Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_multi_spectrum_daniell (g13cd)

## Purpose

nag_tsa_multi_spectrum_daniell (g13cd) calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

## Syntax

[xg, yg, ng, ifail] = g13cd(nxy, mtxy, pxy, mw, ish, pw, l, xg, yg, 'kc', kc)
[xg, yg, ng, ifail] = nag_tsa_multi_spectrum_daniell(nxy, mtxy, pxy, mw, ish, pw, l, xg, yg, 'kc', kc)

## Description

The supplied time series may be mean and trend corrected and tapered as in the description of nag_tsa_uni_spectrum_daniell (g13cb) before calculation of the unsmoothed sample cross-spectrum
 $fxy* ω = 12πn ∑ t=1 n yt expiωt × ∑ t=1 n xt exp-iωt$
for frequency values ${\omega }_{j}=\frac{2\pi j}{K}$, $0\le {\omega }_{j}\le \pi$.
A correction is made for bias due to any tapering.
As in the description of nag_tsa_uni_spectrum_daniell (g13cb) for univariate frequency window smoothing, the smoothed spectrum is returned as a subset of these frequencies,
 $νl=2π lL, l=0,1,…,L/2$
where [ ] denotes the integer part.
Its real part or co-spectrum $cf\left({\nu }_{l}\right)$, and imaginary part or quadrature spectrum $qf\left({\nu }_{l}\right)$ are defined by
 $fxy νl = cf νl + iqf νl = ∑ ωk < πM w~k fxy* νl+ωk$
where the weights ${\stackrel{~}{w}}_{k}$ are similar to the weights ${w}_{k}$ defined for nag_tsa_uni_spectrum_daniell (g13cb), but allow for an implicit alignment shift $S$ between the series:
 $w~k=wkexp-2π iSk/L.$
It is recommended that $S$ is chosen as the lag $k$ at which the cross-covariances ${c}_{xy}\left(k\right)$ peak, so as to minimize bias.
If no smoothing is required, the integer $M$, which determines the frequency window width $\frac{2\pi }{M}$, should be set to $n$.
The bandwidth of the estimates will normally have been calculated in a previous call of nag_tsa_uni_spectrum_daniell (g13cb) for estimating the univariate spectra of ${y}_{t}$ and ${x}_{t}$.

## 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:     $\mathrm{nxy}$int64int32nag_int scalar
$n$, the length of the time series $x$ and $y$.
Constraint: ${\mathbf{nxy}}\ge 1$.
2:     $\mathrm{mtxy}$int64int32nag_int scalar
Whether the data is to be initially mean or trend corrected.
${\mathbf{mtxy}}=0$
For no correction.
${\mathbf{mtxy}}=1$
For mean correction.
${\mathbf{mtxy}}=2$
For trend correction.
Constraint: $0\le {\mathbf{mtxy}}\le 2$.
3:     $\mathrm{pxy}$ – double scalar
The proportion of the data (totalled over both ends) to be initially tapered by the split cosine bell taper.
A value of $0.0$ implies no tapering.
Constraint: $0.0\le {\mathbf{pxy}}\le 1.0$.
4:     $\mathrm{mw}$int64int32nag_int scalar
$M$, the frequency width of the smoothing window as $\frac{2\pi }{M}$.
A value of $n$ implies that no smoothing is to be carried out.
Constraint: $1\le {\mathbf{mw}}\le {\mathbf{nxy}}$.
5:     $\mathrm{ish}$int64int32nag_int scalar
$S$, the alignment shift between the $x$ and $y$ series. If $x$ leads $y$, the shift is positive.
Constraint: $-{\mathbf{l}}<{\mathbf{ish}}<{\mathbf{l}}$.
6:     $\mathrm{pw}$ – double scalar
$p$, the shape parameter of the trapezium frequency window.
A value of $0.0$ gives a triangular window, and a value of $1.0$ a rectangular window.
If ${\mathbf{mw}}={\mathbf{nxy}}$ (i.e., no smoothing is carried out) then pw is not used.
Constraint: if ${\mathbf{mw}}\ne {\mathbf{nxy}}$, $0.0\le {\mathbf{pw}}\le 1.0$.
7:     $\mathrm{l}$int64int32nag_int scalar
$L$, the frequency division of smoothed cross spectral estimates as $\frac{2\pi }{L}$.
Constraints:
• ${\mathbf{l}}\ge 1$;
• l must be a factor of kc.
8:     $\mathrm{xg}\left({\mathbf{kc}}\right)$ – double array
The nxy data points of the $x$ series.
9:     $\mathrm{yg}\left({\mathbf{kc}}\right)$ – double array
The nxy data points of the $y$ series.

