For a bivariate time series, g13cef calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra.
The routine may be called by the names g13cef or nagf_tsa_multi_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 g13ccforg13cdf;
${f}_{xx}\left(\omega \right)$ and ${f}_{yy}\left(\omega \right)$ are the univariate spectrum estimates for the two series as obtained using g13caforg13cbf.
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 g13caforg13cbf.
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:
$$T=\frac{2F}{d-2+2F}\text{,}$$
where $d$ is the degrees of freedom associated with the univariate spectrum estimates. The value of $T$ is returned by the routine.
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
5Arguments
1: $\mathbf{xg}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the ng univariate spectral estimates, ${f}_{xx}\left(\omega \right)$, for the $x$ series.
2: $\mathbf{yg}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the ng univariate spectral estimates, ${f}_{yy}\left(\omega \right)$, for the $y$ series.
3: $\mathbf{xyrg}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the real parts, $cf\left(\omega \right)$, of the ng bivariate spectral estimates for the $x$ and $y$ series. The $x$ series leads the $y$ series.
4: $\mathbf{xyig}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the imaginary parts, $qf\left(\omega \right)$, of 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 method of smoothing. For rectangular, Bartlett, Tukey or Parzen smoothing windows, the same cut-off point of lag window and the same frequency division of the spectral estimates must be used. For the trapezium frequency smoothing window, 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.
5: $\mathbf{ng}$ – IntegerInput
On entry: the number of spectral estimates in each of the arrays xg, yg, xyrg and xyig. It is also the number of cross amplitude spectral and squared coherency estimates.
Constraint:
${\mathbf{ng}}\ge 1$.
6: $\mathbf{stats}\left(4\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the four associated statistics for the univariate spectral estimates for the $x$ and $y$ series. ${\mathbf{stats}}\left(1\right)$ contains the degrees of freedom, ${\mathbf{stats}}\left(2\right)$ and ${\mathbf{stats}}\left(3\right)$ contain the lower and upper bound multiplying factors respectively and ${\mathbf{stats}}\left(4\right)$ contains the bandwidth.
Constraints:
${\mathbf{stats}}\left(1\right)\ge 3.0$;
$0.0<{\mathbf{stats}}\left(2\right)\le 1.0$;
${\mathbf{stats}}\left(3\right)\ge 1.0$.
7: $\mathbf{ca}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the ng cross amplitude spectral estimates $\hat{A}\left(\omega \right)$ at each frequency of $\omega $.
8: $\mathbf{calw}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the ng lower bounds for the ng cross amplitude spectral estimates.
9: $\mathbf{caup}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the ng upper bounds for the ng cross amplitude spectral estimates.
10: $\mathbf{t}$ – Real (Kind=nag_wp)Output
On exit: the critical value for the significance of the squared coherency, $T$.
11: $\mathbf{sc}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the ng squared coherency estimates, $\hat{W}\left(\omega \right)$ at each frequency $\omega $.
12: $\mathbf{sclw}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the ng lower bounds for the ng squared coherency estimates.
13: $\mathbf{scup}\left({\mathbf{ng}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the ng upper bounds for the ng squared coherency estimates.
14: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: if more than one failure of the types $2$, $3$, $4$ and $5$ occurs then the failure type which occurred at lowest frequency is returned in ifail. However the actions indicated above are also carried out for failures at higher frequencies.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ng}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ng}}\ge 1$.
On entry, ${\mathbf{stats}}\left(1\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{stats}}\left(1\right)\ge 3.0$.
On entry, ${\mathbf{stats}}\left(2\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{stats}}\left(2\right)>0.0$.
On entry, ${\mathbf{stats}}\left(2\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{stats}}\left(2\right)\le 1.0$.
On entry, ${\mathbf{stats}}\left(3\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{stats}}\left(3\right)\ge 1.0$.
${\mathbf{ifail}}=2$
A bivariate spectral estimate is zero. For this frequency the cross amplitude spectrum and shared coherency and their bounds are set to zero.
${\mathbf{ifail}}=3$
A univariate spectral estimate is negative. For this frequency the cross amplitude spectrum and shared coherency and their bounds are set to zero.
${\mathbf{ifail}}=4$
A univariate spectral estimate is zero. For this frequency the cross amplitude spectrum and shared coherency and their bounds are set to zero.
${\mathbf{ifail}}=5$
A calculated value of the squared coherency exceeds $1.0$. For this frequency the squared coherency is reset to $1.0$ and this value for the squared coherency is used in the formulae for the calculation of bounds for both the cross amplitude spectrum and squared coherency. This has the consequence that both squared coherency bounds are $1.0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
All computations are very stable and yield good accuracy.
8Parallelism and Performance
g13cef is not threaded in any implementation.
9Further Comments
The time taken by g13cef is approximately proportional to ng.
10Example
This example reads the set of univariate spectrum statistics, the two univariate spectra and the cross spectrum at a frequency division of $\frac{2\pi}{20}$ for a pair of time series. It calls g13cef to calculate the cross amplitude spectrum and squared coherency and their bounds and prints the results.