Wavelet Analysis of Covariance with Application to Atmospheric Time Series

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Wavlet analysis of covariance with application to atmospheric time series

Brandon Whichter Peter Guttorp Donald 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. 024

Wavelet analysis of covariance with application

to atmospheric time series

Brandon Whitcher

EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

Peter Guttorp

University of Washington, Department of Statistics, Box 354322, Seattle, WA

98195-4322, USA.

Donald B. Percival

University of Washington, Applied Physics Lab oratory,Box 355640, Seattle, WA

98195-5640, USA; and MathSoft, Inc., 1700 Westlake Avenue North, Seattle,

Washington 98109-9891, USA.

Short title:

Abstract. Multi-scale analysis of univariate time series has app eared in the

literature at an ever increasing rate. Here we intro duce the multi-scale analysis of

covariance between two time series using the discrete wavelet transform. The wavelet

covariance and wavelet correlation are de ned and applied to this problem as an

alternative to traditional cross-sp ectrum analysis. The wavelet covariance is shown

to decomp ose the covariance between two stationary pro cesses on a scale by scale

basis. Asymptotic normality is established for estimators of the wavelet covariance

and correlation. Both quantities are generalized into the wavelet cross-covariance and

cross-correlation in order to investigate p ossible lead/lag relationships. A thorough

analysis of El-Nino{Southern ~ Oscillation events and the Madden{Julian oscillation is

p erformed using a 35+ year record. We show how p otentially complicated patterns of

cross-correlation are easily decomp osed using the wavelet cross-correlation on a scale

by scale basis, where each wavelet cross-correlation series is asso ciated with a sp eci c

physical time scale.

Some key words: Con dence intervals; Cross-correlation; Cross-covariance; Madden{

Julian oscillation; Maximal overlap discrete wavelet transform; Southern Oscillation Index.

1. Intro duction

The bivariate relationship between two time series is often of crucial interest in

atmospheric science. For example, the Madden{Julian oscillation MJO [Madden and

Julian, 1971] was found using bivariate sp ectral analysis between the station pressure

and zonal wind comp onents at Canton Island 2.8 S, 171.7 W { sp eci cally the

co-sp ectrum and magnitude squared coherence. This oscillation has b een do cumented

as having a p erio d anywhere from 30{60 days and has app eared in many studies in

the Indian Ocean and tropical Paci c Ocean; see Madden and Julian [1994] for a

review. This apparent broadband nature of the oscillation has b een hyp othesized as

b eing nonstationary, so the broad p eak observed in previous sp ectral analyses might

be attributed to the fading in-and-out of the oscillation over the measured time series.

Temp oral variations in the MJO and its relationship with El Nino{Southern ~ Oscillation

ENSO events have previously b een investigated using traditional cross-sp ectral

analysis.

Given the nonstationary nature of the MJO, analyzing it through a transform

which captures events b oth lo cally in time and frequency is app ealing. The short-time

Fourier transform is one such technique, where the discrete Fourier transform is applied

to subsets of the time series via a moving window resulting in a grid-like partition of

the time-frequency plane. The discrete wavelet transform DWT is another technique,

where the time-frequency plane is partitioned such that high-frequencies are given small

windows in time and low-frequencies are given large windows in time via an adaptive

windowing of the time series. See Kumar and Foufoula-Georgiou [1994] and Kumar

[1996] for descriptions of time-frequency/time-scale analysis. An non-decimated version

of the orthonormal DWT { the maximal overlap DWT MODWT { has proven useful

in analyzing various geophysical pro cesses [Percival and Guttorp, 1994; Percival and

Mofjeld, 1997].

In this pap er, we deal with a formulation of the wavelet covariance and correlation

between two time series based up on the MODWT and Daub echies family of wavelets

with compact supp ort [Daubechies, 1992, Sec. 6.2]. We show that the wavelet covariance

o ers a scale by scale decomp osition of the usual covariance between time series. We

then derive statistical prop erties of estimators of the wavelet covariance and correlation.

Both quantities are generalized into the wavelet cross-covariance and cross-correlation

in order to investigate p ossible lead/lag relationships. We demonstrate the use of these

results in an analysis of the relationship between a Southern Oscillation Index SOI

time series and the station pressure series from Truk Island 7.4 N, 151.8 W.

1.1. Relationship to Previous Work

Previous work on the time-varying nature of the MJO includes Anderson et al.

[1984] who ltered atmospheric relative angular momentum 4 years and the 850{

200 mb shear of the zonal wind at Truk Island 25 years with a lter designed to pass

the frequency band corresp onding to p erio ds of 32{64 days. They noted that, with

resp ect to the Truk Island series, a p ossible asso ciation with increased amplitude of the

oscillation during the 1956{57, 1972{73, and 1976{77 ENSO warm events but noted

that the duration of these increases were much longer than the ENSO events. Madden

[1986] p erformed a seasonally varying cross-sp ectral analysis on nearly twenty time

series of rawinsonde data from tropical stations around the world. The MJO app ears

strongest during Decemb er{February and weakest during June{August, and that it is

always stronger in the western Paci c and Indian o ceans than elsewhere. Madden and

Julian [1994] note the broadband nature of the oscillation by comparing the station

pressure sp ectra for Truk Island 7.4 N, 151.8 W during two time spans { 1967 to 1979

and 1980 to 1985. The MJO app ears to have a 26-day p erio d in the early 1980s.

The in uence of ENSO events has also b een hyp othesized to a ect the p erio d of

the MJO. Gray [1988] p erformed a correlation analysis between daily station pressure

data from Truk 7 N, 152 W, Balb oa 9 N, 80 W, Darwin 12 S, 131 E and Gan

1 S, 73 E, with seasonal sea surface temp erature anomalies on a 5 grid. The data

were partitioned into ENSO and non-ENSO years; in non-ENSO years a strong seasonal

shift in frequency was found at all sites except Truk Island. Kuhnel [1989] investigated

the characteristics of a 40{50 day oscillation in cloudiness for the Australo{Indonesian

region. Using data on a 10 by5 grid, regions in the eastern Indian Ocean and western

Paci c Ocean were found to have a pronounced 40{50 day p eak with no obvious seasonal

variation. Another region in the Indian Ocean 5{15 S, 95{100 E showed a stronger

oscillation in the March{June p erio d. Regions around 5{15 S over northern Australia

and in the Paci c Ocean showed a much stronger 40{50 day oscillation during the

Australian monso on season from December to March, than the rest of the year. The

40{50 day cloud amount oscillation did not app ear to b e a ected bywarm ENSO events.