### Optional Input Parameters

1:     $\mathrm{kc}$int64int32nag_int scalar
Default: the dimension of the arrays xg, yg. (An error is raised if these dimensions are not equal.)
The dimension of the arrays xg and yg. the order of the fast Fourier transform (FFT) used to calculate the spectral estimates. kc should be a product of small primes such as ${2}^{m}$ where $m$ is the smallest integer such that ${2}^{m}\ge 2n$, provided $m\le 20$.
Constraints:
• ${\mathbf{kc}}\ge 2×{\mathbf{nxy}}$;
• kc must be a multiple of l. The largest prime factor of kc must not exceed $19$, and the total number of prime factors of kc, counting repetitions, must not exceed $20$. These two restrictions are imposed by the internal FFT algorithm used.

### Output Parameters

1:     $\mathrm{xg}\left({\mathbf{kc}}\right)$ – double array
The real parts of the ng cross spectral estimates in elements ${\mathbf{xg}}\left(1\right)$ to ${\mathbf{xg}}\left({\mathbf{ng}}\right)$, and ${\mathbf{xg}}\left({\mathbf{ng}}+1\right)$ to ${\mathbf{xg}}\left({\mathbf{kc}}\right)$ contain $0.0$. The $y$ series leads the $x$ series.
2:     $\mathrm{yg}\left({\mathbf{kc}}\right)$ – double array
The imaginary parts of the ng cross spectral estimates in elements ${\mathbf{yg}}\left(1\right)$ to ${\mathbf{yg}}\left({\mathbf{ng}}\right)$, and ${\mathbf{yg}}\left({\mathbf{ng}}+1\right)$ to ${\mathbf{yg}}\left({\mathbf{kc}}\right)$ contain $0.0$. The $y$ series leads the $x$ series.
3:     $\mathrm{ng}$int64int32nag_int scalar
The number of spectral estimates, $\left[L/2\right]+1$, whose separate parts are held in xg and yg.
4:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{nxy}}<1$, or ${\mathbf{mtxy}}<0$, or ${\mathbf{mtxy}}>2$, or ${\mathbf{pxy}}<0.0$, or ${\mathbf{pxy}}>1.0$, or ${\mathbf{mw}}<1$, or ${\mathbf{mw}}>{\mathbf{nxy}}$, or ${\mathbf{pw}}<0.0$ and ${\mathbf{mw}}\ne {\mathbf{nxy}}$, or ${\mathbf{pw}}>1.0$ and ${\mathbf{mw}}\ne {\mathbf{nxy}}$, or ${\mathbf{l}}<1$, or $\left|{\mathbf{ish}}\right|\ge {\mathbf{l}}$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{kc}}<2×{\mathbf{nxy}}$, or kc is not a multiple of l, or kc has a prime factor exceeding $19$, or kc has more than $20$ prime factors, counting repetitions.
${\mathbf{ifail}}=3$
This indicates that a serious error has occurred. Check all array subscripts in calls to nag_tsa_multi_spectrum_daniell (g13cd). Seek expert help.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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.

nag_tsa_multi_spectrum_daniell (g13cd) carries out an FFT of length kc to calculate the sample cross spectrum. The time taken by the function for this is approximately proportional to ${\mathbf{kc}}×\mathrm{log}\left({\mathbf{kc}}\right)$ (but see function document nag_sum_fft_realherm_1d (c06pa) for further details).

