COMPARISON OF DIFFERENT WAVELET BASES IN THE CASE OF WAVELETS EXPANSIONS OF RANDOM PROCESSES Olga Polosmak

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1 4 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 COMPAISON OF DIFFEENT WAVELET BASES IN THE CASE OF WAVELETS EXPANSIONS OF ANDOM POCESSES Olga Polosa Abstrat: I the paper wavelets expasios of rado proesses are studied The atter is that although it is eough iforatio for wavelets expasios of deteriisti futios, for rado proesses suh theory is wea ad it should be developed The paper ivestigates uifor overgee of wavelet expasios of Gaussia rado proesses The overgee is obtaied uder siple geeral oditios o proesses ad wavelets whih a be easily verified Appliatios of the developed tehique are show for several wavelet bases So, oditios of uifor overgee for Battle-Learie wavelets ad Meyer wavelets expasios of Gaussia rado proesses are preseted Aother useful i various oputatioal appliatios thig is the rate of overgee, espeially if we are iterested i the optiality of the stohasti approxiatio or the siulatios A expliit estiate of the rate of uifor overgee for Battle-Learie wavelets ad Meyer wavelets expasios of Gaussia rado proesses is obtaied ad opared Keywords: rado proesses, wavelets expasio, uifor overgee, Battle-Learie wavelets, Meyer wavelets, Gaussia proesses ACM Classifiatio Keywords: G3 Probability ad Statistis - Stohasti proesses Itrodutio Wavelet aalysis is a exitig effetive ethod for solvig diffiult probles i atheatis, physis, eoois, ediie ad egieerig The ost atual issues of appliatio of wavelet aalysis related with sigal proessig ad siulatio, audio ad iage opressio, oise reoval, the idetifiatio of short-ter ad global patters, spetral aalysis of the sigal Fro a pratial poit of view, ultiresolutio aalysis provides a effiiet basis for the expasio of stohasti proesses Wavelet represetatios ould be used to overt the proble of aalyzig a otiuoustie rado proess to that of aalyzig a rado sequee, whih is uh sipler This approah is widely used i statistis to estiate a urve give observatios of the urve plus soe oise, i tie series aalysis for soothig futioal data, i siulatio studies of various futioals defied o realizatios of a rado proess, et eetly, a osiderable attetio was give to wavelet orthooral series represetatios of stohasti proesses Soe results, appliatios, ad referees o overgee of wavelet expasios of rado proesses i various spaes a be foud i [Atto et al, 00; Bardet et al, 00; Didier et al, 008; Kozaheo et al, 0, 03; Kozaheo, Polosa, 008], ust to etio a few I the paper we study uifor overgee of wavelet deopositios whih is required for various pratial appliatios (but ost ow results i the ope literature oer the ea-square overgee of wavelets expasios) So we osider statioary Gaussia rado proesses X () t ad their approxiatios by sus of wavelet futios

2 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, X ():= t () t (), t (), =0 where := ( 0, 0,, ), futios 0 (), t () t are wavelet bases (i the papper we osider Battle- Learie ad Meyer wavelets) I diret uerial ipleetatios we always osider truated series lie (), where the uber of ters i the sus is fiite by appliatio reasos (this aes it possible to fid a expliit estiate of the rate of uifor overgee for wavelets expasios of rado proesses) The rate of overgee is very useful otio i various oputatioal appliatios But this questio has bee studied very little I this paper our fous is o the Battle-Learie ad Meyer wavelet bases This is doe with the ai to show that all our results are ot oly theoretial, but they a be used i pratie Usig the progra Wolfra Matheatia, we get the overgee rate for Battle-Learie ad Meyer wavelet deopositios of Gaussia rado proesses The orgaizatio of this artile is the followig I the seod setio we itrodue the eessary bagroud fro wavelet theory ad a theore o uifor overgee i probability of the wavelet expasios of statioary Gaussia rado proesses, obtaied i [Kozaheo et al, 0] I the third setio we give soe otios about the Meyer wavelet bases ad obtai oditios of uifor overgee for this wavelets The ext setio otais the rate of overgee i the spae C([0, T ]) of Meyer wavelet deopositios of statioary Gaussia rado proesses I the setio 5 we give soe otios about the Battle-Learie wavelet bases ad obtai oditios of uifor overgee for this wavelets The ext setio otais the rate of uifor overgee of Battle-Learie wavelet deopositios of statioary Gaussia rado proesses Colusios are ade i setio 7 Wavelet epresetatio of ado Proesses Let ( x), x be a futio fro the spae L ( ) suh that (0) 0 ad ( y) is otiuous at 0, where iyx ( y)= e ( x) dx is the Fourier trasfor of Suppose that the followig assuptio holds true: ( y ) =( ae ), Z there exists a futio 0( x) L([0, ]), suh that ( ) 0 x has the period ad 0 ( y)= y/ y/ ( ae ) I this ase the futio ( x) is alled the f -wavelet Let ( x) be the iverse Fourier trasfor of the futio y y ( ) = y y 0 exp i iyx The the futio ( x)= e ( y) dy is alled the -wavelet / / Let ( x)= ( x), ( x)= ( x),, Z

