g13 Chapter Contents
g13 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_tsa_spectrum_bivar_cov (g13ccc)

## 1  Purpose

nag_tsa_spectrum_bivar_cov (g13ccc) calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows: rectangular, Bartlett, Tukey or Parzen.

## 2  Specification

 #include #include
 void nag_tsa_spectrum_bivar_cov (Integer nxy, NagMeanOrTrend mtxy_correction, double pxy, Integer iw, Integer mw, Integer ish, Integer ic, Integer nc, double cxy[], double cyx[], Integer kc, Integer l, double xg[], double yg[], Complex g[], Integer *ng, NagError *fail)

## 3  Description

The smoothed sample cross spectrum is a complex valued function of frequency $\omega$, ${f}_{xy}\left(\omega \right)=cf\left(\omega \right)+iqf\left(\omega \right)$, 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 ${w}_{\mathit{k}}={w}_{-\mathit{k}}$, for $\mathit{k}=0,1,\dots ,M-1$, is the smoothing lag window as defined in the description of nag_tsa_spectrum_univar_cov (g13cac). The alignment shift $S$ is recommended to be chosen as the lag $k$ at which the cross-covariances ${c}_{xy}\left(k\right)$ peak, so as to minimize bias.
The results are calculated for frequency values
 $ωj=2πjL, j=0,1,…,L/2,$
where $\left[\right]$ denotes the integer part.
The cross-covariances ${c}_{xy}\left(k\right)$ may be supplied by you, or constructed from supplied series ${x}_{1},{x}_{2},\dots ,{x}_{n}$; ${y}_{1},{y}_{2},\dots ,{y}_{n}$ 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_spectrum_univar_cov (g13cac) 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_spectrum_univar_cov (g13cac) for estimating the univariate spectra of ${y}_{t}$ and ${x}_{t}$.

