Simulating Gaussian Stationary Processes with Unbounded Spectra Brandon Whitcher EURANDOM P.O. Box 513 5600 MB Eindhoven The Netherlands
[email protected]
June 5, 2000 Abstract
We propose a new method for simulating a Gaussian process, whose spectrum diverges at one or multiple frequencies in [0 21 ] (not necessarily at zero). The method utilizes a generalization of the discrete wavelet transform, the discrete wavelet packet transform (DWPT), and only requires explicit knowledge of the spectral density function of the process { not its autocovariance sequence. An orthonormal basis is selected such that the spectrum of the wavelet coecients is as at as possible across speci c frequency intervals, thus producing approximately uncorrelated wavelet coecients. We compare this method to a popular time-domain technique based on the Levinson-Durbin recursions. Simulations show that the DWPT-based method performs comparably to the time-domain technique for a variety of sample sizes and processes { at signi cantly reduced computational time. The degree of approximation and reduction in computer time may be adjusted through selection of the orthonormal basis and wavelet lter. ;
Some key words: Autocovariance, Discrete wavelet packet transform; Gegenbauer process; Orthonormal basis; Seasonal persistent process; Time series.
1
1 Introduction The use of fractional autoregressive integrated moving-averages (ARIMAs) as models for time series exhibiting long-range dependence is widely accepted. Such time series are characterized by P an autocovariance sequence fs g that slowly decays and, hence, diverges; i.e., 1 =0 js j = 1. Let
fXt g be a stochastic process whose dth order backward di erence (1 ? B )d Xt = t
(1.1)
is a stationary process, where ? 12 < d < 21 and B is the backward di erence operator. This process is stationary and invertible. If ft g is a Gaussian white noise process with variance 2 , then fXt g is a simple example of a fractional ARIMA process, a fractional ARIMA(0; d; 0), which we refer to as a fractional di erence process. The spectral density function (SDF) of fXt g is given by SX (f )
2 j2 sin(f )j2d for
? 12 < f < 12 ;
so that SX (f ) ! 1 as f ! 0 and, thus, the SDF diverges at zero frequency. Further introductions to fractional di erence and related processes can be found in, e.g., Granger and Joyeux (1980) and Hosking (1981). Both time- and frequency domain techniques have been established for the simulation of such long-memory processes (see Hosking (1984), Percival (1992) and references therein), the partitioning of the time-frequency plane by the discrete wavelet transform (DWT) makes it a natural alternative to the discrete Fourier transform. This has been investigated by, e.g., Wornell (1993), Masry (1993) and McCoy and Walden (1996), who found that wavelet coecients of a long-memory process are approximately uncorrelated. Hence, simulation may be performed by simply generating Gaussian random variables with zero mean and variance proportional to the integrated spectrum over octave bands. Applying the inverse DWT produces a realization of the desired long-memory process. 1
The time-domain technique is readily adaptable to processes whose spectral energy is not dominated in the lower frequencies as long as the autocovariance sequence is known. An example of such a process is the MA(1) time series model Xt = t + t?1 which is dominated by higher frequencies in its spectrum as ! ?1. Frequency-domain techniques have diculties because they must explicitly deal with the asymptote in the SDF. Percival (1992) gives an approximate frequencydomain technique to simulate stationary power-law processes by approximating the variance of its sample mean, thus determining the variance for the Gaussian random variable associated with zero frequency. This result was derived solely for stationary power-law processes and it is not known at this time if it can be easily generalized when the asymptote occurs at a frequency f 6= 0. Even if the method of Percival (1992) was available to us, an additional problem is that we are not guaranteed that the asymptote occurs exactly on a Fourier frequency (which is always the case for pure power-law processes) and therefore minor re-shaping of the spectrum would be needed. The method proposed here avoids these diculties by assuming a piecewise constant spectrum across variable-length frequency intervals. The DWT cannot adapt to a general SDF, instead we may make use of the discrete wavelet packet transform (DWPT), which creates a redundant collection of wavelet coecients at each level of the transform organized in a binary tree structure (Wickerhauser 1994, Ch. 7). To determine an appropriate orthonormal basis, we propose a method based on achieving the most shallow spectrum for each level of the DWPT, thus producing the least correlated wavelet coecients for a broader class of stochastic processes. Simulation may be performed through a method similar to that for long-memory processes. The technique proposed by McCoy and Walden (1996) may be considered a special case of the method given here. Implementation of the DWPT simulation method involves several choices for the researcher. First, an appropriate wavelet family must be selected. While the Daubechies extremal phase and 2
least asymmetric wavelet families have a closed form expression for their squared gain function, the new family of minimum-bandwidth discrete-time (MBDT) wavelets (Morris and Peravali 1999) is a better approximation to an ideal band-pass lter and outperforms Daubechies wavelets in the simulation studies presented here. Once a wavelet family has been selected, we must determine an appropriate length for the lter. By studying the SDF of wavelet coecients, lters of length 8 or longer are required to produce approximately uncorrelated coecients. Balancing computational eciency and performance, a length 16 MBDT wavelet appears to be sucient for the processes we investigated. That is, it consistently performed better than shorter wavelets and there was no clear advantage to using longer ones. As with the DWT, only dyadic length vectors (i.e., vectors of length N = 2k for some positive integer k) may be synthesized in a straightforward manner using the DWPT. If one wishes to simulate a non-dyadic length series simply select the next largest dyadic length, generate a realization and then discard an appropriate portion of it. There is very little increase in computation time given the speed of the DWPT-based method. In Section 2 we introduce the seasonal persistent process as an example of a time series with unbounded spectrum at a single frequency. The DWPT is brie y de ned in Section 3, our basis selection technique is illustrated and simulation procedure outlined. Simulation results are presented in Section 4 where the DWPT method is compared to an exact time-domain technique that utilizes the Levinson-Durbin recursions and their computation time is illustrated. The wavelet-based procedure is also extended to the case of two spectral asymptotes. Conclusions are presented and generalizations given at the end.
2 Seasonal Persistent Processes A simple generalization of the model given in Equation 1.1 was mentioned, in passing, by Hosking (1981) and allows the singularity in the spectrum to be located at any frequency 0 f 21 . Such 3
a process has been referred to as a Gegenbauer process (Gray et al. 1989) and also a seasonal persistent process (SPP) (Andel 1986). We prefer the latter term because it more accurately and concisely describes the content of the time series. That is, a sinusoid of particular frequency is associated with the singularity present in the spectrum thus causing a persistent oscillation in the process. Gray et al. (1989) t a seasonal long-memory model to the Wolfer sunspot data, where short-range dependence was allowed through tting ARMA components. Recent attention has also appeared in the economics literature, where Ooms (1995) t seasonal long-memory models to the U.S. gross national product and a time series of Danish shipping records. Arteche and Robinson (1999) discuss various models for seasonal long-memory and propose a semi-parametric estimation procedure based on the periodogram. Let fYt g be a stochastic process such that ?
