g13ce:: Time Series Analysis (NAG Toolbox)

Estimates of the cross amplitude spectrum

A (ω)

and squared coherency

W (ω)

are calculated for each frequency

ω

\begin{array}{l} A (ω) = |f_{x y} (ω)| = \sqrt{c f {(ω)}^{2} + q f {(ω)}^{2}} and \\ W (ω) = \frac{{|f_{x y} (ω)|}^{2}}{f_{x x} (ω) f_{y y} (ω)}, \end{array}

where

$c f (ω)$ and $q f (ω)$ are the co-spectrum and quadrature spectrum estimates between the series, i.e., the real and imaginary parts of the cross spectrum $f_{x y} (ω)$ as obtained using nag_tsa_multi_spectrum_lag (g13cc) or nag_tsa_multi_spectrum_daniell (g13cd);
$f_{x x} (ω)$ and $f_{y y} (ω)$ are the univariate spectrum estimates for the two series as obtained using nag_tsa_uni_spectrum_lag (g13ca) or nag_tsa_uni_spectrum_daniell (g13cb).

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 nag_tsa_uni_spectrum_lag (g13ca) or nag_tsa_uni_spectrum_daniell (g13cb).

Upper and lower

95 %

confidence limits for the cross amplitude are given approximately by

A (ω) [1 \pm (1.96 / \sqrt{d}) \sqrt{W {(ω)}^{- 1} + 1}],

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 (ω)

with the critical value

T

derived from the upper

5 %

point of the

F

-distribution on

(2, d - 2)

degrees of freedom:

T = \frac{2 F}{d - 2 + 2 F},

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

ω

, i.e., that both the true cross amplitude and coherency are zero, may be rejected at the

5 %

level if

W (ω) > T

. Tests at two frequencies separated by more than the bandwidth may be taken to be independent.

The confidence limits on

A (ω)

are strictly appropriate only at frequencies for which the coherency is significant. The same applies to the confidence limits on

W (ω)

which are however calculated at all frequencies using the approximation that

arctanh (\sqrt{W (l)})

is Normal with variance

1 / d

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 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.

Accuracy

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_spectrum_bivar (g13ce) to calculate the cross amplitude spectrum and squared coherency and their bounds and prints the results.

function g13ce_example


fprintf('g13ce 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];

% Calculate cross-amplitude spectrum
[ca, calw, caup, t, sc, sclw, scup, ifail] = ...
  g13ce( ...
	 xg, yg, xyrg, xyig, stats);

% Display results
fprintf('      Cross amplitude spectrum\n\n');
fprintf('                    Lower     Upper\n');
fprintf('          Value     bound     bound\n');
for j=1:ng
  fprintf('%5d%10.4f%10.4f%10.4f\n', j-1, ca(j), calw(j), caup(j))
end
fprintf('\nSquared coherency test statistic = %12.4f\n\n', t);
fprintf('         Squared coherency\n\n');
fprintf('                    Lower     Upper\n');
fprintf('          Value     bound     bound\n');
for j=1:ng
  fprintf('%5d%10.4f%10.4f%10.4f\n', j-1, sc(j), sclw(j), scup(j))
end

g13ce example results

      Cross amplitude spectrum

                    Lower     Upper
          Value     bound     bound
    0    6.5499    3.9277   10.9228
    1    1.2382    0.7364    2.0820
    2    0.1299    0.0755    0.2236
    3    0.0099    0.0049    0.0197
    4    0.0005    0.0001    0.0017
    5    0.0004    0.0001    0.0015
    6    0.0003    0.0001    0.0010
    7    0.0002    0.0001    0.0007
    8    0.0001    0.0000    0.0018

Squared coherency test statistic =       0.1926

         Squared coherency

                    Lower     Upper
          Value     bound     bound
    0    0.9593    0.9185    0.9799
    1    0.9018    0.8093    0.9507
    2    0.7679    0.5811    0.8790
    3    0.3674    0.1102    0.6177
    4    0.0797    0.0000    0.3253
    5    0.0750    0.0000    0.3182
    6    0.1053    0.0000    0.3610
    7    0.0952    0.0000    0.3475
    8    0.0122    0.0000    0.1912

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

NAG Toolbox: nag_tsa_multi_spectrum_bivar (g13ce)

▸▿ Contents

Purpose

Syntax

Description