Asymptotic decorrelation of between-scale wavelet coefficients of generalized fractional process

$Page 1: Asymptotic decorrelation of between-scale wavelet coefficients of generalized fractional process$
Digital Signal Processing 16 (2006) 320–329

www.elsevier.com/locate/dsp

Asymptotic decorrelation of between-scale wavelet coefficients ofgeneralized fractional process

Alex Gonzaga a,b,∗, Akira Kawanaka b

a Department of Physical Sciences and Mathematics, University of the Philippines, Padre Faura St., Manila 1000, Philippinesb Department of Electrical and Electronics Engineering, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan

Available online 9 December 2005

Abstract

Recent interest on the wavelet transform of digital random signals with long-memory is significantly due to the approximatedecorrelation of their wavelet coefficients, which simplifies system identification and estimation. In this paper, we show that fora fairly general model of long-memory across-scale autocovariances of wavelet coefficients converge rapidly to zero, and wedetermine the rate of converge. The result provides useful groundwork for wavelet-based processing of long-memory randomsignals.© 2005 Elsevier Inc. All rights reserved.

Keywords: Wavelet coefficients; Long-memory process; GARMA(p,d,u, q) process; Generalized fractional process

1. Introduction

There is ample evidence that the phenomenon of long-memory occurs in various areas of human endeavor such as intelecommunications [1–3], video traffic [4], economics [5], medicine [6], and hydrology [7]. A long-memory processis usually defined as a stationary process for which the autocorrelation function at large k satisfies ρ(k) ∼ Cρk2d∗−1 ask → ∞, where Cρ �= 0 and 0 < d∗ < 0.5. In this case, autocorrelations are not summable and decay to zero slowly ata hyperbolic rate. If −0.5 < d∗ < 0, autocorrelations are summable and we call the process an intermediate-memoryprocess. An equivalent statement for the power spectrum of Yt is given by SY (f ) ∼ Cf |f |−2d∗

, as f → 0, which hasa pole at the origin when 0 < d∗ < 0.5 and zero if −0.5 < d∗ < 0.

A fairly general model of long-memory and intermediate-memory is the Gegenbauer autoregressive moving av-erage (GARMA) process defined in [8]. It generalizes the definition of long-memory and intermediate-memory byincluding a parameter that accounts for persistent cyclic behavior of a random signal. It allows the power spectrumto have pole, not only at 0, but at any frequency in the interval [0,0.5]. This model has been shown to represent wellsome random signals such as the Bellcore Ethernet trace data [9], atmospheric measurements [10], and economicdata [11].

* Corresponding author. Fax: +81 3 3238 3321.E-mail addresses: [email protected], [email protected] (A. Gonzaga).

1051-2004/$ – see front matter © 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2005.11.003

A. Gonzaga, A. Kawanaka / Digital Signal Processing 16 (2006) 320–329 321

A GARMA(p,d,u, q) process is the output of the system function

H(z) = Θ(z)

Φ(z)

(1 − 2uz−1 + z−2)−d (1)

driven by a stationary white noise input with mean 0 and variance σ 2. The rational function

Θ(z)

Φ(z)= 1 + θ1z

−1 + · · · + θqz−q

1 − φ1z−1 − · · · − φpz−p, θq �= 0, φp �= 0,

is the autoregressive moving average, ARMA(p,q) system, such that Θ(z) and Φ(z) have no common zeros, andtheir zeros lie inside the unit circle, which implies that the system is causal and invertible. On the other hand, theGegenbauer system, (1 − 2uz−1 + z−2)−d , d �= 0, can be written as

(1 − 2uz−1 + z−2)−d =

∞∑n=0

Cdn (u)z−n, (2)

where Cdn (u) is the Gegenbauer polynomial defined by

Cdn (u) =

(n/2)∑k=0

(−1)k(2u)n−2kΓ ((d) − k + n)

k!(n − 2k)!Γ (d),

and (n/2) is the largest integer less than or equal to n/2. The parameter u provides information about the periodicmovement in the random signal. When the input is a stationary white noise, the output is called a Gegenbauer process,which is stationary if d < 0.5 and |u| < 1 or if d < 0.25 and |u| = 1; it is invertible if −0.5 < d and |u| < 1 or−0.25 < d and |u| = 1 [8]. If we set u = 1 in (1), we obtain the system function of the well-known autoregressivefractionally integrated moving average (ARFIMA(p,2d, q)) process, which has a power spectrum with pole at theorigin when 0 < d < 0.5. It is called a fractionally differenced process when p = q = 0 and u = 1.