1 ? 2 B + B 2 Yt = t
(2.2)
is a stationary process, then fYt g is an SPP, where Gray et al. (1989) showed that fYt g is stationary if j j < 1 and < 21 or j j = 1 and < 41 and invertible if j j < 1 and > ? 21 or j j = 1 and > ? 41 .
Clearly, the de nition of an SPP also includes a fractional di erence process. When = 1 we have that fYt g is a fractional di erence process given by Equation 1.1 with fractional di erence parameter d = 2. If ft g is a Gaussian white noise process, then fYt g has the following in nite moving average representation Yt =
1 X k
=0
() ; Ck; t?k
(2.3)
() is a Gegenbauer polynomial (Rainville 1960, Ch. 17). The SDF of fY g is given by where Ck; t
SY (f ) =
2 f2j cos(2f ) ? jg2 ; for
4
? 12 < f < 12 ;
(2.4)
so that SY (f ) becomes unbounded at frequency fG (cos?1 )=(2), sometimes called the Gegenbauer frequency. The autocovariance sequence of an SPP may be expressed via s =
Z 1=2
?1=2
SY (f ) cos(2f ) df:
(2.5)
An explicit solution is known only for special cases (Andel 1986). In practice the autocovariance sequence may be obtained through numeric integration of Equation 2.5. Gray et al. (1994) showed that the autocorrelation sequence of an SPP may be approximated via r
2?1 cos(2fG ) as ! 1:
(2.6)
Two sequences are related via as ! 1 if lim !1f = g = c where c is a nite nonzero constant. The result in Equation 2.6 was given in Hosking (1982) without proof. When the fractional di erence parameter is large in magnitude, the asymptotic approximation matches the true autocorrelation sequence quite well { only slightly over-estimating it. This apparent overestimation is even more pronounced when is relatively small in magnitude, and persists for a large number of lags.
3 Simulation via the Discrete Wavelet Packet Transform 3.1 The Discrete Wavelet Packet Transform
The orthonormal discrete wavelet transform (DWT) produces a length N vector of wavelet coe-
W that are obtained by projecting a length N vector of observations (time series) X onto a speci c set of basis functions; i.e., W = W X where W is an N N orthonormal matrix de ning
cients
the DWT. To implement the DWT, and all future transforms, we assume a dyadic length N = 2J
X
for . The basis functions used in constructing W di er from sinusoids used in the discrete Fourier transform (DFT) by having compact support, thus allowing wavelet coecients to capture features that are local in both time and frequency. 5
The DWT is known to approximately decorrelate long-memory processes (Tew k and Kim 1992; Wornell 1996). In signal processing terms, it does this through band-pass ltering the process in such a way that the spectrum in each pass-band is approximately constant. Using this property, McCoy and Walden (1996) outlined a procedure for simulating a fractional di erence process using the DWT, where only the overall variance and band-pass variances of the process are required. Explicit knowledge of the covariance structure is not necessary in order to implement their simulation procedure. The DWT has a very speci c band-pass structure that partitions the spectrum of a long-memory process by halving the frequency bands { producing ner partitions where the spectrum is rapidly changing. This is done through a succession of ltering and downsampling operations which is a reformulation of the matrix operation previously used to de ne the DWT; see, e.g., Percival and Walden (2000, Ch. 4) for more information. In order to exploit the approximate decorrelation property for SPPs we need to generalize the partitioning scheme of the DWT. This is easily done by performing the discrete wavelet packet transform (DWPT); see, e.g., Wickerhauser (1994, Ch. 7) and Percival and Walden (2000, Ch. 6). Instead of one particular ltering sequence, the DWPT executes all possible ltering combinations to obtain a wavelet packet tree. Let T = f(j; n) j j = 0; : : : ; J ; n = 0; : : : ; 2j ? 1g be the collection of all doublets (j; n) that form the indices of the nodes of a wavelet packet tree. An orthonormal basis B T is obtained when a collection of DWPT coecients is chosen, whose ideal band-pass frequencies are disjoint and cover [0; 21 ]. The set B is simply a collection of doublets (j; n) which correspond to an orthonormal basis. Thus, a vector of DWPT coecients is obtained via
WB = WBX, where WB is an N N
orthonormal matrix de ning the DWPT for the orthonormal basis B. Let h0 ; : : : ; hL?1 be the unit scale wavelet (high-pass) lter coecients from a Daubechies compactly supported wavelet family (Daubechies 1992) of even length L. In the future, we will 6
denote the Daubechies family of extremal phase compactly supported wavelets with D(L) and the Daubechies family of least asymmetric compactly supported wavelets with LA(L). The scaling (low-pass) coecients may be computed via the quadrature mirror relationship gl = (?1)l+1 hL?l?1 ; l = 0; : : : ; L ? 1:
Now de ne un;l
gl ; if n mod 4 = 0 or 3; hl ; if n mod 4 = 1 or 2;
to be the appropriate wavelet packet lter at a given node of T . Let
X be a length N = 2
J
vector of observations and
W
j;n
, (j; n) 2 T , denote the vector of
?
wavelet coecients associated with the frequency interval j;n 2jn+1 ; 2nj+1 +1 ; see Figure 1. Here we assume that a complete wavelet packet table will be constructed down to level J . Let Wj;n;t denote the tth element of the length Nj N=2j vector
W ?1 b n2 c, we compute W j
;
j;n;t
Wj;n;t
?1
L X l
=0
W
j;n
. Given the vector of DWPT coecients
via un;l Wj ?1;b n2 c;2t+1?l mod Nj?1 ; t = 0; 1; : : : ; Nj ? 1:
Since the wavelet coecients from the previous level j ? 1 are being utilized here, constructing the current series of wavelet coecients only requires a convolution with L nonzero terms. To start the recursion set
W0 0 = X. As with the DWT, the DWPT is most eciently computed using a ;
pyramid algorithm that has O(N log N ) operations, the same number as the fast Fourier transform.
