g13cac calculates the smoothed sample spectrum of a univariate time series using one of four lag windows – rectangular, Bartlett, Tukey or Parzen window.

2 Specification

#include <nag.h>

void	g13cac (Integer nx, Integer mtx, double px, Integer iw, Integer mw, Integer ic, Integer nc, double c[], Integer kc, Integer l, Nag_LoggedSpectra lg_spect, Integer nxg, double xg[], Integer ng, double stats[], NagError fail)

The function may be called by the names: g13cac, nag_tsa_uni_spectrum_lag or nag_tsa_spectrum_univar_cov.

3 Description

The smoothed sample spectrum is defined as

\hat{f} (ω) = \frac{1}{2 π} (C_{0} + 2 \sum_{k = 1}^{M - 1} w_{k} C_{k} \cos (ω k)),

where

M

is the window width, and is calculated for frequency values

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

where

[]

denotes the integer part.

The autocovariances

C_{k}

may be supplied by you, or constructed from a time series

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

, as

C_{k} = \frac{1}{n} \sum_{t = 1}^{n - k} x_{t} x_{t + k},

the fast Fourier transform (FFT) being used to carry out the convolution in this formula.

The time series may be mean or trend corrected (by classical least squares), and tapered before calculation of the covariances, the tapering factors being those of the split cosine bell:

\begin{array}{l} \frac{1}{2} (1 - \cos (π (t - \frac{1}{2}) / T)), & 1 \leq t \leq T \\ \frac{1}{2} (1 - \cos (π (n - t + \frac{1}{2}) / T)), & n + 1 - T \leq t \leq n \\ 1, & otherwise, \end{array}

where

T = [\frac{n p}{2}]

and

p

is the tapering proportion.

The smoothing window is defined by

w_{k} = W (\frac{k}{M}), k \leq M - 1,

which for the various windows is defined over

0 \leq α < 1

rectangular:

W (α) = 1

Bartlett:

W (α) = 1 - α

Tukey:

W (α) = \frac{1}{2} (1 + \cos (π α))

Parzen:

\begin{array}{l} W (α) = 1 - 6 α^{2} + 6 α^{3}, & 0 \leq α \leq \frac{1}{2} \\ W (α) = 2 {(1 - α)}^{3}, & \frac{1}{2} < α < 1 . \end{array}

The sampling distribution of

\hat{f} (ω)

is approximately that of a scaled

χ_{d}^{2}

variate, whose degrees of freedom

d

is provided by the function, together with multiplying limits

m u

m l

from which approximate

95 %

confidence intervals for the true spectrum

f (ω)

may be constructed as

[m l \times \hat{f} (ω), m u \times \hat{f} (ω)]

. Alternatively, log

\hat{f} (ω)

may be returned, with additive limits.

The bandwidth

b

of the corresponding smoothing window in the frequency domain is also provided. Spectrum estimates separated by (angular) frequencies much greater than

b

may be assumed to be independent.

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: $nx$ – Integer Input

On entry:

n

, the length of the time series.

Constraint:

nx \geq 1

2: $mtx$ – Integer Input

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

ic = 0

), mtx must specify whether the data are to be initially mean or trend corrected.

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

Constraint: if

ic = 0

0 \leq mtx \leq 2

If covariances are supplied (

ic \neq 0

), mtx is not used.

3: $px$ – double Input

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

ic = 0

), px must specify the proportion of the data (totalled over both ends) to be initially tapered by the split cosine bell taper.

If covariances are supplied

(ic \neq 0)

, px must specify the proportion of data tapered before the supplied covariances were calculated and after any mean or trend correction. px is required for the calculation of output statistics. A value of

0.0

implies no tapering.

Constraint:

0.0 \leq px \leq 1.0

4: $iw$ – Integer Input

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: $mw$ – Integer Input

On entry:

M

, the ‘cut-off’ point of the lag window. Windowed covariances at lag

M

or greater are zero.

Constraint:

1 \leq mw \leq nx

6: $ic$ – Integer Input

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

$ic = 0$: Covariances are to be calculated.
$ic \neq 0$: Covariances are to be supplied.

7: $nc$ – Integer Input

On entry: the number of covariances to be calculated in the function or supplied in the call to the function.

Constraint:

mw \leq nc \leq nx

8: $c [nc]$ – double Input/Output

On entry: if

ic \neq 0

, c must contain the nc covariances for lags from

0

(nc - 1)

, otherwise c need not be set.

On exit: if

ic = 0

, c will contain the nc calculated covariances.

ic \neq 0

, the contents of c will be unchanged.

9: $kc$ – Integer Input

On entry: if

ic = 0

, kc must specify the order of the fast Fourier transform (FFT) used to calculate the covariances.

ic \neq 0

, that is covariances are supplied, kc is not used.

