Revised Chapter 15 in Specifying and Diagnostically Testing Econometric Models (Edition 3)
© by Houston H. Stokes 8 February 2015. All rights reserved. Preliminary Draft
Chapter 15
Spectral Analysis of Time Series ..................................................... 1
15.0 Introduction ............................................................. 1
15.1 A Brief Treatment of Spectral Analysis Theory ..................................... 2
Table 15.1 Values and Names Calculated by the B34S spectral Command ..................... 13
15.2 Examples ............................................................... 14
Table 15.2 Program to Generate AR(1) models ........................................ 14
Figure 15.1 AR(1) Model where .9 ............................................ 15
Figure 15.2 AR(1) Model where .9 ..........................................
Table 15.3 Program to Analyze Gas Furnace Data using Spectral Methods ......................
Figure 15.3 Spectral analysis of GASIN .................................
Figure 15.4 Spectral analysis of GASOUT ................................
Figure 15.5 Cross spectral analysis of GASIN-GASOUT part 1 .............
Figure 15.5 Cross spectral analysis of GASIN-GASOUT part 2 .............................
15.3 Matrix Command Implementation .............................................
Table 15.4 Matrix Command Implementation of Spectral Analysis ...........................
Table 15.5 Cross Spectral Analysis with the Matrix Command ..............................
Table 15.6 Verification that the sin and cosine vectors are orthogonal .........................
Table 15.7 Inverse Spectral Examples ..............................................
15.4 Wavelet Analysis .........................................................
Table 15.8 Wavelet basis functions supported .........................................
Table 15.9 Empirically derived factors for four wavelet bases ..............................
Table 15.10 Wavelet Filter For The Nino Series .......................................
Figure 15.6 Raw Nino series and three wavelet smoothed series .............................
15.5 Use of Normalized Cumulative Periodogram to test for white noise ......................
Table 15.12 Calculating the Normalized Cumulative Periodogram .............................
Table 15.13 Testing series for white noise using the Normalized Cumulative Periodogram .............
Figure 15.7 Normalized Cumulative Periodogram for gasout series ............................
Figure 15.8 Normalized Cumulative Periodogram for white noise series .........................
15.6 Forecasting using spectral methods.............................................
Table 15.14 A Subroutine to Calculate Forecasts using the FFT of a series ........................
Table 15.15 Using the FFT to Forecast ...............................................
15.7 Conclusion..............................................................
16
17
22
23
24
25
25
25
26
27
28
32
33
34
36
39
41
41
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43
44
44
45
47
53
Spectral Analysis of Time Series
15.0 Introduction
For single series the B34S spectral command calculates the Fourier cosine and sine
coefficients, the periodogram and, if weights are supplied, the spectrum. The spectral command
provides substantially more capability than the spectral option available under the B34S bjiden and
bjest sentences. For multiple series the cspectral paragraph calculates the real part of the cross
periodogram, the imaginary part of the cross periodogram, the cospectral density estimate, the
15-1
15- 2
Chapter 15
quadrature-spectrum estimate, the amplitude, the coherency squared and the phase spectrum. The
output from this command is similar to that produced with the SAS/ETS spectra command.1 After a
brief discussion of the theory, the spectrum is calculated for generated AR data with positive and
negative coefficients. While spectral analysis usually proceeds by a Fourier decomposition of the
series into cosine and sine coefficients, it is possible to use OLS methods to make these calculations.
The advantage of the OLS approach is that it highlights the relationship between time and frequency
domain approaches. The matrix command also contains substantial spectral programming capability
that includes spectral, cspectral and fft commands together with complex variable manipulation
capability. Use of these features will be discussed later.
Wavelet analysis provides means by which a series can be filtered in a manner that has
certain advantages over the windowed Fourier transformation (WFT). Torrence and Compo (1998)
stress the WFT has the disadvantage of being inefficient since it "imposes a scale or 'response
interval' T into the analysis, a problem not found with the wavelet transformation. They advocate
using wavelets as a means by which a data series can be filtered from noise. Hastie-TibshiraniFriedman (2001, 149) note the ability of wavelets to represent a series and provide a means by which
a sparse representation can be found. "Wavelets bases are very popular in signal processing and
compression, since they are able to represent both smooth and/or locally bumpy functions in an
efficient way – a phenomenon dubbed time and frequency localization. In contrast, the traditional
Fourier basis allows only frequency localization." In section 15.4 the Torrence-Compo wavelet
implementation is discussed and examples shown.
15.1 A Brief Treatment of Spectral Analysis Theory
The basic idea behind spectral analysis is to decompose the variance in a series by frequency.
Assume the model
yt yt 1 ut .
(15.1-1)
If 0 , it will be shown that the series contains low-frequency information, while if 0 , the
series contains high-frequency information. Spectral quantities, such as the periodogram and
spectrum, can be calculated using frequency-domain methods or time domain methods such as OLS.
Assume an autocovariance-generating function (see Hamilton (1994) equation [3.6.1])
g y ( z)
j
j
z j,
(15.1-2)
1 Jenkins - Watts (1968), Anderson (1971) and Bloomfield (1976) provide good references for spectral methods.
Hamilton (1994, Chap. 6) provides a very concise treatment, which is further summarized here. For a more modern
reference, see Wei (2006, Chapters 11-13)
15-2
where z = complex scalar, and covariances are summable. For spectral models assume
z cos() i sin() ei .
(15.1-3)
The population spectrum S y is equal to the autocovariance-generating function g y ( z ) evaluated at
z e i and divided by 2π
S y ( ) (2 ) 1 g y (ei w ) (2 ) 1 j ei j .
(15.1-4)
j
Consider an MA(1) model. Once we get the covariance-generating function, from this function we
can get the spectrum. Assume a model such as
yt et et 1,
(15.1-5)
which was discussed in Chapter 7. Here
E ( yt ) E (et ) E (et 1 )
(15.1-6)
E ( yt ) 2 E (et et 1 ) 2
E ( et2 2 et et 1 2et21 )
0
2
2
(15.1-7)
2
E ( yt )( yt 1 ) E (et et 1 )(et 1 et 2 )
E (et et 1 et21 et et 2 2et 1et 2 )
(15.1-8)
0 et21 0 0
Using (15.1-2), (15.1-7) and (15.1-8) the autocovariance- generating function for this model becomes
g y ( z ) [ 2 ] z 1 [(1 2 ) 2 ]z 0 [ 2 ]z1
2 [ z 1 1 2 z ]
(15.1-9)
(1 z )(1 z ).
2
1
For the general MA(q) model
g y ( z) 2 (1 1z 2 z 2 ,
, q z q )(1 1z 1 2 z 2 ,
15-3
, q z q ).
(15.1-10)
15-4
Chapter 15
For the AR(1) model
yt (1 B ) 1 et
(15.1-11)
g y ( z) [ 2 /(1 1z)(1 1z 1 )].
(15.1-12)
In general, for the ARMA(p,q) model
g y ( z ) [ 2 (1 1 z 2 z 2 ,
, q z q )(1 1z 1 2 z 2 ,
, q z q )]/
(1 1 z 2 z 2 ,
, p z p )(1 1 z 1 2 z 2 ,
, p z p )].
[
(15.1-13)
We now show how to write the spectrum in terms of z of the MA(1) model.
From (15.1-3), (15.1-4) and (15.1-9)
S y ( ) (2 ) 1 2 (1 e i )(1 ei )
(2 ) 1 2 (1 e i ei 2 ).
(15.1-14)
Spectral Analysis of Time Series
15-5
Since
e i ei cos( ) i sin( ) cos( ) i sin( )
2 cos( )
(15.1-15)
S y () (2 )1 2 [1 2 2 cos()].
(15.1-16)
Since cos( ) goes from 1 to -1 as goes from 0 to , equation (15.1-16) indicates that if
0 ( 0) , the spectrum monotonically decreases (increases) as frequency increases from 0 to
.
The spectrum of the AR(1) model in equation (15.1-11) can be calculated from the
covariance-generating function (15.1-12) using (15.1-15) as
S y (2 ) 1 2 /[(1 e i )(1 ei )]
(2 ) 1 2 /[(1 e i ei 2 )]
(15.1-17)
(2 ) /[(1 2 cos( ))].
1
2
2
Equation (15.1-17) shows that if 0 ( 0) , the spectrum is monotonically decreasing
(increasing) over the range [0, ] since the denominator increases (decreases).
From (15.1-13) the spectrum for the ARMA(p,q) process becomes
S y ( ) [ 2 (1 1 z 2 z 2 ,
, q z q )(1 1z 1 2 z 2 ,
, q z q )]/
[2 (1 1 z 2 z 2 ,
, p z p )(1 1 z 1 2 z 2 ,
, p z p )].
(15.1-18)
Following Hamilton (1994), it can be shown that if
(1 1z 2 z 2 ,
, q z q ) (1 1z)(1 2 z),
,(1 q z)
(15.1-19)
(1 1z 2 z 2 ,
, q z q ) (1 1z)(1 2 z),
,(1 q z)
(15.1-20)
then (15.1-17) can be written as
15-6
Chapter 15
q
S y ( ) ( / 2 )
(1
2
j
(1
2
j
j 1
p
2
j 1
2 j cos( ))
.
(15.1-21)
2 j cos( ))
The spectrum contains all the information concerning the autocovariances. Assuming the sequence
of autocovariances j is summable over the range j , (i. e., the covariances die out)
Hamilton (1994, Appendix 6.A) proves that
S
y
( )eik d k
(15.1-22)
j
or, alternatively,
S y cos(k ) d k .
(15.1-23)
j
For k=0, equation (15.1-22) becomes
S y ( ) d 0 ,
(15.1-24)
j
which indicates that the area under the population spectrum between to is the variance. Since
the spectrum is symmetric, the portion of the variance of yt that is attributed to frequencies less than
is 2 S y ( ) d. .
j 0
Results for a sample of T observations follow directly from the above results for the
population. From (15.1-4) we get
Pˆy ( ) (2 ) 1
T 1
j T 1
ˆ j ei j ,
(15.1-25)
where Pˆy ( ) is the sample periodogram.2 Since
Pˆy ( ) is called the sample periodogram. The sample periodogram can be smoothed to form the sample
spectrum Sˆ ( ) . Weights are selected to smooth out noise in the estimated sample periodogram. In B34S there
2
y
must be an odd number of weights. All weights are normalized to sum to (1/(4*π)). WEIGHTS(1 1 1)$ implies
Spectral Analysis of Time Series
ei j cos( j ) i sin( j )
15-7
(15.1-26)
and for a covariance stationary process j j , we can write (15.1-25) as
Pˆy ( ) (2 ) 1
T 1
ˆ [cos( j ) i sin( j )]
j T 1
(2 ) 0 [cos(0) i sin(0)] (2 )
1
1
T 1
ˆ [cos( j) i sin( j)],
(15.1-27)
j
j 1
which since cos(0) 1, sin(0) 0, sin( ) sin( ) and cos( ) cos( ) , sin(0) =0, sin(-θ) = sin(θ) and cos(-θ)=cos(θ) can be written
T 1
Pˆy ( ) (2 ) 1[ˆ0 2 ˆ j cos( j )].
(15.1-28)
j 1
The sample periodogram can be estimated using Fourier methods or OLS. While the OLS approach
is slower, it highlights what is being estimated with spectral analysis. The sample equivalent of the
spectral representation theorem that states that any covariance-stationary process yt can be written as
yt [ ( ) cos( t ) ( )sin(t )] d
(15.1-29)
0
is
M
yt ˆ [ˆ j cos( j (t 1)) ˆj sin( j (t 1))] et .
(15.1-30)
j 1
If T is odd, there will be M=(T-1)/2 frequencies in equation (15.1-30), where
1 2 / T , 2 4 / T , and M 2 M / T (T 1) / T . If equation (15.1-30) is estimated M
times, then the M sets of coefficients ˆ and ˆ can be used to recover the sample periodogram and
j
j
other quantities. The reason for this is that all the right-hand-side variables are orthogonal and hence
the regression can be done in sequences of models with two variables on the right plus the constant.
Define Pˆi (i ) as the sample periodogram of series i (here yt ) evaluated at frequency j , then
rectangular weighting, while WEIGHTS(1 2 1)$ implies triangular weighting.
15-8
Chapter 15
Pˆi ( i ) (T / 8 )(ˆ 2j ˆj2 ).
(15.1-31)
The sample variance3 involves summing the periodogram values and is
T
M
i 1
j 1
(1/ T ) ( yt y )2 .5[ (ˆ 2j ˆ j2 )].
(15.1-32)
Since
T
ˆ j (2 / T ) yt cos[ j (t 1)]
(15.1-33)
t 1
T
ˆ j (2 / T ) yt sin[ j (t 1)],
(15.1-34)
t 1
(15.1-31) can be written as
T
Pˆi ( j ) (1/ 2 T )[( yt cos[ j (t 1)]
(15.1-35)
t 1
It can be shown that
2Pˆi () / Si () 2 (2),
(15.1-36)
which implies that
E[ Pˆi ()] Si ().
(15.1-37)
Since 2 (2) has a mean of 2 and a 95% confidence interval of .05 - 7.4, Pˆi ( ) is not a good
estimate of the population spectrum. Another problem is that (15.1-28) requires that as many
parameters ( i ) have to be estimated as observations. The solution is to weight Pˆi ( ) to form an
estimate of the sample spectrum for series i, where h = the number of weights minus 1 divided by 2.
Sˆi ( )
h
w
m h
Pˆ ( ).
m h 1 i
3 Note that the large sample variance formula is used.
(15.1-38)
Spectral Analysis of Time Series
15-9
In B34S the WEIGHTS sentence requires that the user set an odd number of weights to be
used for smoothing the spectrum. The weights are then normalized to sum to 1/(4*π) or .079577 . Up
to 99 weights can be supplied. If the supplied sentence was
weights(1 2 3 2 1) $
then triangular weights of .0088419, .017684, .026526, .017684 and .0088419 would be used, while
if the sentence was
weights(1 1 1 1 1)$
then there would be five weights of (1/ 20 ) or .015915. Note that both sets of weights sum to
(1/ 4 ) or .079577.
The B34S and SAS have slightly different parameterizations of the spectral quantities. The
values calculated by B34S (and SAS) are listed in Table 15.1. Note that j in equations (15.1-33)
and (15.1-34) is 2k / T in Table 15.1 in the calculation of ˆ and ˆ , which are called COSj(k) and
j
j
SINj(k), respectively. B34S and SAS normalize (ˆ ˆ 2 ) by T/2 in place of (T/8π) listed in
equation (15.1-31). If the data contain cycles greater than π, then these will be seen as having cycles
with a range of 0 to π. For the lowest frequency 1 2 / T , the corresponding period is T. In words
2
this means that if there are T data points, it will be impossible to detect a cycle longer than T. If two
series are added, then the autocovariance-generating function of the sum is the sum of the
autocovariance-generating functions of each series.
