NAG Library Function Document
nag_tsa_spectrum_univar_cov (g13cac)
1
Purpose
nag_tsa_spectrum_univar_cov (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> |
#include <nagg13.h> |
void |
nag_tsa_spectrum_univar_cov (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) |
|
3
Description
The smoothed sample spectrum is defined as
where
is the window width, and is calculated for frequency values
where
denotes the integer part.
The autocovariances
may be supplied by you, or constructed from a time series
, as
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:
where
and
is the tapering proportion.
The smoothing window is defined by
which for the various windows is defined over
by
rectangular:
Bartlett:
Tukey:
Parzen:
The sampling distribution of
is approximately that of a scaled
variate, whose degrees of freedom
is provided by the function, together with multiplying limits
,
from which approximate
confidence intervals for the true spectrum
may be constructed as
. Alternatively, log
may be returned, with additive limits.
The bandwidth of the corresponding smoothing window in the frequency domain is also provided. Spectrum estimates separated by (angular) frequencies much greater than 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:
– IntegerInput
-
On entry: , the length of the time series.
Constraint:
.
- 2:
– IntegerInput
-
On entry: if covariances are to be calculated by the function (
),
mtx must specify whether the data are to be initially mean or trend corrected.
- For no correction.
- For mean correction.
- For trend correction.
Constraint:
if
,
If covariances are supplied (
),
mtx is not used.
- 3:
– doubleInput
-
On entry: if covariances are to be calculated by the function (
),
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
,
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
implies no tapering.
Constraint:
.
- 4:
– IntegerInput
-
On entry: the choice of lag window.
- Rectangular.
- Bartlett.
- Tukey.
- Parzen.
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the ‘cut-off’ point of the lag window. Windowed covariances at lag or greater are zero.
Constraint:
.
- 6:
– IntegerInput
-
On entry: indicates whether covariances are to be calculated in the function or supplied in the call to the function.
- Covariances are to be calculated.
- Covariances are to be supplied.
- 7:
– IntegerInput
-
On entry: the number of covariances to be calculated in the function or supplied in the call to the function.
Constraint:
.
- 8:
– doubleInput/Output
-
On entry: if
,
c must contain the
nc covariances for lags from
to
, otherwise
c need not be set.
On exit: if
,
c will contain the
nc calculated covariances.
If
, the contents of
c will be unchanged.
- 9:
– IntegerInput
-
On entry: if
,
kc must specify the order of the fast Fourier transform (FFT) used to calculate the covariances.
If
, that is covariances are supplied,
kc is not used.
Constraint:
.
- 10:
– IntegerInput
-
On entry: , the frequency division of the spectral estimates as . Therefore it is also the order of the FFT used to construct the sample spectrum from the covariances.
Constraint:
.
- 11:
– Nag_LoggedSpectraInput
-
On entry: indicates whether unlogged or logged spectral estimates and confidence limits are required.
- Unlogged.
- Logged.
Constraint:
or .
- 12:
– IntegerInput
-
On entry: the dimension of the array
xg.
Constraints:
- if , ;
- if , .
- 13:
– doubleInput/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,
, for
in
to
respectively (logged if
). The elements
, for
contain
.
- 14:
– Integer *Output
-
On exit: the number of spectral estimates,
, in
xg.
- 15:
– doubleOutput
-
On exit: four associated statistics. These are the degrees of freedom in , the lower and upper confidence limit factors in and respectively (logged if ), and the bandwidth in .
- 16:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument 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, and : .
On entry, and : .
On entry, .
Constraint: , , or .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: if , .
- NE_INT_3
-
On entry, , and .
Constraint: if , .
On entry, , and .
Constraint: if , .
- 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- 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
nag_tsa_spectrum_univar_cov (g13cac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_spectrum_univar_cov (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.
nag_tsa_spectrum_univar_cov (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
is approximately proportional to
(but see
Section 9 in
nag_sum_fft_realherm_1d (c06pac) for further details).
10
Example
This example reads a time series of length . It selects the mean correction option, a tapering proportion of , the Parzen smoothing window and a cut-off point for the window at lag . It chooses to have auto-covariances calculated and unlogged spectral estimates at a frequency division of . It then calls nag_tsa_spectrum_univar_cov (g13cac) to calculate the univariate spectrum and statistics and prints the autocovariances and the spectrum together with its confidence multiplying limits.
10.1
Program Text
Program Text (g13cace.c)
10.2
Program Data
Program Data (g13cace.d)
10.3
Program Results
Program Results (g13cace.r)