## Example

This example reads two time series of length $296$. It selects mean correction and a 10% tapering proportion. It selects a $2\pi /16$ frequency width of smoothing window, a window shape parameter of $0.5$ and an alignment shift of $3$. It then calls nag_tsa_multi_spectrum_daniell (g13cd) to calculate the smoothed sample cross spectrum and prints the results.
```function g13cd_example

fprintf('g13cd example results\n\n');

% Problem size
nxy  = int64(296);

% Control parameters
mtxy = int64(1);
pxy  = 0.1;
iw   = int64(4);
mw   = int64(16);
ish  = int64(3);
kc   = int64(640);
l    = int64(80);
pw = 0.5;

% Series
xg = zeros(kc, 1);
xg(1:nxy) = ...
[-0.109; 0.000; 0.178; 0.339; 0.373; 0.441; 0.461; 0.348; 0.127;-0.180;
-0.588;-1.055;-1.421;-1.520;-1.302;-0.814;-0.475;-0.193; 0.088; 0.435;
0.771; 0.866; 0.875; 0.891; 0.987; 1.263; 1.775; 1.976; 1.934; 1.866;
1.832; 1.767; 1.608; 1.265; 0.790; 0.360; 0.115; 0.088; 0.331; 0.645;
0.960; 1.409; 2.670; 2.834; 2.812; 2.483; 1.929; 1.485; 1.214; 1.239;
1.608; 1.905; 2.023; 1.815; 0.535; 0.122; 0.009; 0.164; 0.671; 1.019;
1.146; 1.155; 1.112; 1.121; 1.223; 1.257; 1.157; 0.913; 0.620; 0.255;
-0.280;-1.080;-1.551;-1.799;-1.825;-1.456;-0.944;-0.570;-0.431;-0.577;
-0.960;-1.616;-1.875;-1.891;-1.746;-1.474;-1.201;-0.927;-0.524; 0.040;
0.788; 0.943; 0.930; 1.006; 1.137; 1.198; 1.054; 0.595;-0.080;-0.314;
-0.288;-0.153;-0.109;-0.187;-0.255;-0.299;-0.007; 0.254; 0.330; 0.102;
-0.423;-1.139;-2.275;-2.594;-2.716;-2.510;-1.790;-1.346;-1.081;-0.910;
-0.876;-0.885;-0.800;-0.544;-0.416;-0.271; 0.000; 0.403; 0.841; 1.285;
1.607; 1.746; 1.683; 1.485; 0.993; 0.648; 0.577; 0.577; 0.632; 0.747;
0.999; 0.993; 0.968; 0.790; 0.399;-0.161;-0.553;-0.603;-0.424;-0.194;
-0.049; 0.060; 0.161; 0.301; 0.517; 0.566; 0.560; 0.573; 0.592; 0.671;
0.933; 1.337; 1.460; 1.353; 0.772; 0.218;-0.237;-0.714;-1.099;-1.269;
-1.175;-0.676; 0.033; 0.556; 0.643; 0.484; 0.109;-0.310;-0.697;-1.047;
-1.218;-1.183;-0.873;-0.336; 0.063; 0.084; 0.000; 0.001; 0.209; 0.556;
0.782; 0.858; 0.918; 0.862; 0.416;-0.336;-0.959;-1.813;-2.378;-2.499;
-2.473;-2.330;-2.053;-1.739;-1.261;-0.569;-0.137;-0.024;-0.050;-0.135;
-0.276;-0.534;-0.871;-1.243;-1.439;-1.422;-1.175;-0.813;-0.634;-0.582;
-0.625;-0.713;-0.848;-1.039;-1.346;-1.628;-1.619;-1.149;-0.488;-0.160;
-0.007;-0.092;-0.620;-1.086;-1.525;-1.858;-2.029;-2.024;-1.961;-1.952;
-1.794;-1.302;-1.030;-0.918;-0.798;-0.867;-1.047;-1.123;-0.876;-0.395;
0.185; 0.662; 0.709; 0.605; 0.501; 0.603; 0.943; 1.223; 1.249; 0.824;
0.102; 0.025; 0.382; 0.922; 1.032; 0.866; 0.527; 0.093;-0.458;-0.748;
-0.947;-1.029;-0.928;-0.645;-0.424;-0.276;-0.158;-0.033; 0.102; 0.251;
0.280; 0.000;-0.493;-0.759;-0.824;-0.740;-0.528;-0.204; 0.034; 0.204;
0.253; 0.195; 0.131; 0.017;-0.182;-0.262];
yg = zeros(kc, 1);
yg(1:nxy) = ...
[53.8; 53.6; 53.5; 53.5; 53.4; 53.1; 52.7; 52.4; 52.2; 52.0; 52.0; 52.4;
53.0; 54.0; 54.9; 56.0; 56.8; 56.8; 56.4; 55.7; 55.0; 54.3; 53.2; 52.3;
51.6; 51.2; 50.8; 50.5; 50.0; 49.2; 48.4; 47.9; 47.6; 47.5; 47.5; 47.6;
48.1; 49.0; 50.0; 51.1; 51.8; 51.9; 51.7; 51.2; 50.0; 48.3; 47.0; 45.8;
45.6; 46.0; 46.9; 47.8; 48.2; 48.3; 47.9; 47.2; 47.2; 48.1; 49.4; 50.6;
51.5; 51.6; 51.2; 50.5; 50.1; 49.8; 49.6; 49.4; 49.3; 49.2; 49.3; 49.7;
50.3; 51.3; 52.8; 54.4; 56.0; 56.9; 57.5; 57.3; 56.6; 56.0; 55.4; 55.4;
56.4; 57.2; 58.0; 58.4; 58.4; 58.1; 57.7; 57.0; 56.0; 54.7; 53.2; 52.1;
51.6; 51.0; 50.5; 50.4; 51.0; 51.8; 52.4; 53.0; 53.4; 53.6; 53.7; 53.8;
53.8; 53.8; 53.3; 53.0; 52.9; 53.4; 54.6; 56.4; 58.0; 59.4; 60.2; 60.0;
59.4; 58.4; 57.6; 56.9; 56.4; 56.0; 55.7; 55.3; 55.0; 54.4; 53.7; 52.8;
51.6; 50.6; 49.4; 48.8; 48.5; 48.7; 49.2; 49.8; 50.4; 50.7; 50.9; 50.7;
50.5; 50.4; 50.2; 50.4; 51.2; 52.3; 53.2; 53.9; 54.1; 54.0; 53.6; 53.2;
53.0; 52.8; 52.3; 51.9; 51.6; 51.6; 51.4; 51.2; 50.7; 50.0; 49.4; 49.3;
49.7; 50.6; 51.8; 53.0; 54.0; 55.3; 55.9; 55.9; 54.6; 53.5; 52.4; 52.1;
52.3; 53.0; 53.8; 54.6; 55.4; 55.9; 55.9; 55.2; 54.4; 53.7; 53.6; 53.6;
53.2; 52.5; 52.0; 51.4; 51.0; 50.9; 52.4; 53.5; 55.6; 58.0; 59.5; 60.0;
60.4; 60.5; 60.2; 59.7; 59.0; 57.6; 56.4; 55.2; 54.5; 54.1; 54.1; 54.4;
55.5; 56.2; 57.0; 57.3; 57.4; 57.0; 56.4; 55.9; 55.5; 55.3; 55.2; 55.4;
56.0; 56.5; 57.1; 57.3; 56.8; 55.6; 55.0; 54.1; 54.3; 55.3; 56.4; 57.2;
57.8; 58.3; 58.6; 58.8; 58.8; 58.6; 58.0; 57.4; 57.0; 56.4; 56.3; 56.4;
56.4; 56.0; 55.2; 54.0; 53.0; 52.0; 51.6; 51.6; 51.1; 50.4; 50.0; 50.0;
52.0; 54.0; 55.1; 54.5; 52.8; 51.4; 50.8; 51.2; 52.0; 52.8; 53.8; 54.5;
54.9; 54.9; 54.8; 54.4; 53.7; 53.3; 52.8; 52.6; 52.6; 53.0; 54.3; 56.0;
57.0; 58.0; 58.6; 58.5; 58.3; 57.8; 57.3; 57];

[xg, yg, ng, ifail] = ...
g13cd( ...
nxy, mtxy, pxy, mw, ish, pw, l, xg, yg);

fprintf('                      Returned sample spectrum\n\n');
fprintf('%23s%22s%22s\n', 'Real  Imaginary', 'Real  Imaginary', ...
'Real  Imaginary');
fprintf('%21s%22s%22s\n', 'Lag    part     part', '  Lag    part     part', ...
'  Lag    part     part');
result = [double([0:ng-1]); xg(1:ng)'; yg(1:ng)'];
for j = 1:3:ng
fprintf('%4d%9.4f%9.4f', result(:,j:min(j+2,ng)));
fprintf('\n');
end