3 44 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 It is ow that the faily of futios { 0;, N 0} is a orthooral basis i L ( ) (see, for exaple, [Hardle et al, 998]) A arbitrary futio f( x) L ( ) a be represeted i the for f ( x)= ( x) ( x), 0 0 Z =0Z = f ( x) ( x) dx, = f( x) ( x) dx 0 0 The represetatio () is alled a wavelet represetatio The series () overges i the spae L ( ) ie < 0 Z =0Z The itegrals 0 ad ay also exist for futios fro Lp ( ) ad other futio spaes Therefore it is possible to obtai the represetatio () for futio lasses whih are wider tha L ( ) Let {, BP, } be a stadard probability spae Let X (), t t be a rado proess suh that EX ()=0 t for all t It is possible to obtai represetatios lie () for rado proesses, if their saple traetories are i the spae L ( ) However the aority of rado proesses do ot possess this property For exaple, saple paths of statioary proesses are ot i L ( ) (as) We ivestigate a represetatio of the id () for X () t with ea-square itegrals = X( t) ( t) dt, = X ( t) ( t) dt 0 0 t Cosider the approxiats X, () of X () t defied by () Assuptio S [Hardle et al, 998] For the futio there exists a dereasig futio ( x), x 0 suh that (0) <, ( x) ( x ) (ae) ad ( x ) dx < Let X () t be a statioary separable etered Gaussia rado proess suh that its ovariae futio ts (, )= t ( s) is otiuous Let the f -wavelet ad the orrespodig -wavelet be otiuous futios ad the assuptio S holds true for both ad Theore below guaratees the uifor overgee of X, () to X () t t Theore [Kozaheo et al, 0] Suppose that the followig oditios hold: There exist ( u), '( u), ad (0) = 0, '(0) = 0; := sup ( u) <, := sup '( u) <, '( u) L( ), := sup ' ( u) < ; u u 3 ( u) 0 ad ( u) 0 whe u ; 4 There exist 0< < ad > suh that u l( u ) ( u) du <, l( u ) ( u) du < ; ()

4 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, There exists ( z ) ad sup z ( )< ; z 6 ( z) dz < ad ( ) 4 ( z) z dz < p for p =0, The X, () () X uiforly i probability o eah iterval [0, T ] whe, 0 ad for all 0 Coditios of Uifor Covergee for Meyer Wavelets Deopositios of Gaussia ado Proesses Meyer wavelets ( x) ad ( x) ab be give as iverse Fourier trasfors of the futios ( y) ad ( y) respetively The expressios of ( y) ad ( y) are followig: where ad where ( y)= y 4 h( ), y 3 4 0, y > 3, y 3 hy ( )= 0, y [, ] [, ] 3 3 ( y)= 0, y, 3 y 4 g( ), y 3 3 y i y 4 8 e h( ), y, , y > 3 iy g( y)= e h * ( y ) The futios ( x) ad ( x) are C beause their Fourier trasfors have a opat support Wavelet ( x) has a ifiite uber of vaishig oets [Mallat, 998], so ( ) (0) = 0, 0 Theore Let X () t be a statioary separable etered Gaussia rado proess suh that its ovariae futio ts (, )= t ( s) is otiuous Let ad be Meyer wavelets Suppose that the followig oditios hold: (3) (4) (5)

