Bayesian Separation of Non-Stationary Mixtures of Dependent Gaussian Sources

Transcription

1 Bayea earao of No-aoary Mure of Deede Gaua ource Dez Geçağa rca. uruoğu Ayşı rüzü Boğazç Uvery ecrca ad ecroc geerg DearmeBebek3434 Đabu Turkey ITI ogo Nazoae dee cerche va G. Moruzz 564 Pa Iay Abrac. I h ork e rooe a ove aroach o erform Deede omoe Aay DA. DA ca be hough a he earao of ae deede ource from her oberved mure hch a more reac mode ha Ideede omoe Aay IA here he ource are aumed o be deede. I geera he ource ca be aoemoray deede ad he mg yem may be o-aoary. ere e rooe a DA agorhm ha combe coce of arce fer ad Markov ha Moe aro MM mehod order o earae o-aoary mure of aay deede Gaua ource. eyord: Bayea ource earao Parce Fer Markov ha Moe aro. INTODUTION ource earao robem ha aay araced may reearcher from dffere dce uch a eecommucao bomedce audo eech ad arohyc. Th are a a reu of he eed o vegae dffere ad reeva roere of he dered ga hch are geeray hdde a he med ad oy obervao. I he a decade reearch o ource earao a may focued o he deedece aumo of he ource ha are med hece caed a he IA []. oever he hyca ord he deedece aumo cao aay hod ad he deedece houd be ake o accou. I eraure here a med umber of referece coderg he robem of deede ource [-3]. Moreover he caca IA aroache he earao robem geeray haded a a bd mehod. oever hyca ord he reearcher do have ome a ror koedge abou her ecfc robem ad ca ao make ue of h formao. Therefore ead of bd roceg h a ror formao ca be eoed [4-5]. I ource earao robem aoher vegao oc he aoary of he ga. If he mg yem coa over me ad he ource are aoary he MM mehod ca be aed. oe [3] rooe MM mehod for earag boh deede ad deede ource for uch cae. O he oher had for o-aoary acao aoher Bayea aroach ko a arce fer [6-7] have bee dey ued for roceg he daa Th ork uored aray by TÜBĐTA-N rojec umber: 7 ad aray by Boğazç Uvery cefc eearch Fud rojec umber: 4A. The reearcher a uored by NATO-TÜBĐTA A feoh hroughou h reearch a ITI-N Iay.

2 equeay a ooed o he bach aure of he MM mehod. I order o mode he o-aoare arce fer have bee aed o he earao of auoregreve A ource hoe A coeffce are me-varyg [8] ad he ource are deede a a gve me a. I [9-] oher o-aoary IA robem are oved by arce fer for o-gaua ource here he mg yem are me-varyg. I h ork e rooe a ove mehod order o earae o deede Gaua ource from her o-aoary mure. I h aroach e ued arce fer o emae he coeffce of mg marce ad MM o earae he ource. I h aroach arce fer ca be ued for emag ay oaoary mg mar. Thu dffere from he mehod [3 h.3] here eca form of o-aoary mg marce are emaed by MM. PATI FIT Parce fer are ued order o equeay udae a ror koedge abou ome redeermed ae varabe by ug he obervao daa. I geera hee ae varabe are he hdde varabe a o-gaua ad oear ae-ace modeg yem. uch a yem ca be gve by he foog equao: θ θ v f h θ here θ v ad reree ae obervao roce ad obervao oe reecvey. ere he objecve o equeay oba he a oeror drbuo of he ae varabe obaed va he obervao daa gahered u o ha me.e. θ. Drbuo are aromaed erm of arce a foo [6-7]: : : : : θ : : θ δ θ here θ : δ. deoe he egh he h arce ad he roecker dea oeraor reecvey. The arce ha ake ace equao are dra by a mehod ko a he equea Imorace amg [6-7] ad he correodg Imorace Wegh for each of hem deoed by hch defed a foo: θ θ θ 3 q θ θ : : here q. fuco caed a he Imorace Fuco ad drag ame from h robaby dey fuco df eaer ha ha of orga drbuo [67].

3 MM MTOD FO PAATING TATIONAY MITU I h eco bref backgroud formao reeed abou he MM mehod ha uzed for he earao of aoary mure of ource [3]. Th mehod aume ha he mg yem ad he ource are aoary. The foog mode ued for he mure: 4 here deoe he mure overa coa mea mg mar ource ad he oe marce reecvey. Thee marce are rereeed a foo: m '... ' here. deoe he raoo oeraor. I h ork he ' ource are aay deede Gaua rocee ad each ource emoray ucorreaed. Uke he ource he addve oe comoe are boh aay ad emoray deede. Becaue 4 reree a mure mode a a gve me a a umber of obervao ay form he foog mar oao for he emao from h amed bach [3]: e N 5 here N... ad e deoe a - dmeoa vecor of oe [3]. By augmeg he ad he e vecor 5 u o a more comac form a foo: here ad Z e Z N 6. The he kehood fuco gve a foo: Z e r Z Z 7 here deoe he dagoa covarace mar of he oe vecor ad r. reree he race oeraor. Drbuo 7 ko a he Mar-Norma drbuo [3]. For he mode of 6 he foog cojugae ror drbuo ca be ued for he mode arameer [3] here MN ad IW ad for Mar Norma ad Ivered Whar drbuo reecvey. I he foog equao η V B deoe he hyerarameer hrough hch he a ror formao eoed.

4 MN r m IW r IW r MN r m B B V V e e e e η η 8 Above deoe he covarace mar of he ource ad o coraed o be dagoa ad a of eeme are ef free o mode he deedece beee he ource. Afer ome agebra he foog oeror codoa are obaed o be uzed he Gbb amg hch he referred MM mehod here [3]: [ ] e e e e e V B Z Z G G Z Z Z Z Z r r r r m η 9 By cycg hrough he oeror codoa gve above Gbb amg ca be erformed order o emae he mode arameer [3]. T POPOD MTOD I eraure MM echque are geeray ued for bach roceg here he mode arameer are aumed o o chage h he oberved daa bock []. Tha hy he arce ferg mehod have bee deveoed order o make he emao equeay cae of o-aoare. I h ork he objecve o earae deede Gaua ource from her mure here he mg yem

5 me-varyg uke he cearo gve he revou eco. Tha he eeme of he mg mar 5 chage over me. o 5 ca be re a foo: N e here.... o e rooe o ue arce ferg o emae he me-varyg eeme of he mg mar. ere aumed ha here o coa mea he mure.e. ad e have a ror formao abou he ac of he ource ad he oe o ha e ca form formave ror for hee. ve f a he ac of he ource ad he oe are ko a ror earag he me evouo of he mg mar ad he ource eed o e of ae he arce ferg mode.e. { } θ of a ay me a. ere he vecor ha formed by cocaeag he ro of he mar oe by oe. ce arce fer emae he jo df gve by ource mu be egraed ou from h jo df order o oba he marga df emae of he mg mar. From 3 ee ha he morace egh of each arce ereed erm of he kehood fuco f a ror rao are ued for he ae [6-7]. The morace egh emao 3 ake he foog form: θ To oba he marga morace egh of he kehood fuco egraed: d From ee ha codoa dey ha a Gaua drbuo. ce he ource are ao Gaua drbued he egrao ha ao a Gaua drbuo. Thu ead of emag for each arce oy mea ad varace emao ca be foud a foo ug Moe aro egrao: { } { } { } { } { } { } { } j j j j d d ˆ ˆ Σ 3 here ame j ca be eay dra from he Gaua df gve by ad he oa umber of arce. By hee oerao become a foo: 4

6 here Gaua hoe mea ad covarace marce ca be foud by 3. I addo he a ror ae rao hch dcued above ca be re a foo: v 5 here v N Q ad Q dagoa. ad 5 f o he geera ae-ace formuao of he arce ferg ho. avg foud he me evouo of he mg mar eeme by egrag he ource a oued above ource ao eed o be eraced from he mure. ad he mg mar coeffce bee coa over me he MM mehod gve above coud have bee ued for emag he ource. oever h o he cae our cearo. Thu e rooe o aume ha he eeme of he mg mar do o chage coderaby over ma bock of daa. I h cae oe may ue MM hee bock hoever he choce of he arorae ror drbuo of he mg mar eeme are a a robem. Th due o he requreme of ug formave ror drbuo for he arameer gve 8 [3. 56]. ce e ca oba emae of hee mg mar eeme by ug he arce ferg cheme hch eaed above e rooe o ue hee emae order o form ome aromae formave ror for he mg mar eeme hch are deoed by 8 ad 9. Tabe. Peudo-code PATI FIT PAT I. Iaze he arce N P for... here P a dagoa covarace mar. θ : v ad N II. Dra e ame { } v N Q N P yerarameer: IW V η III. acuae he morace egh by egrag ou he ource: here N ; ˆ Σˆ N ; ˆ ˆ Σ deoe he evauao of he obervao daa a he Gaua df hoe mea ad covarace marce are gve by 3 IV. Normaze he morace egh: V. eame a each erao from { } morace egh equa o each oher. VI. Go o e II ad reea.... ad make he uormazed

7 MM PAT I. mae he Mmum Mea quare rror MM emae of a foo: d : : : : : ˆ II. For daa bock of ze M cacuae he mea of each mg mar eeme foud above ad ue ha a he mg mar ror he fr equao of 9: ˆ M M For oher arameer ue he ror gve 8. III. The cyce hrough he Gbb erao by ug he oeror codoa 9: ar h he a yce: Dcard he varae of bur erod ad emae he arameer a foo: PIMNT To verfy he erformace he foog cearo muaed o he comuer: : co P Q P m hyerarameer N N a π a a a η π B V By ug hee formave ror boh he mg mar eeme ad he ource are emaed afacory a ho Fgure.

8 FIGU. a Obervao: b ource ad MM mae c Obervao: d ource: ad MM mae e Fr mg mar eeme ad emae: a f ecod mg mar eeme ad emae: a g: Zoomed aveform of ource ad MM mae for bock ze of h: Zoomed aveform of ource ad MM mae for bock ze of

9 ONUION AND FUTU WO I h ork e rooe a ove mehod o earae o-aoary mure of aay deede Gaua ource. ere arce fer uzed o emae he o-aoary mg mar he Gbb amg ued o erac he ource. By muao oberved ha he ga o Ierferece ao raed aromaey o 6 db from db a a reu of he earao. The ucce of he rooed DA agorhm a romg reu hch ca be ued fuure acao here he deedece aumo of IA avoded for more reac hyca robem modeg. FN. A. yvare J. arhue. Oja Ideede omoe Aay Joh Wey & o.. A.. Barro The Ideedece Aumo: Deede omoe Aay Advace Ideede omoe Aay eded by M. Groam rger. 3. D. B. oe Muvarae Bayea ac Mode for ource earao ad ga Umg hama & a/ uh A Bayea aroach o ource earao Proceedg of he Fr Ieraoa Workho o Ideede omoe Aay ad ga earao: IA'99 Auo Frace Ja A. Mohammad-Djafar A Bayea aroach o ource earao 9 h Ieraoa Workho o Mamum roy ad Bayea Mehod Idaho UA Aug A. Douce N. de Frea N. J. Gordo equea Moe aro Mehod Pracce rger- Verag. 7. A. Douce. God. Adreu O equea Moe aro amg mehod for Bayea ferg ac ad omug Adreu. God A arce fer for mode baed audo ource earao Proceedg of he Ieraoa Workho o Ideede omoe Aay ad ga earao: IA' ek Fad Jue.. 9. A. Ahmed ga earao Ph.D. The ambrdge Uvery.. A. Ahmed. Adreu A. Douce P. J. W. ayer O-e o-aoary IA ug mure mode IAP vero. ober Parce Fer for No-aoary IA Advace Ideede omoe Aay eded by. M. Groam rger... P. ober G. aea Moe aro aca Mehod rger-verag D. B. oe A Bayea Aroach o Bd ource earao Joura of Ierdcary Mahemac Vo. 5 No