NAG Library Routine Document

G13CCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

G13CCF calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows: rectangular, Bartlett, Tukey or Parzen.

2 Specification

SUBROUTINE G13CCF (

NXY, MTXY, PXY, IW, MW, ISH, IC, NC, CXY, CYX, KC, L, NXYG, XG, YG, NG, IFAIL)

INTEGER	NXY, MTXY, IW, MW, ISH, IC, NC, KC, L, NXYG, NG, IFAIL
REAL (KIND=nag_wp)	PXY, CXY(NC), CYX(NC), XG(NXYG), YG(NXYG)

3 Description

The smoothed sample cross spectrum is a complex valued function of frequency

ω

f_{x y} (ω) = c f (ω) + i q f (ω)

, defined by its real part or co-spectrum

c f (ω) = \frac{1}{2 π} \sum_{k = - M + 1}^{M - 1} w_{k} C_{x y} (k + S) \cos (ω k)

and imaginary part or quadrature spectrum

q f (ω) = \frac{1}{2 π} \sum_{k = - M + 1}^{M - 1} w_{k} C_{x y} (k + S) \sin (ω k)

where

w_{k} = w_{- k}

, for

k = 0, 1, \dots, M - 1

, is the smoothing lag window as defined in the description of G13CAF. The alignment shift

S

is recommended to be chosen as the lag

k

at which the cross-covariances

c_{x y} (k)

peak, so as to minimize bias.

The results are calculated for frequency values

ω_{j} = \frac{2 π j}{L}, j = 0, 1, \dots, [L / 2],

where

[]

denotes the integer part.

The cross-covariances

c_{x y} (k)

may be supplied by you, or constructed from supplied series

x_{1}, x_{2}, \dots, x_{n}

;

y_{1}, y_{2}, \dots, y_{n}

c_{x y} (k) = \frac{\sum_{t = 1}^{n - k} x_{t} y_{t + k}}{n}, k \geq 0

c_{x y} (k) = \frac{\sum_{t = 1 - k}^{n} x_{t} y_{t + k}}{n} = c_{y x} (- 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 G13CAF 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 G13CAF 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 Parameters

1: $NXY$ – INTEGERInput

On entry:

n

, the length of the time series

x

and

y

Constraint:

NXY \geq 1

2: $MTXY$ – INTEGERInput

On entry: if cross-covariances are to be calculated by the routine (

IC = 0

), MTXY must specify whether the data is to be initially mean or trend corrected.

$MTXY = 0$: For no correction.
$MTXY = 1$: For mean correction.
$MTXY = 2$: For trend correction.

If cross-covariances are supplied

(IC \neq 0)

, MTXY is not used.

Constraint: if

IC = 0

MTXY = 0

1

2

3: $PXY$ – REAL (KIND=nag_wp)Input

On entry: if cross-covariances are to be calculated by the routine (

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

(IC \neq 0)

, PXY is not used.

Constraint: if

IC = 0

0.0 \leq PXY \leq 1.0

4: $IW$ – INTEGERInput

On entry: the choice of lag window.

$IW = 1$: Rectangular.
$IW = 2$: Bartlett.
$IW = 3$: Tukey.
$IW = 4$: Parzen.

Constraint:

1 \leq IW \leq 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

(- MW + ISH)

or less, and at lags

(MW + ISH)

or greater are zero.

Constraints:

$MW \geq 1$ ;
$MW + |ISH| \leq NXY$ .

6: $ISH$ – INTEGERInput

On entry:

S

, the alignment shift between the

x

and

y

series. If

x

leads

y

, the shift is positive.

Constraint:

- MW < ISH < 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.

$IC = 0$: Cross-covariances are to be calculated.
$IC \neq 0$: 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:

MW + |ISH| \leq NC \leq NXY

9: $CXY (NC)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

IC \neq 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

(NC - 1)

IC = 0

, CXY need not be set.

On exit: if

IC = 0

, CXY will contain the NC calculated cross-covariances.

IC \neq 0

, the contents of CXY will be unchanged.

10: $CYX (NC)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

IC \neq 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

(NC - 1)

IC = 0

, CYX need not be set.

On exit: if

IC = 0

, CYX will contain the NC calculated cross-covariances.

IC \neq 0

, the contents of CYX will be unchanged.

11: $KC$ – INTEGERInput

On entry: if

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} \geq n + NC

IC \neq 0

, that is if covariances are supplied, KC is not used.

Constraint:

KC \geq NXY + 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 π}{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} \geq 2 M - 1

Constraint:

L \geq 2 \times 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 $IC = 0$ , $NXYG \geq \max (KC, L)$ ;
otherwise $NXYG \geq L$ .

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

XG (1)

XG (NG)

, and

XG (NG + 1)

XG (NXYG)

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

YG (1)

YG (NG)

, and

YG (NG + 1)

YG (NXYG)

contain

0.0

. The

y

series leads the

x

series.

16: $NG$ – INTEGEROutput

On exit: the number,

[L / 2] + 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 ​ 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 ​ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NXY < 1$ ,
or	$MTXY < 0$ and $IC = 0$ ,
or	$MTXY > 2$ and $IC = 0$ ,
or	$PXY < 0.0$ and $IC = 0$ ,
or	$PXY > 1.0$ and $IC = 0$ ,
or	$IW \leq 0$ ,
or	$IW > 4$ ,
or	$MW < 1$ ,
or	$MW + \|ISH\| > NXY$ ,
or	$\|ISH\| \geq MW$ ,
or	$NC < MW + \|ISH\|$ ,
or	$NC > NXY$ ,
or	$NXYG < \max (KC, L)$ and $IC = 0$ ,
or	$NXYG < L$ and $IC \neq 0$ .

$IFAIL = 2$

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

This error only occurs when

IC = 0

$IFAIL = 3$

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

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

G13CCF 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 G13CCF is therefore dependent on the choice of these values. The time taken for an FFT of length

n

is approximately proportional to

n \log (n)

(but see Section 9 in C06PAF 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 G13CCF to calculate the cross spectrum and then prints the cross-covariances and cross spectrum.

NAG Library Routine DocumentG13CCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

G13CCF