The ability of the DWT to capture variability in b oth time and scale can provide

insight into the nature of atmospheric phenomena such as the MJO, but we must make

use of the DWTs of two time series in such a way that these bivariate prop erties can

be brought out. Hudgins [1992] intro duced the concepts of the wavelet cross sp ectrum

and wavelet cross correlation in his thesis, b oth in terms of the continuous wavelet

transform CWT. In a subsequent pap er, Hudgins et al. [1993] applied these concepts

to atmospheric turbulence. They found the bivariate wavelet techniques provided

a b etter analysis of the data over traditional Fourier metho ds { esp ecially at low

frequencies. Liu [1994] de ned a cross wavelet sp ectrum, equivalent to that of Hudgins

[1992], and complex-valued wavelet coherency which was constructed using the co-

and quadrature wavelet sp ectra. These quantities were used to reveal new insights on

o cean wind waves; such as wave group parameterizations, phase relations, and wave

breaking characteristics. Lindsay et al. [1996] de ned the sample wavelet covariance for

the DWT and MODWT along with con dence intervals based on large sample results.

They applied this metho dology to the surface temp erature and alb edo of ice pack in

the Beaufort Sea. Recently, Torrence and Compo [1998] discussed the cross-wavelet

sp ectrum, which is complex valued, and the cross-wavelet p ower, which is the magnitude

of their cross-wavelet sp ectrum, in terms of the CWT. They also intro duced con dence

intervals for their cross-wavelet power and compared the SOI with Nino3 ~ sea surface

temp erature readings.

In this pap er, we extend the notion of wavelet covariance for the MODWT

and de ne the wavelet cross-covariance and wavelet cross-correlation. The wavelet

cross-covariance is shown to decomp ose the pro cess cross-covariance on a scale by scale

basis for sp eci c typ es of nonstationary sto chastic pro cesses. Asymptotic normality is

proven for the MODWT-based estimators of wavelet cross-covariance.

1.2. Outline

The MODWT, a non-decimated variation of the DWT, is describ ed in Section 2. In

Section 3, the wavelet cross-covariance and cross-correlation are de ned in terms of the

MODWT co ecients at a particular scale of the transform. Central limit theorems are

stated for estimators of b oth quantities based on a nite sample of MODWT co ecients

pro ofs are given in the App endix. Explicit metho ds for computing approximate

1001 2p con dence intervals are provided in Section 4. Fisher's z -transformation

is utilized in order to keep the con dence interval of the wavelet correlation b ounded by

1 for small sample sizes.

In Section 5, we p erform a wavelet analysis of covariance with a daily Southern

Oscillation Index and daily station pressure readings from Truk Island. Sp ectral analysis

shows two p eaks in the magnitude squared coherence between the two time series

around the established MJO range of frequencies, and cross-correlation analysis provides

a complicated correlation structure. The wavelet cross-correlation decomp oses the usual

cross-correlation pro ducing several patterns asso ciated with physical time scales. Using

the wavelet cross-correlation, the relationship between the two time series is more easily

shown.

2. Maximal Overlap Discrete Wavelet Transform

The DWT of a time series X is now a reasonably well-established metho d for

analyzing its multi-scale features. In this section we provide a brief intro duction to

a slight variation of the DWT, called the maximal overlap DWT MODWT. The

MODWT gives up orthogonality through not subsampling in order to gain features

such as translation-invariance and the ability to analyze any sample size. Here, we

follow previous de nitions of the MODWT by Percival and Guttorp [1994] and Percival

and Mofjeld [1997].

~ ~ ~

Let fh g fh ;::: ; h g denote a the wavelet lter co ecients from a

1 1;0 1;L1

Daub echies compactly supp orted wavelet family [Daubechies, 1992, Sec. 6.2] and let

fg~ g fg~ ;::: ;g~ g be the corresp onding scaling lter co ecients, de ned via the

1 1;0 1;L1

m+1

following quadrature mirror relationship g~ = 1 h . By de nition, the

1;m 1;L1m

~ ~

wavelet lter fh g is asso ciated with unit scale, is normalized such that h = 1=2

and is orthogonal to its even shifts. For any sample size N L and with h =0 for

1;m

m L, let

N 1

i2mk=N

H = h e ; k =0;::: ;N 1;

1;k 1;m

m=0

be the discrete Fourier transform DFT of fh g, and let G denote the DFT of fg~ g.

1 1;k 1

j 1

Now de ne the wavelet lter fh g for scale 2 as the inverse DFT of

j j

j 2

e e e

j 1

G ; k =0;::: ;N 1: H = H

j;k

1;2 k mo d N

l =0

The wavelet lter asso ciated with scale has length minfN; L g where L

j j j

2 1L 1+1. When N >L , de ne h =0 for m L . Also, de ne the scaling

j j;m j

lter fg~ g for scale 2 as the inverse DFT of

J J

J 1

e e

G l ; k =0;::: ;N 1: G =

J;k

1;2 k mo d N

l =0

In order to construct a j th order partial MODWT, we let fX g fX ;::: ;X g

k 0 N 1

b e the DFT of X for arbitrary N . The vector of MODWT co ecients W , j =1;::: ;J

is de ned to b e the inverse DFT of fH X g and is asso ciated with changes of scale .

j;k k j

The vector of MODWT scaling co ecients V is de ned similarly by the inverse DFT

of fG X g and is asso ciated with averages of scale 2 and higher. For time series of

J;k k J

dyadic length, the MODWT may be subsampled and rescaled to obtain an orthonormal

DWT. In practice, apyramid scheme similar to that of the DWT is utilized to compute

the MODWT; see Percival and Guttorp [1994] and Percival and Mofjeld [1997].