Up until now the analysis has been in terms of one series. We now assume two series, xt
and yt , and develop a spectral representation . Expanding the notation, define the population cross
spectrum as
S x y () (1/ 2 ) x( jy){cos( j ) i sin( j )}.
(15.1-39)
j
Equation (15.1-40) can be broken into two parts, the population cospectrum cx y ( ) defined as
cx y () (1/ 2 ) x( jy) cos( j)
j
and the population quadrature spectrum defined as
(15.1-40)
15-10
Chapter 15
qx y () (1/ 2 ) x( jy) sin( j ),
(15.1-41)
j
where
S x y ( ) cx y ( ) i qx y ( ).
(15.1-42)
It can be shown that the covariance between x and y is
S
() d E ( yt y )( xt x )
(15.1-43)
() E ( yt y )( xt x )
(15.1-44)
() d 0.
(15.1-45)
xy
or
c
xy
since
q
xy
The population cospectrum, cx y ( ) , measures the portion of the covariance between xt and yt that is
attributable to cycles of frequency . Looking now at the sample, define
T
ˆ y j (2 / T ) yt cos[ j (t 1)]
(15.1-46)
ˆy j (2 / T ) yt sin[ j (t 1)]
(15.1-47)
ˆ x j (2 / T ) xt cos[ j (t 1)]
(15.1-48)
ˆx j (2 / T ) xt sin[ j (t 1)]
(15.1-49)
t 1
T
t 1
T
t 1
T
t 1
The sample covariance between xt and yt can be written as
T
M
t 1
j 1
(1/ T ) ( yt y )( xt x ) (1/ 2) (ˆ y jˆ x j ˆx jˆy j ),
(15.1-49)
which implies that the portion of the sample covariance between xt and yt that is due to common
dependence on cycles of frequency is (1/ 2)(ˆ ˆ ˆ ˆ ) . The sample cross periodogram is
j
xj
yj
xj
yj
Spectral Analysis of Time Series
15-11
the sum of a real cˆx y ( j ) and imaginary component qˆ x y ( j )
Pˆx y ( j ) cˆx y ( j ) i qˆ x y ( j )
(15.1-50)
where
cˆx y ( j ) (T / 8 )(ˆ x jˆ y j ˆx jˆy j )
(15.1-51)
qˆ x y ( j ) (T / 8 )(ˆ x jˆ y j ˆx jˆy j ).
(15.1-52)
The formulas used by B34S and SAS scales by (T/2) in place of (T/8π) in equation (15.1-51) and
(15.1-52) or RPxy (k ) 4 cˆx y ( j ) and IPx y (k ) qˆ x y ( j ).
While the real part of the sample cross periodogram measures to what degree xt and yt have
common cycles, i. e., cycles at the same frequency, the imaginary component of the sample cross
periodogram corrects for whether these cycles are in phase or out of phase. If both series shared
common cycles, but these cycles were out of phase, the result would be that the contemporaneous
covariance would be low. In a manner similar to equation (15.1-38), the real and imaginary parts of
the sample cross periodogram can be weighted to form an estimate of the sample cospectral density
estimate Cˆ x y ( j ) and sample quadrature spectrum Qˆ x y ( j ).
Cˆ x y ( )
Qˆ x y ( )
h
w
m h 1
m h
h
w
m h
cˆx y ( )
(15.1-53)
qˆ ( ).
(15.1-54)
m h 1 x y
The population coherence hˆx y ( j ) , measures the correlation between xt and yt by frequency, is
hˆ ( ) [(Cˆ ( ))2 (Qˆ ( )) 2 ]/[ Sˆ ( ) Sˆ ( )],
(15.1-55)
xy
j
xy
j
xy
j
x
j
y
j
while the amplitude Aˆ x y ( j ), is
Aˆ x y ( j ) [(Cˆ x y ( j ))2 (Qˆ x y ( j ))2 ].5.
(15.1-56)
Using (15.1-56), (15.1-55) can be rewritten as
hˆx y ( j ) [ Aˆ x y ( j )]2 / [ Sˆx ( j ) Sˆ y ( j )].
(15.1-57)
15-12
Chapter 15
The phase ˆ x y ( j ) between xt and yt is
ˆ x y ( j ) arctan[Qˆ x y ( j ), Cˆ x y ( j )].
(15.1-58)
In the next section generated data and the Box-Jenkins (1976) gas furnace data are used to illustrate
these spectral magnitudes. Table 15.1 provides a summary of the formulas used in these calculations
and the b34s names. Before moving to this discussion it is important to fully document the
differences between SAS and B34S.
The B34S spectral command will exactly replicate the SAS procedure spectra with two
exceptions. In contrast with SAS, B34S does not print the zero frequency data point (where
PERIOD = ). There is a "bug" in SAS concerning how weighting is done for the end points of the
quadrature spectrum. The SAS Institute has acknowledged this "bug" and provides a switch that will
provide the correct results. Since the quadrature spectrum is an intermediate step toward calculating
the amplitude, the coherency squared and the phase, these values differ also. To illustrate the
problem, assume the Box-Jenkins (1976) gas furnace data is run with WEIGHTS (1 1 1). Here
the weights are all .026525. The sum of the three weights is .079575, which is (1/4π). For frequency
.00378, .006757, .01014, and .013551 the IP values are -33.07, -39.08, -35.49 and -12.04,
respectively. The QS values should be -2.791, -2.855, -2.298 and -1.586. SAS produces -1.914, 2.855, -2.298 and -1.586. The number of values differing at the beginning and at the ending are the
number of weights - 2 or 1 in this case. At the end points, the correct way to calculate the QS value
is to fold. This is illustrated next.
QS(1) =(.026525)*(-33.07)+(.026525)*(-33.07)+(.026525)*(-39.08)
QS(2) =(.026525)*(-33.07)+(.026525)*(-39.08)+(.026525)*(-35.49)
QS(3) =(.026525)*(-39.08)+(.026525)*(-35.49)+(.026525)*(-12.04).
SAS uses the above formulas for QS(2) and QS(3) but for QS(1) adds in the IP value of 0.0 for
frequency 0.0, giving
QS(1) =(.026525)*(0.0)+(.026525)*(-33.07)+(.026525)*(-39.08).
4
4 The SAS command ALTW gives the “correct” answer but was not made the default to maintain compatibility with
older versions of SAS and replicate the old answers. This appears to be a self serving decision on the part of SAS.
Spectral Analysis of Time Series
15-13
Table 15.1 Values and Names Calculated by the B34S spectral Command
_________________________________________________________________
FREQ(k)
Frequency from 0 to π if NOBSPP is set = 0. Cycles per
observation is NOBSPP*FREQ/2π, where NOBSPP is the number of
observations per unit time. Using the default setting
of NOBSPP = 1, FREQ is in the range 0 to .5. The
number of frequencies calculated goes from 1 to K,
where if T is even, K = (T/2) - 1
PERIOD(k)
Period or wavelength ( 1 / FREQ ).
COSj(k)
Cosine transform of Xjt. Defined over range 1 to K.
T (2/T)*X *cos(k*(i-1)*2*π/T).
COSj(k)= Σi=1
ji
SINj(k)
Sine transform of Xjt. Defined over range 1 to K.
T (2/T)*X *sin(k*(i-1)*2*π/T).
SINj(k)= Σi=1
ji
Pj(k)
Periodogram of Xjt. Defined over range 1 to K.
Pj(k) = (T/2)*((COSj(k)**2) + (SINj(k)**2)).
Sj(k)
Spectral density estimate of Xjt. Defined over range 1 to
K as Sj(k) = Σvi=-v w(i)*Pj(k+i), where v=(p-1)/2 if the WEIGHTS
sentence (containing v elements) is present and Sj(k) = Pj(k) if
it is not.
RPmn(k)
Real part of cross periodogram of Xmt and Xnt. Defined over
range 1 to K as
Rpmn(k)= (T/2)*((COSm(k)*COSn(k))+(SINm(k)*SINn(k))).
IPmn(k)
Imaginary part of cross periodogram of Xmt and Xmt. Defined over
over range 1 to K as
IPmn(k) = (T/2)*((COSn(k)*SINm(k))-(SINn(k)*COSm(k))).
CSmn(k)
Cospectral density estimate (real part of cross spectrum)
or the weighted real part of the cross periodogram. Defined over
range 1 to K as CSmn(k) = Σvi=-v w(i) * RPmn(k+i), where v=(p-1)/2 if
the WEIGHTS sentence (containing v elements) is present and as
Csmn(k) = RPmn(k) if it is not.
QSmn(k)
Quadrature-spectrum (imaginary part of cross spectrum) or the
weighted imaginary part of the cross periodogram. Defined over
range 1 to K as QSmn(k) = Σvi=-v w(i)*IPmn(k+i), where v=(p-1)/2 if the
weights sentence (containing v elements) is present and QSmn(k)=
Ipmn(k) if it is not.
Amn(k)
Amplitude (modulus of cross-spectrum). Defined over range
1 to K as Amn(k) = ((CSmn(k)**2) + (QSmn(k)**2))**.5.
Kmn(k)
Coherency squared. Defined over range 1 to K as
Kmn(k)=(Amn(k)**2)/(Sn(k)*Sm(k)). If the WEIGHTS sentence
is not present, Kmn(k) reduces to 1.0 for all frequencies.
PHmn(k)
Phase spectrum in radians. Defined over range 1 to K as
PHmn(k) = arctan(QSmn(k),CSmn(k)).
15-14
Chapter 15
15.2 Examples
Table 15.2 shows the statements to generate AR(1) models of the form of equation (15.1-1),
where in Model 1, Φ = .9, while in Model 2, Φ = -.9. The programming setup illustrates the use of
the Macro facility. If the B34SLET statement setting PLOT1 = no is changed to PLOT1 = yes,
hard copy graphs will be produced.
Table 15.2 Program to Generate AR(1) models
____________________________________________________________
%b34slet
b34sexec
b34sexec
b34sexec
b34sexec
plot1 = no $
options open('_JUNK.FSV') disp=unknown unit(44)$ b34srun$
options clean(44)$
b34srun$
options gfactor(.8)$ b34srun$
data noob=300 maxlag=1 heading('Sample AR(1) Data')$
build noise x1 x2$ gen noise= rn()$
gen x1 = lp(1,1,noise) ar(.9) ma(1.0) values(0.0) $
gen x2 = lp(1,1,noise) ar(-.9) ma(1.0) values(0.0) $
b34srun$
b34sexec spectral list(sin,cos,p,s) nobspp=0
scafname=spec scaunit=44 output(all)$
weights( 1 2 3 4 3 2 1)$ var x1$ b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
plottype=xyplot$
plot=(freq sin_1) gposition(1) title("Sine Transform Model 1")$
plot=(freq cos_1) gposition(2) title("Cos Transform Model 1")$
plot=(freq p_1 ) gposition(3) title("Periodogram Model 1")$
plot=(freq s_1 ) gposition(4) title("Spectrum Model 1")$
b34srun$
%b34sif(&plot1.eq.YES.or.&plot1.eq.yes)%then$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
print gport('fig1.wmf') plottype=xyplot$
plot=(freq sin_1) gposition(1) title("Sine Transform Model 1")$
plot=(freq cos_1) gposition(2) title("Cos Transform Model 1")$
plot=(freq p_1 ) gposition(3) title("Periodogram Model 1")$
plot=(freq s_1 ) gposition(4) title("Spectrum Model 1")$
b34srun$
%b34sendif$
b34sexec options clean(44)$ B34SRUN$
b34sexec spectral list(sin,cos,p,s) nobspp=0
scafname=spec scaunit=44 output(all)$
weights( 1 2 3 4 3 2 1)$ var x2$ b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
plottype=xyplot$
plot=(freq sin_1) gposition(1) title("Sine Transform Model 2")$
plot=(freq cos_1) gposition(2) title("Cos Transform Model 2")$
plot=(freq p_1 ) gposition(3) title("Periodogram Model 2")$
plot=(freq s_1 ) gposition(4) title("Spectrum Model 2")$
b34srun$
%b34sif(&plot1.eq.YES.or.&plot1.eq.yes)%then$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
print gport('fig2.wmf') plottype=xyplot$
plot=(freq sin_1) gposition(1) title("Sine Transform Model 2")$
plot=(freq cos_1) gposition(2) title("Cos Transform Model 2")$
Spectral Analysis of Time Series
15-15
plot=(freq p_1 ) gposition(3) title("Periodogram Model 2")$
plot=(freq s_1 ) gposition(4) title("Spectrum Model 2")$
b34srun$
%b34sendif$
Graphics produced by the code in Table 15.2 are listed in Figures 15.1 and 15.2. Note that as
predicted by theory, all the information in the series where Φ = .9 is at low frequency while when Φ
= .-9, the information is at high frequency. The rough appearance of the periodogram is smoothed by
the WEIGHTS (1 2 3 2 1).
Figure 15.1 AR(1) Model where .9
15-16
Figure 15.2 AR(1) Model where .9
Chapter 15
Spectral Analysis of Time Series
15-17
The next example uses the Box-Jenkins (1976) gas furnace data. First the periodogram is
estimated with OLS methods as suggested in equation (15.1-30). The code in Table 15.3 provides an
example that uses the MACRO OLSSPEC, which is called as
%b34smcall %olsspec(y=gasout noob=296 ntest=20)$
to produce estimates to the periodogram using the reg command. Next, the spectral command is
used to estimate, list and plot spectral and cross-spectral values. These are discussed below. The
rather long but complete command file is provided to show how all figures and tables were
produced. It is to be stressed that the OLS method of estimating the periodogram is not the preferred
way to proceed for production work, but has much to recommend as a method for understanding the
periodogram as just representing the explained sum of squares at a frequency.