3.2 Selecting an Orthonormal Basis Although several methods exist for choosing a particular collection of nodes to construct an orthonormal basis B (see Chen et al. (1999) for an overview), a simple one is as follows. After the initial high- and low-pass ltering operation, simply apply a successive high- and low-pass ltering operation to the portion of the spectrum which contains the Gegenbauer frequency. The idea 7
behind this is that we want to obtain band-pass spectra which are as at as possible, hence ner partitioning of the frequency axis is necessary where the spectrum is rapidly changing. Figure 2b shows the \ideal basis" described above in terms of the time-scale plane down to level J = 7 for an SPP with spectrum shown in Figure 2a; i.e.,
B = f(1; 1); (2; 1); (3; 0); (4; 3); (5; 4); (6; 11); (7; 20); (7; 21)g: When synthesizing fractional di erence processes, as in McCoy and Walden (1996), the basis associated with the DWT is sucient for producing approximately uncorrelated wavelet coecients at each scale. One inherent property of any wavelet lter is zero mean, which corresponds to its squared gain function (modulus squared of its DFT) having value 0 at frequency zero. In fact, the Daubechies families of wavelet lters considered here have L=2 vanishing moments. This property is key to producing approximately uncorrelated wavelet coecients for so-called long memory processes and is distinctly lacking when the asymptote is allowed to vary throughout frequencies fG 2 [0; 21 ]. The immediate consequence is the ideal basis in Figure 2b will produce approximately
uncorrelated wavelet coecients only when the wavelet lter is a `good enough' approximation to an ideal band-pass lter. Since the length of a Daubechies wavelet lter determines its frequency localization, if we allow L ! 1 then the ideal basis will guarantee approximately uncorrelated wavelet coecients. To illustrate the poor frequency localization of nite length wavelet lters, the SDF of
W1 1 ;
is given for three di erent Daubechies wavelet lters of length L 2 f2; 4; 8g in Figure 3 using the SPP ( = 0:4; fG = 121 ; = 0:866) described in Figure 2a . These results apply exclusively to the Daubechies extremal phase and least asymmetric families of wavelet lters since their squared gain functions are identical. Even though the ideal pass-band is 41 jf j 21 , and fG =
1 12
is outside
this interval, the poor approximation to an ideal band-pass lter is apparent for the Daubechies 8
length 2 and 4 wavelet lters. The asymptote is still present in the spectrum of
W1 1, and hence ;
these wavelet lters are insucient for producing approximately uncorrelated wavelet coecients for this particular process. The Daubechies wavelet lter of length 8 allows only a very small spike of energy from the singularity into the spectrum, thus producing approximately uncorrelated wavelet coecients at this scale. One way to overcome the poor approximation of the wavelet lters to that of ideal band-pass lters would be to adaptively select an orthonormal basis where the squared gain function of the wavelet lter associated with
W
j;n
is suciently small at the Gegenbauer frequency. Let us de ne
Uj;n(f ) jUj;n(f )j2 to be the squared gain function for the wavelet packet lter fuj;n;lg, where Uj;n(f ) is the DFT of uj;n;l
?1
L X k
=0
un;k uj ?1;b n2 c;l?2j?1 k ; l = 0; : : : ; Lj ? 1;
with u1;0;l gl , u1;1;l hl and Lj = (2j ? 1)(L ? 1) + 1 (Percival and Walden 2000, Ch. 6). The partition of the time-scale plane would therefore depend upon the choice of wavelet lter, overall depth of the partition being inversely proportional to the length of the wavelet lter. Hence, the basis selection procedure involves selecting the combination of wavelet basis functions that achieve
Uj;n(fG) < ", for some " > 0 at the minimum level j . For our purposes, only the location of the Gegenbauer frequency is used to select the orthonormal basis. The strength of the persistence present, and thus the rate of increase around the singularity in the spectrum, is not considered. This mimics the relationship between the orthonormal basis of the DWT and long-memory processes. An immediate question to ask is, can Uj;n(fG ) < " always be achieved? For the SPPs provided here, the minimum value of the squared gain function Uj;n(fG ) rapidly achieves zero as the level of the transform increases. Based on a length 16 Daubechies
9
extremal phase or least asymmetric wavelet lter we nd that, for unit level, min fU1;n ( 121 )g = 3:4 10?6 ;
0n1
min fU1;n (0:352)g = 8:7 10?3 ; and
0n1
min fU1;n (0:016)g = 2:1 10?17 :
0n1
If the level is increased, say to J = 7, we have min fU7;n ( 121 )g = 5:8 10?15 ;
0n27 ?1
min fU7;n (0:352)g = 7:4 10?25 ; and
0n27 ?1
min fU7;n (0:016)g = 6:0 10?72 :
0n27 ?1
Hence, moderate thresholds like " = 0:01 are attained for all levels of the DWPT using Daubechies wavelet lters of sucient length. Figure 4 gives an orthonormal basis, according to the criterion Uj;n(fG ) < 0:01, for Daubechies wavelet lters (extremal phase or least asymmetric) of various lengths L 2 f2; 4; 8; 16g applied to the SPP ( = 0:4; fG = 121 ; = 0:866) shown in Figure 2a. As to be expected, the shorter wavelet lters L 2 f2; 4g are poor approximations to an ideal band-pass lter and produce an orthonormal basis which is nothing like the ideal basis shown in Figure 2b. As the length of the wavelet lter increases the frequencies captured outside the nominal pass-band are reduced, and thus, it better mimics an ideal band-pass lter. The Daubechies wavelet lter with L = 8 produces a basis that follows the general shape of the ideal one, but su ers around the Gegenbauer frequency. The longest wavelet lter (L = 16), and therefore the best approximation to an ideal band-pass lter, generates an orthonormal basis that closely follows the underlying spectrum with only a few anomalies.
10
3.3 Simulation Procedure The band-pass variance Bj;n for an SPP, with spectrum given in Equation 2.4, in the frequency interval 2jn+1 jf j 2nj+1 +1 is Bj;n 2 4?
Z
n+1 2j +1
2 n j cos(2f ) ? j2 df: 2j +1
(3.7)
As in McCoy and Walden (1996), we replace the true SDF at each frequency band with a constant Sj;n = Sj;n(f ), for all f , such that the band-pass variances are equal. This step assumes the SDF
is slowly varying across the frequency interval j;n (c.f. Section 3.2). Integrating the constant spectrum over j;n gives Z
n+1 2j +1
Sj;n df = Sj;n2?j ?1 : n 2j +1
Equating this to the band-pass variance gives 2Sj;n2?j ?1 = Bj;n =) Sj;n = 2j Bj;n:
(3.8)
The variance of Wj;n;t is therefore given by Sj;n because of the band-pass nature of the transform. Let us consider simulating a length M = 2JM SPP with known parameters and . It is common when using Fourier-based methods to simulate a series several times longer than required and then use a segment of the appropriate length. We recommend using simulation lengths of M = 2N or M = 4N where N = 2J is the desired length of the simulated series. For non-dyadic sample sizes,
one can simply increase the length of the simulation to the minimum dyadic length that exceeds the desired sample size and then subsample it appropriately. The DWPT-based simulation procedure can be implemented as follows: 1. We must determine the appropriate orthonormal basis B from the wavelet packet tree T . This is done by computing, for levels j = 1; : : : ; JM ? 1, the squared gain function of the 11
desired wavelet lter at the Gegenbauer frequency. Applying the criterion Uj;n(fG ) < ", (j; n) 2 T , and orthogonalizing yields the doublets to be included in B. At the nal level JM , all remaining doublets corresponding to uncovered frequency intervals must be included in order to complete the orthonormal basis. 2. We need to calculate the band-pass variances Bj;n, (j; n) 2 B. They are de ned in Equation 3.7 and may be calculated via numeric integration. We use Equation 3.8 to obtain Sj;n. Each DWPT coecient Wj;n;t, (j; n) 2 B, t = 1; : : : ; 2JM ?j is a independent Gaussian random variable with zero mean and variance Sj;n. 3. Once the DWPT coecients have been generated we may organize them into the vector and apply the inverse DWPT via
WB
X = WB WB to produce a length M portion of the desired b
T
SPP. By generating a uniform random integer on the interval [1; M ? N +1], we may obtain a length N segment Xb ; : : : ; Xb+N ?1 . The numeric integration for all band-pass variances (except the one such that fG 2 j;n for (j; n) 2 B) required in Step 2 is easily computed using the routine QAGS from QUADPACK (Piessons et al. 1983), since the SDF is bounded on all frequency intervals. The nal band-pass variance will capture the unbounded portion of the spectrum, where numeric integration using QAGS may still be performed by splitting the integral at the Gegenbauer frequency.