The power spectrum of a GARMA(p,d,u, q) process is given by

SX(f ) = σ 2z

∣∣∣∣Θ(e−i2πf )

Φ(e−i2πf )

∣∣∣∣2∣∣2(

cos(2πf ) − u)∣∣−2d (3)

where f ∈ (−0.5,0.5], and v = cos−1(u)/2π ∈ [0,0.5] is called the Gegenbauer frequency at which the power spec-trum becomes unbounded when 0 < d < 0.5. It provides an alternative approach in finding the autocovariance functionby evaluating the following integral

γ (k) = 2

0.5∫0

SX(f ) cos(2πkf )df, (4)

which can be computed using any software that allows singularities in the integrand. From [12], if u ∈ (−1,1) theautocovariance function at lag k of a GARMA(0, d,u,0) is given by

γ (k) = σ 2

2√

πΓ (1 − 2d)

[2 sin(v)

]0.5−2d[P 2d−0.5

k−0.5 (u) + (−1)kP 2d−0.5k−0.5 (−u)

],

where P ba (x) are the associated Legendre functions of the first kind. If u = 1, we have the autocovariance of a frac-

tionally differenced process given by

γ (k) = (−1)kσ 2Γ (1 − 4d)

Γ (k − 2d + 1)Γ (1 − k − 2d), |d| < 0.25.

If u = −1, the autocovariance function is given by

γ (k) = σ 2Γ (1 − 4d), |d| < 0.25.

Γ (k − 2d + 1)Γ (1 − k − 2d)

322 A. Gonzaga, A. Kawanaka / Digital Signal Processing 16 (2006) 320–329

The lack of simple expression for the autocovariance function, particularly for u ∈ (−1,1), makes the analysis ofGegenbauer process difficult. It could be simplified by applying a decorrelating transformation such as a wavelettransform.

In this paper, we deal with GARMA(p,d,u, q) process for which the Gegenbauer component is stationary andinvertible, the ARMA component is causal and invertible, and Θ(z) and Φ(z) are coprime. We call this process ageneralized fractional process. We derive the rate of convergence to zero of across-scale autocovariance function ofwavelet coefficients of this process. An example is presented to verify the theoretical results.

This paper is organized as follows. Our main results and an example are presented in Sections 2 and 3, respectively.Some concluding remarks are given in Section 4.

2. Autocovariance of wavelet coefficients

In this section, we present asymptotic properties of between-scale autocovariance function of wavelet coefficientsof a generalized fractional process. Our main results generalize the between-scale asymptotic properties of waveletcoefficients of fractionally differenced process [13,14] and fractional Brownian motion [15,16].

Let {djt | j = 1,2, . . . , J, t = 0, . . . ,N2−j − 1} be the nonboundary wavelet coefficients of the random signal{Yt }N−1

t=0 , where N = 2J for some integer J . From [17, p. 348],

Cov(djt , dj ′t ′) =1/2∫

−1/2

ei2πf (2j ′(t ′+1)−2j (t+1))Hj (f )H ∗

j ′(f )SY (f )df, (5)

where Hj,L(f ) is the Fourier transform of the level j Daubechies wavelet filter {hj,l} and * denotes the complexconjugation operator. Given the wavelet filter {h�,1}, the scaling filter {g�,1} is defined by g�,1 = (−1)�+1hL−�−1,1 for� = 0, . . . ,L − 1. For j > 1, wavelet and scaling filters {h�,j } and {g�,j } are of the same length Lj = (2j − 1)(L −1) + 1. Their transfer functions, Hj,L(f ) and Gj,L(f ), satisfy

