Computes the multichannel cross-correlation function of two mutually stationary multichannel time series.
X — NOBSX by NCHANX
matrix containing the first time series. (Input)
Each row of
X corresponds to an observation of a
multivariate time series and each column of X corresponds to a univariate time
series.
Y — NOBSY by NCHANY
matrix containing the second time series. (Input)
Each row of
Y corresponds to
an observation of a multivariate time series and each column of Y corresponds to a
univariate time series.
MAXLAG — Maximum
lag of cross-covariances and cross-correlations to be computed.
(Input)
MAXLAG
must be greater than or equal to one and less than the minimum of NOBSX
and NOBSY.
CC — Array of
size NCHANX
by NCHANY
by 2 * MAXLAG
+ 1 containing the cross-correlations between the channels of X
and Y.
(Output)
The cross-correlation between channel i of the X
series and channel j of the Y
series at lag k corresponds to CC(i,
j, k) where i = 1, …, NCHANX,
j = 1, …, NCHANY,
and
k = −MAXLAG,
…, −1, 0,
1, …, MAXLAG.
NOBSX — Number of
observations in each channel of the first time series X.
(Input)
NOBSX must be greater
than or equal to two.
Default: NOBSX = size (X,1).
NCHANX — Number
of channels in the first time series X.
(Input)
NCHANX
must be greater than or equal to one.
Default: NCHANX = size (X,2).
LDX — Leading
dimension of X exactly as specified in the dimension
statement of the calling program. (Input)
LDX must be greater
than or equal to NOBSX.
Default: LDX = size (X,1).
NOBSY — Number of
observations in each channel of the second time series Y.
(Input)
NOBSY must be greater than
or equal to two.
Default: NOBSY = size (Y,1).
NCHANY — Number
of channels in the second time series Y.
(Input)
NCHANY
must be greater than or equal to one.
Default: NCHANY = size (Y,2).
LDY — Leading
dimension of Y
exactly as specified in the dimension statement of the calling
program. (Input)
LDY must be greater
than or equal to NOBSY.
Default:
LDY = size
(Y,1).
IPRINT — Printing
option. (Input)
Default: IPRINT = 0.
IPRINT |
Action |
0 |
No printing is performed. |
1 |
Prints the means and variances. |
2 |
Prints the means, variances, and cross-covariances. |
3 |
Prints the means, variances, cross-covariances, and cross-correlations. |
IMEAN — Option
for computing the means. (Input)
Default: IMEAN = 1.
IMEAN |
Action |
0 |
XMEAN and YMEAN are user-specified. |
1 |
XMEAN and YMEAN are set to the arithmetic means of their respective channels. |
XMEAN — Vector of length NCHANX containing the means of the channels of X. (Input, if IMEAN = 0; output, if IMEAN = 1)
YMEAN — Vector of length NCHANY containing the means of the channels of Y. (Input, if IMEAN = 0; output, if IMEAN = 1)
XVAR — Vector of length NCHANX containing the variances of the channels of X. (Output)
YVAR — Vector of length NCHANY containing the variances of the channels of Y. (Output)
CCV — Array of
size NCHANX
by NCHANY
by 2 * MAXLAG
+ 1 containing the cross-covariances between the channels of X and Y.
(Output)
The cross-covariance between channel i of the X series and channel
j of the Y
series at lag k corresponds to CCV(i,
j, k) where i = 1, …, NCHANX,
j = 1, …, NCHANY,
and
k = −MAXLAG,
…, −1, 0,
1, …, MAXLAG.
LDCCV — Leading
dimension of CCV
exactly as specified in the dimension statement in the calling
program. (Input)
LDCCV must be greater
than or equal to NCHANX.
Default:
LDCCV = size
(CCV,1).
MDCCV — Middle
dimension of CCV
exactly as specified in the dimension statement in the calling
program. (Input)
MDCCV must be greater
than or equal to NCHANY.
Default:
MDCCV = size
(CCV,2).
LDCC — Leading
dimension of CC
exactly as specified in the dimension statement in the calling
program. (Input)
LDCC must be greater
than or equal to NCHANX.
Default: LDCCV = size (CC,1).
MDCC — Middle
dimension of CC
exactly as specified in the
dimension statement in the calling program. (Input)
MDCC must be greater
than or equal to NCHANY.
Default: MDCCV = size (CC,2).
Generic: CALL MCCF (X, Y, MAXLAG, CC [,…])
Specific: The specific interface names are S_MCCF and D_MCCF.
Single: CALL MCCF (NOBSX, NCHANX, X, LDX, NOBSY, NCHANY, Y, LDY, MAXLAG, IPRINT, IMEAN, XMEAN, YMEAN, XVAR, YVAR, CCV, LDCCV, MDCCV, CC, LDCC, MDCC)
Double: The double precision name is DMCCF.
Routine MCCF estimates the multichannel cross-correlation function of two mutually stationary multichannel time series. Define the multichannel time series X by
X = (X1, X2, …, Xp)
where
Xj = (X1j, X2j, …, Xnj)T, j = 1, 2, …, p
with n = NOBSX and p = NCHANX. Similarly, define the multichannel time series Y by
Y = (Y1, Y2, …, Yq)
where
Yj = (Y1j, Y2j, …, Ymj)T, j = 1, 2, …, q
with m = NOBSY and q = NCHANY. The columns of X and Y correspond to individual channels of multichannel time series and may be examined from a univariate perspective. The rows of X and Y correspond to observations of p-variate and q-variate time series, respectively, and may be examined from a multivariate perspective. Note that an alternative characterization of a multivariate time series X considers the columns to be observations of the multivariate time series while the rows contain univariate time series. For example, see Priestley (1981, page 692) and Fuller (1976, page 14).