4 Simulation Results
4.1 Comparison of the DWPT and Hosking Methods
Gray et al. (1989) used Equation 2.3 to simulate an SPP by truncating the in nite sum to 290,000 terms. We abandon this approach for two reasons, it is very computer intensive and it depends on the rate of decay of the series. Woodward et al. (1998) instead used a time domain approach that utilizes the Levinson{Durbin recursions; see, e.g., Hosking (1984) and Percival (1992). The 12
autocovariance sequence is required for this procedure and was calculated via numeric integration. We call this the Hosking method. Figure 5 shows the results of a simulation study (N = 512) demonstrating the ability of the Hosking method to generate SPPs with parameters given in Figures 8{11 from Andel (1986). Note, there is an error in the caption of Figure 11 in Andel (1986), the frequency is given as ! = 2fG = 0:1, but is incorrectly stated as 0.955 whereas it should be 0.995. As to be expected,
the Hosking method produces realizations with excellent second-order properties. The sample ACVS, averaged over the 500 realizations, follows the true ACVS very well for all four processes. The 5% and 95% points are provided to indicate variability of the sample ACVS. The DWPT-based method outlined in Section 3.3 was performed to simulate SPPs (N = 512; M = 2N ) with parameters identical to those in Figure 5. A non-adaptive basis was used
BM = f(log2 M; n) j n = 0; : : : ; 2log2 M ? 1g and the wavelet lter was the minimum-bandwidth discrete-time (MBDT) wavelet with 16 nonzero coecients. We refer to this class of wavelet lters as MB(L); see Morris and Peravali (1999) for details on this family of wavelets. The MBDT wavelets are better approximations to ideal band-pass lters than the Daubechies extremal phase or least asymmetric wavelet families at comparable lengths. Looking at Figure 6, the mean ACVS follows the true ACVS almost as well as the Hosking method with similar variability as demonstrated by the 5% and 95% points. Although graphically comparing the two methods is illuminating, a quantitative comparison between the true ACVS and the averaged sample ACVS is provided by the one-sided least squares di erence LSD(T )
T X
=0
js ? s j2 ;
where fs g is the averaged sample ACVS and fs g is the true ACVS. 13
Tables 1 and 2 provide the value LSD(N=2), where fs g was computed over 500 realizations, from a simulation study comparing the Hosking and DWPT methods for a variety of wavelet lters and sample sizes. As in Figure 6, the basis BM was used for all wavelet lters under the DWPT method. Note that the number of lags included in the computation of the LSD increases with sample size, 64 for N = 128, 128 for N = 256, 256 for N = 512 and so on. Therefore no monotonic improvement may be assumed with increasing N . This is in fact the case for the Hosking method where its LSD(N=2) values appear to lack a pattern across sample sizes. The DWPT method, when utilizing the MB(8), D(16) and LA(16) wavelet lters, performs similarly under all four models. This is because the squared gain functions are identical for the D(16) and LA(16) wavelets and very close to the MB(8) wavelet, even though it is half their length. When using the MB(16) and MB(24) wavelet lters, the DWPT method usually performs better than the other wavelet lters. It is not clear from the simulations that the longer MB(24) wavelet lter outperforms the MB(16). When the simulation size M is increased relative to the desired sample size, marked improvement is seen across all wavelet lters. By doubling the sample size for simulation M = 2N , we see reductions up to a factor of 2-3 times in the LSD for the SPP( = 0:4; fG = = 0:866) and around a factor of 1-2 times for the other SPPs. Mixed results are seen when M = 4N , in some cases the LSD is reduced even further (sometimes approaching the value seen under the Hosking method) while in other cases very little improvement is observed. In general, the best combination appears to be the MB(16) wavelet lter and M = 2N when one balances performance and complexity. Tables 3 and 4 provide results from a duplicate simulation study where an adaptive orthonormal basis was used for the DWPT method, under the criterion Uj;n(fG ) < 0:01. No optimization was performed to arrive at the value " = 0:01, rather it was chosen to illustrate that an adaptive basis which looks quite di erent from the full partitioning scheme produces realizations with a 14
comparable ACVS. The rst two SPPs ( 2 f0:2; 0:4g; = 0:866) have an identical orthonormal basis since fG = 121 ; see Figure 4. The third SPP ( = 0:3; = ?0:6) has an asymptote in its spectrum around fG = 0:352 and looks similar to a re ected version of Figure 4. The fourth SPP ( = 0:3; = 0:995) has its asymptote around fG = 0:016, quite close to zero. Hence, its basis looks very close to the DWT basis used for long-memory processes, with di erences appearing only at the lower frequencies and larger scales of the wavelet packet tree. The one-sided LSD between the true ACVS and average sample ACVS are very similar to those obtained using the basis BM as given in Tables 1 and 2, sometimes producing smaller values. Hence, the method proposed in Section 3.2 is generating an adaptive orthonormal basis that produces approximately uncorrelated wavelet coecients. This is a much more desirable basis because of its parsimony and computational eciency versus simply using all nodes at a particular level of the transform. It also illustrates the ability of the DWPT to decorrelate a broader class of stationary processes than previously known.
4.2 Extension: Multiple Singularities in the Spectrum An obvious extension of this method would be to allow multiple singularities to appear in the SDF of the process. Consider the zero mean k-factor SPP given by k Y ?