Percival and Mofjeld [1997] proved that the MODWT is an energy preserving

transform in the sense that

2 2

e f

: V + W kXk =

J j

j =1

This allows for a scale-based analysis of variance of a time series similar to sp ectral

analysis via the DFT. In a wavelet analysis of variance, the individual wavelet

co ecients are asso ciated with a band of frequencies and sp eci c time scale whereas

Fourier co ecients are asso ciated with a sp eci c frequency only.

3. 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 covariance is shown, and central limit theorems are provided for the

wavelet covariance and correlation.

3.1. De nition and Prop erties of the Wavelet Cross-Covariance

Let fU g f::: ;U ;U ;U ;:::g b e a sto chastic pro cess whose dth order backward

t 1 0 1

di erence 1 B U = Z is a stationary Gaussian pro cess with zero mean and sp ectral

t t

density function S , where d is a non-negative integer. Let

L 1

~ ~

= h U h U ; t = ::: ; 1; 0; 1;::: ; W

j;l t j;l tl

j;t

l =0

be the sto chastic pro cess obtained by ltering fU g with the MODWT wavelet lter

fh g. Percival and Walden [1999, Sec. 8.2] showed that if L 2d, then f g is a W

j;l

j;t

stationary pro cess with zero 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

t 1 0 1 t 1 0 1

pro cesses whose d th and d th order backward di erences are stationary Gaussian

X Y

pro cesses as de ned ab ove, and de ne d max fd ;d g. Let S denote their

X Y XY

cross sp ectrum and, S and S denote their autosp ectra, resp ectively. The cross

X Y

i2f

sp ectrum is a complex valued function de ned to b e S f C e for

XY ;XY

jf j 1=2, where C is the cross covariance sequence given by C Cov fX ;Y g.

;XY ;XY t t+

j 1

The wavelet cross-covariance of fX ;Y g for scale =2 and lag is de ned to be

t t j

o n

X Y

W ; W ; 1 Cov

;XY j

j;t j;t+

X Y

where fW g and fW g are the scale MODWT co ecients for fX g and fY g,

j t t

j;t j;t

resp ectively. When L 2d the MODWT co ecients have mean zero and therefore

X Y

W W = E f g. By setting =0 and Y to X or X to Y , Equation 1

;XY j t t t t

j;t j;t+

2 2

reduces to the wavelet variance for X or Y denoted as, resp ectively, or

t t j j

X Y

[Percival, 1995].

The following theorem demonstrates that the wavelet cross-covariance decomp oses

the covariance between two stationary time series on a scale by scale basis.

Theorem 1 Let fX g and fY g be weakly stationary processes with autospectra S

t t X

X Y

and S , respectively. Let L>2d and de ne V g~ X and V g~ Y ,

Y J;l t J;l t

J;t J;t

which are stationary processes obtained by ltering fX g and fY g using the MODWT

t t

scaling lter fg~ g, respectively. For any integer J 1, we have

J;l

o n

X Y

V ; V + : C =Cov fX ;Y g =Cov

;XY j ;XY t t+

J;t J;t+

j =1

As J ! 1, we have C = , that is, the wavelet cross-covariance

;XY ;XY j

j =1

decomposes the cross-covariance between fX g and fY g on a scale by scale basis.

t t

J +1 J +1

If we think of fg~ g as a low-pass lter covering the nominal band [2 ; 2 ],

this statement is intuitively plausible since the scaling lter fg~ g is capturing smaller

and smaller p ortions of the cross sp ectrum as J !1.

3.2. Estimating the Wavelet Cross-Covariance

Supp ose X ;::: ;X and Y ;::: ;Y can b e regarded as realizations of p ortions

0 N 1 0 N 1

of the pro cesses fX g and fY g, whose d th and d th order backward di erences form

t t X Y

stationary Gaussian pro cesses. As b efore, let d = max fd ;d g.

X Y

f f

Let W = W for those indices t where W is una ected by the b oundary {

j;t j;t j;t

this is true as long as t L 1. Thus, if N L , we can de ne a biased estimator

j j

~ ofthe wavelet cross-covariance based up on the MODWT via

;XY j

X Y

N 1

> f f e e

W W ; =0;::: ;N 1; N

j;l j;l +

l =L 1

X Y

N 1

e f f e

N W W ; = 1;::: ; N 1;

;XY j

j;l j;l +

l =L 1

0; otherwise ;

e e

where N N L +1. The bias is due to the denominator 1=N remaining constant

j j j

for all lags, though it disapp ears at lag = 0. The following theorem establishes

asymptotic normality for the MODWT-based estimator of the wavelet covariance. We

W g with may generalize to the wavelet cross-covariance by simply replacing shifting f

j;t

resp ect to f W g and app ealing to the same theorem as in the pro of of Thereom 1.

j;t

X Y

Theorem 2 If L > 2d, and suppose fW ; W g is a bivariate Gaussian stationary

j;t j;t

R R

1=2 1=2

2 2

process with autospectra satisfying S f < 1 and S f < 1, then the

j;X j;Y

1=2 1=2

estimator ~ is asymptotical ly normal ly distributed with mean and large

XY j XY j

sample variance N S 0. The quantity S 0 is the spectral density function

j; ;XY j;XY

X Y

for f W W g the product of the MODWT coecients.

j;t j;t

Since we are strictly interested in Gaussian pro cesses, we can re-express the variance

of the wavelet covariance see the App endix for details by

Varf ~ g ; 2

XY j

for large N , where

Z Z

1=2 1=2

V S 0 = S f S f df + S f df: 3

j j;XY j;X j;Y

j;X Y

1=2 1=2

Note, S is the cross sp ectrum between two series of scale MODWT co ecients

j;X Y j

whereas S is the autosp ectrum for the pro duct of scale MODWT co ecients.

j;XY

This will allow us to easily construct approximate con dence intervals for the MODWT

estimator of the wavelet covariance.