Hj,L(f ) = H1,L

(2j−1f

) j−2∏l=0

G1,L

(2lf

)and Gj,L(f ) =

j−1∏l=0

G1,L

(2lf

), (6)

for |f | � 1/2, where H1,L(f ) and G1,L(f ) are the transfer functions of {h�,1} and {g�,1}, respectively. From [14], theenergy spectrum of the wavelet filter at scale 1 is given by

∣∣H1,L(f )∣∣2 = 2 sinL(πf )

(L/2)−1∑�=0

(L/2 − 1 + �

�

)cos2�(πf ) (7)

such that∣∣H1,L(f )∣∣2 + ∣∣G1,L(f )

∣∣2 = 2 and∣∣G1,L(f )

∣∣2 = ∣∣H1,L(0.5 − f )∣∣2

(8)

for all f .From [13, Lemma 4.1], as L → ∞, we have

∣∣G1,L(2kf )∣∣ →

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2, |f | ∈[0, 1

2k+2

),

1, |f | = 12k+2

,

0, |f | ∈(

12k+2 , 1

2k+1

]for k = 0,1, . . . , j − 2, and

∣∣H1,L(2j−1f )∣∣ →

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, |f | ∈[0, 1

2j+1

),

1, |f | = 12j+1 ,

2, |f | ∈(

1j+1 , 1

j

].

2 2


Hence for j > j ′, as L → ∞, we have

∣∣Hj ′L(f )∣∣2∣∣G1,L

(2j ′−1f

)∣∣2 →{

1, |f | = 12j ′+1 ,

0, otherwise

and

∣∣H1,L

(2j ′−1f

)∣∣2j−2∏m=0

∣∣G1,L

(2mf

)∣∣2 →{

1, |f | = 12j ′+1 = 1

2j ,

0, otherwise.

These imply that as L → ∞,∣∣Hj ′,L(f )∣∣2∣∣G1,L

(2j ′−1f

)∣∣2 → 0 a.e.

and

∣∣H1,L

(2j ′−1f

)∣∣2j−2∏m=0

∣∣G1,L

(2mf

)∣∣2 → 0 a.e.

on [0,0.5]. We use these facts to prove the ensuing theorems.In Theorem 1, we determine the order of convergence to zero of across-scale covariance function of wavelet co-

efficients of generalized fractional process with d < 0 as the filter length L increases. We show that the covarianceconverges at the rate O(1/L3/4), which is similar to the rate obtained in [13] for fractionally differenced process.We provide a separate proof for the case d > 0, in which the rate of convergence depends on the location of theGegenbauer frequency.

Theorem 1. Let {Yt } be a generalized fractional process and {h�,1, � = 0,1, . . . ,L − 1} the orthonormal Daubechieswavelet filter of length L, then for j > j ′ and d < 0,

∣∣Cov(djt , dj ′t ′)∣∣ = O

(1

L3/4

).

Proof. Let SY (f ) be the power spectrum of a GARMA(0, d,u,0) process, where d < 0. From (5), we have

∣∣Cov(djt , dj ′t ′)∣∣2 � C1

( 1/2∫0

∣∣Hj,L(f )∣∣∣∣Hj ′,L(f )

∣∣SY (f )df

)2

(9)

for some constant C1 independent of L. Since j > j ′ there is a positive integer k � 1 such that j − k = j ′. This and(6) give us

∣∣Cov(djt , dj ′t ′)∣∣ � C1

( 0.5∫0

∣∣H(1)j,L(f )

∣∣∣∣G1,L

(2j ′−1f

)∣∣∣∣Hj ′,L(f )∣∣SY (f )df

)2

,

where |H(1)j,L(f )| = |H1,L(2j−1f )|∏k∈S |G1,L(2kf )|, S = {0, . . . , j ′ − 2, j ′, . . . , j − 2}. By Cauchy–Schwarz in-

equality, we have

∣∣Cov(djt , dj ′t ′)∣∣2 � C1

( 0.5∫0

∣∣Hj ′,L(f )∣∣2∣∣G1,L

(2j ′−1f

)∣∣2SY (f )df

)( 0.5∫0

∣∣H(1)j,L(f )

∣∣2SY (f )df

). (10)

