NAG Library Function Document
nag_tsa_cross_spectrum_bivar (g13cec)
1
Purpose
For a bivariate time series, nag_tsa_cross_spectrum_bivar (g13cec) calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra.
2
Specification
#include <nag.h> |
#include <nagg13.h> |
void |
nag_tsa_cross_spectrum_bivar (const double xg[],
const double yg[],
const Complex xyg[],
Integer ng,
const double stats[],
double ca[],
double calw[],
double caup[],
double *t,
double sc[],
double sclw[],
double scup[],
NagError *fail) |
|
3
Description
Estimates of the cross amplitude spectrum
and squared coherency
are calculated for each frequency
as
where:
and
are the co-spectrum and quadrature spectrum estimates between the series, i.e., the real and imaginary parts of the cross spectrum
as obtained using
nag_tsa_spectrum_bivar_cov (g13ccc) or
nag_tsa_spectrum_bivar (g13cdc).
and
are the univariate spectrum estimates for the two series as obtained using
nag_tsa_spectrum_univar_cov (g13cac) or
nag_tsa_spectrum_univar (g13cbc). The same type and amount of smoothing should be used for these estimates, and this is specified by the degrees of freedom and bandwidth values which are passed from the calls of
nag_tsa_spectrum_univar_cov (g13cac) or
nag_tsa_spectrum_univar (g13cbc).
Upper and lower 95% confidence limits for the cross amplitude are given approximately by
except that a negative lower limit is reset to 0.0, in which case the approximation is rather poor. You are therefore particularly recommended to compare the coherency estimate
with the critical value
derived from the upper 5% point of the
-distribution on
degrees of freedom:
where
is the degrees of freedom associated with the univariate spectrum estimates. The value of
is returned by the function.
The hypothesis that the series are unrelated at frequency , i.e., that both the true cross amplitude and coherency are zero, may be rejected at the 5% level if . Tests at two frequencies separated by more than the bandwidth may be taken to be independent.
The confidence limits on are strictly appropriate only at frequencies for which the coherency is significant. The same applies to the confidence limits on which are however calculated at all frequencies using the approximation that is Normal with variance .
4
References
Bloomfield P (1976) Fourier Analysis of Time Series: An Introduction Wiley
Jenkins G M and Watts D G (1968) Spectral Analysis and its Applications Holden–Day
5
Arguments
- 1:
– const doubleInput
-
On entry: the
ng univariate spectral estimates,
, for the
series.
- 2:
– const doubleInput
-
On entry: the
ng univariate spectral estimates,
, for the
series.
- 3:
– const ComplexInput
-
On entry:
, the
ng bivariate spectral estimates for the
and
series. The
series leads the
series.
Note: the two univariate and the bivariate spectra must each have been calculated using the same amount of smoothing. The frequency width and the shape of the window and the frequency division of the spectral estimates must be the same. The spectral estimates and statistics must also be unlogged.
- 4:
– IntegerInput
-
On entry: the number of spectral estimates in each of the arrays
xg,
yg and
xyg. It is also the number of cross amplitude spectral and squared coherency estimates.
Constraint:
.
- 5:
– const doubleInput
-
On entry: the 4 associated statistics for the univariate spectral estimates for the and series. contains the degrees of freedom, and contain the lower and upper bound multiplying factors respectively and contains the bandwidth.
Constraints:
- ;
- ;
- .
- 6:
– doubleOutput
-
On exit: the
ng cross amplitude spectral estimates
at each frequency of
.
- 7:
– doubleOutput
-
On exit: the
ng lower bounds for the
ng cross amplitude spectral estimates.
- 8:
– doubleOutput
-
On exit: the
ng upper bounds for the
ng cross amplitude spectral estimates.
- 9:
– double *Output
-
On exit: the critical value for the significance of the squared coherency, .
- 10:
– doubleOutput
-
On exit: the
ng squared coherency estimates,
at each frequency
.
- 11:
– doubleOutput
-
On exit: the
ng lower bounds for the
ng squared coherency estimates.
- 12:
– doubleOutput
-
On exit: the
ng upper bounds for the
ng squared coherency estimates.
- 13:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BIVAR_SPECTRAL_ESTIM_ZERO
-
A bivariate spectral estimate is zero.
For this frequency the cross amplitude spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.
- NE_INT_ARG_LT
-
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call
and any array sizes. If the call is correct then please contact
NAG for
assistance.
- NE_REAL_ARG_GT
-
On entry, must not be greater than 1.0: .
- NE_REAL_ARG_LE
-
On entry, must not be less than or equal to 0.0: .
- NE_REAL_ARG_LT
-
On entry, must not be less than 3.0: .
On entry, must not be less than 1.0: .
- NE_SQUARED_FREQ_GT_ONE
-
A calculated value of the squared coherency exceeds one.
For this frequency the squared coherency is reset to one with the result that the cross amplitude spectrum is zero and the contribution to the impulse response function at this frequency is zero.
- NE_UNIVAR_SPECTRAL_ESTIM_NEG
-
A bivariate spectral estimate is negative.
For this frequency the cross amplitude spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.
- NE_UNIVAR_SPECTRAL_ESTIM_ZERO
-
A bivariate spectral estimate is zero.
For this frequency the cross amplitude spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.
7
Accuracy
All computations are very stable and yield good accuracy.
8
Parallelism and Performance
nag_tsa_cross_spectrum_bivar (g13cec) is not threaded in any implementation.
The time taken by
nag_tsa_cross_spectrum_bivar (g13cec) is approximately proportional to
ng.
10
Example
The example program reads the set of univariate spectrum statistics, the 2 univariate spectra and the cross spectrum at a frequency division of for a pair of time series. It calls nag_tsa_cross_spectrum_bivar (g13cec) to calculate the cross amplitude spectrum and squared coherency and their bounds and prints the results.
10.1
Program Text
Program Text (g13cece.c)
10.2
Program Data
Program Data (g13cece.d)
10.3
Program Results
Program Results (g13cece.r)