NAG Library Routine Document

G13CCF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NXY – INTEGERInput

On entry: $n$, the length of the time series $x$ and $y$.

Constraint: $\mathbf{NXY}\ge 1$.

2:     MTXY – INTEGERInput

On entry: if cross-covariances are to be calculated by the routine ($\mathbf{IC}=0$), MTXY must specify 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.

If cross-covariances are supplied $\left(\mathbf{IC}\ne 0\right)$, MTXY is not used.

Constraint: if $\mathbf{IC}=0$, $\mathbf{MTXY}=0$, $1$ or $2$.

3:     PXY – REAL (KIND=nag_wp)Input

On entry: if cross-covariances are to be calculated by the routine ($\mathbf{IC}=0$), 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:     IW – INTEGERInput

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:     MW – INTEGERInput

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:     ISH – INTEGERInput

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:     IC – INTEGERInput

On entry: indicates whether cross-covariances are to be calculated in the routine or supplied in the call to the routine.

$\mathbf{IC}=0$

Cross-covariances are to be calculated.

$\mathbf{IC}\ne 0$

Cross-covariances are to be supplied.

8:     NC – INTEGERInput

On entry: the number of cross-covariances to be calculated in the routine or supplied in the call to the routine.

Constraint: $\mathbf{MW}+\left|\mathbf{ISH}\right|\le \mathbf{NC}\le \mathbf{NXY}$.

9:     CXY(NC) – REAL (KIND=nag_wp) arrayInput/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) – REAL (KIND=nag_wp) arrayInput/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:   KC – INTEGERInput

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:   L – INTEGERInput

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: $\mathbf{L}\ge 2\times \mathbf{MW}-1$. 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:   NXYG – INTEGERInput

On entry: the dimension of the arrays XG and YG as declared in the (sub)program from which G13CCF is called.

Constraints:

• if $\mathbf{IC}=0$, $\mathbf{NXYG}\ge \max \left(\mathbf{KC},\mathbf{L}\right)$;
• if $\mathbf{IC}\ne 0$, $\mathbf{NXYG}\ge \mathbf{L}$.

14:   XG(NXYG) – REAL (KIND=nag_wp) arrayInput/Output

On entry: 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.

On exit: contains the real parts of the NG complex 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 $\mathbf{XG}\left(\mathbf{NXYG}\right)$ contain $0.0$. The $y$ series leads the $x$ series.

15:   YG(NXYG) – REAL (KIND=nag_wp) arrayInput/Output

On entry: 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.

On exit: contains the imaginary parts of the NG complex 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 $\mathbf{YG}\left(\mathbf{NXYG}\right)$ contain $0.0$. The $y$ series leads the $x$ series.

16:   NG – INTEGEROutput

On exit: the number, $\left[\mathbf{L}/2\right]+1$, of complex spectral estimates, whose separate parts are held in XG and YG.

17:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ 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).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{NXY}<1$,
or	$\mathbf{MTXY}<0$ and $\mathbf{IC}=0$,
or	$\mathbf{MTXY}>2$ and $\mathbf{IC}=0$,
or	$\mathbf{PXY}<0.0$ and $\mathbf{IC}=0$,
or	$\mathbf{PXY}>1.0$ and $\mathbf{IC}=0$,
or	$\mathbf{IW}\le 0$,
or	$\mathbf{IW}>4$,
or	$\mathbf{MW}<1$,
or	$\mathbf{MW}+\left|\mathbf{ISH}\right|>\mathbf{NXY}$,
or	$\left|\mathbf{ISH}\right|\ge \mathbf{MW}$,
or	$\mathbf{NC}<\mathbf{MW}+\left|\mathbf{ISH}\right|$,
or	$\mathbf{NC}>\mathbf{NXY}$,
or	$\mathbf{NXYG}<\max \left(\mathbf{KC},\mathbf{L}\right)$ and $\mathbf{IC}=0$,
or	$\mathbf{NXYG}<\mathbf{L}$ and $\mathbf{IC}\ne 0$.

$\mathbf{IFAIL}=2$

On entry,	$\mathbf{KC}<\mathbf{NXY}+\mathbf{NC}$,
or	KC has a prime factor exceeding $19$,
or	KC has more than $20$ prime factors, counting repetitions.

This error only occurs when $\mathbf{IC}=0$.

$\mathbf{IFAIL}=3$

On entry,	$\mathbf{L}<2\times \mathbf{MW}-1$,
or	L has a prime factor exceeding $19$,
or	L has more than $20$ prime factors, counting repetitions.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (g13ccfe.f90)

9.2  Program Data

Program Data (g13ccfe.d)

9.3  Program Results

Program Results (g13ccfe.r)