On Parameter Estimation of the Envelope Gaussian Mixture Model

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1 Australan Communcatons Thory Worsho (AusCTW) On Paramtr Estmaton of th Envlo Gaussan Mtur Modl Lnyun Huang Dartmnt of ECSE Monash Unvrsty, Australa Y Hong Dartmnt of ECSE Monash Unvrsty, Australa Emanul Vtrbo Dartmnt of ECSE Monash Unvrsty, Australa Abstract In many communcaton systms, th Gaussan mtur modl (GMM) s wdly usd to charactrz non-gaussan man-mad and natural ntrfrnc. Th nvlo dstrbuton of such nos modl s oftn rssd as a wghtd sum of Raylgh f n-has and quadratur comonnts of th nos ar dndnt. Instad, n ths ar, a sml and act closd form robablty dnsty functon of th nvlo Gaussan mtur modl (.. th nvlo of ndndnt n-has and quadratur comonnts of coml non-gaussan nos) s obtand. Furthrmor, th roblm of stmatng of th nvlo Gaussan mtur aramtrs s addrssd. Th roosd stmator of wghts and varancs s basd uon th Ectaton-Mamzaton (EM) algorthm. Ind Trms Gaussan mtur nos, Envlo Gaussan Mtur, Mamum Llhood, EM algorthm, aramtr stmaton. I. INTRODUCTION In wrlss communcaton systms, classcal wht Gaussan nos s oftn assumd to b a vry accurat modl. Howvr, n othr cass, such as th undrwatr [] and th owrln communcatons [], th nos may hbt non- Gaussan bhavour and thus t s mortant to consdr dffrnt vrsatl and robust nos/ntrfrnc modls. In 977, Mddlton [3] roosd th Mddlton s Class A nos modl, a Gaussan mtur dnsty modl wth Posson slcton, to dscrb th lctromagntc (EM) ntrfrnc from a varty of nos sourcs. Arzbrgr t al. [] also suggstd that th nos n owrln channls can also b modlld by th Gaussan mtur modl, a aramtrc robablty dnsty functon (df) rssd as summaton of wghtd Gaussan df s. Th nvlo dstrbuton of ths mtur dnsty s oftn rrsntd as wghtd sum of Raylgh undr th assumton that th n-has comonnts and quadratur comonnts of th nos ar dndnt []. In our wor, w assum that both n-has and quadratur comonnts ar ndndnt and dntcally dstrbutd (..d.) random varabls as motvatd by [], []. Hnc, th nvlo dstrbuton of n-has and quadratur nos comonnts wll not rsult n th Raylgh mtur modl, but gv rs to th nvlo Gaussan mtur modl (EGMM). Paramtrs of th nvlo Gaussan mtur modl can b stmatd by mamum llhood stmaton (MLE). Whn closd form rsson cannot b found for MLE, tratv mthods, such as th Nwton-basd and th EM algorthm, ar usd. Ths algorthms tratvly mamz th log llhood functon. In an arlr wor by Sar t al. [], a convntonal mamum llhood stmator usng th quas-nwton mthod s usd. In ths ar, w consdr th mamum llhood stmator va th EM algorthm, a wdly usd mthod oularzd by Dmstr, Lard and Rubn n 977 [5]. Th EM algorthm has bn usd n many aramtrs stmaton roblms, scally n dalng wth th curvd onntal dnsts [6], such as th Mddlton s Class A modl [7], th Raylgh mtur [8] and th Gaussan mtur dnsty [9]. Mamum llhood stmaton va th EM algorthm s consdrd n ths ar for th nvlo Gaussan mtur modl. Th ar s organsd as follows. In Scton II, th modl for non-gaussan channl nos s dscrbd. In Scton III, th nvlo Gaussan mtur dnsty functon s drvd. In Scton IV, th mamum llhood (ML) stmator va th EM algorthm s rsntd. Th rformanc of th EM algorthm s comard to that of convntonal ML stmator usng th quas-nwton mthod n Scton V. II. NON-GAUSSIAN NOISE MODEL W frst dscuss th wdly usd