Partitioning the interval of integration of the first factor in (10), we get

A1 =0.5∫ ∣∣Hj ′,L(f )

∣∣2∣∣G1,L

(2j ′−1f

)∣∣2SY (f )df

0


� 2j ′2−j ′−1∫

0

∣∣H1,L

(2j ′−1f

)∣∣2SY (f )df + 2j ′

j ′−2∑m=0

2−m−1∫2−m−2

∣∣G1,L

(2mf

)∣∣2SY (f )df

� C2

2−j ′−1∫0

∣∣H1,L

(2j ′−1f

)∣∣2∣∣(cos(2πf ) − u)∣∣−2d

df

+ C2

j ′−2∑m=0

2−m−1∫2−m−2

∣∣G1,L

(2mf

)∣∣2∣∣(cos(2πf ) − u)∣∣−2d

df (11)

for some constant C2 independent of L. By change of variable, we get

A1 � C3

1/4∫0

∣∣H1,L(y)∣∣2∣∣(cos

(2−j ′+2πy

) − u)∣∣−2d

dy

+j ′−2∑m=0

C′m

1/2∫1/4

∣∣H1,L(0.5 − y)∣∣2∣∣(cos

(2−m+1πy

) − u)∣∣−2d

dy

for some constants C3 and C′m independent of L. By change of variable in the second integral, we have

A1 � C3

1/4∫0

∣∣H1,L(y)∣∣2∣∣(cos

(2−j+2πy

) − u)∣∣−2d

dy

+j ′−2∑m=0

C′m

1/4∫0

∣∣H1,L(x)∣∣2∣∣(cos

(2−m+1πx − 2−mπ

) − u)∣∣−2d

dx. (12)

From [13], for f ∈ [0,0.25), we have

∣∣H1,L(f )∣∣2 = cos(2πf )

∞∑k=L

(4k − 1)!!(4k)!! sin4k(2πf ). (13)

Hence the first term in (12) becomes

A2 = limε→0+

∞∑k=L

(4k − 1)!!(4k)!!

0.25−ε∫0

cos(2πy) sin4k(2πy)∣∣(cos

(2−j+2πy

) − u)∣∣−2d

dy. (14)

Since d < 0, we have

A2 � limε→0+

(1 + |u|)−2d

∞∑k=L

(4k − 1)!!(4k)!!

0.25−ε∫0

cos(2πy) sin4k(2πy)dy

= limε→0+

(1 + |u|)−2d

2π

∞∑k=L

(4k − 1)!!(4k)!!

cos4k+1(2πε)

4k + 1= (1 + |u|)−2d

2π

∞∑k=L

(4k − 1)!!(4k)!!

1

4k + 1. (15)

Similarly, for the second term in (12), we have for each m

0.25∫ ∣∣H1,L(x)∣∣2∣∣(cos

(2−m+1πx − 2−mπ

) − u)∣∣−2d

dx � (1 + |u|)−2d

2π

∞∑k=L

(4k − 1)!!(4k)!!

1

4k + 1. (16)

0


From [13, Lemma 4.2], (4k − 1)!!/(4k)!! ≈ k−1/2 for large k, where a ≈ b means a/b is bounded both from belowand above. Moreover, the second factor in (10) is bounded for each L since

0.5∫0

∣∣H(1)j,L(f )

∣∣2SY (f )df � 2j−1

0.5∫0

SY (f )df = 2j−1 Var(Yt ). (17)

Thus, from (15)–(17) we have∣∣Cov(djt , dj ′t ′)∣∣2 = O

(1

L3/2

). (18)

The result for a GARMA(p,d,u, q) process follows since the power spectrum of a causal and invertible ARMA(p, q)process is bounded. �

In Theorem 2, we determine the order of convergence to zero of across-scale covariance function of wavelet co-efficients of generalized fractional process with d > 0, as the filter length L increases. We show that the rate ofconvergence depends on the location of the Gegenbauer frequency.

