Integer, Intent (In)	::	ng
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	xg(ng), yg(ng), xyrg(ng), xyig(ng), stats(4)
Real (Kind=nag_wp), Intent (Out)	::	gn(ng), gnlw(ng), gnup(ng), ph(ng), phlw(ng), phup(ng)

C Header Interface

#include <nag.h>

void	g13cff_ (const double xg[], const double yg[], const double xyrg[], const double xyig[], const Integer ng, const double stats[], double gn[], double gnlw[], double gnup[], double ph[], double phlw[], double phup[], Integer ifail)

The routine may be called by the names g13cff or nagf_tsa_multi_gain_bivar.

3 Description

Estimates of the gain

G (ω)

and phase

ϕ (ω)

of the dependency of series

y

on series

x

at frequency

ω

are given by

\begin{array}{l} \hat{G} (ω) = \frac{A (ω)}{f_{x x} (ω)} \\ \hat{ϕ} (ω) = \arccos (\frac{c f (ω)}{A (ω)}), & if ​ q f (ω) \geq 0 \\ \hat{ϕ} (ω) = 2 π - \arccos (\frac{c f (ω)}{A (ω)}), & if ​ q f (ω) < 0 . \end{array}

The quantities used in these definitions are obtained as in Section 3 in g13cef.

Confidence limits are returned for both gain and phase, but should again be taken as very approximate when the coherency

W (ω)

, as calculated by g13cef, is not significant. These are based on the assumption that both

(\hat{G} (ω) / G (ω)) - 1

and

\hat{ϕ} (ω)

are Normal with variance

\frac{1}{d} (\frac{1}{W (ω)} - 1) .

Although the estimate of

ϕ (ω)

is always given in the range

[0, 2 π)

, no attempt is made to restrict its confidence limits to this range.

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: $xg (ng)$ – Real (Kind=nag_wp) array Input: On entry: the ng univariate spectral estimates, $f_{x x} (ω)$ , for the $x$ series.
2: $yg (ng)$ – Real (Kind=nag_wp) array Input: On entry: the ng univariate spectral estimates, $f_{y y} (ω)$ , for the $y$ series.
3: $xyrg (ng)$ – Real (Kind=nag_wp) array Input: On entry: the real parts, $c f (ω)$ , of the ng bivariate spectral estimates for the $x$ and $y$ series. The $x$ series leads the $y$ series.
4: $xyig (ng)$ – Real (Kind=nag_wp) array Input: On entry: the imaginary parts, $q f (ω)$ , 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: $ng$ – Integer Input: On entry: the number of spectral estimates in each of the arrays xg, yg, xyrg and xyig. It is also the number of gain and phase estimates.

Constraint: $ng \geq 1$ .
6: $stats (4)$ – Real (Kind=nag_wp) array Input: On entry: the four associated statistics for the univariate spectral estimates for the $x$ and $y$ series. $stats (1)$ contains the degrees of freedom, $stats (2)$ and $stats (3)$ contain the lower and upper bound multiplying factors respectively and $stats (4)$ holds the bandwidth.

Constraint: $stats (1) \geq 3.0$ .
7: $gn (ng)$ – Real (Kind=nag_wp) array Output: On exit: the ng gain estimates, $\hat{G} (ω)$ , at each frequency $ω$ .
8: $gnlw (ng)$ – Real (Kind=nag_wp) array Output: On exit: the ng lower bounds for the ng gain estimates.
9: $gnup (ng)$ – Real (Kind=nag_wp) array Output: On exit: the ng upper bounds for the ng gain estimates.
10: $ph (ng)$ – Real (Kind=nag_wp) array Output: On exit: the ng phase estimates, $\hat{ϕ} (ω)$ , at each frequency $ω$ .
11: $phlw (ng)$ – Real (Kind=nag_wp) array Output: On exit: the ng lower bounds for the ng phase estimates.
12: $phup (ng)$ – Real (Kind=nag_wp) array Output: On exit: the ng upper bounds for the ng phase estimates.
13: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ 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 $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−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 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.

$ifail = 1$: On entry, $ng = ⟨ value ⟩$ .
Constraint: $ng \geq 1$ .

On entry, $stats (1) = ⟨ value ⟩$ .
Constraint: $stats (1) \geq 3.0$ .

$ifail = 2$: A bivariate spectral estimate is zero. For this frequency the cross amplitude spectrum and shared coherency and their bounds are set to zero.

$ifail = 3$: A univariate spectral estimate is negative. For this frequency the cross amplitude spectrum and shared coherency and their bounds are set to zero.

$ifail = 4$: A univariate spectral estimate is zero. For this frequency the cross amplitude spectrum and shared coherency and their bounds are set to zero.

$ifail = 5$: A calculated value of the squared coherency exceeds $1.0$ . For this frequency the squared coherency is reset to $1.0$ in the formulae for the gain and phase bounds.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

All computations are very stable and yield good accuracy.

8 Parallelism and Performance

g13cff is not threaded in any implementation.

9 Further Comments

The time taken by g13cff is approximately proportional to ng.

10 Example

This example reads the set of univariate spectrum statistics, the two univariate spectra and the cross spectrum at a frequency division of

\frac{2 π}{20}

for a pair of time series. It calls g13cff to calculate the gain and the phase and their bounds and prints the results.

g13cf: FL CL CPP AD

NAG FL Interfaceg13cff (multi_​gain_​bivar)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g13cff (multi_gain_bivar)