Mathematical Background for Wavelet Estimators of Cross-Covariance and Cross-Correlation Brandon Whitcher Peter Guttorp Donald B. Percival NRCSE T e c h n i c a l R e p o r t S e r i e s NRCSE-TRS No. 038 The NRCSE was established in 1996 through a cooperative agreement with the United States Environmental Protection Agency which provides the Center's primary funding. Mathematical Background for Wavelet Estimators of Cross-Covariance and Cross-Correlation Brandon Whitcher EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Peter Guttorp University of Washington, Department of Statistics, Box 354322, Seattle, WA 98195-2933 Donald B. Percival University of Washington, Applied Physics Lab oratory,Box 355640, Seattle, WA 98195-5640 MathSoft, Inc., 1700 Westlake Avenue North, Seattle, Washington 98109-9891 June 8, 1999 Abstract In this technical rep ort we provide mathematical supp ort to the claims in Whitcher, Guttorp, and Percival 1999. First, the decomp osition of covariance by the wavelet covariance is rmly established. Central limit theorems for MODWT estimators of the wavelet covariance and correlation are then provided along with the de nition of the variance for estimators of wavelet covariance under the assumption of Gaussianity. 1 1 Wavelet-Based Estimators of Covariance and Correlation Here we de ne the basic quantities of interest for estimating asso ciation between two time series using the MODWT. The decomp osition of covariance on a scale by scale basis of the wavelet covari- ance is shown, and central limit theorems are provided for the wavelet covariance and correlation. 1.1 De nition and Prop erties of the Wavelet Cross-Covariance Let fU g f::: ;U ;U ;U ;::: g be a sto chastic pro cess whose dth order backward di erence t 1 0 1 d 1 B U = Z is a stationary Gaussian pro cess with zero mean and sp ectral density function t t S , where d is a non-negativeinteger. Let Z L 1 j X U ~ ~ W = h U h U ; t = ::: ;1; 0; 1;::: ; t j;l j;l tl j;t l =0 ~ b e the sto chastic pro cess obtained by ltering fU g with the MODWT wavelet lter fh g. Percival t j;l U W g is a stationary pro cess with zero and Walden 2000, Sec. 8.2 showed that if L 2d, then f j;t mean and sp ectrum given by S . j;U Let fX g f::: ;X ;X ;X ;::: g and fY g f::: ;Y ;Y ;Y ;::: g be sto chastic pro cesses t 1 0 1 t 1 0 1 whose d th and d th order backward di erences are stationary Gaussian pro cesses as de ned X Y ab ove, and de ne d maxfd ;d g. Let S denote their cross sp ectrum and, S and S X Y XY X Y j 1 denote their autosp ectra, resp ectively. The wavelet cross-covariance of fX ;Y g for scale =2 t t j and lag is de ned to b e n o X Y Cov W W ; 1 ; ;XY j j;t j;t+ X Y where f W g and fW g are the scale MODWT co ecients for fX g and fY g, resp ec- j t t j;t j;t tively. The MODWT co ecients have mean zero, when L 2d , and therefore = ;XY j X Y W W g. When = 0 we obtain the wavelet covariance between fX ;Y g, which we E f t t j;t j;t+ denote as = to simplify notation. XY j 0;X Y j Setting = 0 and Y to X or X to Y , Equation 1 reduces to the wavelet variance for X t t t t t 2 2 or Y denoted as, resp ectively, or Percival 1995. The wavelet variance decomp oses t j j X Y the pro cess variance on a scale by scale basis, and the wavelet cross-covariance give a similar decomp osition for the pro cess cross-covariance. Theorem 1 Let fX g and fY g be two weakly