Table 15.3 Program to Analyze Gas Furnace Data using Spectral Methods
______________________________________________________
b34sexec options include('c:\b34slm\gas.b34')$
b34srun$
%b34smacro olsspec$
/$ Needs to be called as %b34smcall(y=yname noob=_ ntest=_ )
/$ y = yname
/$ noob = number of observations i series
/$ ntest must be set to lt noob/2
b34sexec spectral list(sin,cos,p,s) nobspp=0$
var %b34seval(&Y)$ weights(1 2 3 2 1)$
b34srun$
%b34sdo i=1,&ntest$
b34sexec data set $
%b34sif(&i.eq.1)%then$
build sin cos freq$
%b34sendif$
gen freq=(timespi(2.0)/%b34seval(&noob))*%b34seval(&I)$
gen sin=sin((kount()-1.)*freq)$
gen cos=cos((kount()-1.)*freq)$
b34srun$
b34sexec reg$ model %b34seval(&y)=sin cos$ b34srun$
%b34senddo $
%b34smend$
%b34smcall olsspec(y= gasout noob=296 ntest=20)$
b34sexec options open('_junk.fsv') disp=unknown unit(44)$
b34srun$
b34sexec options clean(44)$ b34srun$
%b34slet plot1=yes$
b34sexec spectral
scafname=spec scaunit=44 output=all list=all
nobspp= 1 plotby( freq ) $
weights( 1 2 3 4 3 2 1)$
var gasin gasout $
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
plottype=xyplot$
15-18
Chapter 15
plot=(freq sin_1) gposition(1) title("Sine transform Gasin")$
plot=(freq cos_1) gposition(2) title("Cos transform Gasin")$
plot=(freq p_1 ) gposition(3) title("Periodogram Gasin ")$
plot=(freq s_1 ) gposition(4) title("Spectrum Gasin")$
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
plottype=xyplot$
plot=(freq sin_2) gposition(1) title("Sine Transform Gasout")$
plot=(freq cos_2) gposition(2) title("Cos Transform Gasout")$
plot=(freq p_2 ) gposition(3) title("Periodogram Gasout")$
plot=(freq s_2 ) gposition(4) title("Spectrum Gasout")$
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
plottype=xyplot$
plot=(freq rp_1) gposition(1)
title("Real Part Cross Period Gasin-Gasout")$
plot=(freq ip_1) gposition(2)
title("Imag Part Cross Period Gasin-Gasout")$
plot=(freq cs_1) gposition(3)
title("Cospectral Density Gasin-Gasout")$
plot=(freq qs_1) gposition(4)
title("Quadrature-Spectrum Gasin-Gasout")$
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
Plottype=xyplot$
Plot=(freq a_1) gposition(1) Title("Amplitude Gasin-Gasout")$
Plot=(freq k_1) gposition(2) Title("Coherency Gasin-Gasout")$
Plot=(freq ph_1) gposition(3) Title("Phase Gasin-Gasout")$
b34srun$
/$
graphs to list
b34sexec options gfactor(.8)$ b34srun$
%b34sif(&plot1.eq.yes.or.&plot1.eq.yes)%then$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
print gport('fig3.wmf')
plottype=xyplot$
plot=(freq sin_1) gposition(1) title("Sine Transform Gasin")$
plot=(freq cos_1) gposition(2) title("Cos Transform Gasin")$
plot=(freq p_1 ) gposition(3) title("Periodogram Gasin ")$
plot=(freq s_1 ) gposition(4) title("Spectrum Gasin")$
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
print gport('fig4.wmf')
plottype=xyplot$
plot=(freq sin_2) gposition(1) title("Sine Transform Gasout")$
plot=(freq cos_2) gposition(2) title("Cos Transform Gasout")$
plot=(freq p_2 ) gposition(3) title("Periodogram Gasout")$
plot=(freq s_2 ) gposition(4) title("Spectrum Gasout")$
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
print gport('fig5.wmf')
plottype=xyplot$
plot=(freq rp_1) gposition(1)
title("Real Part Cross Period Gasin-Gasout")$
plot=(freq ip_1) gposition(2)
title("Imag Part Cross Period Gasin-Gasout")$
plot=(freq cs_1) gposition(3)
title("Cospectral Density Gasin-Gasout")$
Spectral Analysis of Time Series
15-19
plot=(freq qs_1) gposition(4)
title("Quadrature-Spectrum Gasin-Gasout")$
b34srun$
b34sexec hrgraphics gformat=(fourgraph) scafname(spec)
print gport('fig6.wmf')
plottype=xyplot$
plot=(freq a_1) gposition(1) title("Amplitude Gasin-Gasout")$
plot=(freq k_1) gposition(2) title("Coherency gasin-gasout")$
plot=(freq ph_1) gposition(3) title("Phase gasin-gasout")$
b34srun$
%b34sendif$
Edited output from running the program in Table 15.3 follows. The periodogram and spectrum for
GASOUT for the first ten frequencies are listed below. The regression output that duplicates these
values is shown next. Only the first three reg command outputs are shown. Note that the model sum
of squares of 442.44887, 406.68593 and 529.75342 are the same as found with the spectral command
under P_1. SIN and COS values from the two sources are listed next and found to be the same.
Periodogram values are generated from the SIN and COS values using (15.1-31) with scaling (T/2),
in place of (T / 8 ) giving identical results from a time prospective using OLS or from a frequency
prospective using Fourier analysis. As noted above, while the OLS approach may be easier to
interpret in that it shows how the periodogram relates to the regression diagnostics, it is substantially
slower. Note that each periorogram value can be calculated. It is left as an exercise for the reader to
validate that the spectral values have been calculated from the periodogram using the scaled
(triangular) weights 1, 2, 3, 2, and 1 from equation (15.1-38). For the output listed below, series 1 is
GASOUT. Note that for observation 1, period = 296. Frequency is 1/period = .0033784. The listed
frequency can be obtained by .003378(1/(2 )) .021227
Obs
1
2
3
4
5
6
7
8
9
10
PERIOD
296.0
148.0
98.67
74.00
59.20
49.33
42.29
37.00
32.89
29.60
REG Command. Version
FREQ
SIN_1
0.2123E-01 -1.714
0.4245E-01 -1.482
0.6368E-01 0.6334
0.8491E-01 0.6519
0.1061
0.5667
0.1274
-0.1568
0.1486
-1.063
0.1698
0.4674
0.1910
-0.4580
0.2123
0.5578E-01
10000000
755
OLS Estimation
Dependent variable
Adjusted R**2
Standard Error of Estimate
Sum of Squared Residuals
Model Sum of Squares
Total Sum of Squares
F( 2,
293)
F Significance
1/Condition of XPX
Number of Observations
Durbin-Watson
0}
0}
0}
P_1
442.4
406.7
529.8
129.4
78.69
18.02
255.9
45.92
262.0
76.91
1 February 1997
Real*8 space available
Real*8 space used
Variable
SIN
{
COS
{
CONSTANT {
COS_1
-0.2243
-0.7423
1.783
0.6703
0.4589
-0.3118
0.7739
-0.3030
-1.249
0.7187
Coefficient
-1.7144070
-0.22433884
53.509122
GASOUT
0.1404460150470348
2.968754524497961
2582.356504031045
442.4488675905768
3024.805371621621
25.10062379103647
0.9999999999132527
0.5555555555555556
296
6.376831838373521E-02
Std. Error
0.24403012
0.24403012
0.17255535
t
-7.0253911
-0.91930800
310.09830
S_1
35.35
33.04
28.14
17.95
11.64
7.946
10.93
11.22
11.78
7.876
15-20
Chapter 15
REG Command. Version
1 February 1997
Real*8 space available
Real*8 space used
10000000
755
OLS Estimation
Dependent variable
Adjusted R**2
Standard Error of Estimate
Sum of Squared Residuals
Model Sum of Squares
Total Sum of Squares
F( 2,
293)
F Significance
1/Condition of XPX
Number of Observations
Durbin-Watson
Variable
SIN
{
COS
{
CONSTANT {
0}
0}
0}
GASOUT
0.1285420898400539
2.989240914807500
2618.119445300439
406.6859263211827
3024.805371621621
22.75659665299046
0.9999999993494489
0.5555555555555556
296
6.275877065018662E-02
Coefficient
-1.4821769
-0.74231363
53.509122
REG Command. Version
GASOUT
0.1695058964630655
2.918139078177818
2495.051954119426
529.7534175021956
3024.805371621621
31.10511407826055
0.9999999999994319
0.5555555555555556
296
6.501243564636745E-02
Coefficient
0.63341038
1.7827524
53.509122
REG Command. Version
t
2.6406452
7.4321747
315.47699
10000000
755
OLS Estimation
Dependent variable
Adjusted R**2
Standard Error of Estimate
Sum of Squared Residuals
Model Sum of Squares
Total Sum of Squares
F( 2,
293)
F Significance
1/Condition of XPX
Number of Observations
Durbin-Watson
0}
0}
0}
Std. Error
0.23986955
0.23986955
0.16961339
1 February 1997
Real*8 space available
Real*8 space used
Variable
SIN
{
COS
{
CONSTANT {
-6.0321201
-3.0210463
307.97308
10000000
755
OLS Estimation
Dependent variable
Adjusted R**2
Standard Error of Estimate
Sum of Squared Residuals
Model Sum of Squares
Total Sum of Squares
F( 2,
293)
F Significance
1/Condition of XPX
Number of Observations
Durbin-Watson
0}
0}
0}
t
1 February 1997
Real*8 space available
Real*8 space used
Variable
SIN
{
COS
{
CONSTANT {
Std. Error
0.24571409
0.24571409
0.17374610
Coefficient
0.65193021
0.67027858
53.509122
GASOUT
3.624381523407638E-02
3.143556705612574
2895.410987090722
129.3943845308991
3024.805371621621
6.547007460527660
0.9983466201852209
0.5555555555555556
296
5.641171059948612E-02
Std. Error
0.25839877
0.25839877
0.18271552
t
2.5229618
2.5939697
292.85482
_________________________________________________________
The second set of problems in the code in Table 15.3 involves estimating a cross spectral
model using the Box-Jenkins (1976) gas furnace data. Here weights of [1, 2, 3, 4, 3, 2, 1] are used.
Actual values are listed for the first 40 frequencies and plots of all values are given. Inspection of
Figures 15.3 and 15.4 and especially 15.5 and 15.6 and listings indicates that GASIN maps to
GASOUT at low frequencies. The weighting smoothes the periodogram values. For output listed
below, series 1 is GASIN and series 2 is GASOUT. Note that here FREQ = 1/PERIOD.
Spectral Analysis of Time Series
Obs
15-21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
PERIOD
296.0
148.0
98.67
74.00
59.20
49.33
42.29
37.00
32.89
29.60
26.91
24.67
22.77
21.14
19.73
18.50
17.41
16.44
15.58
14.80
14.10
13.45
12.87
12.33
11.84
11.38
10.96
10.57
10.21
9.867
9.548
9.250
8.970
8.706
8.457
8.222
8.000
7.789
7.590
7.400
FREQ
0.3378E-02
0.6757E-02
0.1014E-01
0.1351E-01
0.1689E-01
0.2027E-01
0.2365E-01
0.2703E-01
0.3041E-01
0.3378E-01
0.3716E-01
0.4054E-01
0.4392E-01
0.4730E-01
0.5068E-01
0.5405E-01
0.5743E-01
0.6081E-01
0.6419E-01
0.6757E-01
0.7095E-01
0.7432E-01
0.7770E-01
0.8108E-01
0.8446E-01
0.8784E-01
0.9122E-01
0.9459E-01
0.9797E-01
0.1014
0.1047
0.1081
0.1115
0.1149
0.1182
0.1216
0.1250
0.1284
0.1318
0.1351
SIN_1
0.4798
0.3560
-0.4462E-01
-0.1421
-0.9377E-01
-0.6245E-01
0.3529
-0.2504
-0.3567
0.1291
-0.1486
0.3193E-02
0.1735
0.1826
0.6261E-01
0.1643
0.7021E-01
-0.2711
0.9543E-01
0.1108E-01
-0.5102E-01
-0.6070E-02
0.8882E-01
-0.6353E-01
-0.6266E-01
0.1244
-0.1261
0.7276E-01
-0.7566E-01
0.3848E-01
0.1100
-0.8815E-01
-0.6512E-01
-0.8999E-02
0.4304E-01
0.1303
-0.1981E-02
-0.1171E-02
0.4124E-03
-0.5830E-01
SIN_2
-1.714
-1.482
0.6334
0.6519
0.5667
-0.1568
-1.063
0.4674
-0.4580
0.5578E-01
-0.3376
-0.2289E-01
-0.9361
-0.8758
0.8874E-01
0.2742
-0.2818
0.4794E-01
-0.2331
-0.4056
0.2422
-0.3126
-0.8514E-01
0.1641
0.1842E-01
0.4029
-0.2429
0.1863
-0.1554
0.4538E-01
0.1857
-0.2289
-0.2054
-0.6225E-01
-0.1184E-01
0.2045
-0.8518E-02
-0.1834E-01
0.4063E-01
0.2978E-01
COS_1
0.1931
0.3565
-0.5042
-0.2708
-0.2222
0.2150
0.1057
-0.2797E-01
0.3711
-0.1646
0.1131
-0.6891E-01
0.2199
0.2172
-0.1209
-0.1507
0.9312E-01
-0.1195
0.1405
0.9569E-01
-0.2574
0.8152E-01
0.1528E-01
-0.2489
-0.2016
-0.1371
0.1878
-0.6586E-01
0.5779E-01
-0.2020E-01
0.2119E-02
-0.3095E-01
-0.3968E-01
0.6456E-01
0.6325E-01
-0.2427E-01
-0.4682E-01
0.3996E-01
0.2228E-01
0.7638E-01
COS_2
-0.2243
-0.7423
1.783
0.6703
0.4589
-0.3118
0.7739
-0.3030
-1.249
0.7187
-0.4302
0.9868E-01
0.3890
0.3462
0.1638
0.3253
0.2096
-0.6516
0.4178
0.2647
-0.4199
0.1288
0.2173
-0.5095
-0.5336
-0.3746
0.1310
-0.5671E-01
0.1007
0.6868E-02
0.6038E-01
-0.5963E-01
-0.1437
0.6460E-01
0.5566E-01
-0.1395
-0.6563E-01
0.3119E-01
-0.2612E-02
0.1332
P_1
39.60
37.56
37.92
13.84
8.612
7.418
20.08
9.395
39.21
6.479
5.165
0.7043
11.61
11.92
2.745
7.355
2.013
12.99
4.271
1.373
10.19
0.9889
1.202
9.764
6.595
5.072
7.577
1.425
1.342
0.2795
1.792
1.292
0.8606
0.6288
0.8663
2.599
0.3251
0.2366
0.7352E-01
1.367
P_2
442.4
406.7
529.8
129.4
78.69
18.02
255.9
45.92
262.0
76.91
44.26
1.519
152.1
131.3
5.137
26.79
18.26
63.18
33.87
34.72
34.78
16.91
8.063
42.41
42.18
44.78
11.27
5.615
5.072
0.3118
5.645
8.278
9.300
1.191
0.4793
9.069
0.6482
0.1938
0.2453
2.758
S_1
2.976
2.675
2.235
1.714
1.299
1.190
1.227
1.278
1.308
1.049
0.8214
0.6559
0.5668
0.5671
0.5458
0.5440
0.4795
0.5033
0.4622
0.3981
0.4056
0.3545
0.3834
0.4592
0.4519
0.4363
0.3766
0.2564
0.1770
0.1213
0.9122E-01
0.8530E-01
0.8654E-01
0.8751E-01
0.8479E-01
0.8274E-01
0.6805E-01
0.5485E-01
0.4527E-01
0.4053E-01
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
PERIOD
296.0
148.0
98.67
74.00
59.20
49.33
42.29
37.00
32.89
29.60
26.91
24.67
22.77
21.14
19.73
18.50
17.41
16.44
15.58
14.80
14.10
13.45
12.87
12.33
11.84
11.38
10.96
10.57
10.21
9.867
9.548
9.250
8.970
8.706
8.457
8.222
8.000
7.789
7.590
7.400
FREQ
0.3378E-02
0.6757E-02
0.1014E-01
0.1351E-01
0.1689E-01
0.2027E-01
0.2365E-01
0.2703E-01
0.3041E-01
0.3378E-01
0.3716E-01
0.4054E-01
0.4392E-01
0.4730E-01
0.5068E-01
0.5405E-01
0.5743E-01
0.6081E-01
0.6419E-01
0.6757E-01
0.7095E-01
0.7432E-01
0.7770E-01
0.8108E-01
0.8446E-01
0.8784E-01
0.9122E-01
0.9459E-01
0.9797E-01
0.1014
0.1047
0.1081
0.1115
0.1149
0.1182
0.1216
0.1250
0.1284
0.1318
0.1351
RP_1
-128.2
-117.2
-137.2
-40.57
-22.96
-8.471
-43.41
-16.07
-44.43
-16.45
0.2236
-1.017
-11.38
-12.53
-2.110
-0.5905
-0.3994E-01
9.601
5.397
3.084
14.17
1.835
-0.6276
17.22
15.75
15.02
8.177
2.559
2.601
0.2379
3.043
3.259
2.823
0.7001
0.4456
4.444
0.4573
0.1877
-0.6136E-02
1.249
IP_1
-33.07
-39.08
-35.49
-12.04
-12.27
-7.870
-57.05
-13.16
-91.10
-15.09
-15.12
0.1868
-40.45
-37.51
-3.106
-14.02
-6.062
-26.99
-10.75
-6.178
-12.40
-3.655
-3.049
-10.84
-5.497
-1.282
-4.308
-1.206
-0.2014
-0.1748
-0.9250
0.2703
-0.1786
-0.5087
-0.4654
1.955
0.3978E-01
-0.1030
0.1341
1.486
CS_1
-9.691
-8.722
-7.268
-5.301
-3.648
-2.626
-2.184
-1.991
-1.846
-1.429
-0.9536
-0.7065
-0.5322
-0.4658
-0.3087
-0.1028
0.1193
0.3557
0.4434
0.4637
0.5361
0.5457
0.6595
0.8470
0.8913
0.8623
0.6943
0.4569
0.2960
0.2015
0.1829
0.1769
0.1732
0.1567
0.1354
0.1247
0.9227E-01
0.7148E-01
0.5326E-01
0.3813E-01
QS_1
-2.641
-2.475
-2.123
-1.867
-1.722
-2.071
-2.612
-2.837
-2.990
-2.499
-2.084
-1.817
-1.688
-1.639
-1.427
-1.312
-1.088
-1.066
-0.9803
-0.8184
-0.7187
-0.5532
-0.4920
-0.4753
-0.3875
-0.3078
-0.2343
-0.1377
-0.8169E-01
-0.5190E-01
-0.2928E-01
-0.2120E-01
-0.1209E-01
-0.1997E-02
0.1177E-01
0.2623E-01
0.2999E-01
0.3352E-01
0.4042E-01
0.4738E-01
A_1
10.04
9.067
7.571
5.620
4.034
3.344
3.405
3.466
3.514
2.879
2.292
1.950
1.770
1.704
1.460
1.316
1.094
1.124
1.076
0.9407
0.8967
0.7771
0.8228
0.9712
0.9719
0.9156
0.7327
0.4772
0.3071
0.2081
0.1852
0.1781
0.1737
0.1568
0.1359
0.1275
0.9702E-01
0.7895E-01
0.6686E-01
0.6082E-01
K_1
0.9974
0.9952
0.9862
0.9525
0.9058
0.8515
0.9030
0.9216
0.9317
0.9186
0.9066
0.9113
0.9188
0.9226
0.8722
0.8510
0.8554
0.8817
0.8698
0.8410
0.8244
0.8031
0.8003
0.8432
0.8550
0.8610
0.8595
0.8525
0.8571
0.8779
0.9012
0.8645
0.8031
0.7518
0.7172
0.7596
0.7774
0.7553
0.7280
0.6905
PH_1
-2.876
-2.865
-2.857
-2.803
-2.701
-2.474
-2.267
-2.183
-2.124
-2.090
-2.000
-1.942
-1.876
-1.848
-1.784
-1.649
-1.462
-1.249
-1.146
-1.055
-0.9299
-0.7922
-0.6409
-0.5114
-0.4101
-0.3428
-0.3254
-0.2927
-0.2693
-0.2521
-0.1588
-0.1193
-0.6965E-01
-0.1274E-01
0.8671E-01
0.2073
0.3142
0.4384
0.6491
0.8931
Obs
S_2
33.99
30.88
26.01
19.35
13.83
11.03
10.46
10.20
10.13
8.601
7.051
6.359
6.014
5.550
4.477
3.741
2.919
2.845
2.880
2.643
2.405
2.121
2.207
2.436
2.445
2.232
1.659
1.042
0.6215
0.4066
0.4173
0.4302
0.4339
0.3735
0.3037
0.2585
0.1779
0.1505
0.1356
0.1322
15-22
Chapter 15
Figure 15.3 Spectral analysis of GASIN
Spectral Analysis of Time Series
Figure 15.4 Spectral analysis of GASOUT
15-23
15-24
Chapter 15
Figure 15.5 Cross spectral analysis of GASIN-GASOUT part 1
Spectral Analysis of Time Series
25
Figure 15.5 Cross spectral analysis of GASIN-GASOUT part 2
15.3 Matrix Command Implementation
The B34S matrix command, discussed in Chapter 16, but used in many chapters, provides a
more compact and flexible way to do spectral analysis. For an example consult the code in Table
15.4 which when run
Table 15.4 Matrix Command Implementation of Spectral Analysis
_____________________________________________________
b34sexec options ginclude('gas.b34'); b34srun;
b34sexec matrix;
call loaddata;
/;
/; No weighting here px=sx
/;
call spectral(gasin,sinx,cosx,px,sx,freq);
freq2=freq/(2.0*pi()); period=vfam(1.0/afam(freq2));
call tabulate(freq freq2 period sinx cosx px sx);
/;
Chapter 15
26
/; With weights 1 2 3 2 1 px ne sx
/;
call spectral(gasin,sinx,cosx,px,sx,freq:1 2 3 2 1);
call tabulate(freq freq2 period sinx cosx px sx);
call graph(freq2,sx:heading 'Spectrum of Gasin'
:plottype xyplot :file 'sp_gasin.wmf');
b34srun;
produces the periodogram, spectrum and other intermediate steps as well as a graph which is not
shown due to space. The first 20 lines of the tabulation illustrate what is being calculated. The first
tabulation shows that if there is no weighting px=sx. Note that by dividing FREQ by 2 we get the
B34S FREQ value.
=>
=>
CALL SPECTRAL(GASIN,SINX,COSX,PX,SX,FREQ:1 2 3 2 1)$
CALL TABULATE(FREQ FREQ2 PERIOD SINX COSX PX SX)$
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
FREQ
FREQ2
PERIOD
0.2123E-01 0.3378E-02
296.0
0.4245E-01 0.6757E-02
148.0
0.6368E-01 0.1014E-01
98.67
0.8491E-01 0.1351E-01
74.00
0.1061
0.1689E-01
59.20
0.1274
0.2027E-01
49.33
0.1486
0.2365E-01
42.29
0.1698
0.2703E-01
37.00
0.1910
0.3041E-01
32.89
0.2123
0.3378E-01
29.60
0.2335
0.3716E-01
26.91
0.2547
0.4054E-01
24.67
0.2760
0.4392E-01
22.77
0.2972
0.4730E-01
21.14
0.3184
0.5068E-01
19.73
0.3396
0.5405E-01
18.50
0.3609
0.5743E-01
17.41
0.3821
0.6081E-01
16.44
0.4033
0.6419E-01
15.58
0.4245
0.6757E-01
14.80
SINX
COSX
PX
0.4798
0.1931
39.60
0.3560
0.3565
37.56
-0.4462E-01 -0.5042
37.92
-0.1421
-0.2708
13.84
-0.9377E-01 -0.2222
8.612
-0.6245E-01 0.2150
7.418
0.3529
0.1057
20.08
-0.2504
-0.2797E-01
9.395
-0.3567
0.3711
39.21
0.1291
-0.1646
6.479
-0.1486
0.1131
5.165
0.3193E-02 -0.6891E-01 0.7043
0.1735
0.2199
11.61
0.1826
0.2172
11.92
0.6261E-01 -0.1209
2.745
0.1643
-0.1507
7.355
0.7021E-01 0.9312E-01
2.013
-0.2711
-0.1195
12.99
0.9543E-01 0.1405
4.271
0.1108E-01 0.9569E-01
1.373
SX
3.100
2.840
2.341
1.588
1.117
0.9096
1.253
1.421
1.544
1.046
0.7134
0.4780
0.6011
0.6412
0.5340
0.4994
0.4752
0.5329
0.4752
0.4157
Cross spectral analysis can easily be done with the matrix command as is shown with the code listed
in Table 15.5
Table 15.5 Cross Spectral Analysis with the Matrix Command
__________________________________________________
b34sexec options ginclude('gas.b34'); b34srun;
b34sexec matrix;
call loaddata;
* For sample output See Stokes (1997) page 424;
call cspectral(gasin,gasout,sinx,siny,cosx,cosy,px,py,sx,sy,
rp,ip,cs,qs,a,k,ph,freq:1 2 3 4 3 2 1);
freq2=freq/(2.0*pi()); period=vfam(1.0/afam(freq2));
call tabulate(freq2,period,sinx,siny,cosx,cosy,px,py,sx,sy);
call tabulate(freq2,period,rp,ip,cs,qs,a,k,ph);
call graph(freq2,a :heading 'Amplitude':plottype xyplot);
call graph(freq2,k :heading 'Coherence':plottype xyplot);
call graph(freq2,ph:heading 'Phase':plottype xyplot);
b34srun;
Spectral Analysis of Time Series
27
which runs the same problem as was illustrated earlier. Edited output showing the first 20 lines of
each tabulation replicates the calculations shown in the prior section.
=>
CALL LOADDATA$
=>
* FOR SAMPLE OUTPUT SEE STOKES (1997) PAGE 424$
=>
=>
CALL CSPECTRAL(GASIN,GASOUT,SINX,SINY,COSX,COSY,PX,PY,SX,SY,
RP,IP,CS,QS,A,K,PH,FREQ:1 2 3 4 3 2 1)$
=>
FREQ2=FREQ/(2.0*PI())$
=>
PERIOD=VFAM(1.0/AFAM(FREQ2))$
=>
CALL TABULATE(FREQ2,PERIOD,SINX,SINY,COSX,COSY,PX,PY,SX,SY)$
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
=>
FREQ2
PERIOD
0.3378E-02
296.0
0.6757E-02
148.0
0.1014E-01
98.67
0.1351E-01
74.00
0.1689E-01
59.20
0.2027E-01
49.33
0.2365E-01
42.29
0.2703E-01
37.00
0.3041E-01
32.89
0.3378E-01
29.60
0.3716E-01
26.91
0.4054E-01
24.67
0.4392E-01
22.77
0.4730E-01
21.14
0.5068E-01
19.73
0.5405E-01
18.50
0.5743E-01
17.41
0.6081E-01
16.44
0.6419E-01
15.58
0.6757E-01
14.80
SINX
SINY
COSX
COSY
PX
0.4798
-1.714
0.1931
-0.2243
39.60
0.3560
-1.482
0.3565
-0.7423
37.56
-0.4462E-01 0.6334
-0.5042
1.783
37.92
-0.1421
0.6519
-0.2708
0.6703
13.84
-0.9377E-01 0.5667
-0.2222
0.4589
8.612
-0.6245E-01 -0.1568
0.2150
-0.3118
7.418
0.3529
-1.063
0.1057
0.7739
20.08
-0.2504
0.4674
-0.2797E-01 -0.3030
9.395
-0.3567
-0.4580
0.3711
-1.249
39.21
0.1291
0.5578E-01 -0.1646
0.7187
6.479
-0.1486
-0.3376
0.1131
-0.4302
5.165
0.3193E-02 -0.2289E-01 -0.6891E-01 0.9868E-01 0.7043
0.1735
-0.9361
0.2199
0.3890
11.61
0.1826
-0.8758
0.2172
0.3462
11.92
0.6261E-01 0.8874E-01 -0.1209
0.1638
2.745
0.1643
0.2742
-0.1507
0.3253
7.355
0.7021E-01 -0.2818
0.9312E-01 0.2096
2.013
-0.2711
0.4794E-01 -0.1195
-0.6516
12.99
0.9543E-01 -0.2331
0.1405
0.4178
4.271
0.1108E-01 -0.4056
0.9569E-01 0.2647
1.373
PY
SX
442.4
406.7
529.8
129.4
78.69
18.02
255.9
45.92
262.0
76.91
44.26
1.519
152.1
131.3
5.137
26.79
18.26
63.18
33.87
34.72
SY
2.976
2.675
2.235
1.714
1.299
1.190
1.227
1.278
1.308
1.049
0.8214
0.6559
0.5668
0.5671
0.5458
0.5440
0.4795
0.5033
0.4622
0.3981
33.99
30.88
26.01
19.35
13.83
11.03
10.46
10.20
10.13
8.601
7.051
6.359
6.014
5.550
4.477
3.741
2.919
2.845
2.880
2.643
CALL TABULATE(FREQ2,PERIOD,RP,IP,CS,QS,A,K,PH)$
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
FREQ2
PERIOD
0.3378E-02
296.0
0.6757E-02
148.0
0.1014E-01
98.67
0.1351E-01
74.00
0.1689E-01
59.20
0.2027E-01
49.33
0.2365E-01
42.29
0.2703E-01
37.00
0.3041E-01
32.89
0.3378E-01
29.60
0.3716E-01
26.91
0.4054E-01
24.67
0.4392E-01
22.77
0.4730E-01
21.14
0.5068E-01
19.73
0.5405E-01
18.50
0.5743E-01
17.41
0.6081E-01
16.44
0.6419E-01
15.58
0.6757E-01
14.80
RP
IP
-128.2
-33.07
-117.2
-39.08
-137.2
-35.49
-40.57
-12.04
-22.96
-12.27
-8.471
-7.870
-43.41
-57.05
-16.07
-13.16
-44.43
-91.10
-16.45
-15.09
0.2236
-15.12
-1.017
0.1868
-11.38
-40.45
-12.53
-37.51
-2.110
-3.106
-0.5905
-14.02
-0.3994E-01 -6.062
9.601
-26.99
5.397
-10.75
3.084
-6.178
CS
-9.691
-8.722
-7.268
-5.301
-3.648
-2.626
-2.184
-1.991
-1.846
-1.429
-0.9536
-0.7065
-0.5322
-0.4658
-0.3087
-0.1028
0.1193
0.3557
0.4434
0.4637
QS
-2.641
-2.475
-2.123
-1.867
-1.722
-2.071
-2.612
-2.837
-2.990
-2.499
-2.084
-1.817
-1.688
-1.639
-1.427
-1.312
-1.088
-1.066
-0.9803
-0.8184
A
K
10.04
9.067
7.571
5.620
4.034
3.344
3.405
3.466
3.514
2.879
2.292
1.950
1.770
1.704
1.460
1.316
1.094
1.124
1.076
0.9407
PH
0.9974
0.9952
0.9862
0.9525
0.9058
0.8515
0.9030
0.9216
0.9317
0.9186
0.9066
0.9113
0.9188
0.9226
0.8722
0.8510
0.8554
0.8817
0.8698
0.8410
-2.876
-2.865
-2.857
-2.803
-2.701
-2.474
-2.267
-2.183
-2.124
-2.090
-2.000
-1.942
-1.876
-1.848
-1.784
-1.649
-1.462
-1.249
-1.146
-1.055
The graphs, which are not shown, can be easily placed in Word or other software systems.