Constraint:

kc \geq nx + nc

10: $l$ – Integer Input

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 covariances.

Constraint:

l \geq 2 \times mw - 1

11: $lg_spect$ – Nag_LoggedSpectra Input

On entry: indicates whether unlogged or logged spectral estimates and confidence limits are required.

$lg_spect = Nag_Unlogged$: Unlogged.
$lg_spect = Nag_Logged$: Logged.

Constraint:

lg_spect = Nag_Unlogged

Nag_Logged

12: $nxg$ – Integer Input

On entry: the dimension of the array xg.

Constraints:

if $ic = 0$ , $nxg \geq \max (kc, l)$ ;
if $ic \neq 0$ , $nxg \geq l$ .

13: $xg [nxg]$ – double Input/Output

On entry: if the covariances are to be calculated, then xg must contain the nx data points. If covariances are supplied, xg may contain any values.

On exit: contains the ng spectral estimates,

\hat{f} (ω_{i})

, for

i = 0, 1, \dots, [L / 2]

xg [0]

xg [ng - 1]

respectively (logged if

lg_spect = Nag_Logged

). The elements

xg [i - 1]

, for

i = ng + 1, \dots, nxg

contain

0.0

14: $ng$ – Integer * Output

On exit: the number of spectral estimates,

[L / 2] + 1

, in xg.

15: $stats [4]$ – double Output

On exit: four associated statistics. These are the degrees of freedom in

stats [0]

, the lower and upper

95 %

confidence limit factors in

stats [1]

and

stats [2]

respectively (logged if

lg_spect = Nag_Logged

), and the bandwidth in

stats [3]

16: $fail$ – 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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONFID_LIMITS: The calculation of confidence limit factors has failed.
Spectral estimates (logged if requested) are returned in xg, and degrees of freedom and bandwidth in stats.
NE_INT: On entry, $ic = 0$ and $mtx < 0$ : $mtx = ⟨ value ⟩$ .

On entry, $ic = 0$ and $mtx > 2$ : $mtx = ⟨ value ⟩$ .

On entry, $iw = ⟨ value ⟩$ .
Constraint: $iw = 1$ , $2$ , $3$ or $4$ .

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

On entry, $nx = ⟨ value ⟩$ .
Constraint: $nx \geq 1$ .
NE_INT_2: On entry, $l = ⟨ value ⟩$ and $mw = ⟨ value ⟩$ .
Constraint: $l \geq 2 \times mw - 1$ .

On entry, $mw = ⟨ value ⟩$ and $nx = ⟨ value ⟩$ .
Constraint: $mw \leq nx$ .

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

On entry, $nc = ⟨ value ⟩$ and $nx = ⟨ value ⟩$ .
Constraint: $nc \leq nx$ .

On entry, $nxg = ⟨ value ⟩$ and $l = ⟨ value ⟩$ .
Constraint: if $ic \neq 0$ , $nxg \geq l$ .
NE_INT_3: On entry, $kc = ⟨ value ⟩$ , $nx = ⟨ value ⟩$ and $nc = ⟨ value ⟩$ .
Constraint: if $ic = 0$ , $kc \geq (nx + nc)$ .

On entry, $nxg = ⟨ value ⟩$ , $kc = ⟨ value ⟩$ and $l = ⟨ value ⟩$ .
Constraint: if $ic = 0$ , $nxg \geq \max (kc, l)$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $px = ⟨ value ⟩$ .
Constraint: $px \leq 1.0$ .

On entry, $px = ⟨ value ⟩$ .
Constraint: $px \geq 0.0$ .
NE_SPECTRAL_ESTIMATES: One or more spectral estimates are negative.
Unlogged spectral estimates are returned in xg, and the degrees of freedom, unloged confidence limit factors and bandwidth in stats.

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

g13cac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g13cac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

g13cac carries out two FFTs of length kc to calculate the covariances and one FFT of length l to calculate the sample spectrum. The time taken by the function for an FFT of length

n

is approximately proportional to

n \log (n)

(but see Section 9 in c06pac for further details).

10 Example

This example reads a time series of length

256

. It selects the mean correction option, a tapering proportion of

0.1

, the Parzen smoothing window and a cut-off point for the window at lag

100

. It chooses to have

100

auto-covariances calculated and unlogged spectral estimates at a frequency division of

2 π / 200

. It then calls g13cac to calculate the univariate spectrum and statistics and prints the autocovariances and the spectrum together with its

95 %

confidence multiplying limits.

g13ca: FL CL CPP AD

NAG CL Interfaceg13cac (uni_​spectrum_​lag)

▸▿ 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
g13cac (uni_spectrum_lag)