1 =1 f (y 1,,y,X = i)γ i,j f (y +1 j) f (y +2,,y X +1 = j) =1 1 f (y 1,,y,X = i) f (y +1,,y X = i) 9

11 By roposiions A5, A6, A7 of S99 and Markov dependence condiion, we obain = 1 =1 f (y 1,,y,X = i, X +1 = j) γ (k+1) i,j By Subsecions 22 and 23, we have =1 1 f (y 1,,y,X = i) =1 1 f (y 1,,y,X = i, X +1 = j) = =1 0 ( 1) and δ i f (y 1 i)γ i,j f (y 2 j)γ j F 3 Γ F 4 Γ F δ 0 F 1 Γ F 2 Γ F 1Γ i f (y i)γ i,j f (y +1 j)γ j F +2 Γ F δ 0 F 1 Γ F 2 Γ F 2 Γ i f (y 1 i)γ i,j f (y j) =1 1 f (y 1,,y,X = i) = =1 0 ( 1) δ i f (y 1 i)γ i F 2 Γ F 3 Γ F δ 0 F 1 Γ F 2 Γ F 1Γ i f (y i) Γ i F +1 Γ F δ 0 F 1 Γ F 2 Γ F 2 Γ i f (y 1 i)γ i F, which ends he proof roposiion 5 - Given a GHMM {Y ; X }, he expressions of he unknown parameers µ i and σ 2 i of he gaussian random variable Y (i), for any, obained a he (k +1) h ieraion of he EM algorihm, are C y µ (k+1) i = 10 ( ) 1 0 ( ) C (7) and 1 0 ( ) σ 2(k+1) i = µ ³ C y µ (k+1) i 1 ( ) ( ) C (8) 10

12 for any sae i ofhemarkovchain{x },where C = B c ; c = δ 0 F 1 Γ F 2 Γ F 1 Γ i f (y i); y =(y 1,,y ) 0 and he symbol denoes he Hadamard produc roof - By roposiion 3, we have µ (k+1) i = =1 σ 2(k+1) i = =1 α =1 α α (i) β (i) y (i) β (i) ³ (i) β (i) =1 α y µ (k+1) i (i) β (i) 2 (9) Replacing (??) in(??), by roposiion A5 of S99, we obain µ (k+1) i = =1 f (y 1,,y,X =i) y =1 f (y 1,,y,X =i) σ 2(k+1) i = =1 ³ f (y 1,,y,X =i) y µ (k+1) 2 i By Subsecion 22, we have =1 f (y 1,,y,X =i) =1 f (y 1,,y,X = i)y = =1 0 ( ) δ i f (y 1 i)γ i F 2 Γ F 3 Γ F δ 0 F 1 Γ F 2 Γ F 1Γ i f (y i) Γ i F +1 Γ F δ 0 F 1 Γ F 2 Γ F 1 Γ i f (y i) y 1 y y, 11

13 where he symbol denoes he Hadamard produc, =1 ³ f (y 1,,y,X = i) y µ (k+1) i =1 0 ( ) ³ 2 = δ i f (y 1 i)γ i F 2 Γ F 3 Γ F δ 0 F 1 Γ F 2 Γ F 1Γ i f (y i) Γ i F +1 Γ F δ 0 F 1 Γ F 2 Γ F 1 Γ i f (y i) y 1 µ (k+1) i ³ y µ (k+1) i ³ y µ (k+1) i and =1 f (y 1,,y,X = i) = =1 0 ( ) δ i f (y 1 i)γ i F 2 Γ F 3 Γ F δ 0 F 1 Γ F 2 Γ F 1Γ i f (y i) Γ i F +1 Γ F δ 0 F 1 Γ F 2 Γ F 1 Γ i f (y i), which ends he proof If we have a sequence of observaions wih unrecorded daa, we change he srucure of he various join pdf s in he expressions of he formulas in roposiions 4 and 5, puing he w-seps ransiion probabiliies marices in, as relaed in Subsecion ³ 21 Besides we mus muliply every y of formula (??) andevery y µ (k+1) 2 i of formula (??) by he indicaor funcion I, so defined ½ 1 if y has been recorded a ime I = 0 if y has been unrecorded a ime 12