Let
be the row vector containing the means of the channels of X. In particular,
where for j = 1, 2, …, p
Let
be similarly defined. The cross-covariance of lag k between channel i of X and channel j of Y is estimated by
where i = 1, …, p, j = 1, …, q, and K = MAXLAG. The summation on t extends over all possible cross-products with N equal to the number of cross-products in the sum.
Let
be the row vector consisting of the estimated variances of the channels of X. In particular,
where
Let
be similarly defined. The cross-correlation of lag k between channel i of X and channel j of Y is estimated by
1. For a given lag k, the multichannel cross-covariance coefficient is defined as the array of dimension NCHANX by NCHANY whose components are the single-channel cross-covariance coefficients CCV(i, j, k). A similar definition holds for the multichannel cross-correlation coefficient.
2. Multichannel autocovariances and autocorrelations may be obtained by setting the first and second time series equal.
Consider the Wolfer Sunspot Data (Y ) (Box and
Jenkins 1976, page 530) along with data on northern light activity
(X1) and earthquake
activity (X2) (Robinson 1967, page
204) to be a
three-channel time series. Routine MCCF
is used to computed the cross-covariances and cross-correlations between
X1 and Y and
between X2 and Y with lags
from −MAXLAG
= −10
through lag MAXLAG
= 10:
USE GDATA_INT
USE MCCF_INT
IMPLICIT NONE
INTEGER IPRINT, LDCC, LDCCV, LDX, LDY, MAXLAG, MDCC, MDCCV, &
NCHANX, NCHANY, NOBSX, NOBSY
PARAMETER (IPRINT=3, MAXLAG=10, NCHANX=2, NCHANY=1, NOBSX=100, &
NOBSY=100, LDCC=NCHANX, LDCCV=NCHANX, LDX=NOBSX, &
LDY=NOBSY, MDCC=NCHANY, MDCCV=NCHANY)
!
INTEGER IMEAN, NCOL, NROW
REAL CC(LDCC,MDCC,-MAXLAG:MAXLAG), CCV(LDCCV,MDCCV,- &
MAXLAG:MAXLAG), RDATA(100,4), X(LDX,NCHANX), &
XMEAN(NCHANX), XVAR(NCHANX), Y(LDY,NCHANY), &
YMEAN(NCHANY), YVAR(NCHANY)
!
EQUIVALENCE (X(1,1), RDATA(1,3)), (X(1,2), RDATA(1,4))
EQUIVALENCE (Y(1,1), RDATA(1,2))
!
CALL GDATA (8, RDATA, NROW, NCOL)
! USE Default Option to estimate
! channel means
! Compute multichannel CCVF and CCF
CALL MCCF (X, Y, MAXLAG, CC, IPRINT=IPRINT)
!
END
Channel means of X from
MCCF
1
2
63.43
97.97
Channel variances of X
1 2
2643.7 1978.4
Channel means of Y from
MCCF
46.94
Channel variances of
Y
1383.8
Multichannel cross-covariance between X and Y from MCCF
Lag K
= -10
1
-20.51
2 70.71
Lag K
= -9
1
65.02
2 38.14
Lag K
= -8
1
216.6
2 135.6
Lag K
= -7
1
246.8
2 100.4
Lag K
= -6
1
142.1
2 45.0
Lag K
= -5
1
50.70
2 -11.81
Lag K
= -4
1
72.68
2 32.69
Lag K
= -3
1
217.9
2 -40.1
Lag K
= -2
1
355.8
2 -152.6
Lag K
= -1
1
579.7
2 -213.0
Lag K
= 0
1
821.6
2 -104.8
Lag K
= 1
1
810.1
2 55.2
Lag K
= 2
1
628.4
2 84.8
Lag K
= 3
1
438.3
2 76.0
Lag K
= 4
1
238.8
2 200.4
Lag K
= 5
1
143.6
2 283.0
Lag K
= 6
1
253.0
2 234.4
Lag K
= 7
1
479.5
2 223.0
Lag K
= 8
1
724.9
2 124.5
Lag K
= 9
1
925.0
2 -79.5
Lag K
= 10
1
922.8
2 -279.3
Multichannel cross-correlation
between X and Y from
MCCF
Lag K = -10
1
-0.01072
2 0.04274
Lag K
= -9
1
0.03400
2 0.02305
Lag K
= -8
1
0.1133
2 0.0819
Lag K
= -7
1
0.1290
2 0.0607
Lag K
= -6
1
0.07431
2 0.02718
Lag K
= -5
1
0.02651
2 -0.00714
Lag K
= -4
1
0.03800
2 0.01976
Lag K
= -3
1 0.1139
2 -0.0242
Lag K = -2
1 0.1860
2 -0.0923
Lag K
= -1
1 0.3031
2 -0.1287
Lag K = 0
1 0.4296
2 -0.0633
Lag K
= 1
1 0.4236
2 0.0333
Lag K =
2
1 0.3285
2 0.0512
Lag K
= 3
1 0.2291
2 0.0459
Lag K =
4
1 0.1248
2 0.1211
Lag K
= 5
1 0.0751
2 0.1710
Lag K =
6
1 0.1323
2 0.1417
Lag K
= 7
1 0.2507
2 0.1348
Lag K =
8
1 0.3790
2 0.0752
Lag K
= 9
1 0.4836
2 -0.0481
Lag K = 10
1 0.4825
2 -0.1688
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