g13cg:: Time Series Analysis (NAG Toolbox)

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

Description

An estimate of the noise spectrum in the dependence of series

y

on series

x

at frequency

ω

is given by

f_{y ∣ x} (ω) = f_{y y} (ω) (1 - W (ω)),

where

W (ω)

is the squared coherency described in nag_tsa_multi_spectrum_bivar (g13ce) and

f_{y y} (ω)

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

(d - 2)

degrees of freedom.

If the dependence of

y_{t}

x_{t}

can be assumed to be represented in the time domain by the one sided relationship

y_{t} = v_{0} x_{t} + v_{1} x_{t - 1} + \dots + n_{t},

where the noise

n_{t}

is independent of

x_{t}

, then it is the spectrum of this noise which is estimated by

f_{y ∣ x} (ω)

Estimates of the impulse response function

v_{0}, v_{1}, v_{2}, \dots

may also be obtained as

v_{k} = \frac{1}{π} \int_{0}^{π} Re (\frac{\exp (i k ω) f_{x y} (ω)}{f_{x x} (ω)}),

where

Re

indicates the real part of the expression. For this purpose it is essential that the univariate spectrum for

x

f_{x x} (ω)

, and the cross spectrum,

f_{x y} (ω)

, be supplied to this function for a frequency range

ω_{l} = [\frac{2 π l}{L}], 0 \leq l \leq [L / 2],

where

[]

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}

x_{t}

. This will bias the estimates of

v_{k}

in any causal dependence of

y_{t}

x_{t}, x_{t - 1}, \dots

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

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

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.

Further Comments

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 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

On entry,	$ng < 1$ ,
or	$stats (1) < 3.0$ ,
or	$stats (2) \leq 0.0$ ,
or	$stats (2) > 1.0$ ,
or	$stats (3) < 1.0$ ,
or	$n < 1$ .

On entry,	$[l / 2] + 1 \neq ng$ ,
or	l has a prime factor exceeding $19$ ,
or	l has more than $20$ prime factors, counting repetitions.

NAG Toolbox: nag_tsa_multi_noise_bivar (g13cg)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example