stationary processes with autospectra given by S f t t X and S f ,respectively. If we require L>2d, then for any integer J 1 we have Y J n o X X Y C Cov fX ;Y g =Cov V ; V + ; ;XY t t+ ;XY j J;t J;t+ j =1 1 X Y where V g~ X and V g~ Y are obtained by ltering fX g and fY g using the t t t t J;l J;l J;t J;t MODWT scaling lter fg~ g, respectively. As J !1,we have J;l 1 X C = ; ;XY ;XY j j =1 which gives the required decomposition. Pro of of Theorem 1 Before proving Theorem 1, we require the following lemma. o n X Y < for J >J . V ; V Cov Lemma 1 For al l >0, there exists a J such that J;t J;t P 2 J=2 Pro of of Lemma 1 For the orthonormal DWT g =1 and by de nition g~ = g =2 . J;l J;l l J;l P 2 J Therefore wehave g~ =1=2 . Parseval's relation tells us that l J;l Z Z L 1 J 1=2 1=2 X 2 1 2 e e G f df = G f df = g~ = : J J J;l J 2 1=2 1=2 l =0 We know the amplitude sp ectrum A f jS f j is a non-negative real valued function. XY XY Hence, if A is b ounded by some nite number C , then for J >J , XY Z Z o n 1=2 1=2 C X Y e e V ; V G f jS f j df = C G f df = <: Cov J XY J J;t J;t J 2 1=2 1=2 If A cannot be b ounded by any nite number C , there at least exists a constant C such XY R that A f df <=2, using a Leb esgue integral. A rough b ound on the squared gain XY A f C XY e function of the scaling lter for Daub echies wavelets is G f 1, so for all J >J , J Z Z 1=2 e e G f jS f j df G f S f df J XY J XY A f C 1=2 XY Z e + G f jS f j df J XY A f <C XY Z Z e A f df + C G f df XY J A f C A f <C XY XY Z 1=2 C e + C G f df + <: J J 2 2 2 1=2 2 Y X W g with resp ect to fW g to get Without loss of generality, we set = 0 and simply shift f j;t j;t X Y 6=0. Because fW g and fW g are obtained by ltering the pro cesses fX g and fY g with a t t j;t j;t Daub echies compactly supp orted wavelet lter of even length L>2d, resp ectively, we know that X Y e f W g and fW g are stationary pro cesses with autosp ectra de ned by S f H f S f j;X j X j;t j;t Q j 2 j 1 l e e e e and S f H f S f where H f H 2 f G 2 f is the squared gain function for j;Y j Y j l =0 2 ~ fh g. Note, the squared gain functions asso ciated with unit scale for the wavelet and scaling lters j 2 2 e e e e are given by H f jH f j and G f jGf j . X Y The covariance between f W g and fW g is given by j;t j;t Z 1=2 e H f S f df: = j XY XY j 1=2 This is a straightforward generalization of the univariate case; see Whitcher 1998 for more details. X Y The covariance between f V g and fV g is given by j;t j;t Z o n 1=2 X Y e V ; V = G f S f df; Cov J XY J;t J;t 1=2 Q J 1 l e e where G f G 2 f is the squared gain function for fg~ g. Because of the following identity J J l =0 e e for squared gain functions H f +G f = 1 for all f Percival and Walden 2000, Sec. 4.3, wehave Z h i n o 1=2 X Y e e V ; V Cov fX ;Y g = G f +H f S f df =Cov + ; t t XY XY 1 1;t 1;t 1=2 and the case when J = 1 holds. Wenow pro ceed to prove the main assertion by induction. Assume the prop erty holds for J 1; i.e., J 1 n o X X Y Cov fX ;Y g =Cov ; V V + : t t XY j J 1;t J 1;t j =1 So wehave " Z J 2 o n 1=2 Y Y X l e G 2 f S f df = ;V Cov V XY J 1;t J 1;t 1=2 l =0 " Z J 2 i h 1=2 Y l J 1 J 1 e e e G 2 f S f df G 2 f +H 2 f = XY 1=2 l =0 Z h i 1=2 e e = G f +H f S f df J J XY 1=2 n o X Y = Cov V ; V + : XY J J;t J;t The decomp osition of covariance b etween fX ;Y g has now b een established for a nite number of t t scales.