=1
1 ? 2 i B + B 2 i Yt = t ;
i
exhibiting k asymptotes located at the frequencies fi = (cos?1 i )=(2), i = 1; : : : ; k, in its spectrum SY(k) (f ) 2
Qk
=1 f2j cos(2f ) ? i jg
i
?2i , jf j < 1 . This process was introduced by Woodward et al. 2
(1998) under the name `k-factor Gegenbauer process.' For illustration purposes let us consider the 2-factor SPP with 1 = 0:45, 1 = 0:8, 2 = 0:3 and 2 = ?0:8 (Figure 7). In this case, orthonormal basis selection is done by merging the basis selected using f1 = 0:102 with the basis selected using f2 = 0:398. Thinking of each basis as a binary tree of Boolean variables 15
(true if the vector is included in the basis and false if not), one can picture a node-by-node `or' operation being performed between them and then an `orthogonalization' operation that translates two children nodes with the value true into a parent node with value true from the bottom of the tree to the root. This procedure is illustrated in Figure 8 using a length 16 Daubechies wavelet lter (extremal phase or least asymmetric). A small simulation study is provided (Table 5) in order to illustrate the ability of the DWPT method to handle a process whose spectrum exhibits more than one asymptote. As indicated by the LSD for the Hosking method, this 2-factor SPP is more dicult to simulate than any of the previous processes. Its ACVS is compared to that of the single factor SPP ( = 0:4; = 0:866) in Figure 7b. As with the previous simulation studies, both an adaptive and non-adaptive orthonormal basis were used in the DWPT method. Only MBDT wavelets were utilized given their performance with single factor SPPs. Under the non-adaptive basis BM , the combination producing simulations with average ACVS closest to the Hosking method was using the MB(24) wavelet lter and simulation size M > N . Results obtained through an adaptive basis (we used a slightly higher threshold " = 0:001 given the increased diculty encountered by the Hosking method) appear to equal
or improve upon the non-adaptive basis. This particular 2-factor SPP consists of asymptotes separated by a reasonable number of frequencies. There may be problems when the locations of the asymptotes are quite close together. Clearly, there is some advantage to not simply using an orthonormal basis that mimics the behavior of the DFT; i.e., achieving the highest frequency resolution possible. More work is needed in order to identify an `optimal' orthonormal basis, where optimal in this case would be a basis that produces the best average ACVS as a function of ".
16
4.3 Computing Time Computational eciency of an algorithm is an important issue in simulation methodology. The DWPT has O(N log N ) computational complexity, while the Levinson-Durbin recursions require O(N 2 ) operations. Both methods were implemented in S-Plus with the intensive computations
being written in C. Using the function unix.time(), Figure 9 compares the computational time needed to simulate SPPs for a variety of sample sizes on an SGI workstation. It appears the DWPT-based method requires a minimum of a half-second in order to compute one realization of small to moderate length. After N 210 = 1024, the computation time increases at a relatively slow rate and agrees with a similar technique for simulating long-range dependent network trac (Ribeiro et al. 1999). The Hosking method is quite fast for small sample sizes, is comparable around N = 512, then greatly exceeds the DWPT method for larger sample sizes. In fact, S-Plus failed to
successfully execute when N 212 { subsequent simulation times are given by extrapolation. In order to compute the Hosking method for large sample sizes, ecient storage and computation of the partial autocorrelations (N N matrix) would be required.
4.4 Availability of the Programs Programs used to produce the results in this article were written in S-Plus, C, and FORTRAN. They will be publicly available on the Internet at http://lib.stat.cmu.edu/JCGS/ under the title `whitcher'. This is the address for StatLib, a statistical archive maintained by Carnegie{Mellon University.
5 Discussion A new method has been proposed in order to simulate Gaussian stationary processes, whose SDF is unbounded at one or several frequencies in [0; 21 ], through the discrete wavelet packet transform. While not an `exact' method, as compared with the procedure given by Hosking (1984), it performs 17
comparably in Monte Carlo simulations. Given the relative eciency of the DWPT, it is faster to compute than the Hosking method and is `adaptive' in the sense that a variety of orthonormal bases may be selected { giving the user a choice between precision and computational speed. Simulating higher dimensional processes is a natural extension of this methodology. The key feature used here was the ability to select an orthonormal basis that approximately decorrelates the process of interest; e.g. a seasonal persistent process. This allows us to simulate independent and identically distributed Gaussian random variables and rely on the inverse DWPT to mix the wavelet coecients and produce a realization with adequate second-order properties. Stoksik et al. (1995) used the two-dimensional DWT to generate fractal images that were realizations of twodimensional fractional Brownian motion. Directly applying the techniques here in two dimensions appears straightforward on the surface, but further investigation is needed to verify this initial observation. Potential complications include overcoming edge e ects and extending the criterion for selecting a particular wavelet given its two-dimensional Fourier transform. The DWPT method for simulating SPPs gives a new perspective on determining the \appropriate" basis for a wavelet transform. When using the DWPT to analyze (decompose) time series one may select the basis by, e.g., the best basis algorithm (Coifman and Wickerhauser 1992) or matching pursuit (Mallat and Zhang 1993). Saito (1998) de ned a least statistically-dependent basis by using a measure of entropy, which is an additive measure, and applied it to image compression. In this article we were interested in the synthesis of time series using a speci c type of basis { one that minimizes the correlation structure of the DWPT coecients. In the analysis of SPPs, we have a best basis by knowing the Gegenbauer frequency. If we were given a series with an unknown Gegenbauer frequency, we could apply the DWPT and then test the DWPT coecients using a standard test for white noise (e.g., the cumulative periodogram test). A basis would then be selected which produces DWPT coecients with least residual correlation at the lowest level 18
possible. Percival et al. (2000) and Whitcher (2000) have both investigated hypothesis testing procedures for this purpose. Additional research is required to determine how to select a basis of least correlation for modelling observed time series.
6 Acknowledgements The author would like to thank two anonymous referees for comments and suggestions that greatly improved the quality of this work.
References Andel, J. (1986), \Long memory time series models," Kybernetika, 22, 105{123. Arteche, J., and Robinson, P. M. (1999), \Seasonal and cyclical long memory," in Asymptotics, Nonparametrics, and Time Series, ed. S. Ghosh, New York: Marcel Dekker, pp. 115{148.
Chen, S. S. B., Donoho, D. L., and Saunders, M. A. (1999), \Atomic decomposition by basis pursuit," SIAM Journal of Scienti c Computing, 20, 33{61. Coifman, R. R., and Wickerhauser, M. V. (1992), \Entropy-based algorithms for best basis selection," IEEE Transactions on Information Theory, 38, 713{718. Daubechies, I. (1992), Ten Lectures on Wavelets, Philadelphia: Society for Industrial and Applied Mathematics. Granger, C. W. J., and Joyeux, R. (1980), \An introduction to long-memory time series models and fractional di erencing," Journal of Time Series Analysis, 1, 15{29. Gray, H. L., Zhang, N.-F., and Woodward, W. A. (1989), \On generalized fractional processes," Journal of Time Series Analysis, 10, 233{257.