Theorem 2. Let {Yt } be a generalized fractional process and {h�,1, � = 0,1, . . . ,L − 1} the orthonormal Daubechieswavelet filter of length L, then for j > j ′ and d > 0,∣∣Cov(djt , dj ′t ′)

∣∣ = O

(1

L3/4

)if v ∈ [

0,2−j−1],and ∣∣Cov(djt , dj ′t ′)

∣∣ = O

(1

L1/4

)if v /∈ [

0,2−j−1].Proof. From (9) and Cauchy–Schwarz inequality, we have

∣∣Cov(djt , dj ′t ′)∣∣2 � C1

( 0.5∫0

∣∣H (2j ′−1f

)∣∣2j−2∏m=0

∣∣G1,L

(2mf

)∣∣2SY (f )df

)( 0.5∫0

∣∣H(2)j,L(f )

∣∣2SY (f )df

), (19)

where |H(2)j,L(f )|2 = |H1,L(2j−1f )|2 ∏j ′−2

m=0 |G1,L(2mf )|2. Clearly, the second factor in (19) is finite by similar argu-ment to (17). Partitioning the interval of integration of the first factor in (19), we get

A3 =( 0.5∫

0

∣∣H (2j ′−1f

)∣∣2j−2∏m=0

∣∣G1,L

(2mf

)∣∣2SY (f )df

)� C4

(A4 + A5 + A′

6

),

where

A4 =2−j−1∫0

∣∣H1,L

(2j ′−1f

)∣∣2∣∣(cos(2πf ) − u)∣∣−2d

df,

A5 =2−j∫

2−j−1

∣∣H1,L

(2j ′−1f

)∣∣2∣∣(cos(2πf ) − u)∣∣−2d

df,

A′6 =

j−2∑m=0

2−m−1∫2−m−2

∣∣G1,L

(2mf

)∣∣2∣∣(cos(2πf ) − u)∣∣−2d

df,

A6 =2−m−1∫−m−2

∣∣G1,L

(2mf

)∣∣2∣∣(cos(2πf ) − u)∣∣−2d

df

2


for some constant C4 independent of L (Ci will be used to denote a constant independent of L). From [13], forf ∈ [0,2−j ′−1), we have

∣∣H1,L

(2j ′−1f

)∣∣2 = cos(2j ′

πf) ∞∑

k=L

(4k − 1)!!(4k)!! sin4k

(2j ′

πf). (20)

Since j > j ′, [0,2−j−1] ⊂ [0,2−j ′−1) and

A4 =∞∑

k=L

(4k − 1)!!(4k)!!

2−j−1∫0

cos(2j ′

πf)

sin4k(2j ′

πf)∣∣(cos(2πf ) − u

)∣∣−2ddf,

where from [13, Lemma 4.2], (4k − 1)!!/(4k)!! ≈ k−1/2 for large k.If v /∈ [0,2−j−1], then

A4 ≈ C5

∞∑k=L

1

k1/2

2−j−1∫0

cos(2j ′

πf)

sin4k(2j ′

πf)df = C6

∞∑k=L

sin4k+1(2j ′−j−1π)

k1/2(4k + 1)� C6

∞∑k=L

1

k1/2(4k + 1)22k.

(21)

If v ∈ (0,2−j−1), we write

∣∣2(cos(2πf ) − u

)∣∣−2d =∣∣∣∣∣4 sin

(2π(f + v)

2

)sin

(2π(f − v)

2

)∣∣∣∣∣−2d

=∣∣∣∣∣4 sin

(π(f + v)

){ sin(π(f − v))

π(f − v)

}π(f − v)

∣∣∣∣∣−2d

, (22)

such that

A4 ≈ limε→0+ C7

∞∑k=L

1

k1/2

{ v−ε∫0

cos(2j ′

πf)

sin4k(2j ′

πf)(v − f )−2d df

+2−j−1∫v+ε

cos(2j ′

πf)

sin4k(2j ′

πf)(f − v)−2d df

}.

Since |(f − v)|−2d+1 is bounded for d ∈ (0,0.5), integration by parts gives us

A4 ≈ C7

∞∑k=L

1

k1/2

{C8 cos

(2j ′−j−1π

)sin4k

(2j ′−j−1π

) + C9 cos(2j ′

πν)

sin4k(2j ′

πv)}

� C7

∞∑k=L

1

k1/2

{C8

1

22k+ C9

1

22k

}= C10

∞∑k=L

1

k1/222k. (23)

By a similar argument and appropriate change of limits of integration, we obtain the same results for v = 0 andv = 2−j−1.

