CSSWP
Estimates the nonnormalized cross‑spectral density of two stationary time series using a spectral window given the spectral densities and cross periodogram.
Required Arguments
N — Number of observations in each of the appropriately centered and padded time series X and Y. (Input)
N must be greater than or equal to two.
SX — Vector of length NF containing the estimate of the spectral density of the first time series X. (Input)
SY — Vector of length NF containing the estimate of the spectral density of the second time series Y. (Input)
CPREAL — Vector of length ⌊N/2⌋ + 1 containing the real part of the cross periodogram between X and Y. (Input)
The real part of the cross periodogram evaluated at (angular) frequency wk = 2πk/N is given by CPREAL(k + 1), k = 0, 1, …, ⌊N/2⌋.
CPIMAG — Vector of length ⌊N/2⌋ + 1 containing the imaginary part of the cross periodogram between X and Y. (Input)
The imaginary part of the cross periodogram evaluated at (angular) frequency wk = 2πk/N is given by CPIMAG(k + 1), k = 0, 1, …, ⌊N/2⌋.
F — Vector of length NF containing the (angular) frequencies at which the spectral and cross‑spectral densities are estimated. (Input)
ISWVER — Option for version of the spectral window. (Input)
SWVER | Action |
|---|
1 | Modified Bartlett |
2 | Daniell |
3 | Tukey‑Hamming |
4 | Tukey‑Hanning |
5 | Parzen |
6 | Bartlett‑Priestley |
Refer to the “Description” section for further details.
M — Spectral window parameter. (Input)
M must be greater than or equal to one and less than N. For the Parzen spectral window (ISWVER = 5), the spectral window parameter M must be even.
COSPEC — Vector of length NF containing the estimate of the cospectrum. (Output)
QUADRA — Vector of length NF containing the estimate of the quadrature spectrum. (Output)
CRAMPL — Vector of length NF containing the estimate of the cross‑amplitude spectrum. (Output)
PHASE — Vector of length NF containing the estimate of the phase spectrum. (Output)
COHERE — Vector of length NF containing the estimate of the coherence or squared coherency. (Output)
Optional Arguments
NF — Number of (angular) frequencies. (Input)
NF must be greater than or equal to one. Default: NF = size (F,1).
FORTRAN 90 Interface
Generic: CALL CSSWP (N, SX, SY, CPREAL, CPIMAG, F, ISWVER, M, COSPEC, QUADRA, CRAMPL, PHASE, COHERE [, …])
Specific: The specific interface names are S_CSSWP and D_CSSWP.
FORTRAN 77 Interface
Single: CALL CSSWP (N, SX, SY, CPREAL, CPIMAG, NF, F, ISWVER, M, COSPEC, QUADRA, CRAMPL, PHASE, COHERE)
Double: The double precision name is DCSSWP.
Description
Routine CSSWP estimates the nonnormalized cross‑spectral density function of two jointly stationary time series using a spectral window given the modified cross‑periodogram and spectral densities of the appropriately centered and padded data
for t = 1, …, N.
The routine
CPFFT may be used to compute the modified periodograms
and cross periodogram
over the discrete set of nonnegative frequencies
(Here,
⌊a⌋ means the greatest integer less than or equal to
a.) Either routine
SSWP or routine
SWEP may be applied to the periodograms to obtain nonnormalized spectral density estimates
over the set of frequencies
ω = fi, i = 1, …, nƒ
where nƒ = NF. These frequencies are in the scale of radians per unit time. The time sampling interval Δt is assumed to be equal to one. Note that the spectral window or weight sequence used to compute
may differ from that used to compute
The cross‑spectral density function is complex‑valued in general and may be written as
The cospectrum is estimated by
and the quadrature spectrum is estimated by
where the spectral window Wn(θ) is specified by argument ISWVER. The following spectral windows Wn(θ) are available.
Modified Bartlett
where FM(θ) corresponds to the Fejér kernel of order M.
Daniell
Tukey
where DM(θ) represents the Dirichlet kernel. The Tukey‑Hamming window is obtained when
a = 0.23 and the Tukey‑Hanning window is obtained when a = 0.25.