Gray, H. L., Zhang, N.-F., and Woodward, W. A. (1994), \On generalized fractional processes { a correction," Journal of Time Series Analysis, 15, 561{562. 19
Hosking, J. R. M. (1981), \Fractional di erencing," Biometrika, 68, 165{176. Hosking, J. R. M. (1982), \Some models of persistence in time series," in Time Series Analysis: Theory and Practice, ed. O. D. Anderson, (vol. 1), Amsterdam: North Holland, pp. 641{653.
Hosking, J. R. M. (1984), \Modeling persistence in hydrological time series using fractional di erencing," Water Resources Research, 20, 1898{1908. Mallat, S., and Zhang, Z. (1993), \Matching pursuits with time-frequency dictionaries," IEEE Transactions on Signal Processing, 41, 3397{3415.
Masry, E. (1993), \The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion," IEEE Transactions on Information Theory, 39, 260{264. McCoy, E. J., and Walden, A. T. (1996), \Wavelet analysis and synthesis of stationary longmemory processes," Journal of Computational and Graphical Statistics, 5, 26{56. Morris, J. M., and Peravali, R. (1999), \Minimum-bandwidth discrete-time wavelets," Signal Processing, 76, 181{193.
Ooms, M. (1995), \Flexible seasonal long memory and economic time series," Technical Report 9515/A, Econometric Institute, Erasmus University. Percival, D. B. (1992), \Simulating Gaussian random processes with speci ed spectra," Computing Science and Statistics, 24, 534{538.
Percival, D. B., Sardy, S., and A. C. Davison (2000), \Wavestrapping time series: Adaptive wavelet-based bootstrapping," In B. Fitzgerald, R. Smith, A. T. Walden and P. Young (Eds.), Nonlinear and Nonstationary Signal Processing, Cambridge: Cambridge University Press.
(forthcoming) 20
Percival, D. B., and Walden, A. T. (2000), Wavelet Methods for Time Series Analysis, Cambridge: Cambridge University Press. (forthcoming) Piessons, R., de Doncker-Kapenga, E., U berhuber, C. W., and Kahaner, D. K. (1983), QUADPACK: A Subroutine Package for Automatic Integration, Heidelberg: Springer-Verlag.
Rainville, E. D. (1960), Special Functions, New York: The Macmillan Company. Ribeiro, V. J., Riedi, R. H., Crouse, M. S., and Baraniuk, R. G. (1999), \Simulation of nonGaussian long-range dependent trac using wavelets," In ACM SIGMETRICS Conference on the Measurement and Modeling of Computer Systems.
Saito, N. (1998), \The least statistically-dependent basis and its applications," In Conference Record of The Thirty-Second Asilomar Conference on Signals, Systems and Computers, pp.
732{736. Stoksik, M. A., Lane, R. G., and Nguyen D. T. (1995), \Practical synthesis of accurate fractal images," Graphical Models and Image Processing, 57, 206{219. Tew k, A. H., and Kim, M. (1992), \Correlation structure of the discrete wavelet coecients of fractional Brownian motion," IEEE Transactions on Information Theory, 38, 904{909. Whitcher, B. (2000). \Wavelet-based estimation for seasonal long-memory processes," Technical Report 00-14, EURANDOM, The Netherlands. Wickerhauser, M. V. (1994), Adapted Wavelet Analysis from Theory to Software, Wellesley, Massachusetts: A K Peters. Woodward, W. A., Cheng, Q. C., and Gray, H. L. (1998), \A k-factor GARMA long-memory model," Journal of Time Series Analysis, 19, 485{504. Wornell, G. W. (1993), \Wavelet-based representations for the 1=f family of fractal processes," 21
Proceedings of the IEEE, 81, 1428{1450.
Wornell, G. W. (1996), Signal Processing with Fractals: A Wavelet Based Approach, New Jersey: Prentice Hall.
22
DWPT N Hosking M MB(8) D(16) LA(16) MB(16) ( = 0:4; fG = 121 ; = 0:866) 128 5:07 N 17:14 17:48 16:44 14:99 2N 9:46 9:72 9:45 7:01 4N 10:01 5:06 7:75 5:53 256 5:01 N 25:06 26:27 20:87 21:19 2N 18:56 13:84 17:09 17:35 4N 19:83 12:89 9:66 9:63 512 7:56 N 33:43 33:69 32:09 38:57 2N 23:95 27:46 20:96 17:09 4N 25:60 23:79 19:07 13:10 1024 12:32 N 58:24 53:86 49:29 51:40 2N 37:53 32:34 35:42 24:06 4N 42:28 33:51 37:40 35:10 128 256 512 1024
0:0135 N 2N 4N 0:0053 N 2N 4N 0:0048 N 2N 4N 0:0083 N 2N 4N
0:0495 0:0303 0:0193 0:0781 0:0316 0:0213 0:0470 0:0286 0:0290 0:0647 0:0343 0:0214
( = 0:2; fG = 121 ; = 0:866) 0:0620 0:0557 0:0530 0:0380 0:0336 0:0312 0:0342 0:0259 0:0220 0:0677 0:0816 0:0968 0:0323 0:0344 0:0195 0:0224 0:0286 0:0232 0:0461 0:0434 0:0354 0:0312 0:0360 0:0280 0:0239 0:0252 0:0145 0:0677 0:0640 0:0540 0:0303 0:0301 0:0217 0:0238 0:0209 0:0125
MB(24) 15:44 8:52 7:41 22:45 10:30 9:62 34:58 17:64 16:97 55:76 29:66 32:36 0:0570 0:0317 0:0200 0:0804 0:0297 0:0255 0:0495 0:0240 0:0098 0:0566 0:0172 0:0128
Table 1: Simulation results for the SPPs ( = 0:4; fG = 121 ; = 0:866) and ( = 0:2; fG = 121 ; = 0:866). Values represent the least squares di erence (LSD) between the average sample ACVS (500 simulations) and the true ACVS for lags 0; : : : ; N=2. The basis used for the DWPT method is BM = f(log2 M; n) j n = 0; : : : ; 2log2 M ? 1g, which corresponds with the nest partition of the frequency interval for the simulated series of length M .
