For a bivariate time series, g13cec calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra.
The function may be called by the names: g13cec, nag_tsa_multi_spectrum_bivar or nag_tsa_cross_spectrum_bivar.
3Description
Estimates of the cross amplitude spectrum $A\left(\omega \right)$ and squared coherency $W\left(\omega \right)$ are calculated for each frequency $\omega $ as
$cf\left(\omega \right)$ and $qf\left(\omega \right)$ are the co-spectrum and quadrature spectrum estimates between the series, i.e., the real and imaginary parts of the cross spectrum ${f}_{xy}\left(\omega \right)$ as obtained using g13cccorg13cdc. ${f}_{xx}\left(\omega \right)$ and ${f}_{yy}\left(\omega \right)$ are the univariate spectrum estimates for the two series as obtained using g13cacorg13cbc. 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 g13cacorg13cbc.
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 $W\left(\omega \right)$ with the critical value $T$ derived from the upper 5% point of the $F$-distribution on $(2,d-2)$ degrees of freedom:
$$T=\frac{2F}{d-2+2F}$$
where $d$ is the degrees of freedom associated with the univariate spectrum estimates. The value of $T$ is returned by the function.
The hypothesis that the series are unrelated at frequency $\omega $, i.e., that both the true cross amplitude and coherency are zero, may be rejected at the 5% level if $W\left(\omega \right)>T$. Tests at two frequencies separated by more than the bandwidth may be taken to be independent.
The confidence limits on $A\left(\omega \right)$ are strictly appropriate only at frequencies for which the coherency is significant. The same applies to the confidence limits on $W\left(\omega \right)$ which are however calculated at all frequencies using the approximation that $\mathrm{arctanh}\left(\sqrt{W\left(l\right)}\right)$ is Normal with variance $1/d$.
4References
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
On entry: ${f}_{xy}\left(\omega \right)$, the ng bivariate spectral estimates for the $x$ and $y$ series. The $x$ series leads the $y$ 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: $\mathbf{ng}$ – 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.
On entry: the 4 associated statistics for the univariate spectral estimates for the $x$ and $y$ series. ${\mathbf{stats}}\left[0\right]$ contains the degrees of freedom, ${\mathbf{stats}}\left[1\right]$ and ${\mathbf{stats}}\left[2\right]$ contain the lower and upper bound multiplying factors respectively and ${\mathbf{stats}}\left[3\right]$ contains the bandwidth.
On exit: the ng upper bounds for the ng squared coherency estimates.
13: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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, ${\mathbf{ng}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ng}}\ge 1$.
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, ${\mathbf{stats}}\left[1\right]$ must not be greater than 1.0: ${\mathbf{stats}}\left[1\right]=\u27e8\mathit{\text{value}}\u27e9$.
NE_REAL_ARG_LE
On entry, ${\mathbf{stats}}\left[1\right]$ must not be less than or equal to 0.0: ${\mathbf{stats}}\left[1\right]=\u27e8\mathit{\text{value}}\u27e9$.
NE_REAL_ARG_LT
On entry, ${\mathbf{stats}}\left[0\right]$ must not be less than 3.0: ${\mathbf{stats}}\left[0\right]=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{stats}}\left[2\right]$ must not be less than 1.0: ${\mathbf{stats}}\left[2\right]=\u27e8\mathit{\text{value}}\u27e9$.
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.
7Accuracy
All computations are very stable and yield good accuracy.
8Parallelism and Performance
g13cec is not threaded in any implementation.
9Further Comments
The time taken by g13cec is approximately proportional to ng.
10Example
The example program reads the set of univariate spectrum statistics, the 2 univariate spectra and the cross spectrum at a frequency division of $\frac{2\pi}{20}$ for a pair of time series. It calls g13cec to calculate the cross amplitude spectrum and squared coherency and their bounds and prints the results.