3.3. Wavelet Cross-Correlation

We can de ne the wavelet cross-correlation for scale and lag as

o n

X Y

W ; W Cov

j;t j;t+

;XY j

= :

;XY j

n o n o

1=2

X Y

X j Y j

Var W Var W

j;t j;t+

Since this is just a correlation co ecient between two random variables, 1

1 for all ; j. The wavelet cross-correlation is roughly analogous to its

;XY j

Fourier counterpart { the magnitude squared coherence { but it is related to bands

of frequencies scales. Just as the cross-correlation is used to determine lead/lag

relationships b etween two pro cesses, the wavelet cross-correlation will provide a lead/lag

relationship on a scale by scale basis.

3.4. Estimating the Wavelet Cross-Correlation

Since the wavelet cross-correlation is simply made up of the wavelet cross-covariance

for fX ;Y g and wavelet variances for fX g and fY g, the MODWT estimator of the

t t t t

wavelet cross-correlation is simply

;XY j

~ ; 4

;XY j

~ ~

X j Y j

2 2

where ~ is the wavelet covariance, and ~ and ~ are the wavelet

;XY j j j

X Y

variances. When = 0 we obtain the MODWT estimator of the wavelet correlation

between fX ;Y g.

t t

Large sample theory for the cross-correlation is more dicult to come by than for

the cross-covariance. Bril linger [1979] constructed approximate con dence intervals for

the auto and cross-correlation sequences of bivariate stationary time series. We use

his technique to establish a central limit theorem for the MODWT estimated wavelet

cross-correlation. To simplify notation the following theorem gives a central limit

theorem for the wavelet correlation = 0 but easily generalizes to arbitrary lags.

X Y

Theorem 3 Let L>2d, and suppose f W ; W g is a bivariate Gaussian stationary

j;t j;t

process with square integrable autospectra, then the MODWT estimator ~ of the

XY j

wavelet correlation is asymptotical ly normal ly distributed with mean and large

XY j

sample variance given by

N 1

Varf~ g f +

XY j ;X j ;Y j ;XY j ;Y X j

=N 1

2 [ + ]

0;X Y j ;X j ;Y X j ;Y j ;Y X j

2 2 2 2

1 1

+ + ] g; + [

j j j j

;X ;XY ;Y 0;X Y

2 2

X X

g=[2 ] is the scale wavelet autocorrelation W W where E f

j j j ;X j

j;t j;t+j j X

for the process fX g.

4. Con dence Intervals for Wavelet Estimators

With central limit theorems for b oth the wavelet cross-covariance and cross-

correlation, wemaynow explicitly construct an approximate con dence interval CI for

the estimators. For the wavelet cross-correlation, a nonlinear transformation is utilized

in order to b ound it between 1 and reduce the computational complexity of its large

sample variance. Without loss of generality, we provide the following CIs for lag =0.

We may generalize the results presented here for arbitrary lag by simply replacing

X Y X Y

f f f f

fW g and fW g with fW g where t = L + 1; :::; N 1 and fW g where

j;t j;t j;t j;t+

g where t = L 1; :::; N + 1 and t = L 1; :::; N 1 for 0, or fW

j j

j;t

fW g where t = L 1; :::; N 1 for < 0.

j;t+

4.1. Wavelet Cross-covariance

The formula for an approximate 1001 2p CI of the scale MODWT estimator

of the wavelet cross-covariance ~ , starting from Equation 3, is provided; see

;XY j

also Lindsay et al. [1996]. We make use of the p erio dogram and the cross-p erio dogram

to help estimate quantities of interest under the assumption of =0. First, we use the

p X p

b f b

p erio dogram S of fW g as the estimator of S , and similarly for S of

j;X

j;t

j;X j;Y

fW g. Next, we de ne the biased estimator of the auto covariance sequence asso ciated

j;t

with the scale MODWT co ecients of fX g by

j t

N 1jmj

p X X

f f

s^ W W ;

j;m;X

j;l

j;l +jmj

l =L 1

g, the biased estimator of the auto covariance A similar de nition applies to fs^

j;m;Y

sequence asso ciated with the scale MODWT co ecients fW g. Second, we use

j;t+m

p X Y

b f f

the cross-p erio dogram S offW g and fW g, as the estimator of S , and

j;X Y

j;t j;t

j;X Y

the corresp onding biased estimator of the cross covariance sequence asso ciated with the

scale MODWT co ecients by

p X Y

b f f

W W ;

j;m;X Y

j;l j;l +m

where the summation go es from l = L 1;::: ;N m 1 for m 0 and from

l = L m 1;::: ;N 1 for m<0. j

We can use Parseval's relation to develop an estimate of the large sample variance

for the MODWT-based estimator of the wavelet cross-covariance Equation 3

using the auto covariance and cross-covariance sequences of the MODWT co ecients.

Sp eci cally, the integral of the pro duct of the p erio dograms is determined from the

X Y

f f

auto covariance sequences of fW g and fW g via

j;t j;t

N 1 Z

1=2

p p p p p p

b b

; s^ +2 s^ s^ f df =^s f S S

j;m;Y j;m;X j;0;Y j;0;X j;Y j;X

1=2

m=1

and the integral of the squared cross p erio dogram from the cross covariance sequence of

X Y

f f

fW ; W g via

j;t j;t

N 1 Z

i i h h

1=2

2 2

p p

b b

C : f S df =

j;m;X Y j;Y X

1=2

m=N 1

Hence, an estimate for the large sample variance of the MODWT estimator of the

wavelet cross-covariance is given by

e e

N 1 N 1

p p

j j

h i

X X

s^ s^

j;0;X j;0;Y

p p p

e b

V + C : 5 s^ s^ +

j;0;X Y j;m;X j;0;Y

2 2

m=1

m=N 1

The estimator V is unbiased when = 0 [Whitcher, 1998]. Under the assumption

that the sp ectral estimates are close to the true values, an approximate 1001 2p

con dence interval for is

XY j

s s

e e

V V

j j

1 1

~ 1 p ; ; ~ + 1 p

XY j XY j

e e

N N

j j

where pis the p 100 p ercentage p oint for the standard normal distribution.