The OLS approach to obtaining the periodogram works because the sin and cosine transforms are
orthogonal. This can be demonstrated with the code in Table 15.6
Table 15.6 Verification that the sin and cosine vectors are orthogonal
_______________________________________________________
b34sexec options ginclude('gas.b34')$ b34srun$
b34sexec matrix;
28
Chapter 15
call loaddata;
call echooff;
count=dfloat(integers(norows(gasout)))-1.;
ncase=(norows(gasout)/2)-1;
per=array(ncase:);
test=array(ncase,2*ncase:);
s1 =array(norows(gasout):);
c1 =array(norows(gasout):);
base=(2.0*pi())/dfloat(norows(gasout));
do i=1,ncase;
s1=sin(count*base*dfloat(i));
c1=cos(count*base*dfloat(i));
is_1=(i-1)*2+1;
ic_1= is_1+1;
test(,is_1)=s1;
test(,ic_1)=c1;
if(i.le.20)then;
call olsq(gasout s1 c1 :print :qr );
call print(' ':);
modelss=%tss-%rss;
call print('%tss-%rss',modelss:);
endif;
if(i.gt.20)call olsq(gasout s1 c1
:qr );
per(i)=%tss-%rss;
enddo;
call print(per);
call graph(per);
call print('Illustrate Orthagonality of sine and cosine vectors':);
call print(ccf(test));
b34srun;
which will produce a 294 by 294 matrix of correlations of the 147 pairs of sin and cosin vectors
where all cross correlations are close to machine zero except along the diagonal where they are 1.0.
Due to space limits this matrix, which contains 86,436 correlations, is not shown here.
It is possible to obtain the original values of the series from the sin and cosin values from (15.1-30)
with a small error. Table 15.7 shows both a B34S and a SAS setup.
Table 15.7 Inverse Spectral Examples
________________________________________________
%b34slet dosas=1;
b34sexec options ginclude('gas.b34'); b34srun;
/;
/; Testing FFT
/;
b34sexec options ginclude('gas.b34'); b34srun;
b34sexec matrix;
Spectral Analysis of Time Series
call loaddata;
call echooff;
subroutine spectest(x,testx,error);
call spectral(x,sinx,cosx,px,sx,freq1);
count=dfloat(integers(norows(x)))-1.
;
/; Test 100% recovery of actual value
adj_freq=freq1/(2.*pi());
period=1.0/afam(adj_freq);
do i=1,norows(x);
sum1=sinx*sin(count(i)*freq1);
sum2=cosx*cos(count(i)*freq1);
test(i)=sum1+sum2;
enddo;
call print('mean(x)',mean(x):);
adj=mean(x);
testx=afam(test)+adj;
error_1=x-testx;
call tabulate(x,testx,error_1);
return;
end;
call
call
call
call
call
spectest(gasin, yhat,error);
names(all);
olsq(yhat gasin :print);
spectest(gasout,yhat,error);
olsq(yhat gasout :print);
/; Now look at fft
cfft=fft(gasin);
test=fft(cfft:back)/dfloat(norows(gasin));
error=gasin-test;
call tabulate(gasin,test,error);
b34srun;
%b34sif(&dosas.ne.0)%then;
b34sexec options open('testsas.sas') unit(29) disp=unknown$ b34srun$
b34sexec options clean(29) $ b34seend$
b34sexec pgmcall idata=29 icntrl=29$
sas
$
* sas commands next ;
pgmcards$
proc spectra out=specgas coef p s; var gasin gasout; weights 1 1 1; run;
proc means data=specgas; run;
proc print data=specgas; run;
b34sreturn$
b34srun $
b34sexec options close(29)$ b34srun$
/$ the next card has to be modified to point to sas location
/$ be sure and wait until sas gets done before letting b34s resume
29
Chapter 15
30
/$ ***************************************************************
b34sexec options dodos('start /w /r
sas testsas' )
dounix('sas testsas' )
$ b34srun$
b34sexec options npageout noheader
writeout('
','output from sas',' ',' ')
writelog('
','output from sas',' ',' ')
copyfout('testsas.lst')
copyflog('testsas.log')
/;dodos('erase testsas.sas','erase testsas.lst','erase testsas.log')
dounix('rm
testsas.sas','rm
testsas.lst','rm
testsas.log')
$ b34srun$
b34sexec options header$ b34srun$
%b34sendif;
When the code in Table 15.7 is run it produces edited output that shows the recovery of GASIN and
GASOUT from the sin and cosin values. Note the very small error which is validated with OLS.
mean(x)
Obs
-5.683445945945946E-02
X
TESTX
ERROR_1
1
-0.1090
-0.1090
0.7633E-15
2
0.000
-0.2720E-14 0.2720E-14
3
0.1780
0.1780
0.4496E-14
4
0.3390
0.3390
0.3497E-14
5
0.3730
0.3730
0.1499E-14
6
0.4410
0.4410
-0.1998E-14
7
0.4610
0.4610
-0.4441E-15
8
0.3480
0.3480
-0.1665E-15
9
0.1270
0.1270
-0.1998E-14
10
-0.1800
-0.1800
-0.4496E-14
11
-0.5880
-0.5880
-0.6550E-14
12
-1.055
-1.055
-0.7327E-14
13
-1.421
-1.421
-0.4441E-14
14
-1.520
-1.520
-0.1110E-14
15
-1.302
-1.302
-0.2442E-14
16
-0.8140
-0.8140
-0.1998E-14
17
-0.4750
-0.4750
0.2776E-15
18
-0.1930
-0.1930
0.4441E-14
19
0.8800E-01 0.8800E-01 0.8313E-14
20
0.4350
0.4350
0.1044E-13
………………………………………………………………………………………………….
280
0.2510
0.2510
0.1205E-13
281
0.2800
0.2800
0.6550E-14
282
0.000
0.6939E-16 -0.6939E-16
283
-0.4930
-0.4930
-0.1499E-14
284
-0.7590
-0.7590
-0.3442E-14
285
-0.8240
-0.8240
-0.5440E-14
286
-0.7400
-0.7400
-0.4663E-14
287
-0.5280
-0.5280
0.4885E-14
288
-0.2040
-0.2040
0.1138E-14
289
0.3400E-01 0.3400E-01 -0.3775E-14
290
0.2040
0.2040
0.1360E-14
291
0.2530
0.2530
-0.1094E-13
292
0.1950
0.1950
-0.8438E-14
293
0.1310
0.1310
-0.1887E-14
294
0.1700E-01 0.1700E-01 0.2387E-14
295
-0.1820
-0.1820
0.1016E-13
296
-0.2620
-0.2620
0.1221E-14
Ordinary Least Squares Estimation
Dependent variable
Centered R**2
Adjusted R**2
YHAT
1.000000000000000
1.000000000000000
Spectral Analysis of Time Series
Residual Sum of Squares
Residual Variance
Standard Error
Total Sum of Squares
Log Likelihood
Mean of the Dependent Variable
Std. Error of Dependent Variable
Sum Absolute Residuals
1/Condition XPX
Maximum Absolute Residual
Number of Observations
Variable
GASIN
CONSTANT
mean(x)
Obs
Lag
Coefficient
0
1.0000000
0
0.38515884E-15
53.50912162162162
5.020523454753602E-26
1.707661039031837E-28
1.306775052957408E-14
339.4936188885142
9043.711769338292
-5.683445945945921E-02
1.072765504078466
2.840377339427970E-12
0.8358143402344875
4.363176486776865E-14
296
SE
0.70922662E-15
0.76061639E-15
t
0.14099866E+16
0.50637726
X
TESTX
ERROR_1
1
53.80
53.80
0.2132E-13
2
53.60
53.60
0.7105E-14
3
53.50
53.50
0.7105E-14
4
53.50
53.50
0.000
5
53.40
53.40
-0.7105E-14
6
53.10
53.10
0.7105E-14
7
52.70
52.70
0.000
8
52.40
52.40
-0.7105E-14
9
52.20
52.20
0.9237E-13
10
52.00
52.00
0.9237E-13
11
52.00
52.00
0.1066E-12
12
52.40
52.40
0.8527E-13
13
53.00
53.00
0.7816E-13
14
54.00
54.00
0.1066E-12
15
54.90
54.90
0.1137E-12
16
56.00
56.00
0.1066E-12
17
56.80
56.80
0.7105E-14
18
56.80
56.80
-0.7105E-14
19
56.40
56.40
0.000
20
55.70
55.70
-0.1421E-13
……………………………………………………………………………………………………….
280
54.40
54.40
0.1066E-12
281
53.70
53.70
-0.7105E-14
282
53.30
53.30
-0.3553E-13
283
52.80
52.80
-0.2842E-13
284
52.60
52.60
-0.3553E-13
285
52.60
52.60
-0.5684E-13
286
53.00
53.00
-0.2842E-13
287
54.30
54.30
0.7105E-14
288
56.00
56.00
-0.4263E-13
289
57.00
57.00
0.9237E-13
290
58.00
58.00
0.9948E-13
291
58.60
58.60
0.1066E-12
292
58.50
58.50
0.9237E-13
293
58.30
58.30
0.9948E-13
294
57.80
57.80
0.7816E-13
295
57.30
57.30
0.1208E-12
296
57.00
57.00
0.5684E-13
Ordinary Least Squares Estimation
Dependent variable
Centered R**2
Adjusted R**2
Residual Sum of Squares
Residual Variance
Standard Error
Total Sum of Squares
Log Likelihood
Mean of the Dependent Variable
Std. Error of Dependent Variable
Sum Absolute Residuals
F( 1,
294)
YHAT
0.9999997207772101
0.9999997198274727
8.445943587647710E-04
2.872769927771330E-06
1.694924755784554E-03
3024.804527027029
1469.512199422020
53.50912162163348
3.202120339382677
0.4999998603887690
1052922356.462301
31
Chapter 15
32
F Significance
1/Condition XPX
Maximum Absolute Residual
Number of Observations
Variable
GASOUT
CONSTANT
Lag
0
0
Coefficient
0.99999972
0.14940968E-04
1.000000000000000
1.214609942288382E-06
1.691397594697719E-03
296
SE
0.30817805E-04
0.16519738E-02
t
32448.765
0.90443130E-02
The same calculation is done with the inverse FFT. Note the small errors:
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
290
291
292
293
294
295
296
GASIN
TEST
ERROR
-0.1090
-0.1090
0.3747E-15
0.000
-0.2497E-14 0.2497E-14
0.1780
0.1780
0.4302E-14
0.3390
0.3390
0.3386E-14
0.3730
0.3730
0.1554E-14
0.4410
0.4410
-0.1277E-14
0.4610
0.4610
-0.1554E-14
0.3480
0.3480
-0.9992E-15
0.1270
0.1270
-0.2415E-14
-0.1800
-0.1800
-0.4136E-14
-0.5880
-0.5880
-0.5995E-14
-1.055
-1.055
-0.6217E-14
-1.421
-1.421
-0.3775E-14
-1.520
-1.520
-0.2220E-15
-1.302
-1.302
-0.2220E-15
-0.8140
-0.8140
0.000
-0.4750
-0.4750
0.1721E-14
-0.1930
-0.1930
0.4191E-14
0.8800E-01 0.8800E-01 0.5773E-14
0.4350
0.4350
0.4829E-14
0.2040
0.2040
0.2530
0.2530
0.1950
0.1950
0.1310
0.1310
0.1700E-01 0.1700E-01
-0.1820
-0.1820
-0.2620
-0.2620
-0.2720E-14
-0.5607E-14
-0.5079E-14
-0.2914E-14
-0.9957E-15
-0.6106E-15
-0.6106E-15
From the sin and cosine coefficients the original series was recovered. The OLS model tested the
conversion and found the coefficient of Y and on Yˆ was very close to 1.0, and, as expected, was
close to being a perfect fit.
15.4 Wavelet Analysis
Wavelets provide a means by which a series can be filtered to remove noise.5 The first step is
to estimate a discrete Fourier transform of a series xi i 1, , N where there are N periods at
frequency k, xˆ k , where
5 This section discusses the approach to wavelet estimation suggested by Torrence and Compo (1998) which have
provided the Fortran code that is used for all calculations. Consult this article for future information. Where ever
possible, the notion in this section follows their article.
Spectral Analysis of Time Series
xˆk
1
N
N 1
x e
n 0
33
2 ikn / N
(15.4-1)
n
Define the wavelet function as ( s ) . The wavelet transform is the inverse Fourier transform of the
product
N 1
Wn ( s ) xˆkˆ * ( s k )e
i k n t
(15.4-2)
k 0
where k
2 k
N
2 k
for k and
N t
2
N t
otherwise.
The wavelet transforms in (15.4-2) are
2 s ˆ
normalized to have unit energy ˆ ( sk )
0 ( sk ) where the unscaled transforms ˆ 0 have
t
.5
been normalized to have unit energy or |ˆ 0 ( ') |2 d ' 1 . At each scale s,
N 1
|ˆ ( s ) | N .
2
k 0
k
Table 15.8 lists the wavelet basis functions and their properties and has been taken from Torrence
and Compo (1998) Table 1. H ( ) = Heaviside step function which =1 if 0 and 0 otherwise.
The DOG is the only real valued wave function of the three.
Table 15.8 Wavelet basis functions supported
Name
0 ( )
Morlet
( 0 frequency)
.25ei e
Paul
ˆ 0 ( s )
0
2
/2
2m i mm!
(1 i ) ( m 1)
(2m)!
.25 H ()e( s )
0
2
/2
2m
H ( )( s ) m e s
m(2m 1)!
