hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_tsa_multi_spectrum_lag (g13cc)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_tsa_multi_spectrum_lag (g13cc) calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows: rectangular, Bartlett, Tukey or Parzen.

Syntax

[cxy, cyx, xg, yg, ng, ifail] = g13cc(nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'nc', nc, 'xg', xg, 'yg', yg)
[cxy, cyx, xg, yg, ng, ifail] = nag_tsa_multi_spectrum_lag(nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'nc', nc, 'xg', xg, 'yg', yg)

Description

The smoothed sample cross spectrum is a complex valued function of frequency ω, fxyω=cfω+iqfω, defined by its real part or co-spectrum
cfω=12π ∑k=-M+1 M-1wkCxyk+Scosωk  
and imaginary part or quadrature spectrum
qfω=12π ∑k=-M+ 1 M- 1wkCxyk+Ssinω k  
where wk=w-k, for k=0,1,…,M-1, is the smoothing lag window as defined in the description of nag_tsa_uni_spectrum_lag (g13ca). The alignment shift S is recommended to be chosen as the lag k at which the cross-covariances cxyk peak, so as to minimize bias.
The results are calculated for frequency values
ωj=2πjL,  j=0,1,…,L/2,  
where  denotes the integer part.
The cross-covariances cxyk may be supplied by you, or constructed from supplied series x1,x2,…,xn; y1,y2,…,yn as
cxyk=∑t=1 n-kxtyt+kn,  k≥0  
cxyk=∑t= 1-knxtyt+kn=cyx-k,   k< 0  
this convolution being carried out using the finite Fourier transform.
The supplied series may be mean and trend corrected and tapered before calculation of the cross-covariances, in exactly the manner described in nag_tsa_uni_spectrum_lag (g13ca) for univariate spectrum estimation. The results are corrected for any bias due to tapering.
The bandwidth associated with the estimates is not returned. It will normally already have been calculated in previous calls of nag_tsa_uni_spectrum_lag (g13ca) for estimating the univariate spectra of yt and xt.

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:     nxy – int64int32nag_int scalar
n, the length of the time series x and y.
Constraint: nxy≥1.
2:     mtxy – int64int32nag_int scalar
If cross-covariances are to be calculated by the function (ic=0), mtxy must specify whether the data is to be initially mean or trend corrected.
mtxy=0
For no correction.
mtxy=1
For mean correction.
mtxy=2
For trend correction.
If cross-covariances are supplied ic≠0, mtxy is not used.
Constraint: if ic=0, mtxy=0, 1 or 2.
3:     pxy – double scalar
If cross-covariances are to be calculated by the function (ic=0), pxy must specify 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.
If cross-covariances are supplied ic≠0, pxy is not used.
Constraint: if ic=0, 0.0≤pxy≤1.0.
4:     iw – int64int32nag_int scalar
The choice of lag window.
iw=1
Rectangular.
iw=2
Bartlett.
iw=3
Tukey.
iw=4
Parzen.
Constraint: 1≤iw≤4.
5:     mw – int64int32nag_int scalar
M, the ‘cut-off’ point of the lag window, relative to any alignment shift that has been applied. Windowed cross-covariances at lags -mw+ish or less, and at lags mw+ish or greater are zero.
Constraints:
  • mw≥1;
  • mw+ish≤nxy.
6:     ish – int64int32nag_int scalar
S, the alignment shift between the x and y series. If x leads y, the shift is positive.
Constraint: -mw<ish<mw.
7:     ic – int64int32nag_int scalar
Indicates whether cross-covariances are to be calculated in the function or supplied in the call to the function.
ic=0
Cross-covariances are to be calculated.
ic≠0
Cross-covariances are to be supplied.
8:     cxync – double array
If ic≠0, cxy must contain the nc cross-covariances between values in the y series and earlier values in time in the x series, for lags from 0 to nc-1.
If ic=0, cxy need not be set.
9:     cyxnc – double array
If ic≠0, cyx must contain the nc cross-covariances between values in the y series and later values in time in the x series, for lags from 0 to nc-1.
If ic=0, cyx need not be set.
10:   kc – int64int32nag_int scalar
If ic=0, kc must specify the order of the fast Fourier transform (FFT) used to calculate the cross-covariances. kc should be a product of small primes such as 2m where m is the smallest integer such that 2m≥n+nc.
If ic≠0, that is if covariances are supplied, kc is not used.
Constraint: kc≥nxy+nc. 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.
11:   l – int64int32nag_int scalar
L, the frequency division of the spectral estimates as 2πL . Therefore it is also the order of the FFT used to construct the sample spectrum from the cross-covariances. l should be a product of small primes such as 2m where m is the smallest integer such that 2m≥2M-1.
Constraint: l≥2×mw-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.