4.2. Wavelet Cross-correlation

We now use the large sample theory develop ed in Section 3.4 to construct an

approximate CI for the MODWT estimator of the wavelet cross-correlation. Given

the non-normality of the correlation co ecient for small sample sizes, a nonlinear

transformation is sometimes required { Fisher's z -transformation [Fisher, 1915; Kotz,

Johnson, and Read, 1982, Volume 3]. Let

1 1+

h = tanh log

2 1

de ne the transformation. For the estimated correlation co ecient ^, based on n

indep endent samples, n 3h^ h has approximately a N 0; 1 distribution.

An approximate 1001 2p CI for based on the MODWT is therefore

XY j

9 9 8 8

3 2

= = < <

1 1

1 p 1 p

5 4

q q

; tanh h[~ ] + tanh h[~ ]

XY j XY j

; ; : :

b b

N 3 N 3

j j

j 0 0

= dL 21 2 e is the number of DWT co ecients and L where N = N L

j j

asso ciated with scale . We are using the number of wavelet co ecients as if ~

j XY j

had b een computed using the DWT b ecause, under the assumptions of Fisher's

z -transformation, the denominator should consist of the numb er of indep endent samples

used in constructing the correlation co ecient. If the autosp ectra and cross sp ectrum

of the two pro cesses fX ;Y g are slowly varying over each o ctave band of the DWT,

t t

the assumption of uncorrelated DWT co ecients is justi ed; see McCoy and Walden

[1996] and the references contained therein. This prop erty simply do es not hold for the

MODWT co ecients. If an equivalent degrees of freedom argument were available for

the wavelet cross-covariance, as was used when establishing CIs for the wavelet variance

based on the MODWT [Percival, 1995], this could be used instead of N .

5. Application

5.1. Intro duction

The relationship between ENSO events and the MJO is a topic which could b ene t

by using wavelet techniques. To investigate how these two atmospheric phenomena

interact, we analyze two time series. The rst one b eing the Southern Oscillation Index

SOI [Walker, 1928], which is an indicator of ENSO and usually de ned to be the

di erence between monthly averages of the station pressure series from climate stations

at Darwin, Australia 130.8 E, 12.4 S and Tahiti, French Polynesia 149 W, 14 S.

We deviate slightly from the usual de nition of the SOI by intro ducing a daily

version of it. We obtained daily pressure readings from Darwin, Australia, and Tahiti,

French Polynesia, starting in 1 June 1957 and continuing through 31 December 1992

N = 12; 998 and di erenced them; see Figure 1. The distance of the stations from

the equator is apparent in the strong annual comp onent in the time series. The

measurements in the summer and winter of 1983 app ear to be higher than those in

adjacent years. This approximately corresp onds to a large ENSO event in the early

1980s. We also obtained daily station pressure readings from Truk Island 7.4 N,

151.8 W as an indicator of the MJO. This series also exists from 1 June 1957 to

31 December 1992; see Figure 1. Unlike the SOI, there is no apparent annual trend

given its close proximity to the equator. Missing values in b oth series were replaced by

one-step-ahead predictions from an ARIMA3,1,0 mo del applied to the series [Jones,

1980].

5.2. Time-Domain and Sp ectral Analysis

Here we analyze the SOI and Truk Island station pressure series using standard

time and frequency domain techniques. The cross-correlation sequence is typically

estimated by utilizing the p erio dogram-based estimates of the auto covariance sequences

for fX g and fY g, and their cross-covariance sequence. The estimated cross-correlation

t t

sequence for the SOI and Truk Island series is shown in Figure 2. The maximum o ccurs

at a lag of +1 days. We also observe the characteristic broad-band p eak commonly

found in atmospheric time series from this region, with a approximate range of 35{55

day lags.

A bivariate sp ectral analysis of these data Figure 3 provides some insightinto the

p ossible relationship between ENSO events and the MJO. A Parzen smo othing window

was applied to the p erio dogram with the window parameter chosen such that the

sp ectral window bandwidth was 0:0081 day , as in Madden and Julian [1971]. The lag

window co-sp ectrum between the SOI and Truk Island station pressure series exhibits a

large p eak at the lower frequencies, which include the annual and inter-annual cycles,

1 1

and two distinct p eaks centered at 0:0163 day and 0:0273 day in the frequency

range of the MJO. We can test, at the level of signi cance, the null hyp othesis of

zero mean squared coherence MSC by checking the estimated MSC, on a frequency

2= 2

by frequency basis, against 1 and rejecting if the estimated MSC exceeds

it [Koopmans, 1974, p. 284]. The parameter is the number of equivalent degrees of

freedom asso ciated with the sp ectral estimates; 189 using Table 269 in Percival and

Walden [1993]. We see that b oth p eaks are signi cant at the 5 and 1 levels for a

broad range of frequencies.

5.3. Wavelet Analysis

Daily measurements allow us to apply the MODWT and analyze the sub-series

j +1 j

which corresp ond to ltered series with approximate pass-band 1=2 jf j 1=2 .

Due to the approximate bandpass nature of the MODWT, with the approximation

improving as the length of the wavelet lter increases, it is unnecessary to remove any

annual or semiannual comp onents a similar argument is made when bandpass ltering

atmospheric time series in Anderson et al. [1984], which should b e roughly captured in

the and scales. The MJO is known to o ccur with p erio ds of around 30{60 days.

7 8

We therefore exp ect to see it in scale , asso ciated with changes of 16 days and an

approximate pass-band of 1=64 jf j 1=32.

A partial MODWT J = 10 was applied to each series using the Daub echies least

asymmetric wavelet lter of length L = 8 { which we denote by LA8. Figures 4

and 5 give the MODWT co ecients for the Truk Island station pressure series and SOI,

resp ectively. Each vector has b een circularly shifted in order to obtain an approximate

zero-phase lter, allowing us to align features across scales; see Percival and Mofjeld

[1997] for more details. For the Truk Island series, we observe only a slight annual trend

in the W series and the large disruption in the early 1980s app ears to primarily a ect

scales through . The W series, asso ciated with frequencies in the range of the

7 10 5

MJO, app ears to fade in and out in magnitude with no apparent pattern. The SOI

multiresolution analysis exhibits a strong annual trend where the disturbance in the

early 1980s a ects the scales through . Again, the scale asso ciated with the MJO

8 1

W exhibits numerous bursts across time.

