# NAG CL Interfaceg13cdc (multi_​spectrum_​daniell)

Settings help

CL Name Style:

## 1Purpose

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

## 2Specification

 #include
 void g13cdc (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)
The function may be called by the names: g13cdc, nag_tsa_multi_spectrum_daniell or nag_tsa_spectrum_bivar.

## 3Description

The supplied time series may be mean and trend corrected and tapered as in the description of g13cbc before calculation of the unsmoothed sample cross-spectrum
 $f xy * (ω) = 1 2πn { ∑ t=1 n y t exp(iωt)} × { ∑ t=1 n x t exp(-iωt)}$
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 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 g13cbc, but allow for an implicit alignment shift $S$ between the series:
 $w ~ k = w k exp(−2πiSk/L) .$
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 g13cbc for estimating the univariate spectra of ${y}_{t}$ and ${x}_{t}$.

## 4References

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

## 5Arguments

1: $\mathbf{nxy}$Integer Input
On entry: the length of the time series $x$ and $y$, $n$.
Constraint: ${\mathbf{nxy}}\ge 1$.
2: $\mathbf{mt_correction}$NagMeanOrTrend Input
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: $\mathbf{pxy}$double Input
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: $\mathbf{mw}$Integer Input
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: $\mathbf{is}$Integer Input
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: $\mathbf{pw}$double Input
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: $\mathbf{l}$Integer Input
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: $\mathbf{kc}$Integer Input
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 the internal FFT algorithm used.
9: $\mathbf{x}\left[{\mathbf{kc}}\right]$const double Input
On entry: the nxy data points of the $x$ series.
10: $\mathbf{y}\left[{\mathbf{kc}}\right]$const double Input
On entry: the nxy data points of the $y$ series.
11: $\mathbf{g}$Complex ** 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: $\mathbf{ng}$Integer * Output
On exit: the number of spectral estimates, $\left[L/2\right]+1$, whose separate parts are held in g.
13: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.

## 7Accuracy

The FFT is a numerically stable process, and any errors introduced during the computation will normally be insignificant compared with uncertainty in the data.

## 8Parallelism and Performance

g13cdc is not threaded in any implementation.

g13cdc 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 c06pac for further details).

## 10Example

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 g13cdc to calculate the smoothed sample cross spectrum and prints the results.

### 10.1Program Text

Program Text (g13cdce.c)

### 10.2Program Data

Program Data (g13cdce.d)

### 10.3Program Results

Program Results (g13cdce.r)