Optional Input Parameters

1:     nc – int64int32nag_int scalar
Default: the dimension of the arrays cxy, cyx. (An error is raised if these dimensions are not equal.)
The number of cross-covariances to be calculated in the function or supplied in the call to the function.
Constraint: mw+ish≤nc≤nxy.
2:     xgnxyg – double array
If the cross-covariances are to be calculated, then xg must contain the nxy data points of the x series. If covariances are supplied, xg need not be set.
3:     ygnxyg – double array
If cross-covariances are to be calculated, yg must contain the nxy data points of the y series. If covariances are supplied, yg need not be set.

Output Parameters

1:     cxync – double array
If ic=0, cxy will contain the nc calculated cross-covariances.
If ic≠0, the contents of cxy will be unchanged.
2:     cyxnc – double array
If ic=0, cyx will contain the nc calculated cross-covariances.
If ic≠0, the contents of cyx will be unchanged.
3:     xgnxyg – double array
Contains the real parts of the ng complex spectral estimates in elements xg1 to xgng, and xgng+1 to xgnxyg contain 0.0. The y series leads the x series.
4:     ygnxyg – double array
Contains the imaginary parts of the ng complex spectral estimates in elements yg1 to ygng, and ygng+1 to ygnxyg contain 0.0. The y series leads the x series.
5:     ng – int64int32nag_int scalar
The number, l/2+1, of complex spectral estimates, whose separate parts are held in xg and yg.
6:     ifail – int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,nxy<1,
ormtxy<0 and ic=0,
ormtxy>2 and ic=0,
orpxy<0.0 and ic=0,
orpxy>1.0 and ic=0,
oriw≤0,
oriw>4,
ormw<1,
ormw+ish>nxy,
orish≥mw,
ornc<mw+ish,
ornc>nxy,
ornxyg<maxkc,l and ic=0,
ornxyg<l and ic≠0.
   ifail=2
On entry,kc<nxy+nc,
orkc has a prime factor exceeding 19,
orkc has more than 20 prime factors, counting repetitions.
This error only occurs when ic=0.
   ifail=3
On entry,l<2×mw-1,
orl has a prime factor exceeding 19,
orl has more than 20 prime factors, counting repetitions.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   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.

Further Comments

nag_tsa_multi_spectrum_lag (g13cc) carries out two FFTs of length kc to calculate the sample cross-covariances and one FFT of length L to calculate the sample spectrum. The timing of nag_tsa_multi_spectrum_lag (g13cc) is therefore dependent on the choice of these values. The time taken for an FFT of length n is approximately proportional to nlogn (but see Further Comments in nag_sum_fft_realherm_1d (c06pa) for further details).

Example

This example reads two time series of length 296. It then selects mean correction, a 10% tapering proportion, the Parzen smoothing window and a cut-off point of 35 for the lag window. The alignment shift is set to 3 and 50 cross-covariances are chosen to be calculated. The program then calls nag_tsa_multi_spectrum_lag (g13cc) to calculate the cross spectrum and then prints the cross-covariances and cross spectrum.