Figure 6 shows the estimated wavelet correlation b etween the SOI and Truk station

pressure series at a lag of zero days. The wavelet correlation app ears to be signi cantly

di erent from zero in fact, p ositive for all scales except and . The signi cant

6 7

correlation for scale , and larger magnitude with resp ect to neighb oring scales, lends

credibility to the hyp othesis of an asso ciation b etween the uctuations in ENSO activity

and the MJO.

If we are to investigate a p ossible lead/lag relationship b etween the two series, then

the wavelet cross-correlation must be estimated for various lags. Figure 7 shows the

estimated wavelet cross-correlation between the SOI and Truk Island station pressure

series. Con dence intervals not displayed may be computed from Section 4.2. The

large p ositive p eak in the rst ve scales is at a lag of 1 day for scales and , a lag

1 2

of2days for scales , a lag of 4 days for scale and zero days for scale . The higher

3 4 5

scales do not show any apparent trend when lo oking at lags up to 240 days.

There is an obvious asymmetry in the wavelet cross-correlation for scales and

higher. The analysis here was p erformed so that at a lag of +1 day, the SOI time series

leads the Truk station pressure series. As already noted for scale , the largest p ositive

cross-correlation o ccurs at lag zero. The largest negative cross-correlation is at a lag

of 20 days, approximately half the p erio d of the MJO, and the second largest p ositive

cross-correlation o ccurs at a lag of 41 days, approximately a full p erio d of the MJO.

Even though the SOI consists of a di erence between two stations with di erent spatial

lo cations, there is still a strong 40 day oscillation which is correlated and in phase

with the Truk station pressure series. It is also interesting to note that the wavelet

cross-correlation for the rst three scales is essentially an odd function. A p ossible

interpretation is that common short-term weather patterns are causing these small

scale disturbances. In contrast, the wavelet cross-correlation for scales and are

6 7

essentially even functions. Patterns in higher scales lower frequencies corresp ond to

the annual and inter-annual trends.

Although direct comparison between Figure 7 and Figure 2 is not appropriate,

b ecause the wavelet correlation do es not decomp ose the correlation between two

stationary pro cesses, the wavelet covariance do es decomp ose the covariance b etween two

time series. Since the wavelet correlation is simply the wavelet covariance standardized

at each scale, the shap e of each wavelet cross-correlation is the same even though the

magnitudes are o . Hence, we may make a rough comparison between the two, keeping

in mind the facts just stated.

The rst obvious di erence is the fact that usual cross-correlation is p ositive for all

negative lags. Lo oking at Figure 7, we see that the wavelet cross-correlation for scales

and are all p ositive and contribute to this feature, whereas for p ositive lags they are

close to zero and allow the annual scale to dominate. The two dips on either side of

the p eak at a lag of +1 days is the sup erp osition of the rst six scales in Figure 7. The

subsequent p eak around a lag of +40 days is a result of the anti-correlation for scales

and reducing the annual cross-correlation comp onent for smaller lags. This leads

6 8

to a di erent interpretation than what is seen by lo oking at the usual cross-correlation

sequence. The correlation structure, when applied to all scales simultaneously, results in

a quite complex lo oking cross-correlation sequence. When broken up with the wavelet

transform a few simple, yet distinct, patterns app ear which may be asso ciated with

known atmospheric phenomena.

6. Discussion

We have intro duced a new analysis technique for bivariate Gaussian time series

which utilizes the MODWT. Certain nonstationary time series, with dth order

stationary backward di erences, are easily handled by this metho dology. As in the

univariate case with the wavelet variance, natural estimators of the cross-covariance and

cross-correlation are de ned. The wavelet cross-covariance and wavelet cross-correlation

\decomp ose" their classical counterparts on a scale by scale basis. Thus, complicated

patterns of asso ciation between time series are broken down into several much simpler

patterns { each one asso ciated with a physical time scale. With any good statistical

analysis a measure of variability is also required, and the central limit theorems here

enable approximate con dence intervals to b e calculated.

A thorough analysis between the Southern Oscillation Index and a station pressure

series from Truk Island was p erformed in order to gain insightinto p otential interactions

between ENSO events and the MJO. Whereas conventional time and frequency domain

techniques provide results which are dicult to interpret, the wavelet cross-correlation

nicely displays how the asso ciation between the two pro cesses changes with scale.

Short-term weather patterns changes of 1, 2 and 4 days exhibit symmetric wavelet

cross-correlations. The fth scale, asso ciated with the MJO, shows that the SOI an

indicator of ENSO activity is correlated with Truk Island station pressure series an

indicator of the MJO and they are roughly in phase. This scale exhibits the strongest

magnitude correlations when compared with scales shorter than those asso ciated with

the semi-annual and annual frequencies.

References

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missing observations. Technometrics 22 3, 389{395.

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Received

App endix A: Pro ofs of Theorems

o n

X Y

Lemma 1 For al l >0, there exists a J such that V ; V Cov < for J >J .

J;t J;t

Pro of of Lemma 1 For the orthonormal DWT g = 1 and by de nition

J;l

J=2 2 J

g~ = g =2 . Therefore we have g~ =1=2 . Parseval's relation tells us that

J;l J;l

J;l

Z Z

L 1

1=2 1=2

e e

: G f df = G f df = g~ =

J J

J;l

1=2 1=2

l =0

We know the amplitude sp ectrum A f jS f j is a non-negative real valued

XY XY

function. Hence, if A is b ounded by some nite number C , then for J >J ,

Z Z

o n

1=2 1=2

X Y

e e

V ; V G f jS f j df = C G f df = Cov <:

J XY J

J;t J;t

1=2 1=2

If A cannot be b ounded by any nite number C , there at least exists a constant

C such that A f df <=2, using a Leb esgue integral. A rough b ound on

A f C

the squared gain function of the scaling lter for Daub echies wavelets is G f 1, so

for all J >J ,

Z Z Z

1=2

e e e

G f S f df G f jS f j df + G f jS f j df

J XY J XY J XY

1=2 A f

XY XY

Z Z

A f df + C G f df

XY J

A f C A f

XY XY

1=2

G f df <: + C +

2 2 2

1=2

Pro of of Theorem 1 Without loss of generality, we set = 0 and simply

Y X X Y

shift f W g with resp ect to fW g to get 6= 0. Because fW g and fW g

j;t j;t j;t j;t

are obtained by ltering the pro cesses fX g and fY g with a Daub echies compactly

t t

supp orted wavelet lter of even length L>2d, resp ectively, we know that fW g and

j;t

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 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

j;Y j Y j

l =0

function for fh g. Note, the squared gain functions asso ciated with unit scale for the

2 2

e e e e

wavelet and scaling lters 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

1=2

= H f S f df:

XY j j XY

1=2

This is a straightforward generalization of the univariate case; see Whitcher [1998] for

X Y

more details. The covariance between f g and f g is given by V V

j;t j;t

o n

1=2

X Y

G f S f df; V ; V = Cov

J XY

J;t J;t

1=2

J 1

e e

where G f G 2 f is the squared gain function for fg~ g. Because of the

J J

l =0

e e

following identity for squared gain functions H f + G f = 1 for all f [Percival and

Walden, 1999, Sec. 4.3], we have

h i n o

1=2

X Y

e e

Cov fX ;Y g = G f +H f S f df =Cov V ; V + ;

t t XY XY 1

1;t 1;t

1=2

and the case when J = 1 holds. We now pro ceed to prove the main assertion by

induction. Assume the prop erty holds for J 1; i.e.,

J 1

n o

X Y

Cov fX ;Y g =Cov V ; V + :

t t XY j

J 1;t J 1;t

j =1

So we have

J 2

o n

1=2

Y X

;V = G 2 f S f df Cov V

J 1;t J 1;t

1=2

l =0

J 2

h i

1=2

l J 1 J 1

e e e

G 2 f S f df G 2 f +H 2 f =

1=2

l =0

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 between fX ;Y g has now b een established for a nite

t t

number of scales. From Lemma 1, as J ! 1 the remaining covariance between the

scaling co ecients go es to zero. Hence, the theorem is established.

Pro of of Theorem 2 Without loss of generality, we set = 0 and simply shift

Y X

f W g with resp ect to fW g to get 6= 0. With L > 2d, b oth series of MODWT

j;t j;t

X Y

co ecients f g and f g have zero mean. Square integrability of the autosp ectra W W

j;t j;t

implies that fs g ! S and fs g ! S ; i.e., the auto covariance

j; ;X j;X j; ;Y j;Y

sequences and autosp ectra are Fourier transform pairs. Because L>2d, the squared gain

function for Daub echies wavelet lters guarantees we have S 0 = 0 = s .

j;X j; ;X

g and, therefore, fs g and fs g are absolutely A similar statement holds for fW

j; ;X j; ;Y

j;t

summable.

Let S f H f S f denote the MODWT ltered cross sp ectrum.

j;X Y j XY

From the magnitude squared coherence b eing b ounded by unity, and using the

Cauchy{Schwarz inequality, we know that

Z Z

1=2 1=2

jS f j df S f S f df

j;X Y j;X j;Y

1=2 1=2

1=2

Z Z

1=2 1=2

2 2

S f df S f df < 1:

j;X j;Y

1=2 1=2

So the cross-covariance sequence and cross sp ectrum asso ciated with scale are also

a Fourier pair and, again, by using a Daub echies wavelet lters with L>2d, we have

X Y

W W ; g is absolutely S 0 = 0. Therefore, the cross-covariance sequence for f

j;X Y

j;t j;t

summable.

We rst note that the MODWT estimate of the wavelet covariance ~ is

XY j

XY X Y

essentially a sample mean for the time series W W W cf. Equation 1.

j;t j;t j;t

This pro cess also has an absolutely summable cumulant sequence by Theorem 2.9.1 of

Bril linger [1981, p. 38]. Then Theorem 4.4.1 of Bril linger [1981, p. 94] tells us that

~ is asymptotically normal with mean and large sample variance given

XY j XY j

W evaluated at f =0. by N S 0, where S 0 is the sp ectral density for

j;XY j;XY

j;t

j 2

Derivation of Equation 2 Since we are exclusively interested in Gaussian

pro cesses, S 0 may be re-expressed as a function of the auto and cross sp ectra of

j;XY

X Y

the wavelet co ecients fW g and fW g. The variance of the estimated MODWT

j;l j;l

wavelet covariance at scale can be computed directly via

N 1 N 1

n o

X X

X Y X Y

f f f f

Varf ~ g = Cov W W ; W W

XY j

j;m j;m

j;l j;l

l =L 1 m=L 1

j j

N 1

n o

j j 1

X Y X Y

f f f f

1 = Cov W W ; W W

j;l j;l j;l + j;l +

e e

N N

j j

=N 1

N 1

1 j j

1 s ; A1

j; ;X Y

e e

N N

j j

=N 1

where s is the auto covariance sequence for the pro duct of the scale MODWT

j; ;X Y j

co ecients with resp ect to fX g and fY g.

t t

Using the Isserlis theorem and prop erties of the Fourier transform, the sp ectrum

R R

1=2 1=2

of fZ g at f = 0 is S 0 = S f S f df + S f df [Whitcher,

t Z U V

1=2 1=2

1998]. Since we have the Fourier relationship fs g ! S , we necessarily have

;Z Z

S 0 = s , when f = 0. Re-examining Equation A1 and utilizing Ces aro

Z ;Z

summability [Titchmarsh, 1939, p. 411], we can say

N 1

j j

s 1 N Varf ~ g = lim lim

j; ;X Y j XY j

e e

N !1 N !1

j j

=N 1

s = S 0; =

j; ;X Y

j;XY

where

Z Z

1=2 1=2

S f df S f S f df + S 0 =

j;X j;Y j;XY

j;X Y

1=2 1=2

Pro of of Theorem 3 Since L>2d, we have that b oth sets of wavelet co ecients

X Y X Y

2 2

f f f f

fW g and fW g have mean zero. Let us de ne A [W ] , B [W ] , and

j;t j;t

j;t j;t j;t j;t

Y X

f f

, and subsequently de ne their sample means W C W

j;t

j;t j;t

N 1

A ; A = ~

j j j;t

t=L 1

N 1

B B = ~ ; and

j j;t j

t=L 1

N 1

C C =~ :

j j;t XY j

t=L 1

The vector-valued pro cess fA ;B ;C g has an absolutely summable joint cumulant

j;t j;t j;t

sequence by Theorem 2.9.1 of Bril linger [1981, p. 38]. Hence, from Theorem 4.4.1 of

