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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_multi_noise_bivar (g13cg)

## Purpose

For a bivariate time series, nag_tsa_multi_noise_bivar (g13cg) calculates the noise spectrum together with multiplying factors for the bounds and the impulse response function and its standard error, from the univariate and bivariate spectra.

## Syntax

[er, erlw, erup, rf, rfse, ifail] = g13cg(xg, yg, xyrg, xyig, stats, l, n, 'ng', ng)
[er, erlw, erup, rf, rfse, ifail] = nag_tsa_multi_noise_bivar(xg, yg, xyrg, xyig, stats, l, n, 'ng', ng)

## Description

An estimate of the noise spectrum in the dependence of series $y$ on series $x$ at frequency $\omega$ is given by
 $fy∣xω=fyyω1-Wω,$
where $W\left(\omega \right)$ is the squared coherency described in nag_tsa_multi_spectrum_bivar (g13ce) and ${f}_{yy}\left(\omega \right)$ is the univariate spectrum estimate for series $y$. Confidence limits on the true spectrum are obtained using multipliers as described for nag_tsa_uni_spectrum_lag (g13ca), but based on $\left(d-2\right)$ degrees of freedom.
If the dependence of ${y}_{t}$ on ${x}_{t}$ can be assumed to be represented in the time domain by the one sided relationship
 $yt=v0xt+v1xt-1+⋯+nt,$
where the noise ${n}_{t}$ is independent of ${x}_{t}$, then it is the spectrum of this noise which is estimated by ${f}_{y\mid x}\left(\omega \right)$.
Estimates of the impulse response function ${v}_{0},{v}_{1},{v}_{2},\dots \text{}$ may also be obtained as
 $vk=1π∫0πReexpikωfxyω fxxω ,$
where $\mathrm{Re}$ indicates the real part of the expression. For this purpose it is essential that the univariate spectrum for $x$, ${f}_{xx}\left(\omega \right)$, and the cross spectrum, ${f}_{xy}\left(\omega \right)$, be supplied to this function for a frequency range
 $ωl=2πlL , 0≤l≤L/2,$
where $\left[\right]$ denotes the integer part, the integral being approximated by a finite Fourier transform.
An approximate standard error is calculated for the estimates ${v}_{k}$. Significant values of ${v}_{k}$ in the locations described as anticipatory responses in the argument array rf indicate that feedback exists from ${y}_{t}$ to ${x}_{t}$. This will bias the estimates of ${v}_{k}$ in any causal dependence of ${y}_{t}$ on ${x}_{t},{x}_{t-1},\dots \text{}$.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{xg}\left({\mathbf{ng}}\right)$ – double array
The ng univariate spectral estimates, ${f}_{xx}\left(\omega \right)$, for the $x$ series.
2:     $\mathrm{yg}\left({\mathbf{ng}}\right)$ – double array
The ng univariate spectral estimates, ${f}_{yy}\left(\omega \right)$, for the $y$ series.
3:     $\mathrm{xyrg}\left({\mathbf{ng}}\right)$ – double array
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:     $\mathrm{xyig}\left({\mathbf{ng}}\right)$ – double array
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:     $\mathrm{stats}\left(4\right)$ – double array
The four associated statistics for the univariate spectral estimates for the $x$ and $y$ series. ${\mathbf{stats}}\left(1\right)$ contains the degree 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:
• ${\mathbf{stats}}\left(1\right)\ge 3.0$;
• $0.0<{\mathbf{stats}}\left(2\right)\le 1.0$;
• ${\mathbf{stats}}\left(3\right)\ge 1.0$.
6:     $\mathrm{l}$int64int32nag_int scalar
$L$, the frequency division of the spectral estimates as $\frac{2\pi }{L}$. It is also the order of the FFT used to calculate the impulse response function. l must relate to the parameter ng by the relationship.
Constraints:
• ${\mathbf{ng}}=\left[L/2\right]+1$;
• The largest prime factor of l must not exceed $19$, and the total number of prime factors of l, counting repetitions, must not exceed $20$. These two restrictions are imposed by the internal FFT algorithm used.
7:     $\mathrm{n}$int64int32nag_int scalar
The number of points in each of the time series $x$ and $y$. n should have the same value as nxy in the call of nag_tsa_multi_spectrum_lag (g13cc) or nag_tsa_multi_spectrum_daniell (g13cd) which calculated the smoothed sample cross spectrum. n is used in calculating the impulse response function standard error (rfse).
Constraint: ${\mathbf{n}}\ge 1$.

### Optional Input Parameters

1:     $\mathrm{ng}$int64int32nag_int scalar
Default: the dimension of the arrays xg, yg, xyrg, xyig. (An error is raised if these dimensions are not equal.)
The number of spectral estimates in each of the arrays xg, yg, xyrg, xyig. It is also the number of noise spectral estimates.
Constraint: ${\mathbf{ng}}\ge 1$.

