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

2 Specification

#include <nag.h>

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.

3 Description

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_{x y}^{*} (ω) = \frac{1}{2 π n} {\sum_{t = 1}^{n} y_{t} \exp (i ω t)} \times {\sum_{t = 1}^{n} x_{t} \exp (- i ω t)}

for frequency values

ω_{j} = \frac{2 π j}{K}

0 \leq ω_{j} \leq π

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

where [ ] denotes the integer part.

Its real part or co-spectrum

c f (ν_{l})

, and imaginary part or quadrature spectrum

q f (ν_{l})

are defined by

f_{x y} (ν_{l}) = c f (ν_{l}) + i q f (ν_{l}) = \sum_{| ω_{k} | < π / M} {\tilde{w}}_{k} f_{x y}^{*} (ν_{l} + ω_{k})

where the weights

{\tilde{w}}_{k}

are similar to the weights

w_{k}

defined for g13cbc, but allow for an implicit alignment shift

S

between the series:

{\tilde{w}}_{k} = w_{k} \exp (−2 π i S k / L) .

It is recommended that

S

is chosen as the lag

k

at which the cross-covariances

c_{x y} (k)

peak, so as to minimize bias.

If no smoothing is required, the integer

M

which determines the frequency window width

\frac{2 π}{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}

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: $nxy$ – Integer Input

On entry: the length of the time series

x

and

y

n

Constraint:

nxy \geq 1

2: $mt_correction$ – NagMeanOrTrend Input

On entry: whether the data are to be initially mean or trend corrected.

mt_correction = Nag_NoCorrection

for no correction,

mt_correction = Nag_Mean

for mean correction,

mt_correction = Nag_Trend

for trend correction.

Constraint:

mt_correction = Nag_NoCorrection

Nag_Mean

Nag_Trend

3: $pxy$ – 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 \leq pxy \leq 1.0

4: $mw$ – Integer Input

On entry: the frequency width,

M

, of the smoothing window as

2 π / M

A value of

n

implies that no smoothing is to be carried out.

Constraint:

1 \leq mw \leq nxy

5: $is$ – Integer Input

On entry: the alignment shift,

S

, between the

x

and

y

series. If

x

leads

y

, the shift is positive.

Constraint:

- l < is < l

6: $pw$ – 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.

mw = nxy

(i.e., no smoothing is carried out) then pw is not used.

Constraint:

0.0 \leq pw \leq 1.0

mw \neq nxy

7: $l$ – Integer Input

On entry: the frequency division,

L

, of smoothed cross spectral estimates as

2 π / L

Constraint:

l \geq 1

l must be a factor of kc (see below).

8: $kc$ – 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} \geq 2 n

, provided

m \leq 20

Constraints:

$kc \geq 2 \times nxy$ ;
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: $x [kc]$ – const double Input

On entry: the nxy data points of the

x

series.

10: $y [kc]$ – const double Input

On entry: the nxy data points of the

y

series.

11: $g$ – Complex ** Output

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

g [0]

g [ng - 1]

. 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: $ng$ – Integer * Output

On exit: the number of spectral estimates,

[L / 2] + 1

, whose separate parts are held in g.

13: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_CONS: On entry, $kc = ⟨ value ⟩$ while $l = ⟨ value ⟩$ . These arguments must satisfy kc% $l \neq 0$ when $l > 0$ .

On entry, $kc = ⟨ value ⟩$ while $nxy = ⟨ value ⟩$ . These arguments must satisfy $kc \geq 2$ *nxy when $nxy > 0$ .

On entry, $l = ⟨ value ⟩$ while $is = ⟨ value ⟩$ . These arguments must satisfy $‖ is ‖ < l$ when $l > 0$ .
NE_2_INT_ARG_GT: On entry, $mw = ⟨ value ⟩$ while $nxy = ⟨ value ⟩$ . These arguments must satisfy $mw \leq nxy$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: 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, $l = ⟨ value ⟩$ .
Constraint: $l \geq 1$ .

On entry, $mw = ⟨ value ⟩$ .
Constraint: $mw \geq 1$ .

On entry, $nxy = ⟨ value ⟩$ .
Constraint: $nxy \geq 1$ .
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: $pw = ⟨ value ⟩$ .

On entry, pxy must not be greater than 1.0: $pxy = ⟨ value ⟩$ .
NE_REAL_ARG_LT: On entry, pw must not be less than 0.0: $pw = ⟨ value ⟩$ .

On entry, pxy must not be less than 0.0: $pxy = ⟨ value ⟩$ .
NE_TOO_MANY_FACTORS: kc has more than 20 prime factors.

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

Background information to multithreading can be found in the Multithreading documentation.

g13cdc is not threaded in any implementation.

9 Further Comments

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

kc \times \log (kc)

(but see function document c06pac for further details).

10 Example

The example program reads 2 time series of length

296

. It selects mean correction and a 10% tapering proportion. It selects a

2 π / 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.

g13cd: FL CL CPP AD PY MB

NAG CL Interfaceg13cdc (multi_​spectrum_​daniell)

▸▿ 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 CL Interface
g13cdc (multi_spectrum_daniell)