A ; B ; C g are asymptotically Bril linger [1981, p. 94] the vector of sample means f

j j j

2 2

normally distributed with mean vector f ; ; g, and large sample

j j XY j

X Y

variance given by N S 0, where S is the 3 3 sp ectral matrix for

j;AB C j;AB C

fA ;B ;C g.

j;t j;t j;t

The MODWT estimator of the wavelet correlation ~ is essentially a function

XY j

of these sample means g A ; B ; C , where g x; y ; z z= xy . App ealing to Mann and

j j j

Wald [1943], we have that ~ is asymptotically normally distributed with mean

XY j

and large sample variance

XY j

1 2 2 2 2

N g_ ; ; S 0g _ ; ; A2

j j XY j j;AB C j j XY j

j X Y X Y

where g_ ; ; is the gradient of g ; ; . Now let us re-express Equation A2 into the

desired result using the fact that we are only interested in Gaussian pro cesses. Because

we are evaluating S atf =0,it is in fact a symmetric matrix of the form

j;AB C

3 2

S 0 S 0 S 0

j;AA j;AB j;AC

7 6

; S 0 =

7 6

S 0 S 0 S 0

j;AB C

j;AB j;B B j;B C

5 4

S 0 S 0 S 0

j;AC j;B C j;C C

where the elements of the matrix are

Z Z

1=2 1=2

2 2

f df; f df; S 0 = 2 S S 0 = 2 S

j;B B j;AA

j;Y j;X

1=2 1=2

Z Z

1=2 1=2

f df; S 0 = S f S f df + S

j;C C j;X j;Y

j;X Y

1=2 1=2

1=2

S 0 = 2 S f S f df;

j;AB j;X Y j;Y X

1=2

S f S f df; and S 0 = 2

j;X j;Y X j;AC

1=2

S 0 = 2 S f S f df:

j;B C j;Y j;Y X

1=2

The gradient is explicitly given by

2 2

g_ ; ; =

j j XY j

X Y

XY j XY j

p p p

;

2 2 2 2 2 2 2 2

2 2

j j j j j j j j

X X Y X X Y Y Y

and, through matrix multiplication and application of Parseval's relation to each auto

and cross sp ectra in S 0, we may express Equation A2 as

j;AB C

N 1

2 2

j j

XY XY

2s + 2C C

j; ;X Y j; ;Y X

j; ;X

6 2 4 4

4 2

j j j j

X Y X Y

=N 1

2 2

2s s s + C + +

j; ;X j; ;Y

j; ;Y j; ;X Y

6 2 2 2

j j j j

Y Y X X

XY j XY j

2s C 2s C :

j; ;X j; ;Y X j; ;Y j; ;Y X

4 2 2 4

j j j j

X Y X Y

Each of the auto covariance terms are equivalent to the wavelet auto covariance for

scale de ned by letting X = Y in Equation 1 and each cross-covariance term is

j t t

equivalent to the wavelet cross-covariance for scale . This yields the desired result.

j 2

List of Figures

1. Station pressure series for Truk Island 7.4 N, 151.8 W and the Southern

Oscillation Index. The \staggered" lo ok of the Truk Island series prior to 1971 is

the result of rounding to the nearest millibar.

2. Estimated cross-correlation sequence for the Southern Oscillation Index and Truk

Island station pressure series for lags up to 240 days.

3. Estimated lag window co-sp ectrum and magnitude squared coherence MSC

between the Southern Oscillation Index and Truk Island station pressure series. A

Parzen lag window, with sp ectral window bandwidth 0:0081 day , was applied

to the p erio dogram. The dotted and dashed lines corresp ond to the 5 and 1

levels of signi cance test for non-zero MSC, resp ectively.

4. MODWT co ecients for the Truk Island station pressure series using the LA8

f f f

wavelet lter. The wavelet co ecient vectors W ; W ;::: ; W are asso ciated

1 2 10

with variations on scales of 1; 2;::: ;1024 days and the scaling co ecient vector

V is asso ciated with variations of 2048 days or longer.

5. MODWT co ecients for the daily Southern Oscillation Index using the LA8

wavelet lter. The wavelet and scaling co ecients have the same interpretation as

in Figure 4.

6. MODWT estimated wavelet correlation between the Southern Oscillation Index

and Truk Island station pressure series.

7. MODWT estimated wavelet cross-correlation between the Southern Oscillation

Index and Truk Island station pressure series for lags up to 240 days. Con dence

intervals may be computed from Section 4.2. The p ositive p eak in the fth scale

is at a lag of0days. 5 31

Truk 000 1005 1010 1015 SOI Pressure (mb) 0 -10 0 5 2

1960 1970 1980 1990

Time

Figure 1.

0.25 0.20 0.15 0.10 ccs 0.05 0.0 -0.05

-200 -100 0 100 200

Lag (days)

Figure 2. 32

Magnitude Squared Coherence .0 0.10 0.20 0 Co-spectrum 0 20406080

0.0 0.02 0.04 0.06 0.08

frequency

Figure 3. 33

W10

V10

1960 1970 1980 1990

time (years)

Figure 4. 34

W10

V10

1960 1970 1980 1990

time (years)

Figure 5. 35

- -

------Wavelet Correlation -1.0 -0.5 0.0 0.5 1.0

1 2 4 8 16 32 64 128 256 512

Scale (days)

Figure 6. 36

d10

-200 -100 0 100 200

Lag (days)

Figure 7.