Now, suppose that v /∈ [2−j−1,2−j ]. Clearly, [2−j−1,2−j ] ⊂ [0,2−j ′−1) except possibly at the endpoint 2−j whenj − j ′ = 1. We then have

A5 ≈ limδ→0+ C11

∞∑k=L

1

k1/2

2−j −δ∫2−j−1

cos(2j ′

πf)

sin4k(2j ′

πf)df

= C12

∞∑ 1

k1/2

{sin4k+1(2j ′−jπ) − sin4k+1(2j ′−j−1π)

4k + 1

}= C13

∞∑ 1

k1/2(4k + 1). (24)

k=L k=L


If v ∈ (2−j−1,2−j ), by analogous argument to (23), we have

A5 ≈ C14

∞∑k=L

1

k1/2

{C15 cos

(2j ′−j−1π

)sin4k

(2j ′−j−1π

) + C16 cos(2j ′−jπ

)sin4k

(2j ′−jπ

)+ C17 cos

(2j ′

πv)

sin4k(2j ′

πv)}

� C14

∞∑k=L

1

k1/2

{C15

1

22k+ C16

1

22k+ C17 cos

(2j ′

πv)

sin4k(2j ′

πv)}

. (25)

By a similar argument and appropriate change of limits of integration, we obtain the same results for v = 2−j−1

and v = 2−j . The series in (25) converges since cos(2j ′πv) sin(2j ′

πv) = 0 if v = 2−j and j ′ − j = −1; otherwisesin(2j ′

πv) < 1 on v ∈ [2−j−1,2−j ].From [13], for f ∈ (2−m−2,2−m−1], we have

∣∣G1,L

(2mf

)∣∣2 = − cos(2m+1πf

) ∞∑k=L

(4k − 1)!!(4k)!! sin4k

(2m+1πf

),

where m = 0,1, . . . , j − 2. Hence, for v /∈ [2−m−2,2−m−1], we have

A6 ≈ limδ→0+ C18

∞∑k=L

1

k1/2

2−m−1∫2−m−2+δ

cos(2m+1πf

)sin4k

(2m+1πf

)df = C18

∞∑k=L

1

k1/2

{1

4k + 1

}. (26)

If v ∈ (2−m−2,2−m−1), by analogous argument to (23), we have

A6 ≈ C19

∞∑k=L

1

k1/2

{cos

(2m+1πv

)sin4k

(2m+1πv

)}. (27)

By a similar argument and appropriate change of limits of integration, we obtain the same results for v = 2−m−2 andv = 2−m−1. The series in (27) converges since cos(2m+1πv) sin(2m+1πv) = 0 if v = 2−m−1 or v = 2−m−2; otherwisesin(2m+1πv) < 1 for v ∈ (2−m−2,2−m−1).

Thus, if v ∈ [0,2−j−1], from (23), (24), and (26), we have

∣∣Cov(djt , dj ′t ′)∣∣2 = O

(1

L3/2

).

If v /∈ [0,2−j−1], from (21), (24)–(27) we have

∣∣Cov(djt , dj ′t ′)∣∣2 = O

(1

L1/2

).

The result for a GARMA(p,d,u, q) process follows since the power spectrum of a causal and invertible ARMA(p,q)process is bounded. �

The preceding theorems allow us to approximate the across-scale autocovariance of a generalized fractional processby zero for an appropriate choice of wavelet filter length L. This approximation does not depend on the length ofthe discrete signal but the length of the wavelet filter, which is under the control of the analyst. For a fractionallydifferenced process, a special class of GARMA(p,d,u, q) process, |Cov(djt , dj ′t ′)| decreases to zero at the rateO(1/(L3/4)) [13]. Our results show that if d > 0, the autocovariance function converges at the same rate O(1/(L3/4)).However, when d < 0, the rate of convergence is dependent on the location of the Gegenbauer frequency.

