g13 Chapter Contents
g13 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_tsa_spectrum_bivar (g13cdc)

## 1  Purpose

nag_tsa_spectrum_bivar (g13cdc) calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

## 2  Specification

 #include #include
 void nag_tsa_spectrum_bivar (Integer nxy, NagMeanOrTrend mt_correction, double pxy, Integer mw, Integer is, double pw, Integer l, Integer kc, const double x[], const double y[], Complex **g, Integer *ng, NagError *fail)

## 3  Description

The supplied time series may be mean and trend corrected and tapered as in the description of nag_tsa_spectrum_univar (g13cbc) before calculation of the unsmoothed sample cross-spectrum
 $f xy * ω = 1 2πn ∑ t=1 n y t expiωt × ∑ t=1 n x t exp -i ω t$
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_spectrum_univar (g13cbc) for univariate frequency window smoothing, the smoothed spectrum is returned at a subset of these frequencies,
 $ν l = 2πl L , l = 0 , 1 , … , L/2$
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
 $f xy ν l = cf ν l + iqf ν l = ∑ ω k < π/M w ~ k f xy * ν l + ω k$
where the weights ${\stackrel{~}{w}}_{k}$ are similar to the weights ${w}_{k}$ defined for nag_tsa_spectrum_univar (g13cbc), but allow for an implicit alignment shift $S$ between the series:
 $w ~ k = w k exp -2 π iSk / L .$
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_spectrum_univar (g13cbc) 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: the length of the time series $x$ and $y$, $n$.
Constraint: ${\mathbf{nxy}}\ge 1$.
2:     mt_correctionNagMeanOrTrendInput
On entry: whether the data are to be initially mean or trend corrected. ${\mathbf{mt_correction}}=\mathrm{Nag_NoCorrection}$ for no correction, ${\mathbf{mt_correction}}=\mathrm{Nag_Mean}$ for mean correction, ${\mathbf{mt_correction}}=\mathrm{Nag_Trend}$ for trend correction.
Constraint: ${\mathbf{mt_correction}}=\mathrm{Nag_NoCorrection}$, $\mathrm{Nag_Mean}$ or $\mathrm{Nag_Trend}$.
3:     pxydoubleInput
On entry: 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:     mwIntegerInput
On entry: the frequency width, $M$, of the smoothing window as $2\pi /M$.
A value of $n$ implies that no smoothing is to be carried out.
Constraint: $1\le {\mathbf{mw}}\le {\mathbf{nxy}}$.
5:     isIntegerInput
On entry: the alignment shift, $S$, between the $x$ and $y$ series. If $x$ leads $y$, the shift is positive.
Constraint: $-{\mathbf{l}}<{\mathbf{is}}<{\mathbf{l}}$.
6:     pwdoubleInput
On entry: the shape argument, $p$, 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: $0.0\le {\mathbf{pw}}\le 1.0$ if ${\mathbf{mw}}\ne {\mathbf{nxy}}$.
7:     lIntegerInput
On entry: the frequency division, $L$, of smoothed cross spectral estimates as $2\pi /L$.
Constraint: ${\mathbf{l}}\ge 1$.
l must be a factor of kc (see below).
8:     kcIntegerInput
On entry: 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 nag_fft_real (c06eac) and nag_fft_hermitian (c06ebc) which perform the FFT.
9:     x[kc]const doubleInput
On entry: the nxy data points of the $x$ series.
10:   y[kc]const doubleInput
On entry: the nxy data points of the $y$ series.
11:   gComplex **Output
On exit: the complex vector which 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.
The memory for this vector is allocated internally. If no memory is allocated to g (e.g., when an input error is detected) then g will be NULL on return. If repeated calls to this function are required then NAG_FREE should be used to free the memory in between calls.
12:   ngInteger *Output
On exit: the number of spectral estimates, $\left[L/2\right]+1$, whose separate parts are held in g.
13:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_CONS
On entry, ${\mathbf{kc}}=〈\mathit{\text{value}}〉$ while ${\mathbf{l}}=〈\mathit{\text{value}}〉$. These arguments must satisfy kc%${\mathbf{l}}\ne 0$ when ${\mathbf{l}}>0$.
On entry, ${\mathbf{kc}}=〈\mathit{\text{value}}〉$ while ${\mathbf{nxy}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{kc}}\ge 2$*nxy when ${\mathbf{nxy}}>0$.
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$ while ${\mathbf{is}}=〈\mathit{\text{value}}〉$. These arguments must satisfy $‖{\mathbf{is}}‖<{\mathbf{l}}$ when ${\mathbf{l}}>0$.
NE_2_INT_ARG_GT
On entry, ${\mathbf{mw}}=〈\mathit{\text{value}}〉$ while ${\mathbf{nxy}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{mw}}\le {\mathbf{nxy}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument mt_correction had an illegal value.
NE_FACTOR_GT
At least one of the prime factors of kc is greater than 19.
NE_INT_ARG_LT
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\ge 1$.
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_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_REAL_ARG_GT
On entry, pw must not be greater than 1.0: ${\mathbf{pw}}=〈\mathit{\text{value}}〉$.
On entry, pxy must not be greater than 1.0: ${\mathbf{pxy}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARG_LT
On entry, pw must not be less than 0.0: ${\mathbf{pw}}=〈\mathit{\text{value}}〉$.
On entry, pxy must not be less than 0.0: ${\mathbf{pxy}}=〈\mathit{\text{value}}〉$.
NE_TOO_MANY_FACTORS
kc has more than 20 prime factors.

## 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.

nag_tsa_spectrum_bivar (g13cdc) carries out an FFT of length kc to calculate the sample cross spectrum. The time taken by nag_tsa_spectrum_bivar (g13cdc) for this is approximately proportional to ${\mathbf{kc}}×\mathrm{log}\left({\mathbf{kc}}\right)$ (see nag_fft_real (c06eac) for further details).

## 9  Example

The example program reads 2 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 argument of 0.5 and an alignment shift of 3. It then calls nag_tsa_spectrum_bivar (g13cdc) to calculate the smoothed sample cross spectrum and prints the results.

### 9.1  Program Text

Program Text (g13cdce.c)

### 9.2  Program Data

Program Data (g13cdce.d)

### 9.3  Program Results

Program Results (g13cdce.r)