5 46 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 There exists ( z ) ad sup z ( )< ; z ( z) dz < ad ( ) 4 ( z) z dz < p for p =0, The X, () t () t X uiforly i probability o eah iterval [0, T ] whe, 0 ad for all N 0 Proof Stateet of this Theore follows fro Theore, if we tae ito aout that assuptios ) 4) of Theore hold for the Meyer wavelets Ideed, Meyer wavelet ( x) has a ifiite uber of vaishig oets [Mallat, 998], so ( ) (0) = 0, 0 Now we a use suh fat that Fourier trasfors of Meyer wavelets have a opat support, so we have fulfillet of oditios 3) ad 4) of Theore Aother fat that Fourier trasfors of Meyer wavelets is ties otiuously differetiable, the assuptio of Theore holds true Covergee ate i the Spae C[0, T ] of the Meyer Wavelets epresetatios of ado Proesses I the paper [Kozaheo et al, 03] a expliit estiate of the rate of uifor overgee for wavelets expasios of Gaussia rado proesses is obtaied I this setio our fous is o the Meyer wavelet bases So overgee rate i the spae C[0, T ] for the Meyer wavelets deopositios of Gaussia rado proesses is studied Theore 3 [Kozaheo et al, 03] Let X(), t t [0, T] be a separable Gaussia statioary rado proess Let assuptios of Theore hold true for X() t The ( u 8 u ( )) Psup X( t) X, ( t) > u exp, t[0, T] where u >8 ( ), A B C := / / =0 0 A, B, ad C are ostats whih deped oly o the ovariae futio of X () t ad the wavelet basis Expliit expressios for A, B, ad C are give i the proof of the theore Fro the paper [Kozaheo et al, 03] A, B, C, are followig: / A:= B 6A 4 A 3/ = / 3/ = B:= B 6A 4 A

6 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, C:= ( ) 3A B 3 = AB A A B = 3 I the ase T =3 we a get: For the ovariae futio B = '( u) du T ( u) du 34956, ( ) B = '( u) du T ( u) du 00, ( ) 3 l = =, l = = 3/ = = 6 ( )=exp{ } 4 we a obtai the value of the followig expressios: 9 := ( ) A z dz ( z) dz 355, 4 3 A = '( z) z dz ( z) z dz30, A := ( z) dz , 4 A := ( z) z dz , := ( v ) dv 00 The we a alulate ostats for the expressio : A , B 539, C I the paper [Kozaheo et al, 03] eeded forulas for alulatio of the followig expressio are give: ( ) := l( T ), T where := i,, > So for =06 ad =05 we a evaluate followig ostats: = B B B 0 /

7 48 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 ( ) B := q A QK , / 0 / =0 ( ) B := qq q A QK 00885, / / =0 / 3 3 B := q A ( K ) Q 3080 K := (l 5) 0 T (l 5) 00 0 := ( v) dv<, := l( v ) ( v) dv< := ( v) dv<, := l( v ) ( v) dv < 3 ( ) = = l = l QQ = 59, q q A K((l5) ) 3 6 := 880, l= l A K 6 := 500, = A q 6 := l 5 3 0, A ( K ) = q := K := (l 5) 0 T (l 5) 6087, 0 := ( v) dv<, := l( v ) ( v) dv<, := ( v) dv <, := l( v ) ( v) dv< 3 So, if we tae ito osideratio this alulatio, the is followig: 308 Naturally: ( T) = 5373 l e T