14 4 Applicaion o air polluion daa he foregoing ieraive procedure for he idenificaion of he parameers of GHMMs has been implemened in a GAUSS code As in S99, we esimae he dimension m of he sae-space of he Markov chain by he Akaike Informaion Crierion (AIC) and he Bayesian Informaion Crierion (BIC) Now le us consider he ime series of hourly mean concenraion of SO 2 in he 48 h week of 1998, recorded by he air polluion esing saion placed in via Goisis in Bergamo (for he daa se, see Appendix) I is clear ha some daa have no been recorded or validaed As we have seen, his is no a big rouble o esimae he parameers of he model, because he likelihood funcion may be obained even if some daa are no available In he series of hourly mean concenraions y 1,,y 168, he values y 2, y 26, y 50, y 74 y 98, y 122, y 146, y 152, y 153, y 154, y 155, y 156 have no been recorded; so we have o consider he ransiion probabiliies γ i1,i 3 (2), γ i25,i 27 (2), γ i49,i 51 (2), γ i73,i 75 (2),γ i97,i 99 (2),γ i121,i 123 (2), γ i145,i 147 (2), γ i151,i 157 (6) Hence he likelihood funcions is à L 168 (φ) = δ 0 G 1 Γ =3 25Y G! Ã Γ =27 49Y G! Ã Γ =51 73Y G! Ã Γ =75 97Y G! Ã! Ã! Ã!! Y121 Y145 Y151 Y168 Γ =99 G Γ =123 G Γ =147 G Γ Ã 5 =157 G In he same way, he w-seps ransiion probabiliies will be adoped o obain he explici formulas of he esimaor γ (k+1) i,j, µ (k+1) i, σ 2(k+1) i replacing in vecors A, B, C he marices F 2, F 26, F 50, F 74 F 98, F 122, F 146, F 152, F 153, F 154, F 155, F 156 wih he ideniy marix erforming he EM algorihm we obain he following values of log-likelihood, as a funcion of he number m of saes, and he corresponding values of AIC and BIC: m log-likelihood AIC BIC he saring values of he parameers have been randomly generaed and he maximizaion procedures have been repeaed more han once Given ha he iniial disribuion is non-informaive abou he ransiion probabiliies, 13

15 δ has been assumed known and fixed for any ieraion of he EM algorihm: δ i =1, for i =1;δ i =0, for any i =2,,m Considering boh he AIC and he BIC as model selecion crierion, we nln L ³φ (k+1) o choose a six-sae Markov chain he sequence ³ (80) a he 80 h ieraion o ln L 168 φ converges = , saringfromavalue ³ (0) ln L 168 φ = he esimaions of he parameers of he six gaussian pdf sare i i i 106e µ (80) σ 2(80) he ransiion probabiliy marix of he Markov chain is Γ (80) = from which we have he saionary iniial disribuion δ (80) =(00723; 00972; 00528; 01765; 03757; 02255) 0 heimespeninsaei upon each reurn o i for a Markov chain has a geomeric disribuion wih mean 1/(1 γ i,i ); hence he expeced ime spen in sae i, foranyi =1,,6, is: mean number of hours i spen in sae i Conclusions In his paper some resuls obained in aroli and Spezia (1999) have been updaed, in order o sudy univariae gaussian ime series wih unrecorded 14

16 daa New versions of he expressions of he esimaors γ (k+1) i,j,µ (k+1) i,σ 2(k+1) i of gaussian hidden Markov models (GHMMs) have been obained Furhermore an applicaion of GHMMs o a ime series of an air polluan wih unrecorded daa has been shown In his applicaion, he dimension m of he Markov chain sae-space has been esimaed by wo maximum-penalized-likelihood mehods, he Akaike Informaion Crierion (AIC) and he Bayes Informaion Crierion (BIC) Bu he way o esimae m isyeanopenquesionbecause he consisency of AIC and BIC has no been formally esablished However, we are sudying how o esimae he unknown m hrough some explicaives variables Appendix DaasediscussedinSecion4-Series of hourly mean concenraions, in µg/m 3, of sulphur dioxide (SO 2 ) in he 48 h week of 1998, from Monday he 30 h of November o Sunday he 6 h of December, recorded by he air polluion esing saion placed in via Goisis in Bergamo: hour Mon ue Wed hu Fri Sa Sun ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** References aroli R and Spezia L (1999) Gaussian hidden Markov models: parameers esimaion and applicaions o air polluion daa, SerieEn 94,Isiuo di Saisica, Universià Caolica SC di Milano 15

17 Wu C F J (1983) On he Convergence roperies of he EM Algorihm he Annals of Saisics, 11, Robera aroli Luigi Spezia Isiuo di Saisica Diparimeno di Ingegneria Universià Caolica SC Universià degli Sudi di Bergamo Via Necchi 9 Via Marconi Milano Dalmine (Bg) rparoli@miunicai spezia@unibgi Finio di sampare nel marzo 2000 presso l Isiuo di Saisica dell Universià Caolica del Sacro Cuore di Milano 16