Open in the MATLAB editor: g13cc_example

function g13cc_example


fprintf('g13cc example results\n\n');

% Problem size
nxy  = int64(296);
ic   = int64(0);
nc   = 50;

% Control parameters
mtxy = int64(1);
pxy  = 0.1;
iw   = int64(4);
mw   = int64(35);
ish  = int64(3);
kc   = int64(350);
l    = int64(80);

cxy  = zeros(nc, 1);
cyx  = zeros(nc, 1);

% 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];

[cxy, cyx, xg, yg, ng, ifail] = ...
  g13cc( ...
	 nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'xg', xg, 'yg', yg);

% Display results
fprintf('                  Returned cross covariances\n\n');
fprintf(' Lag     XY       YX   Lag     XY       YX   Lag     XY       YX\n');
result = [double([0:nc-1]); cxy'; cyx'];
for j = 1:3:nc
  fprintf('%4d%9.4f%9.4f', result(:,j:min(j+2,nc)));
  fprintf('\n');
end
fprintf('\n                      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



g13cc example results

                  Returned cross covariances

 Lag     XY       YX   Lag     XY       YX   Lag     XY       YX
   0  -1.6700  -1.6700   1  -2.0581  -1.3606   2  -2.4859  -1.1383
   3  -2.8793  -0.9926   4  -3.1473  -0.9009   5  -3.2239  -0.8382
   6  -3.0929  -0.7804   7  -2.7974  -0.7074   8  -2.4145  -0.6147
   9  -2.0237  -0.5080  10  -1.6802  -0.4032  11  -1.4065  -0.3159
  12  -1.2049  -0.2554  13  -1.0655  -0.2250  14  -0.9726  -0.2238
  15  -0.9117  -0.2454  16  -0.8658  -0.2784  17  -0.8180  -0.3081
  18  -0.7563  -0.3257  19  -0.6750  -0.3315  20  -0.5754  -0.3321
  21  -0.4701  -0.3308  22  -0.3738  -0.3312  23  -0.3023  -0.3332
  24  -0.2665  -0.3384  25  -0.2645  -0.3506  26  -0.2847  -0.3727
  27  -0.3103  -0.3992  28  -0.3263  -0.4152  29  -0.3271  -0.4044
  30  -0.3119  -0.3621  31  -0.2837  -0.2919  32  -0.2568  -0.2054
  33  -0.2427  -0.1185  34  -0.2490  -0.0414  35  -0.2774   0.0227
  36  -0.3218   0.0697  37  -0.3705   0.1039  38  -0.4083   0.1356
  39  -0.4197   0.1805  40  -0.3920   0.2460  41  -0.3241   0.3319
  42  -0.2273   0.4325  43  -0.1216   0.5331  44  -0.0245   0.6199
  45   0.0528   0.6875  46   0.1074   0.7329  47   0.1448   0.7550
  48   0.1713   0.7544  49   0.1943   0.7349

                      Returned sample spectrum

        Real  Imaginary       Real  Imaginary       Real  Imaginary
 Lag    part     part  Lag    part     part  Lag    part     part
   0  -6.5500   0.0000   1  -5.4267  -1.9842   2  -3.1323  -2.7307
   3  -1.2649  -2.3998   4  -0.2102  -1.7520   5   0.3411  -1.1903
   6   0.6063  -0.7420   7   0.6178  -0.3586   8   0.4391  -0.1008
   9   0.2422   0.0061  10   0.1233   0.0409  11   0.0574   0.0529
  12   0.0174   0.0452  13  -0.0008   0.0289  14  -0.0058   0.0161
  15  -0.0051   0.0084  16  -0.0027   0.0040  17  -0.0010   0.0015
  18  -0.0006   0.0006  19  -0.0005   0.0003  20  -0.0003   0.0003
  21  -0.0003   0.0004  22  -0.0003   0.0003  23  -0.0003   0.0002
  24  -0.0004   0.0001  25  -0.0004  -0.0000  26  -0.0003  -0.0001
  27  -0.0002  -0.0001  28  -0.0001   0.0001  29  -0.0002   0.0003
  30  -0.0003   0.0002  31  -0.0002   0.0001  32  -0.0001   0.0000
  33  -0.0000  -0.0000  34   0.0001  -0.0001  35   0.0001  -0.0002
  36   0.0001  -0.0001  37   0.0001  -0.0001  38   0.0001  -0.0001
  39   0.0001  -0.0001  40   0.0001   0.0000
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015