The results show that between-scale correlations of wavelet coefficients of a generalized fractional process convergerapidly to zero as the filter length L increases. It is not true for within-scale correlations, which could be relativelylarge even for large values of L. From [18], Daubechies’ wavelet filters converge monotonically to an ideal high-pass


Fig. 1. Absolute maximum values of between-scale correlations of wavelet coefficients.

filter and Daubechies’ scaling filters converge monotonically to an ideal low-pass filter as the filter length L increaseswithout bound. Thus, for an appropriate choice of large L, we may write the autocovariance in the form

Cov(dj,t , dj,t+s) ≈ 2j+1

2−j∫2−(j+1)

cos(2j+1πsf

)SY (f )df = ∣∣Rdj

(s)∣∣.

Clearly, for fixed j , |Rdj(s)| is relatively large when v ∈ [2−j−1,2−j ] and 0 < d < 0.5. However, as v approaches

zero the effect of singularity diminishes due to the decreasing length of the interval [2−j−1,2−j ]. Hence, for appro-priate choice of filter length L, within-scale correlations of wavelet coefficients of a fractionally differenced processare approximately zero. This approximation is not valid for long-memory processes with pole not close to the zerofrequency.

3. An example

Consider a GARMA(1,0.25, u,1), where v = 0.35, Φ1 = 0.5, and θ1 = 0.6 using Daubechies wavelets D(L),L = 2,4, . . . ,10. Wavelet coefficients for scales 1� j ′ < j � 3 are obtained and the maximum absolute value ofCov(djt , dj ′t ′) for all possible values of t and t ′. Numerical computations show that within-scale covariances non-monotonically decrease to zero. In Fig. 1, if j = 2 and j ′ = 1 the maximum correlation for the Haar wavelet is 0.1223,which decreases to 0.0006 when L = 20. The approximation to zero is fairly accurate even for small values of L. Fasterdecorrelation can be observed for larger |j − j ′|.

4. Concluding remarks

In this paper, we derived the rate of convergence to zero of between-scale covariances of wavelet coefficients of ageneralized fractional process. Results show that discrete wavelet transform is an effective between-scale decorrelatorof a generalized fractional process. The approximation does not depend on the length of the signal but the lengthof the filter, which is under the control of the analyst. Numerical computations show that this approximation can befairly accurate even for small values of L. However, within-scale autocorrelations may not be negligible, especiallyfor scale j for which the Gegenbauer frequency v ∈ [2−j−1,2−j ]. Modeling within-scale wavelet coefficients maybe necessary. Alternatively, appropriate choice of the wavelet basis functions that provide optimal decorrelation for aparticular location of singularity of a GARMA(p,d,u, q) process may be desirable.


References

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(1992) 904–909.[17] D.B. Percival, A.T. Walden, Wavelet Methods for Time Series Analysis, Cambridge Univ. Press, Cambridge, UK, 2000.[18] M.J. Lai, On the digital filter associated with Daubechies’ wavelets, IEEE Trans. Signal Process. 43 (1995) 2203–2205.

Alex Gonzaga was born in Samar, Philippines, in 1972. He received his B.S. degree in mathematics from the Philippine NormalUniversity, Manila, in 1993, his M.S. degree in mathematics from De La Salle University, Manila, in 1995, and his Ph.D. degreein statistics from the University of the Philippines, Diliman, in 2001. Since 1995 he has been teaching mathematics and statisticsin the University of the Philippines, Manila, where he is currently an associate professor. His research interests include waveletanalysis of random signals, time series analysis, and graph theory.

Akira Kawanaka was born in Hiroshima, Japan, in 1955. He received his B.E. and Ph.D. degrees in electrical and electronicsengineering from Sophia University, Tokyo, Japan, in 1977 and 1982, respectively. He was a research associate of the Faculty ofScience and Technology at Sophia University in 1982, and was also a research associate at the Institute of Industrial Science, TheUniversity of Tokyo, from 1983 to 1987. Since 1987, he has been with the Department of Electrical and Electronics Engineering,Sophia University, where he is a professor. His research interests include signal processing, image and 3D model data compression,and pattern recognition.

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Asymptotic decorrelation of between-scale wavelet coefficients of generalized fractional process