NAG Library Function Document

nag_tsa_spectrum_bivar_cov (g13ccc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_tsa_spectrum_bivar_cov (g13ccc) calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows: rectangular, Bartlett, Tukey or Parzen.

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_spectrum_bivar_cov (Integer nxy, NagMeanOrTrend mtxy_correction, double pxy, Integer iw, Integer mw, Integer ish, Integer ic, Integer nc, double cxy[], double cyx[], Integer kc, Integer l, double xg[], double yg[], Complex g[], Integer ng, NagError fail)

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 nag_tsa_spectrum_univar_cov (g13cac). 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 nag_tsa_spectrum_univar_cov (g13cac) 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 nag_tsa_spectrum_univar_cov (g13cac) 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: nxy – IntegerInput

On entry:

n

, the length of the time series

x

and

y

Constraint:

nxy \geq 1

2: mtxy_correction – NagMeanOrTrendInput

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

ic = 0

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

$mtxy_correction = Nag_NoCorrection$: For no correction.
$mtxy_correction = Nag_Mean$: For mean correction.
$mtxy_correction = Nag_Trend$: For trend correction.

If cross-covariances are supplied

(ic \neq 0)

, mtxy_correction should be set to

mtxy_correction = Nag_NoCorrection

Constraint: if

ic = 0

mtxy_correction = Nag_NoCorrection

Nag_Mean

Nag_Trend

3: pxy – doubleInput

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

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 function or supplied in the call to the function.

$ic = 0$: 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 function or supplied in the call to the function.

Constraint:

mw + |ish| \leq nc \leq nxy

9: cxy[nc] – doubleInput/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] – doubleInput/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: xg[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array xg must be at least

$\max (kc, l)$ , when $ic = 0$ ;
$l$ , when $ic \neq 0$ .

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

ic = 0

) xg must contain the nxy data points of the

x

series. If covariances are supplied (

ic \neq 0

) xg may contain any values.

On exit: contains the real parts of the ng complex spectral estimates in elements

xg [0]

xg [ng - 1]

, and

xg [ng]

xg [\dim - 1]

contain

0.0

. The

y

series leads the

x

series.

14: yg[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array yg must be at least

$\max (kc, l)$ , when $ic = 0$ ;
$l$ , when $ic \neq 0$ .

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

ic = 0

) yg must contain the nxy data points of the

y

series. If covariances are supplied (

ic \neq 0

) yg may contain any values.

On exit: contains the imaginary parts of the ng complex spectral estimates in elements

yg [0]

yg [ng - 1]

, and

yg [ng]

yg [\dim - 1]

contain

0.0

. The

y

series leads the

x

series.

15: g[ $l / 2 + 1$ ] – ComplexOutput

On exit: the complex vector that contains the ng cross spectral estimates in elements

g [0]

g [ng - 1]

. The

y

series leads the

x

series.

16: ng – Integer *Output

On exit: the number,

[l / 2] + 1

, of complex spectral estimates.

17: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $ic = 0$ and $mtxy_correction \neq Nag_NoCorrection$ , $Nag_Mean$ or $Nag_Trend$ : $mtxy_correction = ⟨value⟩$ .
On entry, $iw = ⟨value⟩$ .
Constraint: $iw = 1$ , $2$ , $3$ or $4$ .
On entry, $mw = ⟨value⟩$ .
Constraint: $mw \geq 1$ .
On entry, $nxy = ⟨value⟩$ .
Constraint: $nxy \geq 1$ .
NE_INT_2: On entry, $ish = ⟨value⟩$ and $mw = ⟨value⟩$ .
Constraint: $|ish| \leq mw$ .
On entry, $l = ⟨value⟩$ and $mw = ⟨value⟩$ .
Constraint: $l \geq 2 \times mw - 1$ .
On entry, $nc = ⟨value⟩$ and $nxy = ⟨value⟩$ .
Constraint: $nc \leq nxy$ .
NE_INT_3: On entry, $kc = ⟨value⟩$ , $nxy = ⟨value⟩$ and $nc = ⟨value⟩$ .
Constraint: if $ic = 0$ , $kc \geq nxy + nc$ .
On entry, $mw = ⟨value⟩$ , $ish = ⟨value⟩$ and $nxy = ⟨value⟩$ .
Constraint: $mw + |ish| \leq nxy$ .
On entry, $nc = ⟨value⟩$ , $mw = ⟨value⟩$ and $ish = ⟨value⟩$ .
Constraint: $nc \geq mw + |ish|$ .
NE_INT_REAL: On entry, $pxy = ⟨value⟩$ .
Constraint: if $ic = 0$ , $pxy \leq 1.0$ .
On entry, $pxy = ⟨value⟩$ .
Constraint: if $ic = 0$ , $pxy \geq 0.0$ .
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_PRIME_FACTOR: kc has a prime factor exceeding $19$ , or more than 20 prime factors (counting repetitions): $kc = ⟨value⟩$ .
l has a prime factor exceeding $19$ , or more than 20 prime factors (counting repetitions): $l = ⟨value⟩$ .

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

nag_tsa_spectrum_bivar_cov (g13ccc) 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 nag_tsa_spectrum_bivar_cov (g13ccc) 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 nag_sum_fft_realherm_1d (c06pac) 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 nag_tsa_spectrum_bivar_cov (g13ccc) to calculate the cross spectrum and then prints the cross-covariances and cross spectrum.

NAG Library Function Documentnag_tsa_spectrum_bivar_cov (g13ccc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_tsa_spectrum_bivar_cov (g13ccc)