```
```g13cd example results

Returned sample spectrum

Real  Imaginary       Real  Imaginary       Real  Imaginary
Lag    part     part  Lag    part     part  Lag    part     part
0  -6.1563   0.0000   1  -5.5905  -2.0119   2  -3.2711  -2.7963
3  -1.1803  -2.3264   4  -0.2061  -1.8132   5   0.3434  -1.1357
6   0.6200  -0.7351   7   0.5967  -0.3449   8   0.4523  -0.0984
9   0.2391   0.0177  10   0.1129   0.0402  11   0.0564   0.0523
12   0.0134   0.0443  13  -0.0032   0.0299  14  -0.0057   0.0148
15  -0.0057   0.0069  16  -0.0033   0.0038  17  -0.0011   0.0012
18  -0.0004   0.0001  19  -0.0004   0.0002  20  -0.0003   0.0001
21  -0.0001   0.0002  22  -0.0002   0.0003  23  -0.0002   0.0002
24  -0.0002   0.0000  25  -0.0004   0.0000  26  -0.0002  -0.0002
27  -0.0001  -0.0000  28  -0.0001   0.0002  29  -0.0001   0.0002
30  -0.0002   0.0003  31  -0.0002   0.0001  32  -0.0001   0.0000
33  -0.0000  -0.0000  34   0.0000  -0.0001  35   0.0001  -0.0001
36   0.0001  -0.0001  37   0.0001  -0.0001  38   0.0000  -0.0001
39   0.0000  -0.0001  40   0.0001   0.0000
```