23
N Hosking M
128 256 512 1024
128 256 512 1024
0:086 N 2N 4N 0:139 N 2N 4N 0:181 N 2N 4N 0:121 N 2N 4N 1:472 N 2N 4N 1:173 N 2N 4N 1:147 N 2N 4N 2:071 N 2N 4N
DWPT MB(8) D(16) LA(16) MB(16) MB(24) ( = 0:3; fG = 0:352; = ?0:6) 0:262 0:215 0:277 0:251 0:171 0:274 0:272 0:320 0:173 0:167 0:279 0:221 0:280 0:148 0:128 0:466 0:457 0:336 0:222 0:205 0:482 0:408 0:292 0:192 0:161 0:438 0:291 0:376 0:184 0:141 0:615 0:752 0:745 0:449 0:458 0:565 0:543 0:653 0:261 0:290 0:604 0:569 0:572 0:211 0:169 0:853 0:748 0:846 0:375 0:308 1:037 0:949 0:871 0:475 0:414 0:998 0:979 0:907 0:495 0:459 1:274 2:017 1:373 1:708 1:037 1:830 2:435 2:517 3:651 8:153 6:718 5:607
( = 0:3; fG = 0:016; = 0:995) 2:101 1:245 1:127 1:328 1:403 1:688 1:260 1:831 1:550 1:571 1:749 1:127 1:597 2:047 1:211 1:639 1:737 1:118 2:319 2:777 2:923 2:037 2:951 1:653 2:221 3:212 2:314 5:844 5:523 5:805 7:613 6:028 6:076 6:977 6:391 5:094
1:307 1:101 1:065 1:285 1:536 1:272 2:351 2:269 1:771 5:657 5:587 5:621
Table 2: Simulation results for the SPPs ( = 0:3; fG = 0:352; = ?0:6) and ( = 0:3; fG = 0:016; = 0:995). Values represent the least squares di erence (LSD) between the average sample ACVS (500 simulations) and the true ACVS for lags 0; : : : ; N=2. The basis used for the DWPT method is BM = f(log2 M; n) j n = 0; : : : ; 2log2 M ? 1g, which corresponds with the nest partition of the frequency interval for the simulated series of length M .
24
N Hosking M
128
5:07
256
5:01
512
7:56
1024 12:32
128 256 512 1024
N 2N 4N N 2N 4N N 2N 4N N 2N 4N
0:0135 N 2N 4N 0:0053 N 2N 4N 0:0048 N 2N 4N 0:0083 N 2N 4N
MB(8) 15:47 10:11 9:42 20:91 15:35 16:30 30:72 29:62 29:70 57:09 32:48 31:30 0:0494 0:0376 0:0267 0:0793 0:0384 0:0214 0:0493 0:0391 0:0203 0:0684 0:0329 0:0191
DWPT D(16) LA(16) MB(16) ( = 0:4; fG = 121 ; = 0:866) 16:87 14:83 14:91 9:26 10:07 7:66 7:31 8:91 7:20 28:03 24:15 23:87 15:37 15:21 12:56 14:40 15:52 11:06 40:18 36:27 36:14 18:73 26:18 19:04 23:65 14:80 13:73 66:24 63:46 52:60 40:65 33:01 29:46 29:52 32:74 27:27 ( = 0:2; fG = 121 ; = 0:866) 0:0547 0:0478 0:0551 0:0443 0:0413 0:0267 0:0290 0:0247 0:0159 0:0806 0:0912 0:0794 0:0334 0:0250 0:0310 0:0393 0:0320 0:0226 0:0503 0:0426 0:0356 0:0365 0:0403 0:0260 0:0212 0:0185 0:0185 0:0796 0:0741 0:0709 0:0339 0:0345 0:0190 0:0172 0:0262 0:0174
MB(24) 16:12 5:36 6:44 23:14 13:34 15:53 37:48 17:94 17:08 52:31 33:88 35:78 0:0478 0:0326 0:0253 0:0819 0:0355 0:0207 0:0510 0:0427 0:0197 0:0695 0:0284 0:0345
Table 3: Simulation results for the SPPs ( = 0:4; fG = 121 ; = 0:866) and ( = 0:2; fG = 121 ; = 0:866). Values represent the least squares di erence (LSD) between the average sample ACVS (500 simulations) and the true ACVS for lags 0; : : : ; N=2. The DWPT method uses an adaptive orthonormal basis depending on the location of the Gegenbauer frequency and length of the wavelet lter. For the rst SPP see Figure 4 for the basis functions B(L) , L 2 f8; 16g.
25
N Hosking M
128 256 512 1024
128 256 512 1024
0:086 N 2N 4N 0:139 N 2N 4N 0:181 N 2N 4N 0:121 N 2N 4N 1:472 N 2N 4N 1:173 N 2N 4N 1:147 N 2N 4N 2:071 N 2N 4N
DWPT MB(8) D(16) LA(16) MB(16) MB(24) ( = 0:3; fG = 0:352; = ?0:6) 0:308 0:265 0:295 0:178 0:095 0:318 0:305 0:250 0:206 0:222 0:292 0:352 0:316 0:183 0:250 0:456 0:391 0:368 0:306 0:201 0:485 0:427 0:416 0:273 0:212 0:432 0:342 0:333 0:263 0:182 0:728 0:853 0:661 0:641 0:526 0:573 0:470 0:542 0:287 0:185 0:679 0:578 0:526 0:350 0:254 0:955 0:847 0:823 0:448 0:329 0:963 0:805 0:787 0:490 0:394 0:979 0:932 0:897 0:447 0:469 ( = 0:3; fG = 0:016; = 0:995) 1:306 0:749 1:241 1:984 0:881 1:458 0:772 1:331 1:646 1:818 1:458 1:316 1:551 1:647 2:268 2:013 2:593 1:031 2:085 1:810 1:750 1:411 2:179 1:469 2:852 3:300 3:263 3:181 1:965 2:478 2:891 1:984 2:907 2:546 4:368 2:905 6:135 5:336 5:512 5:951 7:565 6:421 5:731 5:439 5:235 6:765 5:141 6:251
1:399 1:023 0:672 1:670 1:579 1:729 2:640 2:947 2:630 8:755 8:544 8:267
Table 4: Simulation results for the SPPs ( = 0:3; fG = 0:352; = ?0:6) and ( = 0:3; fG = 0:016; = 0:995). Values represent the least squares di erence (LSD) between the average sample ACVS (500 simulations) and the true ACVS for lags 0; : : : ; N=2. The DWPT method uses an adaptive orthonormal basis depending on the location of the Gegenbauer frequency and length of the wavelet lter.