two-trm Gaussan mtur modl for th n-has and quadratur nos amltuds Y and Z. Both n-has and quadratur nos amltuds ar consdrd as ndndnt random varabls wth th followng two-trm Gaussan mtur dnsty functon: GM(a,, ) = a ( a) + () whr al a al s th mng coffcnt and, ar th varancs of th two Gaussan comonnts. An aml whr th abov modl can b usd s whn bacground nos s always rsnt and mulsv nos vnts occur wth robablty ( a). Th frst Gaussan comonnt can b sn as th nomnal bacground nos wth varanc. Th scond comonnt rrsnts th combnaton of th bacground nos and th mulsv nos, whn mulsv nos vnts occur. Snc both bacground nos and mulsv nos ar assumd to b Gaussan random varabls, th sum of th two wll also b a Gaussan random varabl. Th two-trm Gaussan mtur modl was also consdrd as an aromaton of Mddlton s Class A nos modl [],[] and has bn usd tnsvly n both modllng th //$3. IEEE 7

2 Australan Communcatons Thory Worsho (AusCTW) owrln nos [] and th undrwatr communcatons nos []. In many ractcal cass, a small numbr K of Gaussan comonnts (.g., or 3) ar suffcnt to accuratly modl th nos wthout ovrfttng. In gnral w hav GM(, ) = = y () whr = { } K =, ar mng coffcnts of ach Gaussan dnsty and = =. Th varancs of th Gaussan df s ar = { }K =, and w assum that > > >... >. III. ENVELOPE GAUSSIAN MIXTURE Lt us now consdr th nos nvlo random varabl X as a functon of th n-has and quadratur nos comonnts Y and Z. W start wth th cas whr f Y (y) and f Z (z) ar two-trm Gaussan mturs as n () and Y and Z ar..d. random varabls. Thn X can b rssd as: X = Y + Z (3) Gvn th jont df f Y,Z (y, z), th cumulatv dnsty functon (cdf), F X (), s dfnd as: Z Z F X () = f Y,Z (y, z)dydz () Y +Z al W can thn fnd f X () by dffrntatng F X () drctly usng dffrnton rul du to Lbntz [], w hav: f X () = Z f Y,Z ( z,z) z +f Y,Z ( z,z) dz (5) Snc Y and Z ar assumd to b ndndnt, th two trms f Y,Z ( z,z) and f Y,Z ( z,z) wll rsult n: f Y,Z ( z,z)=f Y,Z ( z,z) = a a( a) + z +z a( a) + z z + Substtutng (6) nto (5), ylds: f X () = + Z a( a) al a z z ( a) z + z + z z z +z (6) ( a) z dy (7) Th ntgral (7) can b comutd n closd form by lttng z = sn () and dz = cos ()d to yld: f X () = a a( a) + + ( a) + I (8) whr I ( ) = Z cos () d s th modfd Bssl functon of th frst nd of zro-th ordr. In (8), frst two trms follow a Raylgh dstrbuton. Th trm wth modfd Bssl functon dstngushd th nvlo dstrbuton of two-trm Gaussan mtur from th two-trm Raylgh mtur. Th nvlo of two-trm Gaussan mtur s shown n Fg. for dffrnt valus of a and and th rato of two varancs c = / =. On can obsrv that whn th valu of mng coffcnt a =.7, th tal of th df s aromatly lnarly dcayng from th a valu (dashd ln n Fg. ). Whn th rato of th two varancs /!, smlar to th cas of a two-trm Raylgh mtur modl, th nvlo of two-trm Gaussan mtur (8) turns nto a sngl Raylgh dstrbuton. Th ffct of varyng th rato of / s llustratd n Fg., n whch th dashd gry ln s th Raylgh df wth varanc =. Howvr, whn, th nvlo Gaussan mtur s sgnfcantly dffrnt from th Raylgh mtur as shown n Fg. 3. Th two-trm Raylgh mtur modl oftn hbts a mor ronouncd mut-modal bhavour than th nvlo of two-trm Gaussan mtur. Assumng that th n-has and quadratur comonnts ar not dntcally dstrbutd and follow dffrnt GMM wth K and L comonnts,., f Y (y) = f Z (z) = [ y ] q [ y ] = LX [ z ] [ z ] = y [ y ] (9) y [ z ] () thn a mor gnral nvlo Gaussan mtur dstrbuton f X () can b wrttn as: LX [ y ] [ z ] f X () = [ = = y] [ z] I [ z ] [ y ] [ + z ] [ y ] () Howvr, n many ractcal cass, th n-has and quadratur nos comonnts can b dscrbd by th sam dstrbuton wth th sam aramtrs, and th nvlo Gaussan mtur dnsty smlfs to: f X () = = = I + IV. MAXIMUM LIKELIHOOD ESTIMATION VIA THE EM ALGORITHM () In ths scton, w adot th wll-nown two-st tratv mthod calld th EM Algorthm that fnds th mamum llhood or mamum a ostror stmats of aramtrs 8

3 Australan Communcatons Thory Worsho (AusCTW) robablty dnsty a =. a =. a =.7 a =.9 robablty dnsty Raylgh mtur Envlo Gaussan mtur. 6 8 Fg.. Envlo of two-trm Gaussan mtur df wth mng coffcnt. al a al.9 and c = / =. robablty dnsty Raylgh c = c = 5 c = 5 c = Fg.. Envlo of two-trm Gaussan mtur df wth mng coffcnt a =.5and c = / =, 5, 5, 3. Th nvlo Gaussan mtur convrgs to th Raylgh dstrbuton as c! (dashd gry curv). n statstcal modls n whch obsrvatons ar tratd as ncomlt data [3]. Paramtrs of mtur dnsts, such as th Raylgh mtur modl, th Mddlton s Class A modl and th Gaussan mtur can b stmatd by usng th EM algorthm. Hnc t s natural to rdct that aramtrs of th nvlo Gaussan mtur dnsty can b stmatd by usng th sam algorthm.. Th EM algorthm for th nvlo of Gaussan mtur modl s rsntd n And A. Not that n th rvous scton, Y and Z ar usd to rrsnt th n-has and quadratur nos amltuds. Hr, thy wll b usd as th latnt varabls [3] n th EM algorthm. Gvn a data st X = {,..., N }, w assum that all data samls ar..d.. Lt ( ) b th df that s Fg. 3. Comarson of nvlo of two-trm Gaussan mtur df wth two-trm Raylgh mtur both wth mng coffcnt a =.7 and c = / =5. govrnd by th st of aramtrs,, to b stmatd. W hav: NY (X ) = ( n ) n= L( X) = log (X ) = NX log ( n ) (3) n= whr L( X) s th log llhood functon. Our uros s to fnd th aramtrs whch mamz L( X) such that: ˆ ML = arg ma L( X) () Undr th assumton that th nvlo Gaussan mtur modl s tan, ( n ) s rlacd wth () and th ncomlt data log llhood functon s gvn by: L( X) = NX n log n= I = = n + n (5) In ordr to solv ths quaton, th EM algorthm s utlzd by ntroducng th latnt varabls. Howvr, unl clustrng roblms n Gaussan mtur or Raylgh mtur n whch only on st of latnt varabls s ntroducd, hr w mloy two sts of latnt varabls Y = {y } K = and Z = {z } K = as P bnary ndcator varabls (.. y {, }, z {, }, y =and P z =). Th valu y ndcats whch Gaussan comonnt n (9) gnrats th -th n-has nos saml and smlarly z for th quadratur nos samls. It s mortant to not that snc n-has nos samls and quadratur nos samls ar assumd to b ndndntly gnratd, y and z ar also ndndnt. Th roduct of ths two latnt varabls, y z, s bnary and forms a -dmnsonal ndcator functon (.. y z {, } and P P y z =). 9

4 Australan Communcatons Thory Worsho (AusCTW) W us th EM algorthm to stmat th aramtrs = {, }. Th ctaton st and mamzaton st ar dfnd as follows: E-st: Comut Q( () ), E[L( X) Y,Z, () ] M-st: Dtrmn = (+) mamzng Q( () ) whr () s th stmaton of at -th traton of th EM algorthm. W call X th ncomlt data and w assum that th comlt data S =(X, Y, Z) ncluds th bnary latnt varabls Y and Z. Thn th jont dnsty functon (, y, z) s: (, y, z) =(y, z)( y, z) =(y)(z)( y, z) (6) snc Y and Z ar ndndnt. Th roorton of th nos samls that ar gnratd by -th or -th Gaussan comonnt s or and thrfor th jont dstrbutons ovr Y and Z ar scfd n trms of th mng coffcnt and, such that (y = ) = and (z = ) =. Both y and z ar ndcator varabls, thrfor w can wrt jont dstrbuton n th followng form: (y, z) = = = [ ] y z (7) Smlarly, th condtonal dstrbuton of X gvn artcular valus for Y and Z s th nvlo Gaussan mtur comonnt that s: ( y, z) = = = = = [(, Thn w hav: () = X X (y, z)( y, z) y z = [ ( )] y z (8), )] y z (9) Ths s an quvalnt formulaton of th mtur modl nvolvng two lct latnt varabls. By dalng wth th comlt obsrvaton X, Y and Z, w can smlfy th log llhood functon usng (9). Rlacng ( n ) n (5) wth th quvalnt formulaton found n (9), w hav: L( X) = NX n= = = + log ( n y z {log + log, Usng (), th Q functon, bcoms: Q( () ), E[L( X) Y,Z, () ] = NX n= = = + log ( n E[y (n), )} () z (n) ]{log + log )} () whr E[y (n) z (n) ] = n,, s th condtonal robablty of X gvn Y and Z, w also call t soft assgnmnt (or rsonsblty) whch can b found by usng Bays thorm: n,, = E[y (n) z (n) ]=(y =,z = n ) = (y = )(z = )( y =,z = ) ( n ) = ( n, ) t= s t ( n s, t ) () s= Th mamzaton st of th EM algorthm fnds th rsson for and. W ntroduc th Lagrang multlr wth th constrant P = P = and solv th followng st of quatons: ( X X n,, log + n= = = for =,...,K. Summng lft sd ovr or gvs = N whch rsults n: = NX n,, () n= = Th varancs and can b found by solvng th st of K X N al log n W gt: n,, n= = = n + log I = P N n= = n,, n ( ( P N n= = + n = =,...,K(5) n,, n)) (6) whr ( ) = I( ) I and I ( ) ( ) s th modfd Bssl functon of frst nd of frst ordr. Equatons (6) ar non-lnar quatons and thrfor can only b solvd numrcally. V. SIMULATION RESULTS A smulaton of th roosd stmator va th EM algorthm s rformd, n whch w consdr an nvlo Gaussan mtur modl wth unnown varancs and mng coffcnts, howvr w assum that th numbr of comonnts s nown. Random data samls ar randomly gnratd from th dstrbuton wth nown aramtrs. An altrnatv mthod n stmatng aramtrs of th nvlo Gaussan mtur aart from th EM algorthm s th quas-nwton mthod (S [] for dtal). Th rformanc of th EM algorthm wll b comard wth th rformanc of th quas- Nwton mthod wth th BGFS st udat for th nvlo Gaussan mtur modl []. Th EM algorthm and th quas-nwton mthod wll trmnat whn th chang n log llhood functon s lss than stong crtron,. Th EM algorthm s mor oftn usd n aramtr stmaton of th curvd onntal famls, snc th quas-nwton mthod s mor comlcatd to mlmnt. On th othr hand, th EM algorthm njoys gratr smlcty and stablty but has 3

5 Australan Communcatons Thory Worsho (AusCTW) slowr convrgnc sd. Hr w focus on comarng thr convrgnc sd and accuracy n trms of traton numbrs and Man Squar Error (MSE) btwn th stmatd df and normalsd data hstograms. Our frst aml tas 5 data samls whch ar gnratd by th nvlo of a two-trm Gaussan mtur dstrbuton wth aramtrs =.3, =.7, =and =. Fg. llustrats th normalsd data hstogram wth N = 5 bns of gnratd data samls, togthr wth th stmatd nvlo Gaussan mtur df and tru nvlo Gaussan mtur df. Th MSE btwn th nvlo Gaussan mtur df and th normalsd data hstogram s dfnd as follows: MSE = N NX (Ŷ Y ) (7) = whr Y s th frquncy dnsty of th -th bn and Ŷ s th stmatd robablty dnsty tan at th mdont of - th bn. Th MSE btwn th stmatd df and hstogram s and th aramtrs stmatd ar ˆ =.335, ˆ =.69698, ˆ =.399 and ˆ =9.966, whch show that th ML stmator va th EM algorthm ylds good rformanc. Eaml shown n Fg. 5 also llustrats that aramtrs of th nvlo Gaussan mtur modl hav bn corrctly stmatd by th EM algorthm. In fact, for amls wth dstrbutons that ar wll saratd (.. ), both th EM algorthm and th quas- Nwton mthod rform wll n trms of rroducng th tru aramtrs. Howvr, n som ll-condtond cass (.. aml, 5, 7 and 8) whr too many comonnts hav bn ncludd n th modl du to ovrfttng or to smlar valus of varancs, both th EM algorthm and th quas-nwton mthod rform oorly n trms of attanng th tru valus of aramtrs as shown n TABLE I. In addton, aml 6 dos not convrg to th tru valu du to small saml sz. Fortunatly, both mthods may achv som local otma of th log llhood functons n all cass, thrfor th stmatd nvlo Gaussan mtur df s usng both mthods stll hbt cllnt agrmnt wth th hstograms. Dst that th EM algorthm rforms wll n trms of mamzng th log llhood functon, t has rlatvly slowr convrgnc rat dndng on th modls and data sz []. If w only comar th numbrs of tratons to convrgnc, quas-nwton would ta fwr tratons, snc th convrgnc sd of th quas-nwton algorthm s sur lnar, whras th EM algorthm convrgs lnarly []. TABLE I shows that th comutaton tratons for stmaton of th aramtrs of th nvlo of two-trm Gaussan mtur modl wth stong crtron, = 6. It s clar n th TABLE I that th quas-nwton mthod convrgs much fastr than th EM algorthm. Th quas- Nwton mthod, on avrag, too lss than /8 n traton numbrs of th EM algorthm. For aml, 6 and 7, numbr of traton for th EM algorthm can go ovr. Morovr, th comutaton tm r traton of th EM algorthm s robablty dnsty Hstogram Tru PDF Estmatd PDF 6 8 Fg.. Comarson of Normalsd data hstogram, tru nvlo Gaussan mtur df and stmatd nvlo Gaussan mtur df. robablty dnsty Hstogram Tru PDF Estmatd PDF Fg. 5. Normalsd data hstogram gnratd wth mng coffcnt =.3, =., 3 =.3, =, = 9 and 3 = 5. Stong crtron s st to b 6. Numbr of traton tan for th EM algorthm to convrg s 9. Paramtrs stmatd ar ˆ =.35, ˆ =.679, ˆ 3 =.969, ˆ =.967, ˆ = and ˆ3 =.77. much longr than that of th quas-nwton mthod, snc th varancs, and n (6) hav to b found numrcally n ach EM traton. For ths rason, w conclud that, n trms of traton numbrs, th quas-nwton mthod should b rfrrd to th EM algorthm n stmaton aramtrs of th nvlo Gaussan mtur dnsty functons. Howvr, radr should ta not that th EM algorthm s stll an attractv mthod n ths stmaton roblm du to gratr smlcty (.. automatc satsfacton of robablty constrants and monotonc convrgnc wthout th nd to st a st sz). On th othr hand, mlmntaton of th quas-nwton mthod s ndd comlcatd. As Jamshdan and Jnnrch 3

6 Australan Communcatons Thory Worsho (AusCTW) TABLE I COMPARISON OF THE PERFORMANCE OF THE EM ALGORITHM WITH THE PERFORMANCE OF THE QUASI-NEWTON METHOD FOR THE ENVELOPE GAUSSIAN MIXTURE MODEL Tru Paramtrs Estmatd aramtrs va EM Estmatd aramtrs va QN EM QN # Itr Itr < < < [5] hav ontd out, to choos btwn th EM algorthm and th quas-nwton mthod s mor or lss a rsonal choc. VI. CONCLUSION In ths ar, w hav drvd th gnral rsson for th nvlo Gaussan mtur modl. Such modl s dffrnt from th Raylgh mtur and may b ald to dscrb th nvlo of owrln nos and th nvlo of undrwatr communcaton nos. W roosd th EM algorthm to stmat th aramtrs