### Output Parameters

1:     $\mathrm{er}\left({\mathbf{ng}}\right)$ – double array
The ng estimates of the noise spectrum, ${\stackrel{^}{f}}_{y\mid x}\left(\omega \right)$ at each frequency.
2:     $\mathrm{erlw}$ – double scalar
The noise spectrum lower limit multiplying factor.
3:     $\mathrm{erup}$ – double scalar
The noise spectrum upper limit multiplying factor.
4:     $\mathrm{rf}\left({\mathbf{l}}\right)$ – double array
The impulse response function. Causal responses are stored in ascending frequency in ${\mathbf{rf}}\left(1\right)$ to ${\mathbf{rf}}\left({\mathbf{ng}}\right)$ and anticipatory responses are stored in descending frequency in ${\mathbf{rf}}\left({\mathbf{ng}}+1\right)$ to ${\mathbf{rf}}\left({\mathbf{l}}\right)$.
5:     $\mathrm{rfse}$ – double scalar
The impulse response function standard error.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_tsa_multi_noise_bivar (g13cg) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{ng}}<1$, or ${\mathbf{stats}}\left(1\right)<3.0$, or ${\mathbf{stats}}\left(2\right)\le 0.0$, or ${\mathbf{stats}}\left(2\right)>1.0$, or ${\mathbf{stats}}\left(3\right)<1.0$, or ${\mathbf{n}}<1$.
W  ${\mathbf{ifail}}=2$
A bivariate spectral estimate is zero. For this frequency the noise spectrum is set to zero, and the contribution to the impulse response function and its standard error is set to zero.
W  ${\mathbf{ifail}}=3$
A univariate spectral estimate is negative. For this frequency the noise spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.
W  ${\mathbf{ifail}}=4$
A univariate spectral estimate is zero. For this frequency the noise spectrum is set to zero and the contributions to the impulse response function and its standard error are set to zero.
W  ${\mathbf{ifail}}=5$
A calculated value of the squared coherency exceeds $1.0$. For this frequency the squared coherency is reset to $1.0$ with the consequence that the noise spectrum is zero and the contribution to the impulse response function at this frequency is zero.
${\mathbf{ifail}}=6$
 On entry, $\left[{\mathbf{l}}/2\right]+1\ne {\mathbf{ng}}$, or l has a prime factor exceeding $19$, or l has more than $20$ prime factors, counting repetitions.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
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.

## Accuracy

The computation of the noise is stable and yields good accuracy. The FFT is a numerically stable process, and any errors introduced during the computation will normally be insignificant compared with uncertainty in the data.

The time taken by nag_tsa_multi_noise_bivar (g13cg) is approximately proportional to ng.

## Example

This example reads the set of univariate spectrum statistics, the two univariate spectra and the cross spectrum at a frequency division of $\frac{2\pi }{20}$ for a pair of time series. It calls nag_tsa_multi_noise_bivar (g13cg) to calculate the noise spectrum and its confidence limits multiplying factors, the impulse response function and its standard error. It then prints the results.
```function g13cg_example

fprintf('g13cg example results\n\n');

% Data
xg = [ 2.03490;     0.51554;     0.07640;
0.01068;     0.00093;     0.00100;
0.00076;     0.00037;     0.00021];
yg = [21.97712;     3.29761;     0.28782;
0.02480;     0.00285;     0.00203;
0.00125;     0.00107;     0.00191];
xyrg = ...
[-6.54995;     0.34107;     0.12335;
-0.00514;    -0.00033;    -0.00039;
-0.00026;     0.00011;     0.00007];
xyig = ...
[ 0.00000;    -1.19030;     0.04087;
0.00842;     0.00032;    -0.00001;
0.00018;    -0.00016;     0.00000];
ng = numel(xg);

% Statistics
stats = [30.00000;     0.63858;     1.78670;     0.33288];

l = int64(16);
n = int64(296);
[er, erlw, erup, rf, rfse, ifail] = ...
g13cg( ...
xg, yg, xyrg, xyig, stats, l, n);

% Display results
fprintf('           Noise spectrum\n');
for j=1:ng
fprintf('%5d%16.4f\n', j-1, er(j))
end

fprintf('\nNoise spectrum bounds multiplying factors\n');
fprintf('Lower = %10.4f     Upper = %10.4f\n\n', erlw, erup);
fprintf('Impulse response function\n\n');
for j=1:l
fprintf('%5d%16.4f\n', j-1, rf(j))
end
fprintf('\nImpulse response function standard error = %10.4f\n', rfse);

```
```g13cg example results

Noise spectrum
0          0.8941
1          0.3238
2          0.0668
3          0.0157
4          0.0026
5          0.0019
6          0.0011
7          0.0010
8          0.0019

Noise spectrum bounds multiplying factors
Lower =     0.6298     Upper =     1.8291

Impulse response function

0         -0.0547
1          0.0586
2         -0.0322
3         -0.6956
4         -0.7181
5         -0.8019
6         -0.4303
7         -0.2392
8         -0.0766
9          0.0657
10         -0.1652
11         -0.0439
12         -0.0494
13         -0.0384
14          0.0838
15         -0.0814

Impulse response function standard error =     0.0863
```