( m order)
DOG (Derivative of Gaussian)
( 1) m 1 d m 2 / 2
(e
)
m
( m .5) d
( m derivative)
Source Torrence and Compo (1998 table 1)
2
i m
( s ) m e ( s ) / 2
( m .5)
Chapter 15
34
Table 15.9 Empirically derived factors for four wavelet bases
Name
C
j0
Morlet (0 6)
Paul ( m 4)
Marr (DOG m 2)
DOG ( m 6)
.776
1.132
3.541
1.966
2.32
1.17
1.43
1.37
.60
1.5
1.4
0.97
0 (0)
.25 .7511255
1.079
.867
.884
Source Torrence and Compo (1998 table 2)
= reconstruction factor
C
= decorrelation factor for time averaging
j0 = factor for scale averaging
After selecting the wavelet basis function, wavelet analysis first proceeds by selecting the appropriate
scales s to use. Define
s j s0 2 j j , j 0,1,
,J
J j 1 log 2 ( N t / s0 )
(15.4-3)
s0 is the smallest scale and J is the largest. Smaller t values mean more resolution. For the Morlet
wavelet it is usually set as .5. By summing over all scales, the original time series can be
reconstructed from the real part of the wavelet transformation, Wn ( s j ) , as
j t .5 j Wn ( s j )
xn
s.5
C 0 (0) j 0
j
(15.4-4)
Torrence and Compo (1998) outline how these formulas are used. Assume a new wavelet function
1
for time period n 0 so that xn n 0 which implies a Fourier transform xˆ k
from (15.4-1) that
N
is constant for all k. Substituting xˆ k into (15.4-2) gives
W ( s )
1
N
N 1
ˆ
k 0
*
( sk )
which using (15.4-4) can be solved for
(15.4-5)
Spectral Analysis of Time Series
j t.5 J W ( s j )
C
s.5
0 (0) j 0
j
35
(15.4-6)
Total energy is conserved under the wavelet function and should be checked to make sure
appropriate s0 and t values have been selected.
j t
2
N 1 J
| W ( s ) |
2
C N
n
n 0 j 0
j
(15.4-7)
This suggests that all scales should be inspected using (15.4-4) and (15.4-7) to see if the level and the
variance can be accurately recovered. If this is not the case, s0 and j would have to be adjusted.
The time-averaged wavelet spectrum of a certain period is
Wn2 ( s )
1
na
n2
| W ( s) |
2
n
n n1
(15.4-8)
where na n2 n1 1 . When summed over all local wavelet spectrum the global wavelet spectrum
W 2 ( s)
1
N
N 1
| W ( s) |
2
n
n 0
(15.4-9)
is obtained.
The scale-averaged wavelet power is defined as the weighted sum of the wavelet power spectrum
over scales s1 to s2
W
2
n
j t
C
j2
| Wn ( s j ) |2
j j1
sj
(15.4-10)
It is possible to filter a series by using (15.4-4) and summing a range of scales
xn
j t .5 j Wn ( s j )
s.5
C 0 (0) j j
j
2
(15.4-11)
1
One way to think about (15.4-11) is that the larger j1 the more the information in the short periods
(small scale) that is removed. Thus noise that by assumption is of short duration, can be removed
from the data to capture what is hoped is the more fundamental series. The higher sampling rate, the
Chapter 15
36
more of this noise reduction strategy may be required. Experimentation may be required to set the
scales that are appropriate for the series at hand. Once the series has been processed it can be further
analyzed using nonlinear and linear models that hopefully will better capture the underlying
structure..
Wavelet calculations are illustrated using data of 504 observations on the Nino suggested by
Torrence and Compo (1998). Table 15.4-10 shows a setup to completely filter the Nino series.
Table 15.10 Wavelet Filter For The Nino Series
b34sexec options ginclude('wavedata.mac') member(nino3); b34srun;
/;
/; Basic Wavelet test
/;
b34sexec matrix;
call loaddata;
call wavelet(nino :type morlet :settings :s0 .25 :dt .25
:lower 2. :upper 7.9 :jtot 44);
call tabulate(nino,%recon_y);
call olsq(nino %recon_y :print);
call tabulate(%scale,%period,%w_power,%w_phase,
%w_ampl,%signif,%global,%g_sig);
call print(%sa_df %sa_sig);
call print('mean original data ',mean(nino):);
call print('mean of reconstructed data ',mean(%recon_y):);
call print('Variance of original Data
',variance(nino):);
call print('Variance of reconstructed Data ',variance(%recon_y):);
b34srun;
Edited output follows:
B34S 8.11C
Variable
NINO
CONSTANT
(D:M:Y)
5/ 8/07 (H:M:S) 17: 2:31
Label
DATA STEP
# Cases
1 Nino2 sea surface temperature
2
5/ 8/07. h:m:s 17: 2:31.
=>
CALL LOADDATA$
=>
=>
CALL WAVELET(NINO :TYPE MORLET :SETTINGS :S0 .25 :DT .25
:LOWER 2. :UPPER 7.9 :JTOT 44)$
Wavelet Option Settings
Input Variable
Number of Original Observations
Sampling time (dt)
Wavelet used
Wave number (param)
Smallest scale of wavelet s0=%scale(1)
Largest scale of wavelet (%scale(jtot))
Spacing between discrete scales (dj)
Number of scales (jtot)
Number of observation after padding
Lag1 (background autocorrelation)
Significance level
Lower Scale for filter
Upper Scale for filter
Std. Dev.
504 -0.198413E-04
504
1.00000
Number of observations in data file
504
Current missing variable code
1.000000000000000E+31
Data begins on (D:M:Y) 1: 1:1871 ends 1:10:1996.
Frequency is
B34S(r) Matrix Command. d/m/y
Sea Surface Temp
Mean
NINO
504
0.2500000000000000
Morlet
6.000000000000000
0.2500000000000000
430.5389646099018
0.2500000000000000
44
1024
0.7200000000000000
5.000000000000004E-02
2.000000000000000
7.900000000000000
4
0.734328
0.00000
Variance
0.539238
0.00000
PAGE
1
Maximum
Minimum
2.50000
1.00000
-1.85000
1.00000
Spectral Analysis of Time Series
Work array size (nk)
=>
1024
CALL TABULATE(NINO,%RECON_Y)$
The filter is successful at capturing the NINO series as shown by the application of (15.4-4).
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
.
494
495
496
497
498
499
500
501
502
503
504
NINO
-0.1500
-0.3000
-0.1400
-0.4100
-0.4600
-0.6600
-0.5000
-0.8000
-0.9500
-0.7200
-0.3100
-0.7100
-1.040
-0.7700
-0.8600
-0.8400
-0.4100
-0.4900
-0.4800
-0.7200
-1.210
-0.8000
0.1600
0.4600
0.4000
1.000
%RECON_Y
-0.1508
-0.3005
-0.1408
-0.4109
-0.4618
-0.6617
-0.5019
-0.8021
-0.9533
-0.7219
-0.3113
-0.7119
-1.044
-0.7721
-0.8630
-0.8423
-0.4116
-0.4912
-0.4818
-0.7219
-1.214
-0.8021
0.1602
0.4618
0.4009
1.004
.
.
0.3900
-0.1700
1.040
0.7700
0.1200
-0.3500
-0.2200
0.8000E-01
-0.8000E-01
-0.1800
-0.6000E-01
0.3913
-0.1706
1.043
0.7723
0.1206
-0.3513
-0.2204
0.7998E-01
-0.7998E-01
-0.1808
-0.5990E-01
An OLS regression documents the closeness of the fit.
=>
CALL OLSQ(NINO %RECON_Y :PRINT)$
Ordinary Least Squares Estimation
Dependent variable
Centered R**2
Adjusted R**2
Residual Sum of Squares
Residual Variance
Standard Error
Total Sum of Squares
Log Likelihood
Mean of the Dependent Variable
Std. Error of Dependent Variable
Sum Absolute Residuals
F( 1,
502)
F Significance
1/Condition XPX
Maximum Absolute Residual
Number of Observations
Variable
%RECON_Y
CONSTANT
=>
=>
Lag
0
0
Coefficient
0.99689178
-0.19583793E-04
NINO
0.9999998133138651
0.9999998129419804
5.063609377294851E-05
1.008687126951166E-07
3.175983512159919E-04
271.2364998015873
3345.437370731567
-1.984126984126920E-05
0.7343279745169901
0.1518517527012808
2689004765.108992
1.000000000000000
0.6454896073187762
5.326197560795443E-04
504
SE
0.19224375E-04
0.14146955E-04
t
51855.615
-1.3843115
CALL TABULATE(%SCALE,%PERIOD,%W_POWER,%W_PHASE,
%W_AMPL,%SIGNIF,%GLOBAL,%G_SIG)$
This table produces the power and phase and other values by period/scale
Obs
1
2
3
4
5
6
7
8
9
10
%SCALE
0.2500
0.2973
0.3536
0.4204
0.5000
0.5946
0.7071
0.8409
1.000
1.189
%PERIOD
0.2583
0.3071
0.3652
0.4343
0.5165
0.6143
0.7305
0.8687
1.033
1.229
%W_POWER
%W_PHASE
0.6064E-06
116.4
0.1538E-04
116.6
0.3005E-03
118.3
0.3146E-02
122.8
0.1232E-01
132.4
0.2164E-01
143.6
0.2336E-01
141.6
0.5037E-01
135.1
0.2590
148.1
0.3354
173.0
%W_AMPL
%SIGNIF
0.7787E-03
7.230
0.3922E-02 0.8132
0.1734E-01 0.3707
0.5609E-01 0.2774
0.1110
0.2631
0.1471
0.2855
0.1529
0.3365
0.2244
0.4181
0.5089
0.5369
0.5791
0.7035
%GLOBAL
%G_SIG
0.1340E-05
2.689
0.3449E-04 0.3053
0.6882E-03 0.1406
0.7453E-02 0.1064
0.2912E-01 0.1021
0.4269E-01 0.1123
0.5829E-01 0.1342
0.9573E-01 0.1693
0.1767
0.2211
0.2689
0.2948
37
Chapter 15
38
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
=>
1.414
1.682
2.000
2.378
2.828
3.364
4.000
4.757
5.657
6.727
8.000
9.514
11.31
13.45
16.00
19.03
22.63
26.91
32.00
38.05
45.25
53.82
64.00
76.11
90.51
107.6
128.0
152.2
181.0
215.3
256.0
304.4
362.0
430.5
1.461
1.737
2.066
2.457
2.922
3.475
4.132
4.914
5.844
6.949
8.264
9.828
11.69
13.90
16.53
19.66
23.38
27.80
33.06
39.31
46.75
55.60
66.11
78.62
93.50
111.2
132.2
157.2
187.0
222.4
264.5
314.5
374.0
444.8
=
6.4406460
%SA_SIG =
0.43879433
CALL PRINT('mean of reconstructed data
0.3958
0.5479
0.7847
1.276
1.954
2.569
2.270
2.109
2.357
1.703
1.368
1.318
1.605
1.823
1.311
1.424
1.154
1.295
0.6433
0.5865
1.008
0.5386
0.7283
1.363
1.169
0.6206
0.9145
0.1606
0.1404
1.092
2.003
0.2956
0.7419E-03
0.3191E-08
0.3979
0.5392
0.7290
0.9773
1.292
1.676
2.125
2.625
3.157
3.700
4.236
4.753
5.246
5.716
6.167
6.603
7.027
7.440
7.838
8.214
8.560
8.867
9.128
9.341
9.508
9.634
9.726
9.790
9.834
9.863
9.881
9.893
9.899
9.902
',MEAN(%RECON_Y):)$
-2.582799130488190E-07
CALL PRINT('Variance of original Data
Variance of original Data
=>
0.9314
1.236
1.635
2.140
2.758
3.481
4.285
5.130
5.968
6.750
7.442
8.025
8.496
8.865
9.146
9.355
9.509
9.621
9.702
9.760
9.802
9.831
9.852
9.867
9.878
9.885
9.891
9.894
9.897
9.899
9.900
9.901
9.902
9.902
-1.984126984126920E-05
mean of reconstructed data
=>
0.6751
0.9913
0.6674
0.4481
0.1477
0.1260
0.9716
1.722
0.8570
0.4540
0.5562
1.140
1.433
1.508
0.4866
0.6009
0.3603
1.189
0.9217
0.7477
1.003
0.5339
0.8368
1.215
1.116
0.8159
0.9598
0.4140
0.3777
1.045
1.415
0.5437
0.2724E-01
0.5649E-04
CALL PRINT('mean original data ',MEAN(NINO):)$
mean original data
=>
-125.3
-117.8
-159.3
153.5
13.59
58.03
-165.5
-146.5
-140.4
66.75
120.3
131.0
137.4
129.9
119.8
-116.2
-50.34
12.27
14.68
-34.62
-42.79
2.065
91.35
117.7
133.6
173.3
-173.0
-163.0
-114.7
-112.3
-112.3
-112.3
-112.3
-112.3
CALL PRINT(%SA_DF %SA_SIG)$
%SA_DF
=>
0.4558
0.9827
0.4454
0.2008
0.2182E-01
0.1589E-01
0.9440
2.964
0.7344
0.2061
0.3094
1.300
2.055
2.273
0.2368
0.3610
0.1298
1.415
0.8495
0.5591
1.005
0.2850
0.7003
1.477
1.247
0.6657
0.9212
0.1714
0.1426
1.092
2.003
0.2956
0.7419E-03
0.3191E-08
',VARIANCE(NINO):)$
0.5392375741582253
CALL PRINT('Variance of reconstructed Data ',VARIANCE(%RECON_Y):)$
Variance of reconstructed Data
0.5426052999725153
B34S Matrix Command Ending. Last Command reached.
Space available in allocator
Number variables used
Number temp variables used
11856880, peak space used
54, peak number used
28, # user temp clean
61607
54
0
Table 15.11 illustrates application of equation (15.4-11) to filter the noise from the nino3
series using successively higher values of s0 . The results are shown in Figure 15.6 which is best
viewed in color. The series filter3 is the most aggressive at removing noise due to s0 4.5 .
Table 15.11 Smoothing Nino Series by Local Periods
b34sexec options ginclude('wavedata.mac') member(nino3); b34srun;
/;
/; Illustrates filtering. Increasing so => tighter filter
/;
b34sexec matrix;
call loaddata;
/; This setting will closely filter series
call wavelet(nino :type morlet :settings :s0 2.25 :dt .25
:jtot 44);
filter1=%recon_y;
Spectral Analysis of Time Series
39
call wavelet(nino :type morlet :settings :s0 3.5 :dt .25
:jtot 44);
filter2=%recon_y;
call wavelet(nino :type morlet :settings :s0 4.5 :dt .25
:jtot 44);
filter3=%recon_y;
call tabulate(nino,filter1,filter2,filter3);
call graph(nino,filter1,filter2 filter3 :nolabel
:heading 'Raw Nino and smoothed series' :file 'nino.wmf');
b34srun;
Figure 15.7 shows increased smoothing of the Nino series
Raw Nino and smoothed series
2.5
2
1.5
1
N
I
N
O
.5
0
-.5
-1
-1.5
50
100
150
200
250
Obs
300
350
Figure 15.6 Raw Nino series and three wavelet smoothed series
A portion of the output that produced Figure 15.6 is shown.
