# NAG FL Interfaceg13cdf (multi_​spectrum_​daniell)

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## 1Purpose

g13cdf calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

## 2Specification

Fortran Interface
 Subroutine g13cdf ( nxy, mtxy, pxy, mw, ish, pw, l, kc, xg, yg, ng,
 Integer, Intent (In) :: nxy, mtxy, mw, ish, l, kc Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ng Real (Kind=nag_wp), Intent (In) :: pxy, pw Real (Kind=nag_wp), Intent (Inout) :: xg(kc), yg(kc)
#include <nag.h>
 void g13cdf_ (const Integer *nxy, const Integer *mtxy, const double *pxy, const Integer *mw, const Integer *ish, const double *pw, const Integer *l, const Integer *kc, double xg[], double yg[], Integer *ng, Integer *ifail)
The routine may be called by the names g13cdf or nagf_tsa_multi_spectrum_daniell.

## 3Description

The supplied time series may be mean and trend corrected and tapered as in the description of g13cbf before calculation of the unsmoothed sample cross-spectrum
 $fxy* (ω) = 12πn { ∑ t=1 n ytexp(iωt)} × { ∑ t=1 n xtexp(-iωt)}$
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 g13cbf 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 g13cbf, but allow for an implicit alignment shift $S$ between the series:
 $w~k=wkexp(-2πiSk/L).$
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 g13cbf for estimating the univariate spectra of ${y}_{t}$ and ${x}_{t}$.
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: $n$, the length of the time series $x$ and $y$.
Constraint: ${\mathbf{nxy}}\ge 1$.
2: $\mathbf{mtxy}$Integer Input
On entry: 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: $\mathbf{pxy}$Real (Kind=nag_wp) 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: $M$, 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: $\mathbf{ish}$Integer Input
On entry: $S$, 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: $\mathbf{pw}$Real (Kind=nag_wp) Input
On entry: $p$, 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: $\mathbf{l}$Integer Input
On entry: $L$, 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: $\mathbf{kc}$Integer Input
On entry: the order of the fast Fourier transform (FFT) used to calculate the spectral estimates.
Constraints:
• ${\mathbf{kc}}\ge 2×{\mathbf{nxy}}$;
• kc must be a multiple of l.
9: $\mathbf{xg}\left({\mathbf{kc}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: the nxy data points of the $x$ series.
On exit: 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.
10: $\mathbf{yg}\left({\mathbf{kc}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: the nxy data points of the $y$ series.
On exit: 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.
11: $\mathbf{ng}$Integer Output
On exit: the number of spectral estimates, $\left[L/2\right]+1$, whose separate parts are held in xg and yg.
12: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ish}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$.
Constraint: $|{\mathbf{ish}}|<{\mathbf{l}}$.
On entry, ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{l}}\ge 1$.
On entry, ${\mathbf{mtxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mtxy}}\le 2$.
On entry, ${\mathbf{mtxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mtxy}}\ge 0$.
On entry, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mw}}\ge 1$.
On entry, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mw}}\le {\mathbf{nxy}}$.
On entry, ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nxy}}\ge 1$.
On entry, ${\mathbf{pxy}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{pw}}<0.0$, ${\mathbf{mw}}={\mathbf{nxy}}$.
On entry, ${\mathbf{pxy}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mw}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{pw}}>1.0$, ${\mathbf{mw}}={\mathbf{nxy}}$.
On entry, ${\mathbf{pxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pxy}}\le 1.0$.
On entry, ${\mathbf{pxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pxy}}\ge 0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{kc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$.
Constraint: kc must be a multiple of l.
On entry, ${\mathbf{kc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nxy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{kc}}\ge 2×{\mathbf{nxy}}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 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

g13cdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13cdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

g13cdf carries out an FFT of length kc to calculate the sample cross spectrum. The time taken by the routine for this is approximately proportional to ${\mathbf{kc}}×\mathrm{log}\left({\mathbf{kc}}\right)$ (but see routine document c06paf for further details).

## 10Example

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 g13cdf to calculate the smoothed sample cross spectrum and prints the results.

### 10.1Program Text

Program Text (g13cdfe.f90)

### 10.2Program Data

Program Data (g13cdfe.d)

### 10.3Program Results

Program Results (g13cdfe.r)