of th nvlo Gaussan mtur. Fnally, w dscussd th convrgnc sd and accuracy of th EM algorthm by smulatons. Ths was comard wth th quas-nwton algorthm. ACKNOWLEDGEMENT Th authors would l to acnowldg th suort from Monash Faculty of Engnrng Sd Fundng Schm. REFERENCES [] F. Sar, N. Sar, and L. Ml, Modllng of sa cluttr wth Gaussan mturs and stmaton of th cluttr aramtr, Sgnal Procssng and Communcatons Alcatons Conf. (SPCAC ), Proc. vol., , Istanbul, Tury. [] M. Arzbrgr, K. Dostrt, T. Waldc, and M. Zmmrmann, Fundamntal rorts of th low voltag owr dstrbuton grd, n Proc. st Int. Sym. Powr-Ln Communcatons and ts Alcatons (ISPLC 997), Mar. 997,. 5 5, Essn, Grmany. [3] D. Mddlton, Statstcal-hyscal modls of urban rado-nos nvronmnts-part : Foundatons, IEEE Trans. Elctromagn. Comat., vol. EMC-, no., , May 97. [] H. C. Frrra, L. Lam, J. Nwbury, T. G. Swart, Powr Ln Communcatons: Thory and Alcatons for Narrowband and Broadband Communcatons ovr Powr Lns, John Wly and Sons Ltd,. [5] A. P. Dmstr, N. M. Lard, and D. B. Rdn, Mamum llhood from ncomlt data va th EM algorthm, J. Roy. Statst. Soc., Sr. B, vol. 39,. 38, 977. [6] C.F. J. Wu, On th convrgnc rorts of th EM algorthm, Th Annals of Statstcs, vol., no.,. 95 3, 983. [7] S. M. Zabn and H. V. Poor, Effcnt stmaton of Class A nos aramtrs va th EM algorthm, IEEE Trans. Inf. Thory, vol. 37, no.,. 6 7, Jan. 99. [8] J. C. Sabra, F. Com, O. Pujol, J. Maur, P. Radva, and J. Sanchs, Raylgh mtur modl for laqu charactrzaton n ntravascular ultrasound, IEEE Trans. Bomd. Eng., vol. 58, no. 5,. 3 3, May. [9] C. M. Bsho, Nural Ntwors for Pattrn Rcognton. Oford Unv. Prss, 995. [] D. Mddlton, Statstcal-hyscal modls of lctromagntc ntrfrnc, IEEE Trans. Elctromagn. Comat., vol. EMC-9, no. 3,. 6 7, Aug [] D. Mddlton, Canoncal and quas-canoncal robablty modls of Class A ntrfrnc, IEEE Trans. Elctromagn. Comat., vol. EMC-5, no.,. 76 6, May 983. [] A. Paouls, Probablty, Random Varabls and Stochastc Procsss. Nw Yor: McGraw-Bll, 965. [3] C. M. Bsho, Pattrn Rcognton and Machn Larnng. Srngr, Aug. 6. [] M. Watanab and K. Yamaguch, Th EM Algorthm and Rlatd Statstcal Modls, Evanston, IL: Routldg,, ch. 7. [5] J. M. Jamshdan and R. I. Jnnrch, Acclraton of th EM Algorthm by usng Quas-Nwton Mthods, J. R. Statstcs Soc., vol. B59, , 997. APPENDIX A. EM algorthm for th Envlo Gaussan Mtur Modl. Intalz th mng coffcnts and, togthr wth varancs and wth random valus.. E st: Gvn th data saml n, valuat th rsonsblty usng th currnt aramtrs and for all and from to K n,, = E[y (n) z (n) ]=(y =,z = n ) = ( n, ) t= s t ( n s, t ) s= (8) 3. M st: Udat th aramtrs usng th currnt rsonsblty = NX n,, (9) N = n= = P N n= = n,, n ( ( P N n= = n,, n)) (3) for all from to K. ( ) = I( ) I and I ( ) ( ) s th modfd Bssl functon of frst nd of frst ordr. Equaton (3) ar non-lnar quatons and thrfor can only b solvd numrcally.. Evaluat th llhood functon ) ( NX X K L( X) = log ( n n= = =, ) (3) and chc th convrgnc of thr th aramtrs or th llhood functon. If th convrgnc crtron s not satsfd, rturn to St (E st). 3

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Review - Probabilistic Classification Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 http://www.ngr.mun.ca/~charlsr Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw