Eigen analysis The correlation matrix plays a large role in statistical characterization and processing. It was previously shown that R is Hermetian

Transcription

1 ge aalss he orrelato matr plas a large role statstal haraterzato ad proessg It as preosl sho that s ermeta We ll o frther aalze the orrelato matr throgh ege aalss - egeales ad etors - matr dagoalzato - optmm flterg applatos

2 For a ermeta matr, e sh to fd a etor satsfg lear trasformato b does ot hage dreto of for a matr there are egeetors ad egeales,,3, K, o see ths ote I ths a ol be f the ro/olms of I are learl depedet, or f det I

3 3 det I s a th order polomal, the roots of hh are the egeales,,, eah egeetor s assoated th ol oe egeale. the egeetors are ot e a a ample: for a to-sample etor of zero mea hle ose ad

4 ge Propertes If,,, are the egeales of, the mltpl both sdes b k tmes k k ths k k,,, k K are the egeales of k. he egeetors,,, of are learl depedet a for all ozero salars a, a,, a 4

5 he egeales of are real ad oegate o sho ths, ote that mltplg b ges se s poste sem-defte ad > I most ases, s poste defte ad >,, K, 5

6 If,,, are e egeales of, the the orrespodg, egeales,,, are orthogoal. o proe ths, ote that mltflg b ges * Also, se s real ad s ermeta post mltfg b sbstttg ths from * se,,, are e 6

7 7 Dagoalzato of Let,,, be the e egeetors of ad take,,, to be the orthoormal egeetors Defe [,,, ] ad Ωdag,,, he [ ] L O L L K,,,

8 Note frst that se the egeetors are orthoormal I or ths s tar. Net, ote that the set of epressos,, K a be rtte as or Also ote that ths ges ad here dag /,/, K,/ 8

9 9 Note that det det det B A AB ad trae trae BA AB Usg ths, det det det det det hs det det Smlarl trae trae trae trae hs trae

10 A orrelato matr s ll odtoed f ma / m s large hs s er mportat sgal proessg. Cosder the relato odel/ Flter parameters d Sppose ad d are pertrbed sh that δ/ ad δd/d are o the order ε<<. he Sgal statst matres δ εχ ε ma m Codto mber of hs f s ll odtoed small hages or d a lead to bg hages.

11 he dsrete Karhme-Loee rasform KL Cosder the represetato of the sample etor from the proess {} th orrelato matr. Let,,, be the orthoormal egeetors of. he e a epress as here [,,, ] ad [,,, ] Solg the aboe for or proeto of oto

12 Cosder the orrelato betee terms } { } { } { hs otherse } { Sppose e old lke to represet th feer terms N N < ˆ

13 3 hs N N ˆ he error eerg s ge b N N N N } { } { hs to mmze the error selet the egeetors assoated th the largest egeales.

14 he athed Flter Cosder fdg the flter oeffets for to ases: determst sgal ad stohast sgal Let + he b leart Sgal Nose s + 4

15 5 If the flter parameters are ],,, [ K ad the obserato s ],,, [ + K the + ad s Determst Case: he SN at tme a be defed as } { SN s SN } { } {

16 6 Sppose s zero mea ad orrelated, I. he SN f e restrt Note that Shartz s ealt states B B A A B A th ealt ol f B A k hs SN ostat Ad the SN s mamm he k

17 ample: a ommatos sstem seds ot oe of to sgals Smbol 3 4 Smbol We reee + here s..d Gasa k here k s sh that hs / [,,,] 7

18 hs the mplse respose of the flter s For sgal set 3 s For sgal s f SN 8

19 9 athed Flter for stohast sgals As before s + + No defe the SN as s SN } { } { As before, f I ad SN e a ths ealetl mamze

20 eall from Shartz, A B A A B B th ealt ol f A kb hs s mamzed for hs k s a egeale ad s a egeetor! hs ges k SN o set flter, fd the largest egeetor of ad set ma. he SN ma