Integer, Intent (In)	::	nx, mtx, mw, l, kc, lg
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ng
Real (Kind=nag_wp), Intent (In)	::	px, pw
Real (Kind=nag_wp), Intent (Inout)	::	xg(kc)
Real (Kind=nag_wp), Intent (Out)	::	stats(4)

C Header Interface

#include nagmk26.h

void	g13cbf_ ( const Integer nx, const Integer mtx, const double px, const Integer mw, const double pw, const Integer l, const Integer kc, const Integer lg, double xg[], Integer ng, double stats[], Integer ifail)

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:

\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 unsmoothed sample spectrum

f^{*} (ω) = \frac{1}{2 π} {|\sum_{t = 1}^{n} x_{t} \exp (i ω t)|}^{2}

is then calculated for frequency values

ω_{k} = \frac{2 π k}{K}, k = 0, 1, \dots, [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_{r l} = ν_{l} = \frac{2 π l}{L}, l = 0, 1, \dots, [L / 2],

where

K = r \times 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 \geq 2 \times 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

\begin{array}{l} W (α) = 1, & |α| \leq p \\ W (α) = \frac{1 - |α|}{1 - p}, & p < |α| \leq 1 \end{array}

the proportion

p

being supplied by you.

The width of the window is fixed as

2 π / M

by you supplying

M

. A set of averaging weights are constructed:

W_{k} = g \times W (\frac{ω_{k} M}{π}), 0 \leq ω_{k} \leq \frac{π}{M},

where

g

is a normalizing constant, and the smoothed spectrum obtained is

\hat{f} (ν_{l}) = \sum_{|ω_{k}| < \frac{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

\hat{f} (ν_{l}) = f^{*} (ν_{l})

. Otherwise, in order that the smoothing approximates well to an integration, it is essential that

K ≫ M

, 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

L ≃ n

and

K ≃ 4 n

should be adequate for usual smoothing situations when

M < n / 5

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 routine, 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: $nx$ – IntegerInput

On entry:

n

, the length of the time series.

Constraint:

nx \geq 1

2: $mtx$ – IntegerInput

On entry: 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:

0 \leq mtx \leq 2

3: $px$ – Real (Kind=nag_wp)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 px \leq 1.0

4: $mw$ – IntegerInput

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 \leq mw \leq nx

5: $pw$ – Real (Kind=nag_wp)Input

On entry:

p

, the shape parameter of the trapezium frequency window.

A value of

0.0

gives a triangular window, and a value of

1.0

a rectangular window.

mw = nx

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

Constraint:

0.0 \leq pw \leq 1.0

6: $l$ – IntegerInput

On entry:

L

, the frequency division of smoothed spectral estimates as

2 π / L

Constraints:

$l \geq 1$ ;
l must be a factor of kc.

7: $kc$ – IntegerInput

On entry:

K

, the order of the fast Fourier transform (FFT) used to calculate the spectral estimates.

Constraints:

$kc \geq 2 \times nx$ ;
kc must be a multiple of l.

8: $\lg$ – IntegerInput

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

$\lg = 0$: For unlogged.
$\lg \neq 0$: For logged.

9: $xg (kc)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the

n

data points.

On exit: contains the ng spectral estimates

\hat{f} (ω_{i})

, for

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

, in

xg (1)

xg (ng)

(logged if

\lg \neq 0

). The elements

xg (i)

, for

i = ng + 1, \dots, kc

, contain

0.0

10: $ng$ – IntegerOutput

On exit: the number of spectral estimates,

[L / 2] + 1

, in xg.

11: $stats (4)$ – Real (Kind=nag_wp) arrayOutput

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

stats (1)

, the lower and upper

95 %

confidence limit factors in

stats (2)

and

stats (3)

respectively (logged if

\lg \neq 0

), and the bandwidth in

stats (4)

12: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if

ifail \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Note: g13cbf may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $l = 〈value〉$ .
Constraint: $l \geq 1$ .

On entry, $mtx = 〈value〉$ .
Constraint: $mtx \leq 2$ .

On entry, $mtx = 〈value〉$ .
Constraint: $mtx \geq 0$ .

On entry, $mw = 〈value〉$ .
Constraint: $mw \geq 1$ .

On entry, $mw = 〈value〉$ and $nx = 〈value〉$ .
Constraint: $mw \leq nx$ .

On entry, $nx = 〈value〉$ .
Constraint: $nx \geq 1$ .

On entry, $px = 〈value〉$ , $mw = 〈value〉$ and $nx = 〈value〉$ .
Constraint: if $pw < 0.0$ , $mw = nx$ .

On entry, $px = 〈value〉$ , $mw = 〈value〉$ and $nx = 〈value〉$ .
Constraint: if $pw > 1.0$ , $mw = nx$ .

On entry, $px = 〈value〉$ .
Constraint: $px \geq 0.0$ .

On entry, $px = 〈value〉$ .
Constraint: $px \leq 1.0$ .

$ifail = 2$: On entry, $kc = 〈value〉$ and $l = 〈value〉$ .
Constraint: kc must be a multiple of l.

On entry, $kc = 〈value〉$ and $nx = 〈value〉$ .
Constraint: $kc \geq 2 \times nx$ .

$ifail = 4$: 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.

$ifail = 5$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

g13cbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g13cbf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

g13cbf carries out a FFT of length kc to calculate the sample spectrum. The time taken by the routine for this is approximately proportional to

kc \times \log (kc)

(but see Section 9 in c06paf for further details).

10

Example

This example reads a time series of length

131

. It then calls g13cbf to calculate the univariate spectrum and prints the logged spectrum together with

95 %

confidence limits.

NAG Library Routine Document

g13cbf (uni_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

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