Wavelet Option Settings
Input Variable
Number of Original Observations
Sampling time (dt)
Wavelet used
Wave number (param)
Smallest scale of wavelet s0=%scale(1)
Largest scale of wavelet (%scale(jtot))
Spacing between discrete scales (dj)
NINO
504
0.2500000000000000
Morlet
6.000000000000000
2.250000000000000
3874.850681489117
0.2500000000000000
400
450
500
F
I
L
T
E
R
1
F
I
L
T
E
R
2
F
I
L
T
E
R
3
Chapter 15
40
Number of scales (jtot)
Number of observation after padding
Lag1 (background autocorrelation)
Significance level
Lower Scale for filter
Upper Scale for filter
Work array size (nk)
44
1024
0.7200000000000000
5.000000000000004E-02
2.000000000000000
6.000000000000000
1024
=>
FILTER1=%RECON_Y$
=>
=>
CALL WAVELET(NINO :TYPE MORLET :SETTINGS :S0 3.5 :DT .25
:JTOT 44)$
Wavelet Option Settings
Input Variable
Number of Original Observations
Sampling time (dt)
Wavelet used
Wave number (param)
Smallest scale of wavelet s0=%scale(1)
Largest scale of wavelet (%scale(jtot))
Spacing between discrete scales (dj)
Number of scales (jtot)
Number of observation after padding
Lag1 (background autocorrelation)
Significance level
Lower Scale for filter
Upper Scale for filter
Work array size (nk)
NINO
504
0.2500000000000000
Morlet
6.000000000000000
3.500000000000000
6027.545504538625
0.2500000000000000
44
1024
0.7200000000000000
5.000000000000004E-02
2.000000000000000
6.000000000000000
1024
=>
FILTER2=%RECON_Y$
=>
=>
CALL WAVELET(NINO :TYPE MORLET :SETTINGS :S0 4.5 :DT .25
:JTOT 44)$
Wavelet Option Settings
Input Variable
Number of Original Observations
Sampling time (dt)
Wavelet used
Wave number (param)
Smallest scale of wavelet s0=%scale(1)
Largest scale of wavelet (%scale(jtot))
Spacing between discrete scales (dj)
Number of scales (jtot)
Number of observation after padding
Lag1 (background autocorrelation)
Significance level
Lower Scale for filter
Upper Scale for filter
Work array size (nk)
NINO
504
0.2500000000000000
Morlet
6.000000000000000
4.500000000000000
7749.701362978233
0.2500000000000000
44
1024
0.7200000000000000
5.000000000000004E-02
2.000000000000000
6.000000000000000
1024
=>
FILTER3=%RECON_Y$
=>
CALL TABULATE(NINO,FILTER1,FILTER2,FILTER3)$
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
.
NINO
-0.1500
-0.3000
-0.1400
-0.4100
-0.4600
-0.6600
-0.5000
-0.8000
-0.9500
-0.7200
-0.3100
-0.7100
-1.040
-0.7700
-0.8600
-0.8400
-0.4100
-0.4900
-0.4800
-0.7200
-1.210
-0.8000
0.1600
0.4600
0.4000
1.000
2.170
2.500
2.340
0.8000
.
FILTER1
-0.1085
-0.1893
-0.2873
-0.3926
-0.4926
-0.5770
-0.6408
-0.6861
-0.7198
-0.7479
-0.7707
-0.7817
-0.7720
-0.7381
-0.6890
-0.6467
-0.6371
-0.6741
-0.7441
-0.8005
-0.7732
-0.5943
-0.2287
0.3012
0.9105
1.471
1.847
1.943
1.732
1.265
.
FILTER2
FILTER3
-0.1517
-0.1635
-0.2318
-0.2125
-0.3187
-0.2659
-0.4060
-0.3239
-0.4864
-0.3867
-0.5541
-0.4545
-0.6054
-0.5271
-0.6400
-0.6038
-0.6611
-0.6829
-0.6751
-0.7614
-0.6900
-0.8349
-0.7135
-0.8978
-0.7500
-0.9434
-0.7992
-0.9646
-0.8540
-0.9546
-0.9008
-0.9079
-0.9211
-0.8209
-0.8946
-0.6932
-0.8033
-0.5274
-0.6361
-0.3302
-0.3927
-0.1112
-0.8565E-01 0.1170
0.2599
0.3402
0.6087
0.5438
0.9208
0.7143
1.158
0.8407
1.290
0.9155
1.300
0.9355
1.191
0.9021
0.9806
0.8211
.
.
Spectral Analysis of Time Series
41
15.5 Use of Normalized Cumulative Periodogram to test for white noise
Box-Jenkins (1976, 294-298) and Box-Jenkins-Reinsel (2008, 347-350) suggest using the
normalized cumulative periodogram cˆ(i ) defined as
j
cˆ( j ) x
Pˆ ( )
i 1
x
Tsˆ 2
j
,
(15.5-1)
where Pˆ ( ) is defined in (15.1-25) and (15.1-27) and ŝ 2 is the estimated variance of the series, to
check for white noise. The advantage of using (15.5-1) in conjunction with inspection of the ACF is
that one can tell using the plot if the violation of the white noise assumption occurred due to more
low frequency than high frequency information (the plot is above the diagonal) or more high
frequency information than low frequency information (the plot was below the diagonal). Probability
limits for 99%, 95%, 90% and 75% are respectively 1.63, 1.36, 1.22 and 1.02. The below listed
program cperiod listed in Table 15.12 implements this test which is tested using the program
listed in Table 15.13.
Table 15.12 Calculating the Normalized Cumulative Periodogram
subroutine cperiod(x,name,c_period,c_p_freq,idrop);
/;
/; Normalized Cumulative Periodogram
/;
/; Box-Jenkins-Rensel (2008,347-350) suggests calculation of
/; cumulative Periodogram to test detect periodic nonrandomness
/;
/; See Jenkins and Watts (1968, 235)
/;
/; For significance of .95 and .75 lamda = 1.36 and 1.02
/;
.99 and .90 lamda = 1.63 and 1.22
/; band is +- lamda/sqrt(n/2)-1))
/;
/; Command built October 2009 by Houston H. Stokes
/;
/; x
= series to test
/; name = name of series
/; c_period = normalized cumulative periodogram
/; c_p_freq = frequency of normalized cumulative periodogram
/; idrop
= Number of c_period values to drop
/;
/; name of file is 'c_n_period.wmf'
/;
n=dfloat(norows(x));
varx=variance(x);
if(varx.le.0.0d+00)then;
call print('ERROR: Series has no variance':);
go to done;
Chapter 15
42
endif;
p =spectrum(x,freq2);
c_p_freq=freq2;
c_period=cusum(p)/(n*varx);
if(idrop.gt.0)then;
c_p_freq
=dropfirst(freq2,
idrop);
c_period=dropfirst(c_period,idrop);
endif;
diag=dfloat(integers(1,norows(c_period)));
diag=diag/dfloat(norows(c_period));
test=1./dsqrt(((n/2.) -1.));
upper99=diag+((1.63)*test);
lower99=diag-((1.63)*test);
upper95=diag+((1.36)*test);
lower95=diag-((1.36)*test);
/;
/; These bands can be added if desired
/;
/; upper90=diag+((1.22)*test);
/; lower90=diag-((1.22)*test);
upper75=diag+((1.02)*test);
lower75=diag-((1.02)*test);
call
call
call
call
character(cc,'Cumulative Periodogram of ');
character(cc2,name);
ialen(cc2,ii);
expand(cc,cc2,27,27+ii);
call graph(c_p_freq,c_period,diag, upper99 lower99 upper95, lower95
upper75 lower75
:heading cc :pgborder :nocontact :nolabel :plottype xyplot
:file 'n_c_period.wmf');
done continue;
return;
end;
Table 15.13 Testing series for white noise using the Normalized Cumulative Periodogram
b34sexec options ginclude('gas.b34'); b34srun;
/$
/$ Job tests c_period Command
/$
b34sexec matrix;
call loaddata;
call load(cperiod);
idrop=0;
call cperiod(gasout,'gasout',c_period,c_p_freq,idrop);
call dodos('rename n_c_period.wmf fig15.7.wmf' :);
Spectral Analysis of Time Series
43
x=rn(gasout);
call cperiod(x,
'ran()',c_period,c_p_freq, idrop);
call dodos('rename n_c_period.wmf fig15.8.wmf' :);
b34srun$
Figure 15.7 for the gasout series clearly shows that there is a preponderance of low
frequency information making it possible to accept ta the 99% level that the series is not
white noise. Figure 15.8 shows the plot for a white noise series.
Cumulative Periodogram of gasout
1
.8
CDULULUL
_IPOPOPO
PAPWPWPW
EGEEEEEE
R RRRRRR
I 999977
O 995555
D
.6
.4
.2
0
0
.5
1
1.5
C_P_FREQ
2
Figure 15.7 Normalized Cumulative Periodogram for gasout series
2.5
3
Chapter 15
44
Cumulative Periodogram of ran()
1
.8
CDULULUL
_IPOPOPO
PAPWPWPW
EGEEEEEE
R RRRRRR
I 999977
O 995555
D
.6
.4
.2
0
0
.5
1
1.5
C_P_FREQ
2
2.5
3
Figure 15.8 Normalized Cumulative Periodogram for white noise series
15.6 Forecasting using spectral methods
Table 15.14 lists a matrix command subroutine that uses spectral methods to construct
forecasts. Unlike a RATS command of the same name, no attempt is made to smooth the FFT.
Spectral Analysis of Time Series
Table 15.14 A Subroutine to Calculate Forecasts using the FFT of a series
subroutine specfore(data,startf,numf,detrend,forecast,obs,error,actual);
/;
/; Forecast with spectral methods.
/;
/; Based on code developed by Michael Hunstad using
/; regression methods to partially reverse-engineer the RATS
/; specfore command. An improved version of the
/; Hunsted Matlab code is in c:\b34slm\mfiles as
/; specfore.m
/;
/; This implementation by Houston H. Stokes uses
/; a FFT to save space and reduce CPU use. Added capability is
/; provided Unlike the RATS implementation of this technique,
/; the current implementation does not smooth the FFT
/;
/; Rats smooths the data
/;
/; data
=> series to forecast. # of obs = n
/; startf
=> last period before start forecasting
/; numf
=> number of forecasts
/; detrend => Detrend the data if gt 0. =2 print trend OLS Model
/; forecast => Forecast
/; obs
=> Observation number associated with forecast
/; error
=> Defined if startf lt n
/; actual
=> defined if startf lt n
/;
/; Routine developed 8 December 2009 by Houston H. Stokes
/;
nobs=norows(data);
error=missing();
actual=missing();
if(nobs .lt.startf)then;
call epprint('ERROR: In call specfore startf not le nobs of data':);
call epprint('
nobs of data was
',nobs:);
call epprint('
startf was
',startf:);
go to endit;
endif;
series=data(integers(1,startf));
seriesm=mean(series);
series=series-seriesm;
if(detrend.ne.0)then;
trend=dfloat(integers(startf));
if(detrend.eq.1)call olsq(series trend :qr
if(detrend.eq.2)call olsq(series trend :qr :print);
if(klass(series).eq.5)series=series-afam(%yhat);
if(klass(series).eq.1)series=series-vfam(%yhat);
tcoef=%coef(1);
tmean=%coef(2);
endif;
obs=dfloat(integers(startf+1,startf+numf));
45
46
Chapter 15
call spectral(series,sinx,cosx,px,sx,freq);
beta=vfam(catrow(cosx,sinx));
/; zero out cosx for highest freq
ijunk=norows(cosx);
beta(ijunk,1)=0.0;
forecast=vector(numf:);
tt=dfloat(integers(0,numf-1))+dfloat(startf);
do i=1,numf;
c1=cos(afam(freq)*afam(tt(i)));
s1=sin(afam(freq)*afam(tt(i)));
cc=vfam(catrow(c1,s1));
if(detrend.eq.0)forecast(i)=transpose(beta)*cc + sfam(seriesm);
if(detrend.ne.0)forecast(i)=transpose(beta)*cc + sfam(seriesm)
+tmean + (dfloat(startf+i)*tcoef);
enddo;
if(startf.lt.nobs)then;
iend=dmin1(startf+numf,nobs);
nn2 =integers(iend-startf);
actual = data(integers(startf+1,iend));
if(abs(klass(actual)).eq.1)error = actual - forecast(nn2);
if(abs(klass(actual)).eq.5)error = actual - afam(forecast(nn2));
endif;
endit continue;
return;
end;
Table 15.15 shows the use of the specfore command in B34S and RATS. Note the effect of the
trend correction. The MATLAB implementation uses OLS in the place of the FFT and does not
provide the options of trend correction. The advantage of the OLS implementation to get the sine
and cosine coefficients is transparency. The disadvantage is the added computing costs in both
CPU time and space.
