NAG Library Routine Document

G13CBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G13CBF calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

2 Specification

SUBROUTINE G13CBF (

NX, MTX, PX, MW, PW, L, KC, LG, XG, NG, STATS, IFAIL)

INTEGER	NX, MTX, MW, L, KC, LG, NG, IFAIL
REAL (KIND=nag_wp)	PX, PW, XG(KC), STATS(4)

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 Parameters

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. KC should be a multiple of small primes such as

2^{m}

where

m

is the smallest integer such that

2^{m} \geq 2 n

, provided

m \leq 20

Constraints:

$KC \geq 2 \times NX$ ;
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 – 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 parameter you should refer to Section 3.3 in the Essential Introduction 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 parameters 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,	$NX < 1$ ,
or	$MTX < 0$ ,
or	$MTX > 2$ ,
or	$PX < 0.0$ ,
or	$PX > 1.0$ ,
or	$MW < 1$ ,
or	$MW > NX$ ,
or	$PW < 0.0$ and $MW \neq NX$ ,
or	$PW > 1.0$ and $MW \neq NX$ ,
or	$L < 1$ .

$IFAIL = 2$

On entry,	$KC < 2 \times NX$ ,
or	KC is not a multiple of L,
or	KC has a prime factor exceeding $19$ ,
or	KC has more than $20$ prime factors, counting repetitions.

$IFAIL = 3$: This indicates that a serious error has occurred. Check all array subscripts and subroutine parameter lists in calls to G13CBF. Seek expert help.

$IFAIL = 4$: One or more spectral estimates are negative. Unlogged spectral estimates are returned in XG, and the degrees of freedom, unlogged confidence limit factors and bandwidth in STATS.

$IFAIL = 5$: The calculation of confidence limit factors has failed. This error will not normally occur. Spectral estimates (logged if requested) are returned in XG, and degrees of freedom 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 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 8 in C06PAF for further details).

9 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 DocumentG13CBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G13CBF