For a bivariate time series, g13cgc calculates the noise spectrum together with multiplying factors for the bounds and the impulse response function and its standard error, from the univariate and bivariate spectra.

2 Specification

#include <nag.h>

void	g13cgc (const double xg[], const double yg[], const Complex xyg[], Integer ng, const double stats[], Integer l, Integer n, double er[], double erlw, double erup, double rf[], double rfse, NagError fail)

The function may be called by the names: g13cgc, nag_tsa_multi_noise_bivar or nag_tsa_noise_spectrum_bivar.

3 Description

An estimate of the noise spectrum in the dependence of series

y

on series

x

at frequency

ω

is given by

f_{y ∣ x} (ω) = f_{y y} (ω) (1 - W (ω))

where

W (ω)

is the squared coherency described in G13GEF and

f_{y y} (ω)

is the univariate spectrum estimate for series

y

. Confidence limits on the true spectrum are obtained using multipliers as described for G13CAF, but based on

(d - 2)

degrees of freedom.

If the dependence of

y_{t}

x_{t}

can be assumed to be represented in the time domain by the one sided relationship

y_{t} = v_{0} x_{t} + v_{1} x_{t - 1} + \dots + n_{t}

where the noise

n_{t}

is independent of

x_{t}

, then it is the spectrum of this noise which is estimated by

f_{y ∣ x} (ω)

Estimates of the impulse response function

v_{0}, v_{1}, v_{2}, \dots

may also be obtained as

v_{k} = \frac{1}{π} \int_{0}^{π} Re (\frac{\exp (i k ω) f_{x y} (ω)}{f_{x x} (ω)})

where Re indicates the real part of the expression. For this purpose it is essential that the univariate spectrum for

x

f_{x x} (ω)

,and the cross spectrum,

f_{x y} (ω)

be supplied to this function for a frequency range

ω_{l} = [\frac{2 π l}{L}], 0 \leq l \leq [L / 2],

where

[]

denotes the integer part, the integral being approximated by a finite Fourier transform.

An approximate standard error is calculated for the estimates

v_{k}

. Significant values of

v_{k}

in the locations described as anticipatory responses in the argument array rf, indicate that feedback exists from

y_{t}

x_{t}

. This will bias the estimates of

v_{k}

in any causal dependence of

y_{t}

x_{t}, x_{t - 1}, \dots

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: $xg [ng]$ – const double Input

On entry: the ng univariate spectral estimates,

f_{x x} (ω)

, for the

x

series.

2: $yg [ng]$ – const double Input

On entry: the ng univariate spectral estimates,

f_{y y} (ω)

, for the

y

series.

3: $xyg [ng]$ – const Complex Input

On entry:

f_{x y} (ω)

, of the ng bivariate spectral estimates for the

x

and

y

series. The

x

series leads the

y

series.

Note: the two univariate and bivariate spectra must each have been calculated using the same amount of smoothing. The frequency width and the shape of the window and the frequency division of the spectral estimates must be the same. The spectral estimates and statistics must also be unlogged.

4: $ng$ – Integer Input

On entry: the number of spectral estimates in each of the arrays xg, yg and xyg. It is also the number of noise spectral estimates.

Constraint:

ng \geq 1

5: $stats [4]$ – const double Input

On entry: the 4 associated statistics for the univariate spectral estimates for the

x

and

y

series.

stats [0]

contains the degree of freedom,

stats [1]

and

stats [2]

contain the lower and upper bound multiplying factors respectively and

stats [3]

contains the bandwidth.

Constraints:

$stats [0] \geq 3.0$ ;
$0.0 < stats [1] \leq 1.0$ ;
$stats [2] \geq 1.0$ .

6: $l$ – Integer Input

On entry: the frequency division,

L

, of the spectral estimates as

2 π / L

, as input to g13cbc and g13cdc.

Constraints:

$ng = [l / 2] + 1$ ;
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.

7: $n$ – Integer Input

On entry: the number of points in each of the time series

x

and

y

. n should have the same value as nxy in the call of g13ccc or g13cdc which calculated the smoothed sample cross spectrum. n is used in calculating the impulse response function standard error (rfse).

Constraint:

n \geq 1

8: $er [ng]$ – double Output

On exit: the ng estimates of the noise spectrum,

{\hat{f}}_{y ∣ x} (ω)

at each frequency.

9: $erlw$ – double * Output

On exit: the noise spectrum lower limit multiplying factor.

10: $erup$ – double * Output

On exit: the noise spectrum upper limit multiplying factor.

11: $rf [l]$ – double Output

On exit: the impulse response function. Causal responses are stored in ascending frequency in

rf [0]

rf [ng - 1]

and anticipatory responses are stored in descending frequency in

rf [ng]

rf [l]

12: $rfse$ – double * Output

On exit: the impulse response function standard error.

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, $l = 〈value〉$ while $ng = 〈value〉$ . These arguments must satisfy $ng = [l / 2] + 1$ when $ng > 0$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BIVAR_SPECTRAL_ESTIM_ZERO: A bivariate spectral estimate is zero.
For this frequency the noise spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.
NE_FACTOR_GT: At least one of the prime factors of l is greater than $19$ .
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $ng = 〈value〉$ .
Constraint: $ng \geq 1$ .
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, $stats [1]$ must not be greater than 1.0: $stats [1] = 〈value〉$ .
NE_REAL_ARG_LE: On entry, $stats [1]$ must not be less than or equal to 0.0: $stats [1] = 〈value〉$ .
NE_REAL_ARG_LT: On entry, $stats [0]$ must not be less than 3.0: $stats [0] = 〈value〉$ .

On entry, $stats [2]$ must not be less than 1.0: $stats [2] = 〈value〉$ .
NE_SQUARED_FREQ_GT_ONE: A calculated value of the squared coherency exceeds one.
For this frequency the squared coherency is reset to one with the result that the noise spectrum is zero and the contribution to the impulse response function at this frequency is zero.
NE_TOO_MANY_FACTORS: l has more than 20 prime factors.
NE_UNIVAR_SPECTRAL_ESTIM_NEG: A bivariate spectral estimate is negative.
For this frequency the noise spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.
NE_UNIVAR_SPECTRAL_ESTIM_ZERO: A bivariate spectral estimate is zero.
For this frequency the noise spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero.

7 Accuracy

The computation of the noise is stable and yields good 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

g13cgc is not threaded in any implementation.

9 Further Comments

The time taken by g13cgc is approximately proportional to ng.

10 Example

The example program reads the set of univariate spectrum statistics, the 2 univariate spectra and the cross spectrum at a frequency division of

\frac{2 π}{20}

for a pair of time series. It calls g13cgc to calculate the noise spectrum and its confidence limits multiplying factors, the impulse response function and its standard error. It then prints the results.

g13cg: FL CL CPP AD

NAG CL Interfaceg13cgc (multi_​noise_​bivar)

▸▿ 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
g13cgc (multi_noise_bivar)