Spectral Analysis of Time Series
Table 15.15 Using the FFT to Forecast
%b34slet domatlab = 0;
%b34slet dorats
= 0;
%b34slet dob34s1 = 1;
%b34slet file1="'_b34sdat.dat'"$
%b34slet file2="'b34sdata.m'"$
b34sexec options ginclude('b34sdata.mac') member(lydiapnm);
b34srun;
/$ user places RATS commands between
/$
PGMCARDS$
/$
note: user RATS commands here
/$
B34SRETURN$
/$
%b34sif(&dob34s1.ne.0)%then;
b34sexec matrix;
call echooff;
call loaddata;
call load(specfore);
call print(' ':);
call print('Forecast of sales and Advertising':);
nfor=30;
base=60;
call
call
call
call
specfore(sales,
base,nfor,0,fsales1,obs,error1,actual1);
specfore(sales,
base,nfor,2,fsales2,obs,error2,actual2);
specfore(advertis,base,nfor,0,fadd1,obs,error3,actual3);
specfore(advertis,base,nfor,2,fadd2,obs,error4,actual4);
call print(' ':);
call print('With out Trend Correction':);
call tabulate(obs,actual1,fsales1,error1,actual3,fadd1,error3);
call print('With Trend Correction':);
call tabulate(obs,actual2,fsales2,error2,actual4,fadd2,error4);
nn=integers(norows(actual1));
obs
=
obs(nn);
fsales1=fsales1(nn);
fsales2=fsales2(nn);
nn=integers(norows(actual3));
fadd1 =fadd1(nn);
fadd2 =fadd2(nn);
call tabulate(obs actual1,fsales1 fsales2 );
call graph(obs actual1,fsales1 fsales2 :plottype xyplot
:heading 'Sales Forecast out of sample # 2 with trend'
:nolabel :nocontact :pgborder);
call graph(obs actual3 fadd1 fadd2 :plottype xyplot
:heading 'Advertis Forecast out of sample notrend # 2 with trend'
:nolabel :nocontact :pgborder);
47
Chapter 15
48
cc1=ccf(fsales1,actual1);
cc2=ccf(fsales2,actual2);
cc3=ccf(fadd1,actual3);
cc4=ccf(fadd2,actual4);
ss1=sumsq(error1);
ss2=sumsq(error2);
ss3=sumsq(error3);
ss4=sumsq(error4);
call
call
call
call
call
call
call
call
call
print(' ':);
print('Out of
print('Out of
print('Out of
print('Out of
print('Out of
print('Out of
print('Out of
print('Out of
sample
sample
sample
sample
sample
sample
sample
sample
sales
no
trend
sales
with trend
advertis no
trend
sales
with trend
sales forecast no
sales forecast with
adver forecast no
adver forecast with
sumsq
sumsq
sumsq
sumsq
trend
trend
trend
trend
correlation
correlation
correlation
correlation
',ss1:);
',ss2:);
',ss3:);
',ss4:);
',cc1:);
',cc2:);
',cc3:);
',cc4:);
b34srun;
%b34sendif;
%b34sif(&dorats.ne.0)%then;
B34SEXEC OPTIONS OPEN('rats.dat') UNIT(28) DISP=UNKNOWN$ B34SRUN$
B34SEXEC OPTIONS OPEN('rats.in') UNIT(29) DISP=UNKNOWN$ B34SRUN$
B34SEXEC OPTIONS CLEAN(28)$ B34SRUN$
B34SEXEC OPTIONS CLEAN(29)$ B34SRUN$
B34SEXEC PGMCALL$
RATS PASSASTS
PCOMMENTS('* ',
'* Data passed from B34S(r) system to RATS',
'* ') $
PGMCARDS$
*
* see section 7.5 in RATS manual
*
Source(NOECHO) d:\R\specfore.src
* SOURCE(ECHO) d:\R\specfore.src
SET
SERIES = sales
COMPUTE istart
= 60
COMPUTE iend
= 90
*
* @SPECFORE( options ) series start end forecasts
* Computes forecasts using spectral techniques
*
* Parameters:
*
series
: (input) Series to be forecast
*
start end : Range of entries to forecast
*
forecasts
: (output) Series for computed forecasts
*
@SPECFORE(DIFFS=0,SDIFFS=0,TRANS=NONE,CONSTANT) SERIES ISTART IEND FORE
Spectral Analysis of Time Series
SET ERROR = SERIES - FORE
PRINT istart iend SERIES FORE ERROR
B34SRETURN$
B34SRUN $
B34SEXEC OPTIONS CLOSE(28)$ B34SRUN$
B34SEXEC OPTIONS CLOSE(29)$ B34SRUN$
B34SEXEC OPTIONS
/$
dodos('start /w /r rats386 rats.in rats.out ')
dodos('start /w /r rats32s rats.in /run')
dounix('rats
rats.in rats.out')$ B34SRUN$
B34SEXEC OPTIONS NPAGEOUT
WRITEOUT('Output from RATS',' ',' ')
COPYFOUT('rats.out')
dodos('ERASE rats.in','ERASE rats.out','ERASE rats.dat')
dounix('rm
rats.in','rm
rats.out','rm
rats.dat') $
B34SRUN$
%b34sendif;
%b34sif(&domatlab.ne.0)%then;
/$
/$ Builds a MATLAB input file for MATLAB version 6.
/$ Changes made 2 February 2002
/$
/$ Since MATLAB is case sensitive, use lower case for all variable
/$ references that are from b34s. MATLAB users upper case for a matrix
/$ variable
/$
/$ This job assumes user has already loaded data in B34S
/$ The file name for file1 is hard coded in the matlab m file (file2)
/$
/$ User changes this to default matlab file directory
/$
/$
/$ Job runs on linux matlab and windows matlab
/$
/$ When job ends, output will be seen in b34s.out file
/$
/$ User loads data here if it has not occured already
/$
b34sexec options open(%b34seval(&file1)) unit(28) disp=unknown$
b34seend$
b34sexec options clean(28)$ b34seend$
b34sexec options open(%b34seval(&file2)) unit(29) disp=unknown$
b34seend$
b34sexec options clean(29)$ b34seend$
b34sexec pgmcall$
matlab lowercase outfile(%b34seval(&file1))$
pgmcards$
% User MATLAB commands here such as plot(varname)
% x1=test(sales,60,2,1,1);
x1=specfore(sales,60,10,1,1);
% quit is needed since have to get out of matlab automatically
49
Chapter 15
50
% Comment to stay in matlab and see plot
b34sreturn$
b34seend$
b34sexec options close(28)$ b34srun$
b34sexec options close(29)$ b34srun$
b34sexec options
dodos('start /w /r matlab /r b34sdata /logfile matlab.out')
dounix('matlab < b34sdata.m > matlab.out');
b34srun;
b34sexec options writeout('
',
'Output from Matlab ',
'
'); b34srun;
b34sexec options copyfout('matlab.out'); b34srun;
b34sexec options dodos('erase matlab.out')
dounix('rm
matlab.out');
b34srun;
%b34sendif;
When run this job produces
Variable
SALES
ADVERTIS
CONSTANT
# Cases
1
2
3
78
78
78
Mean
Std Deviation
1278.692308
619.3974359
1.000000000
196.6126330
433.9763696
0.000000000
Variance
Number of observations in data file
78
Current missing variable code
1.000000000000000E+31
Data begins on (D:M:Y) 1: 1:1954 ends 1: 6:1960.
Frequency is
B34S(r) Matrix Command. d/m/y
=>
Maximum
38656.52747
188335.4893
0.000000000
1728.000000
1388.000000
1.000000000
12
8/12/09. h:m:s 16:17:39.
CALL ECHOOFF$
Forecast of sales and Advertising
With out Trend Correction
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
OBS
61.00
62.00
63.00
64.00
65.00
66.00
67.00
68.00
69.00
70.00
71.00
72.00
73.00
74.00
75.00
76.00
77.00
78.00
ACTUAL1
1052.
1102.
1355.
1323.
1296.
1127.
1170.
1059.
1116.
1214.
966.0
1089.
814.0
1087.
1180.
1167.
1210.
1092.
FSALES1
1297.
1316.
1730.
1537.
1326.
1262.
1171.
1477.
1633.
1544.
1461.
1085.
1173.
1404.
1621.
1506.
1523.
1339.
ERROR1
-244.8
-214.2
-374.8
-214.2
-29.78
-135.2
-0.7833
-418.2
-516.8
-330.2
-494.8
3.783
-358.8
-317.2
-440.8
-339.2
-312.8
-247.2
ACTUAL3
838.0
994.0
1020.
865.0
819.0
83.00
56.00
224.0
881.0
436.0
160.0
68.00
749.0
857.0
898.0
705.0
489.0
59.00
FADD1
1245.
1385.
946.6
954.4
51.57
74.43
36.57
502.4
1135.
952.4
665.6
163.4
978.6
1309.
1353.
1106.
501.6
158.4
ERROR3
-406.6
-391.4
73.43
-89.43
767.4
8.567
19.43
-278.4
-253.6
-516.4
-505.6
-95.43
-229.6
-452.4
-454.6
-401.4
-12.57
-99.43
With Trend Correction
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
OBS
61.00
62.00
63.00
64.00
65.00
66.00
67.00
68.00
69.00
70.00
71.00
72.00
73.00
74.00
75.00
76.00
77.00
ACTUAL2
1052.
1102.
1355.
1323.
1296.
1127.
1170.
1059.
1116.
1214.
966.0
1089.
814.0
1087.
1180.
1167.
1210.
FSALES2
1028.
1043.
1461.
1264.
1057.
988.7
901.8
1204.
1364.
1271.
1192.
811.7
903.8
1131.
1352.
1233.
1254.
ERROR2
24.21
59.30
-105.8
59.30
239.2
138.3
268.2
-144.7
-247.8
-56.70
-225.8
277.3
-89.79
-43.70
-171.8
-65.70
-43.79
ACTUAL4
838.0
994.0
1020.
865.0
819.0
83.00
56.00
224.0
881.0
436.0
160.0
68.00
749.0
857.0
898.0
705.0
489.0
FADD2
948.2
1084.
650.2
653.0
-244.8
-227.0
-259.8
201.0
838.2
651.0
369.2
-138.0
682.2
1008.
1056.
805.0
205.2
ERROR4
-110.2
-90.04
369.8
212.0
1064.
310.0
315.8
22.96
42.84
-215.0
-209.2
206.0
66.84
-151.0
-158.2
-100.0
283.8
Minimum
772.0000000
39.00000000
1.000000000
Spectral Analysis of Time Series
18
19
20
21
22
23
24
25
26
27
28
29
30
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Out
Out
Out
Out
Out
Out
Out
Out
of
of
of
of
of
of
of
of
78.00
79.00
80.00
81.00
82.00
83.00
84.00
85.00
86.00
87.00
88.00
89.00
90.00
OBS
61.00
62.00
63.00
64.00
65.00
66.00
67.00
68.00
69.00
70.00
71.00
72.00
73.00
74.00
75.00
76.00
77.00
78.00
sample
sample
sample
sample
sample
sample
sample
sample
1092.
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
1066.
979.8
986.7
1152.
1283.
954.8
777.7
974.8
1086.
1393.
1442.
1104.
1018.
ACTUAL1
1052.
1102.
1355.
1323.
1296.
1127.
1170.
1059.
1116.
1214.
966.0
1089.
814.0
1087.
1180.
1167.
1210.
1092.
FSALES1
1297.
1316.
1730.
1537.
1326.
1262.
1171.
1477.
1633.
1544.
1461.
1085.
1173.
1404.
1621.
1506.
1523.
1339.
sales
no
trend
sales
with trend
advertis no
trend
sales
with trend
sales forecast no
sales forecast with
adver forecast no
adver forecast with
sumsq
sumsq
sumsq
sumsq
trend
trend
trend
trend
26.30
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
59.00
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
-143.0
-271.8
85.04
729.2
525.0
-193.8
-189.0
668.2
916.0
893.2
670.0
293.2
-206.0
51
202.0
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
FSALES2
1028.
1043.
1461.
1264.
1057.
988.7
901.8
1204.
1364.
1271.
1192.
811.7
903.8
1131.
1352.
1233.
1254.
1066.
correlation
correlation
correlation
correlation
1804827.578333335
426933.1095972446
2229763.246666659
1847970.135394790
0.5039421385465304
0.5023270654992271
-9.283852553158210E-02
-9.305692521339150E-02
B34S Matrix Command Ending. Last Command reached.
Space available in allocator
Number variables used
Number temp variables used
B34S 8.11D
(D:M:Y)
8856637, peak space used
100, peak number used
2228, # user temp clean
8/12/09 (H:M:S) 16:17:43
9740
108
0
PGMCALL STEP
Output from RATS
*
* Data passed from B34S(r) system to RATS
*
CALENDAR
1954
1
12
ALLOCATE
78
OPEN DATA rats.dat
DATA(FORMAT=FREE,ORG=OBS,
$
MISSING=
0.1000000000000000E+32
) / $
SALES
$
ADVERTIS
$
CONSTANT
SET TREND = T
TABLE
Series
Obs
Mean
Std Error
SALES
78 1278.69230769 196.61263304
ADVERTIS
78 619.39743590 433.97636957
TREND
78
39.50000000
22.66053839
Minimum
Maximum
772.00000000 1728.00000000
39.00000000 1388.00000000
1.00000000
78.00000000
*
* see section 7.5 in RATS manual
*
Source(NOECHO) d:\R\specfore.src
* SOURCE(ECHO) d:\R\specfore.src
SET
SERIES = sales
COMPUTE istart
= 60
COMPUTE iend
= 90
*
* @SPECFORE( options ) series start end forecasts
* Computes forecasts using spectral techniques
*
* Parameters:
*
series
: (input) Series to be forecast
*
start end : Range of entries to forecast
*
forecasts
: (output) Series for computed forecasts
*
@SPECFORE(DIFFS=0,SDIFFS=0,TRANS=NONE,CONSTANT) SERIES ISTART IEND FORE
SET ERROR = SERIES - FORE
PRINT istart iend SERIES FORE ERROR
Lydia Pinkham Monthly Data
PAGE
2
Chapter 15
52
ENTRY
1958:12
1959:01
1959:02
1959:03
1959:04
1959:05
1959:06
1959:07
1959:08
1959:09
1959:10
SERIES
1072
1052
1102
1355
1323
1296
1127
1170
1059
1116
1214
FORE
1240.460004817
1200.763596247
1235.408547587
1352.704220311
1358.394344689
1320.524022842
1280.304763050
1287.241992611
1305.249870237
1326.211286902
1360.412853130
ERROR
-168.4600048169
-148.7635962471
-133.4085475874
2.2957796894
-35.3943446892
-24.5240228422
-153.3047630499
-117.2419926114
-246.2498702365
-210.2112869018
-146.4128531300
Spectral Analysis of Time Series
B34S 8.11D
1959:11
1959:12
1960:01
1960:02
1960:03
1960:04
1960:05
1960:06
1960:07
1960:08
1960:09
1960:10
1960:11
1960:12
1961:01
1961:02
1961:03
1961:04
1961:05
1961:06
(D:M:Y)
966
1089
814
1087
1180
1167
1210
1092
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
8/12/09 (H:M:S) 16:17:43
1300.166743214
1302.188498103
1323.426262875
1330.313780233
1354.704626186
1320.833749223
1303.224404989
1310.609988566
1330.479685398
1337.848002486
1318.335043563
1317.479168750
1317.880136638
1316.633819117
1335.905858936
1328.465844195
1323.632170191
1333.070850317
1322.175915526
1344.757186664
PGMCALL STEP
Lydia Pinkham Monthly Data
53
PAGE
3
-334.1667432144
-213.1884981030
-509.4262628748
-243.3137802326
-174.7046261861
-153.8337492228
-93.2244049893
-218.6099885658
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
Note the out-of-sample correlation of .5039 and .5023 for sales forecasts made without and
with a trend correction. The advertis forecasts, on the other hand were not good, presumably
due to less structure in the underlying series. Experimentation with the spectral forecasting
approach suggests that it is a valuable tool but cannot be used blindly. Unlike an ARIMA model
which eventually will converge to the expected value of the series, this is not the case with a
spectral forecast.
15.7 Conclusion
After a brief survey of spectral analysis theory, the periodogram of two simple AR
models was graphed. Using the gas furnace data the OLS approach to spectral analysis was
contrasted with the frequency domain approach. The matrix command implementation was
shown to be an easily customizable way to used spectral tools to explore the dynamics of a series.
In Stokes (200x) use of the fast fourier transform to filter series is illustrated. Some simple
examples using the FILTER subroutine are presented in Chapter 16 of this book. Wavelet
analysis, that provides a way to study a series by both time and frequency are discussed and a
number of examples shown that analyze the Nino water temperature data. At issue is the
appropriate period to use for the analysis. If noise in the data is assumed to be of short duration,
with wavelet analysis it is possible to remove this short duration information to allow analysis to
proceed with a smoothed series that more accurately reflects the underlying process. Finally
forecasting a series using FFT methods is illustrated.