8 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, X, () t approxiates proess X() t with the reliability of ad auray, if P sup Xt ( ) X ( t) > 0 t T Let =00, we a use the rule of three, the it a be osidered as =06 I our ase, for the 4 ovariae futio ( )=exp{ }, we a alulate the variae =, so =06 The, for Meyer 9 wavelets, usig the progra Wolfra Matheatia, we a obtai suh =00 at =85, =0 0,, =0 with a slight irease 0, is sigifiatly redued Coditios of Uifor Covergee for Battle-Learie Wavelets Expasios of Gaussia ado Proesses Polyoial splie wavelets itrodued by Battle ad Learie are oputed fro splie ultiresolutio approxiatios Let ( x ) ad ( x) be the iverse Fourier trasfors of the futios ( y ) ad ( y ) respetively The expressios of ( y ) ad ( y ) are followig: y i e ( y)=, y S ( y) where S( y)= = ( y ) (7) ad = if is eve ad =0 if odd y y i S ( ) e ( )= y y y S( y) S( ) (8) For the degree splie wavelet ( x) has vaishig oets [Mallat, 998], so ( ) (0) = 0, 0 Wavelet ( x) has a expoetial deay Sie it is a polyoial splie of degree, it is ties otiuously differetiable (see, for exaple, [Mallat, 998]) To he the assuptio of Theore we a use Lea fro the paper [Polosa, 009]: Lea [Polosa, 009] Let ( x) - suh futio, that ( x) dx <, ( x) 0, x, let the derivative ( x) exists, suh that ( x) dx< for soe 0< < Let ( x) ( y) ( x y ), where ={ ( u), u >0} suh ootoe ireasig futio that (0) = 0 (6)

9 50 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 The futio, that ( y) ( y) dy <, y y where ( )= iyx y e ( x ) dx, ydy ( ) < ad ={ ( y), y }, ( y ) > 0 suh ear I the ase of ( u)= u,0<, we a tae y ( )=l( y ), where > Theore 4 Let X () t be a statioary separable etered Gaussia rado proess suh that its ovariae futio ts (, )= t ( s) is otiuous Let ad be Battle-Learie wavelets Suppose that the followig oditios hold: There exists ( z ) ad sup z ( )< ; z ( z) dz < ad ( ) 4 ( z) z dz < p for p =0, The X, () t X () t uiforly i probability o eah iterval [0, T ] whe, 0 ad for all N 0 Proof Stateet of this Theore follows fro Theore, if we tae ito aout that assuptios ) 4) of Theore hold for the Battle-Learie wavelets Ideed, degree - wavelet Battle-Learie has vaishig oets [Mallat, 998], so ( ) (0) = 0,0 Aother fat that it is ties otiuously differetiable, the, usig forulas (3),(5), we have fulfillet of oditios ) 3) of Theore Assuptio 4 follows fro Lea, differetiability of the Battle-Learie wavelets ad ear Covergee ate i the Spae C[0, T ] of the Battle-Learie Wavelets epresetatios of ado Proesses I the previous setio it was give Theore 3 fro the paper [Kozaheo et al, 03] i whih a expliit estiate of the rate of uifor overgee for wavelets expasios of Gaussia rado proesses is obtaied I this setio our fous is o the Battle-Learie wavelet bases So overgee rate i the spae C[0, T ] for the Battle-Learie wavelets deopositios of Gaussia rado proesses is studied Here we obtai all ostats for Theore 3 i the ase of Battle-Learie wavelets: I the ase T =3 we a get: / A:= B 6A 4 A 3/ = / 3/ = B:= B 6A 4 A C:= ( ) 3A B 3 = AB A A B = 3 /

10 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 5 For the ovariae futio B = '( u) du T ( u) du 3977, ( ) B = '( u) du T ( u) du 46406, ( ) 3 l = =, l = = 3/ = = 6 ( )=exp{ } 4 we a obtai the value of the followig expressios: 9 := ( ) A z dz ( z) dz 355, 4 3 A = '( z) z dz ( z) z dz30, A := ( z) dz , 4 A := ( z) z dz , := ( v ) dv 7575 The we a alulate ostats for the expressio : A , B 7547, C 978 I the paper [Kozaheo et al, 03] eeded forulas for alulatio of the followig expressio are give: ( ) := l( T ), T where := i,, > So for =06 ad =05 we a evaluate followig ostats: = B B B 0 ( ) B := q A QK , / 0 / =0 ( ) B := qq q A QK 00885, / / =0 / B := q A ( K ) Q 46 3 K := (l 5) 0

