CSWEP

Estimates the nonnormalized cross‑spectral density of two stationary time series using a weighted cross periodogram 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 ωk = 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 ωk = 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 density is estimated. (Input)

WT — Vector of length NWT containing the weights used to smooth the periodogram. (Input)
The actual weights are the values in WT normalized to sum to 1 with the current periodogram ordinate taking the middle weight for NWT odd or the weight to the right of the middle for NWT 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. (Output)

Optional Arguments

NF — Number of (angular) frequencies. (Input)
NF must be greater than or equal to one.
Default: NF = size (F,1).

NWT — Number of weights. (Input)
NWT must be greater than or equal to one.
Default: NWT = size (WT,1).

FORTRAN 90 Interface

Generic: CALL CSWEP (N, SX, SY, CPREAL, CPIMAG, F, WT, COSPEC, QUADRA, CRAMPL, PHASE, COHERE [, …])

Specific: The specific interface names are S_CSWEP and D_CSWEP.

FORTRAN 77 Interface

Single: CALL CSWEP (N, SX, SY, CPREAL, CPIMAG, NF, F, NWT, WT, COSPEC, QUADRA, CRAMPL, PHASE, COHERE)

Double: The double precision name is DCSWEP.

Description

Routine CSWEP estimates the nonnormalized cross‑spectral density function of two jointly stationary time series using a fixed sequence of weights given the modified cross‑periodogram and spectral densities of the appropriately centered and padded data

 

and

 

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 = ƒi,      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

 

and k(ω) is the integer such that ωk,0 is closest to ω. The sequence of m = NWT weights {wj} for j = ‑⌊m/2⌋, …, (m ‑ ⌊m/2⌋ ‑ 1) satisfies Σjwj = 1. These weights are fixed in the sense that they do not depend on the frequency ω at which to estimate hXY(ω). Usually, m is odd with the weights symmetric about the middle weight w0. If m is even, the weight to the right of the middle is considered w0. The argument WT may contain relative weights since they are normalized to sum to one in the actual computations. The nonnormalized cross‑spectral density estimate is computed over the same set of frequencies as the nonnormalized spectral density estimates. However, the particular weight sequence used to compute

 

need not correspond to either the weight sequence or spectral window 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 via 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 scales of the spectral and cross‑spectral densities and their estimates are 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 multipication 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 CSWEP to these data produces the following results.

 

USE IMSL_LIBRARIES

 

IMPLICIT NONE

INTEGER LDCPM, LDCSM, LDRDAT, N, NDRDAT, NF, NOBS, &

NPAD, NWT

PARAMETER (LDRDAT=100, NDRDAT=4, NF=10, NOBS=100, &

NWT=7, LDCSM=NF, NPAD=NOBS-1, N=NOBS+NPAD, &

LDCPM=N/2+1)

!

INTEGER I, IPVER, NROW, NVAR

REAL COHERE(NF), COSPEC(NF), CPIMAG(LDCPM), &

CPM(LDCPM,10), CPREAL(LDCPM), CRAMPL(NF), &

CSM(LDCSM,9), F(NF), FLOAT, PHASE(NF), PI, PX(LDCPM), &

PY(LDCPM), QUADRA(NF), RDATA(LDRDAT,NDRDAT), &

SX(NF), SY(NF), WT(NWT), X(NOBS), Y(NOBS)

CHARACTER CLABEL1(5)*24, CLABEL2(6)*16, FMT*8, RLABEL(1)*6, &

TITLE1*32, TITLE2*40

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))

!

DATA WT/1.0, 2.0, 3.0, 4.0, 3.0, 2.0, 1.0/

DATA FMT/'(F12.4)'/

DATA CLABEL1/'%/%/ k', '%/%/Frequency', '%/%/Period', &

'Spectral%/Estimate%/of X', 'Spectral%/Estimate%/of Y'/

DATA CLABEL2/'%/ k', '%/Cospectrum', '%/Quadrature', &

'Cross%/Amplitude', '%/Phase', '%/Coherence'/

DATA RLABEL/'NUMBER'/

DATA TITLE1/'Results of the Spectral Analyses'/

DATA TITLE2/'%/Results of the Cross-Spectral Analysis'/

! Initialization

PI = 2.0*ASIN(1.0)

DO 10 I=1, NF

F(I) = PI*FLOAT(I)/FLOAT(NF)

CALL SCOPY (NF, F, 1, CSM(1:,1), 1)

CSM(I,2) = 2.0*FLOAT(NF)/FLOAT(I)

10 CONTINUE

! Robinson data

CALL GDATA (8, RDATA, NROW, NVAR)

! Center on arithmetic means

! Frequency in radians per unit time

! Modified periodogram version

IPVER = 1

! Compute the cross periodogram

CALL CPFFT (X, Y, CPM, IPVER=IPVER)

! Estimate the spectral densities

CALL SWEP (N, PX, F, WT, SX)

CALL SWEP (N, PY, F, WT, SY)

! Estimate the cross-spectral density

CALL CSWEP (N, SX, SY, CPREAL, CPIMAG, F, WT, COSPEC, &

QUADRA, CRAMPL, PHASE, COHERE)

! Print results

!

! Copy results to output matrices

CALL SCOPY (NF, SX, 1, CSM(1:,3), 1)

CALL SCOPY (NF, SY, 1, CSM(1:,4), 1)

CALL SCOPY (NF, COSPEC, 1, CSM(1:,5), 1)

CALL SCOPY (NF, QUADRA, 1, CSM(1:,6), 1)

CALL SCOPY (NF, CRAMPL, 1, CSM(1:,7), 1)

CALL SCOPY (NF, PHASE, 1, CSM(1:,8), 1)

CALL SCOPY (NF, COHERE, 1, CSM(1:,9), 1)

! Call printing routines

CALL WRRRL (TITLE1, CSM, RLABEL, CLABEL1, NF, 4, FMT=FMT)

CALL WRRRL (TITLE2, CSM(1:,5), RLABEL, CLABEL2, NF, 5, FMT=FMT)

!

END

Output

 

Results of the Spectral Analyses

Spectral Spectral

Estimate Estimate

k Frequency Period of X of Y

1 0.3142 20.0000 116.9550 1315.8370

2 0.6283 10.0000 1206.6086 1005.1219

3 0.9425 6.6667 84.8369 317.2589

4 1.2566 5.0000 55.2120 270.2111

5 1.5708 4.0000 46.5748 115.6768

6 1.8850 3.3333 12.4050 250.0125

7 2.1991 2.8571 7.0934 82.6773

8 2.5133 2.5000 3.4091 62.3267

9 2.8274 2.2222 5.6828 12.8970

10 3.1416 2.0000 4.0346 17.6441

 

Results of the Cross-Spectral Analysis

Cross

k Cospectrum Quadrature Amplitude Phase Coherence

1 94.0531 -254.0125 270.8659 1.2162 0.4767

2 702.5118 21.9823 702.8557 -0.0313 0.4073

3 70.2379 -31.4431 76.9547 0.4209 0.2200

4 -1.8715 -36.1639 36.2123 1.6225 0.0879

5 36.6366 -18.5925 41.0843 0.4696 0.3133

6 32.7071 -6.6569 33.3776 0.2008 0.3592

7 9.4887 -9.1692 13.1950 0.7683 0.2969

8 9.2534 -0.3000 9.2583 0.0324 0.4034

9 -0.5568 2.9455 2.9977 -1.7576 0.1226

10 1.7640 -1.8321 2.5433 0.8043 0.0909