Parzen
where M is even. If M is odd, then M + 1 is used instead of M in the above formula.
Bartlett-Priestley
Only one window parameter M may be specified so that only one estimate of hXY (ω) is computed. The nonnormalized cross‑spectral density estimate is computed over the same set of frequencies as the nonnormalized spectral density estimates discussed above. However, the particular spectral window used to compute
need not correspond to either the spectral window or the weight sequence used to compute either
An equivalent representation of hXY(ω) is the polar form defined by
The cross-amplitude spectrum is estimated by
and the phase spectrum is estimated by
Finally, the coherency spectrum is estimated by
The coherence or squared coherency is output.
Comments
1. The periodograms of
X and
Y and cross periodogram between
X and
Y may be computed using the routine
CPFFT. The spectral densities of
X and
Y may then be estimated using any of the routines
SSWD,
SWED,
SSWP, or
SWEP. Thus, different window types and/or weight sequences may be used to estimate the spectral and cross‑spectral densities given either the series or their periodograms. Note that use of the modified periodograms and modified cross periodogram ensures that the scale of the spectral and cross‑spectral densities and their estimates is equivalent.
2 The time sampling interval, TINT, is assumed to be equal to one. This assumption is appropriate for discrete parameter processes. The adjustment for continuous parameter processes (TINT > 0.0) involves multiplication of the frequency vector F by 1/TINT and multiplication of the spectral and cross‑spectral density estimates by TINT.
3. To convert the frequency scale from radians per unit time to cycles per unit time, multiply F by 1/(2π).
Example
Consider the Robinson Multichannel Time Series Data (Robinson 1967, page 204) where X is the Wölfer sunspot number and Y is the northern light activity for the years 1770 through 1869. Application of routine CSSWP to these data produces the following results.
USE IMSL_LIBRARIES
IMPLICIT NONE
INTEGER LDCPM, LDCSM, LDRDAT, N, NDRDAT, NF, NM, &
NOBS, NPAD
PARAMETER (LDRDAT=100, NDRDAT=4, NF=10, NM=2, &
NOBS=100, LDCSM=NF, NPAD=NOBS-1, N=NOBS+NPAD, &
LDCPM=N/2+1)
!
INTEGER I, IPVER, ISWVER, J, JPT, JST, M(NM), NRCOL, NRROW
REAL COHERE(NF), COSPEC(NF), CPIMAG(LDCPM), &
CPM(LDCPM,10), CPREAL(LDCPM), CRAMPL(NF), &
CSM(LDCSM,7*NM+2), F(NF), FLOAT, P(NF), PHASE(NF), &
PI, PX(LDCPM), PY(LDCPM), QUADRA(NF), &
RDATA(LDRDAT,NDRDAT), SX(NF), SY(NF), X(NOBS), Y(NOBS)
CHARACTER CLABEL1(3)*9, CLABEL2(6)*16, FMT*8, RLABEL(1)*6,&
TITLE*80
INTRINSIC FLOAT
!
EQUIVALENCE (X(1), RDATA(1,2)), (Y(1), RDATA(1,3))
EQUIVALENCE (PX(1), CPM(1,3)), (PY(1), CPM(1,6))
EQUIVALENCE (CPREAL(1), CPM(1,9)), (CPIMAG(1), CPM(1,10))
EQUIVALENCE (CSM(1,1), F(1)), (CSM(1,2), P(1))
!
DATA FMT/'(F12.4)'/
DATA CLABEL1/' k', 'Frequency', 'Period'/
DATA CLABEL2/'%/ k', '%/Cospectrum', '%/Quadrature',&
'Cross%/Amplitude', '%/Phase', '%/Coherence'/
DATA RLABEL/'NUMBER'/
! Initialization
PI = 2.0*ASIN(1.0)
DO 10 I=1, NF
F(I) = PI*FLOAT(I)/FLOAT(NF)
P(I) = 2.0*FLOAT(NF)/FLOAT(I)
10 CONTINUE
! Robinson Data
CALL GDATA (8, RDATA, NRROW, NRCOL)
! Center on arithmetic means
! Frequency in radians per unit time
! Modified periodogram version
IPVER = 1
! Compute cross periodogram
CALL CPFFT (X, Y, CPM, IPVER=IPVER)
! Spectral window parameters
M(1) = 10
M(2) = 30
! Compute cross-spectral density
! using the Parzen window
!