## 4  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

## 5  Arguments

1:     nxyIntegerInput
On entry: $n$, the length of the time series $x$ and $y$.
Constraint: ${\mathbf{nxy}}\ge 1$.
2:     mtxy_correctionNagMeanOrTrendInput
On entry: if cross-covariances are to be calculated by the function (${\mathbf{ic}}=0$), mtxy_correction must specify whether the data is to be initially mean or trend corrected.
${\mathbf{mtxy_correction}}=\mathrm{Nag_NoCorrection}$
For no correction.
${\mathbf{mtxy_correction}}=\mathrm{Nag_Mean}$
For mean correction.
${\mathbf{mtxy_correction}}=\mathrm{Nag_Trend}$
For trend correction.
If cross-covariances are supplied $\left({\mathbf{ic}}\ne 0\right)$, mtxy_correction should be set to ${\mathbf{mtxy_correction}}=\mathrm{Nag_NoCorrection}$
Constraint: if ${\mathbf{ic}}=0$, ${\mathbf{mtxy_correction}}=\mathrm{Nag_NoCorrection}$, $\mathrm{Nag_Mean}$ or $\mathrm{Nag_Trend}$.
3:     pxydoubleInput
On entry: if cross-covariances are to be calculated by the function (${\mathbf{ic}}=0$), 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 $\left({\mathbf{ic}}\ne 0\right)$, pxy is not used.
Constraint: if ${\mathbf{ic}}=0$, $0.0\le {\mathbf{pxy}}\le 1.0$.
4:     iwIntegerInput
On entry: the choice of lag window.
${\mathbf{iw}}=1$
Rectangular.
${\mathbf{iw}}=2$
Bartlett.
${\mathbf{iw}}=3$
Tukey.
${\mathbf{iw}}=4$
Parzen.
Constraint: $1\le {\mathbf{iw}}\le 4$.
5:     mwIntegerInput
On entry: $M$, the ‘cut-off’ point of the lag window, relative to any alignment shift that has been applied. Windowed cross-covariances at lags $\left(-{\mathbf{mw}}+{\mathbf{ish}}\right)$ or less, and at lags $\left({\mathbf{mw}}+{\mathbf{ish}}\right)$ or greater are zero.
Constraints:
• ${\mathbf{mw}}\ge 1$;
• ${\mathbf{mw}}+\left|{\mathbf{ish}}\right|\le {\mathbf{nxy}}$.
6:     ishIntegerInput
On entry: $S$, the alignment shift between the $x$ and $y$ series. If $x$ leads $y$, the shift is positive.
Constraint: $-{\mathbf{mw}}<{\mathbf{ish}}<{\mathbf{mw}}$.
7:     icIntegerInput
On entry: indicates whether cross-covariances are to be calculated in the function or supplied in the call to the function.
${\mathbf{ic}}=0$
Cross-covariances are to be calculated.
${\mathbf{ic}}\ne 0$
Cross-covariances are to be supplied.
8:     ncIntegerInput
On entry: the number of cross-covariances to be calculated in the function or supplied in the call to the function.
Constraint: ${\mathbf{mw}}+\left|{\mathbf{ish}}\right|\le {\mathbf{nc}}\le {\mathbf{nxy}}$.
9:     cxy[nc]doubleInput/Output
On entry: if ${\mathbf{ic}}\ne 0$, 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 $\left({\mathbf{nc}}-1\right)$.
If ${\mathbf{ic}}=0$, cxy need not be set.
On exit: if ${\mathbf{ic}}=0$, cxy will contain the nc calculated cross-covariances.
If ${\mathbf{ic}}\ne 0$, the contents of cxy will be unchanged.
10:   cyx[nc]doubleInput/Output
On entry: if ${\mathbf{ic}}\ne 0$, 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 $\left({\mathbf{nc}}-1\right)$.
If ${\mathbf{ic}}=0$, cyx need not be set.
On exit: if ${\mathbf{ic}}=0$, cyx will contain the nc calculated cross-covariances.
If ${\mathbf{ic}}\ne 0$, the contents of cyx will be unchanged.
11:   kcIntegerInput
On entry: if ${\mathbf{ic}}=0$, 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 ${2}^{m}$ where $m$ is the smallest integer such that ${2}^{m}\ge n+{\mathbf{nc}}$.
If ${\mathbf{ic}}\ne 0$, that is if covariances are supplied, kc is not used.
Constraint: ${\mathbf{kc}}\ge {\mathbf{nxy}}+{\mathbf{nc}}$. 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.
12:   lIntegerInput
On entry: $L$, the frequency division of the spectral estimates as $\frac{2\pi }{L}$. 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 ${2}^{m}$ where $m$ is the smallest integer such that ${2}^{m}\ge 2M-1$.
Constraint: ${\mathbf{l}}\ge 2×{\mathbf{mw}}-1$. 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.
13:   xg[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array xg must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{kc}},{\mathbf{l}}\right)$, when ${\mathbf{ic}}=0$;
• ${\mathbf{l}}$, when ${\mathbf{ic}}\ne 0$.
On entry: if the cross-covariances are to be calculated (${\mathbf{ic}}=0$) xg must contain the nxy data points of the $x$ series. If covariances are supplied (${\mathbf{ic}}\ne 0$) xg may contain any values.
On exit: contains the real parts of the ng complex spectral estimates in elements ${\mathbf{xg}}\left[0\right]$ to ${\mathbf{xg}}\left[{\mathbf{ng}}-1\right]$, and ${\mathbf{xg}}\left[{\mathbf{ng}}\right]$ to ${\mathbf{xg}}\left[\mathit{dim}-1\right]$ contain $0.0$. The $y$ series leads the $x$ series.
14:   yg[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array yg must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{kc}},{\mathbf{l}}\right)$, when ${\mathbf{ic}}=0$;
• ${\mathbf{l}}$, when ${\mathbf{ic}}\ne 0$.
On entry: if the cross-covariances are to be calculated (${\mathbf{ic}}=0$) yg must contain the nxy data points of the $y$ series. If covariances are supplied (${\mathbf{ic}}\ne 0$) yg may contain any values.
On exit: contains the imaginary parts of the ng complex spectral estimates in elements ${\mathbf{yg}}\left[0\right]$ to ${\mathbf{yg}}\left[{\mathbf{ng}}-1\right]$, and ${\mathbf{yg}}\left[{\mathbf{ng}}\right]$ to ${\mathbf{yg}}\left[\mathit{dim}-1\right]$ contain $0.0$. The $y$ series leads the $x$ series.
15:   g[${\mathbf{l}}/2+1$]ComplexOutput
On exit: the complex vector that contains the ng cross spectral estimates in elements ${\mathbf{g}}\left[0\right]$ to ${\mathbf{g}}\left[{\mathbf{ng}}-1\right]$. The $y$ series leads the $x$ series.
16:   ngInteger *Output
On exit: the number, $\left[{\mathbf{l}}/2\right]+1$, of complex spectral estimates.
17:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{ic}}=0$ and ${\mathbf{mtxy_correction}}\ne \mathrm{Nag_NoCorrection}$, $\mathrm{Nag_Mean}$ or $\mathrm{Nag_Trend}$: ${\mathbf{mtxy_correction}}=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{iw}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{iw}}=1$, $2$, $3$ or $4$.
On entry, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mw}}\ge 1$.
On entry, ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nxy}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{ish}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$.
Constraint: $\left|{\mathbf{ish}}\right|\le {\mathbf{mw}}$.
On entry, ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{l}}\ge 2×{\mathbf{mw}}-1$.
On entry, ${\mathbf{nc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nc}}\le {\mathbf{nxy}}$.
NE_INT_3
On entry, ${\mathbf{kc}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nc}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{ic}}=0$, ${\mathbf{kc}}\ge {\mathbf{nxy}}+{\mathbf{nc}}$.
On entry, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ish}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mw}}+\left|{\mathbf{ish}}\right|\le {\mathbf{nxy}}$.
On entry, ${\mathbf{nc}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ish}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nc}}\ge {\mathbf{mw}}+\left|{\mathbf{ish}}\right|$.
NE_INT_REAL
On entry, ${\mathbf{pxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{ic}}=0$, ${\mathbf{pxy}}\le 1.0$.
On entry, ${\mathbf{pxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{ic}}=0$, ${\mathbf{pxy}}\ge 0.0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_PRIME_FACTOR
kc has a prime factor exceeding $19$, or more than 20 prime factors (counting repetitions): ${\mathbf{kc}}=⟨\mathit{\text{value}}⟩$.
l has a prime factor exceeding $19$, or more than 20 prime factors (counting repetitions): ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$.

## 7  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.

Not applicable.

## 9  Further Comments

nag_tsa_spectrum_bivar_cov (g13ccc) 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_spectrum_bivar_cov (g13ccc) is therefore dependent on the choice of these values. The time taken for an FFT of length $n$ is approximately proportional to $n\mathrm{log}\left(n\right)$ (but see Section 9 in nag_sum_fft_realherm_1d (c06pac) for further details).

## 10  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_spectrum_bivar_cov (g13ccc) to calculate the cross spectrum and then prints the cross-covariances and cross spectrum.

### 10.1  Program Text

Program Text (g13ccce.c)

### 10.2  Program Data

Program Data (g13ccce.d)

### 10.3  Program Results

Program Results (g13ccce.r)