NAG CL Interface
g13cbc (uni_​spectrum_​daniell)

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1 Purpose

g13cbc calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

2 Specification

#include <nag.h>
void  g13cbc (Integer nx, NagMeanOrTrend mt_correction, double px, Integer mw, double pw, Integer l, Integer kc, Nag_LoggedSpectra lg_spect, const double x[], double **g, Integer *ng, double stats[], NagError *fail)
The function may be called by the names: g13cbc, nag_tsa_uni_spectrum_daniell or nag_tsa_spectrum_univar.

3 Description

The supplied time series may be mean or trend corrected (by least squares), and tapered, the tapering factors being those of the split cosine bell:
1 2 (1-cos( π (t- 1 2 ) T )) , 1 t T 1 2 (1-cos( π (n-t+ 1 2 ) T )) , n + 1 - T t n 1 , otherwise  
where T = [ np 2 ] and p is the tapering proportion.
The unsmoothed sample spectrum
f * (ω) = 1 2 π | t=1 n x t exp(iωt)| 2  
is then calculated for frequency values
ω k = 2πk K , k = 0 , 1 , , [K/2]  
where [ ] denotes the integer part.
The smoothed spectrum is returned as a subset of these frequencies for which K is a multiple of a chosen value r , i.e.,
ω rl = ν l = 2πl L , l = 0 , 1 , , [L/2]  
where K = r × L . You will normally fix L first, then choose r so that K is sufficiently large to provide an adequate representation for the unsmoothed spectrum, i.e., K 2 × n . It is possible to take L=K , i.e., r=1 .
The smoothing is defined by a trapezium window whose shape is supplied by the function
W (α) = 1 , |α| p W (α) = 1 - |α| 1-p , p < |α| 1  
the proportion p being supplied by you.
The width of the window is fixed as 2 π / M by supplying M . A set of averaging weights are constructed:
W k = g × W ( ω k M π ) , 0 ω k π M  
where g is a normalizing constant, and the smoothed spectrum obtained is
f ^ ( ν l ) = | ω k | < π M W k f * ( ν l + ω k ) .  
If no smoothing is required M should be set to n , in which case the values returned are f ^ ( ν l ) = f * ( ν l ) . Otherwise, in order that the smoothing approximates well to an integration, it is essential that KM , and preferable, but not essential, that K be a multiple of M . A choice of L>M would normally be required to supply an adequate description of the smoothed spectrum. Typical choices of Ln and K 4 n should be adequate for usual smoothing situations when M < n / 5 .
The sampling distribution of 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 mu , ml from which approximate 95% confidence intervals for the true spectrum f (ω) may be constructed as [ ml × f ^ ,(ω),mu,×, f ^,(ω)] . Alternatively, log 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: the length of the time series, n .
Constraint: nx1 .
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 or Nag_Trend.
3: px 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 px 1.0 .
4: mw Integer Input
On entry: the value of M which determines the frequency width of the smoothing window as 2 π / M . A value of n implies no smoothing is to be carried out.
Constraint: 1 mw nx .
5: 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.
If mw=nx (i.e., no smoothing is carried out), then pw is not used.
Constraint: 0.0 pw 1.0 . if mwnx .
6: l Integer Input
On entry: the frequency division, L , of smoothed spectral estimates as 2 π / L .
  • l1 ;
  • l must be a factor of kc (see below).
7: kc Integer Input
On entry: the order of the fast Fourier transform (FFT), K , used to calculate the spectral estimates. kc should be a multiple of small primes such as 2 m where m is the smallest integer such that 2 m 2 n , provided m20 .
  • kc 2 × nx ;
  • 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.
8: lg_spect Nag_LoggedSpectra Input
On entry: indicates whether unlogged or logged spectral estimates and confidence limits are required. lg_spect=Nag_Unlogged for unlogged. lg_spect=Nag_Logged for logged.
Constraint: lg_spect=Nag_Unlogged or Nag_Logged.
9: x[kc] const double Input
On entry: the n data points.
10: g double ** Output
On exit: vector which contains the ng spectral estimates f ^ ( ω i ) , for i=0,1,,[L/2], in g[0] to g[ng-1] (logged if lg_spect=Nag_Logged). 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.
11: ng Integer * Output
On exit: the number of spectral estimates, [L/2] + 1 , in g.
12: 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] .
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

On entry, kc=value while l=value . These arguments must satisfy kc% l=0 when l>0 .
On entry, kc=value while nx=value . These arguments must satisfy kc2×nx when nx>0 .
On entry, mw=value while nx=value . These arguments must satisfy mwnx .
Dynamic memory allocation failed.
On entry, argument lg_spect had an illegal value.
On entry, argument mt_correction had an illegal value.
The calculation of confidence limit factors has failed. Spectral estimates (logged if requested) are returned in g, and degrees of freedom and bandwith in stats.
At least one of the prime factors of kc is greater than 19.
On entry, l=value.
Constraint: l1.
On entry, mw=value.
Constraint: mw1.
On entry, nx=value.
Constraint: nx1.
On entry, pw must not be less than 0.0: pw=value .
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.
On entry, pw must not be greater than 1.0: pw=value .
On entry, px must not be greater than 1.0: px=value .
On entry, px must not be less than 0.0: px=value .
One or more spectral estimates are negative. Unlogged spectral estimates are returned in g and the degrees of freedom, unlogged confidence limit factors and bandwith in stats.
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.
g13cbc is not threaded in any implementation.

9 Further Comments

g13cbc carries out a FFT of length kc to calculate the sample spectrum. The time taken by the function for this is approximately proportional to kc×log(kc) (but see Section 9 in c06pac for further details).

10 Example

The example program reads a time series of length 131. It selects the mean correction option, a tapering proportion of 0.2, the option of no smoothing and a frequency division for logged spectral estimates of 2 π / 100 . It then calls g13cbc to calculate the univariate spectrum and prints the logged spectrum together with 95% confidence limits. The program then selects a smoothing window with frequency width 2 π / 30 and shape argument 0.5 and recalculates and prints the logged spectrum and 95% confidence limits.

10.1 Program Text

Program Text (g13cbce.c)

10.2 Program Data

Program Data (g13cbce.d)

10.3 Program Results

Program Results (g13cbce.r)