! Print frequency and period
TITLE = 'Cross-Spectral Analysis Using Parzen Window'
CALL WRRRL (TITLE, CSM, RLABEL, CLABEL1, NF, 2, FMT=FMT)
ISWVER = 5
DO 20 J=1, NM
! Estimate the spectral densities
CALL SSWP (N, PX, F, M(J), SX, ISWVER=ISWVER)
CALL SSWP (N, PY, F, M(J), SY, ISWVER=ISWVER)
! Estimate the cross-spectral density
CALL CSSWP (N, SX, SY, CPREAL, CPIMAG, F, ISWVER, M(J), &
COSPEC, QUADRA, CRAMPL, PHASE, COHERE)
! Copy results to output matrices
JPT = 7*(J-1) + 2
JST = 7*(J-1) + 5
CALL SCOPY (NF, SX, 1, CSM(1:,JPT+1), 1)
CALL SCOPY (NF, SY, 1, CSM(1:,JPT+2), 1)
CALL SCOPY (NF, COSPEC, 1, CSM(1:,JPT+3), 1)
CALL SCOPY (NF, QUADRA, 1, CSM(1:,JPT+4), 1)
CALL SCOPY (NF, CRAMPL, 1, CSM(1:,JPT+5), 1)
CALL SCOPY (NF, PHASE, 1, CSM(1:,JPT+6), 1)
CALL SCOPY (NF, COHERE, 1, CSM(1:,JPT+7), 1)
! Print results
TITLE = '%/Results of the Cross-Spectral Analysis With '// &
'Spectral Window Parameter M = '
WRITE (TITLE(77:78),'(I2)') M(J)
CALL WRRRL (TITLE, CSM(1:,JST), RLABEL, CLABEL2, NF, 5, FMT=FMT)
20 CONTINUE
!
END
Output
Cross-Spectral Analysis Using Parzen Window
k Frequency Period
1 0.3142 20.0000
2 0.6283 10.0000
3 0.9425 6.6667
4 1.2566 5.0000
5 1.5708 4.0000
6 1.8850 3.3333
7 2.1991 2.8571
8 2.5133 2.5000
9 2.8274 2.2222
10 3.1416 2.0000
Results of the Cross-Spectral Analysis With Spectral Window Parameter M = 10
Cross
k Cospectrum Quadrature Amplitude Phase Coherence
1 463.5888 -65.9763 468.2600 0.1414 0.2570
2 286.5450 -75.0209 296.2029 0.2561 0.1710
3 150.1073 -57.8263 160.8604 0.3677 0.1438
4 52.9840 -32.3642 62.0866 0.5483 0.0998
5 21.5435 -15.0888 26.3020 0.6110 0.0794
6 21.4228 -9.8188 23.5658 0.4298 0.1716
7 15.7005 -5.3704 16.5936 0.3296 0.2112
8 8.0118 -1.8887 8.2314 0.2315 0.1272
9 2.7682 0.2007 2.7754 -0.0724 0.0446
10 0.5777 0.1008 0.5864 -0.1727 0.0091
Results of the Cross-Spectral Analysis With Spectral Window Parameter M = 30
Cross
k Cospectrum Quadrature Amplitude Phase Coherence
1 169.7542 -193.4384 257.3615 0.8505 0.1620
2 452.6187 32.3813 453.7755 -0.0714 0.2213
3 94.5221 -90.8159 131.0800 0.7654 0.2629
4 -0.2096 -6.1127 6.1163 1.6051 0.0019
5 27.4711 -22.1946 35.3166 0.6796 0.2492
6 29.1329 -4.0128 29.4080 0.1369 0.3170
7 11.2058 -9.3403 14.5881 0.6948 0.2594
8 8.0017 0.8813 8.0501 -0.1097 0.1928
9 -0.4199 2.2893 2.3275 -1.7522 0.0468
10 0.5570 -1.0767 1.2123 1.0934 0.0678