26
DWPT
Non-adaptive
Adaptive MB(8) MB(16) MB(24) 25:43 11:88 12:06 22:12 15:51 12:46 17:26 11:05 10:71 45:51 38:56 31:26 40:48 29:81 25:80 47:74 16:14 18:30 132:09 115:95 116:65 87:73 48:37 39:03 105:82 42:32 51:25
N Hosking M MB(8) MB(16) MB(24) 128 7:45 N 17:75 12:97 15:40 2N 19:76 13:16 8:86 4N 19:73 10:94 10:96 256 10:26 N 45:77 37:33 40:58 2N 46:77 31:12 21:55 4N 54:18 29:19 17:85 512 19:02 N 125:37 116:43 112:41 2N 85:36 50:75 51:93 4N 105:62 48:11 52:50
Table 5: Simulation results for the 2-factor SPP (1 = 0:45; f1 = 0:102; 1 = 0:8; 2 = 0:3; f2 = 0:398; 2 = ?0:8). Values represent the least squares di erence (LSD) between the average sample ACVS (500 simulations) and the true ACVS for lags 0; : : : ; N=2. For the DWPT method, both a non-adaptive and an adaptive orthonormal basis " = 0:001) were used.
j=0 j=1 j=2 j=3
W0 0 = X ;
W1 0 W1 1 W2 0 W2 1 W2 2 W2 3 W3 0 W3 1 W3 2 W3 3 W3 4 W3 5 W3 6 W3 7 ;
;
;
;
;
;
;
;
;
;
;
;
;
;
W
Figure 1: Wavelet packet table T down to level J = 3. The wavelet coecient vectors j;n are arranged in sequency ordering which leads to a monotonically increasing partition of the frequency interval.
27
50.0 100.0
0
1
3
Level
10.0 5.0
spectrum
2
4
1.0
6
0.5
5
7
0.0
0.1
0.2 0.3 frequency
0.4
0.5
(a) Spectral density function
(b) Wavelet packet table
Figure 2: Ideal basis for a seasonal persistent process. On the left, the spectrum for the SPP ( = 0:4; fG = 121 ; = 0:866) and, on the right, its ideal basis (J = 7) consisting of wavelet coecient vectors { from left to right { 3;0 , 5;4 , 7;20 , 7;21 , 6;11 , 4;3 , 2;1 , and 1;1 .
0.0
0.1
0.2 0.3 frequency
(a) L = 2
0.4
0.5
25
W
0
5
10
15
20
25 20 15 10 5 0
0
5
10
15
20
25
W W W W W W W
0.0
0.1
0.2 0.3 frequency
(b) L = 4
W
0.4
0.5
0.0
0.1
0.2 0.3 frequency
0.4
0.5
(c) L = 8
Figure 3: Spectrum of the wavelet coecient vector 1;1 for an SPP with Gegenbauer frequency fG = 121 . These spectra are obtained using either the Daubechies extremal phase or least asymmetric families of wavelet lters of lengths L 2 f2; 4; 8g. The spectrum is plotted on a linear scale. 28
L=4
0
0
1
1
2
2
3
3
Level
Level
L=2
4
4
5
5
6
6
7
7
L = 16
0
0
1
1
2
2
3
3
4
Level
Level
L=8
4
5
5
6
6
7
7
Figure 4: Adaptive basis selection for an SPP with Gegenbauer frequency fG = 121 . These wavelet packet tables are based on the Daubechies extremal phase or least asymmetric family of wavelet lters with length L 2 f2; 4; 8; 16g.
29
6 4 2 0
autocovariance
-2 -4
0
20
40
60
80
100
120
80
100
120
80
100
120
80
100
120
1.0 0.5 0.0
autocovariance
lag
0
20
40
60
1.0 0.5 -0.5 0.0
autocovariance
1.5
lag
0
20
40
60
2 1 0 -2
-1
autocovariance
3
4
lag
0
20
40
60 lag
Figure 5: Sample autocovariance sequences (ACVS) for the Hosking method averaged over 500 simulations. Four SPPs (N = 512) are displayed with parameters { from top to bottom { ( = 0:4; fG = 121 ; = 0:866), ( = 0:2; fG = 121 ; = 0:866), ( = 0:3; fG = 0:352; = ?0:6) and ( = 0:3; fG = 0:016; = 0:995). The true ACVS s is given by the dashed line and the average sample ACVS s is given by the solid line up to lag = 128. Also shown are the 5% and 95% points (dotted lines).
30
6 4 2 0
autocovariance
-2 -4
0
20
40
60
80
100
120
80
100
120
80
100
120
80
100
120
1.0 0.5 0.0
autocovariance
lag
0
20
40
60
1.0 0.5 -0.5 0.0
autocovariance
1.5
lag
0
20
40
60
2 1 0 -2
-1
autocovariance
3
4
lag
0
20
40
60 lag
Figure 6: Sample autocovariance sequence (ACVS) for the DWPT method averaged over 500 simulations using the MB(16) wavelet lter, M = 2N and non-adaptive basis. The four SPPs (N = 512) are identical to those in Figure 5. The true ACVS s is given by the dashed line and the average sample ACVS s is given by the solid line up to lag = 128. Also shown are the 5% and 95% points (dotted lines).
31
3 1 -1
0
autocovariance
2
50.0 100.0 10.0
spectrum
5.0 1.0
-2
0.5 0.0
0.1
0.2 0.3 frequency
0.4
0.5
0
(a) Spectral density function
20
40
60 Lag
80
100
120
(b) Autocovariance sequence
0
0
1
1
1
2
2
2
3
3
3
Level
0
Level
Level
Figure 7: Spectral density function and autocovariance sequence (ACVS) for the 2-factor SPP (1 = 0:45; f1 = 0:102; 1 = 0:8; 2 = 0:3; f2 = 0:398; 2 = ?0:8). The ACVS from the SPP ( = 0:4; fG = 121 ; = 0:866) is also provided (dotted line) for comparison.
4
4
4
5
5
5
6
6
6
7
7
7
(a) Basis for f1
(b) Basis for f2
(c) 2-factor basis
Figure 8: Orthonormal basis selection for the 2-factor SPP with asymptotes at f1 = 0:102 and f2 = 0:398. An orthonormal basis is determined for each Gegenbauer frequency individually, shown in (a) and (b). In these two cases the criterion Uj;n(fG ) < 0:001 was used. Then the two bases are `merged' together and `orthogonalized' in order to produce an orthonormal basis for the 2-factor SPP (c). 32
10^6
+ • +
DWPT Method Hosking Method
10^5
+ +
+
10^3
+ +
10^2
Time (seconds)
10^4
+
+
10^1
+
• •
+
• •
10^0
+
10^-1
•
•
+•
•
•
•
•
•
•
•
+ + 2^7
2^9
2^11
2^13 2^15 Sample Size
2^17
2^19
Figure 9: Computation time (in seconds) for the DWPT and Hosking methods to simulate one realization of an SPP. All calculations for the DWPT were implemented in S-Plus. For the Hosking method, sample sizes N = 27 ; : : : ; 212 were computed in S-Plus, while larger sample sizes could not be evaluated. Subsequent computation times for the Hosking method were extrapolated from the previous values.
33