11 5 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, 04 3 T (l 5) := ( v) dv<, := l( v ) ( v) dv< := ( v) dv<, := l( v ) ( v) dv < 3 ( ) = = l = l QQ = 59, q q A K((l5) ) 3 6 := 880, l= l A K 6 := 500, = A q A ( K ) q := := l 5 3 0, = 3 3 K := (l 5) 0 T (l 5) 5689, 0 := ( v) dv<, := l( v ) ( v) dv<, := ( v) dv <, := l( v ) ( v) dv< 3 So, if we tae ito osideratio this alulatio, the is followig: 464 Naturally: ( T) = l e T X, () t approxiates proess X() t with the reliability of ad auray, if P sup Xt ( ) X ( t) > 0 t T Let =00, we a use the rule of three, the it a be osidered as =06 I our ase, for the 4 ovariae futio ( )=exp{ }, we a alulate the variae =, so =06 The, for Battle- 9 Learie wavelets, usig the progra Wolfra Matheatia, we a obtai suh =00 at =0, 0, =0, =0 with a slight irease 0, is sigifiatly redued

12 Iteratioal Joural Iforatio Theories ad Appliatios, Vol, Nuber, Colusio Coditios of uifor overgee for Meyer wavelet deopositios ad Battle-Learie wavelet deopositios of statioary Gaussia rado proesses are preseted The rate of overgee i the spae C([0, T ]) of Meyer wavelet deopositios ad Battle-Learie wavelet deopositios of statioary Gaussia rado proesses are obtaied We a olude that both wavelet bases are good for expasio of statioary Gaussia rado proesses, but the Meyer wavelets have soe advatages For the sae auray of the approxiatio i the ase of the Meyer wavelets, we eed fewer ters i the expasio Bibliography [Atto et al, 00] A Atto Wavelet paets of ostatioary rado proesses: otributig fators for statioarity ad deorrelatio A Atto, Y Berthouieu IEEE Tras Ifor Theory 58(), p , 00 [Bardet et al, 00] JM Bardet A wavelet aalysis of the oseblatt proess: haos expasio ad estiatio of the selfsiilarity paraeter JM Bardet, CA Tudor Stohasti Proess Appl, 0(), p 33 36, 00 [Buldygi et al, 000] V V Buldygi Metri haraterizatio of rado variables ad rado proesses V V Buldygi ad Yu V Kozaheo Aeria Matheatial Soiety, Providee I, 000 [Daubehies, 99] I Daubehies Te letures o wavelets Soiety for idustrial ad applied atheatis, Philadelphia, 99 [Didier et al, 008] G Didier Gaussia statioary proesses: adaptive wavelet deopositios, disrete approxiatios ad their overgee G Didier, V Pipiras J Fourier Aal ad Appl, 4, p 03-34, 008 [Hardle et al, 998] W Hardle Wavelet, Approxiatio ad statistial appliatios W Hardle, G Keryaharia, D Piard, A Tsybaov Spriger, New Yor, 998 [Kozaheo et al, 008] Yu V Kozaheo Uifor overgee i probability of wavelet expasios of rado proesses fro L ( ) Yu V Kozaheo, O V Polosa ado operators ad stohasti equatios 6 (4), p 37, 008 [Kozaheo et al, 0] Yu V Kozaheo Uifor overgee of wavelet expasios of Gaussia rado proesses Yu V Kozaheo, A Ya Oleo, O V Polosa Stoh Aal Appl 9(), p 69 84, 0 [Kozaheo et al, 03] Yu V Kozaheo Covergee rate of wavelet expasios of Gaussia rado proesses, Co Statist Theory Methods 4, p , 03 [Kozaheo, 004] Yu V Kozaheo Letures o wavelet aalysis (Uraiia), TViMC Kyiv, 004 [Mallat, 998] Stephae Mallat A wavelet tour of sigal proessig Aadei Press, New Yor, 998 [Polosa, 009] O V Polosa ate of uifor overgee for expasios of rado proesses fro L ( ) o Battle-Learie wavelets Bulleti of Uiversity of Kyiv Series: Physis ad Matheatis, 3, p 36 4, 009 Authors' Iforatio Olga Polosa PhD, assistat of eooi yberetis departet, faulty of eoois, Taras Shevheo atioal uiversity of Kiev, Kiev, Uraie, olgapolosa@yadexru Maor Fields of Sietifi esearh: ado proesses, Wavelet aalysis, Wavelet expasios of stohasti