NAG Library Routine Document

G13CAF

+− 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

G13CAF calculates the smoothed sample spectrum of a univariate time series using one of four lag windows – rectangular, Bartlett, Tukey or Parzen window.

2 Specification

SUBROUTINE G13CAF (

NX, MTX, PX, IW, MW, IC, NC, C, KC, L, LG, NXG, XG, NG, STATS, IFAIL)

INTEGER	NX, MTX, IW, MW, IC, NC, KC, L, LG, NXG, NG, IFAIL
REAL (KIND=nag_wp)	PX, C(NC), XG(NXG), STATS(4)

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 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: if covariances are to be calculated by the routine (

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 – REAL (KIND=nag_wp)Input

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

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 – INTEGERInput

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 – INTEGERInput

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 – INTEGERInput

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

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

7: NC – INTEGERInput

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

Constraint:

MW \leq NC \leq NX

8: C(NC) – REAL (KIND=nag_wp) arrayInput/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 – INTEGERInput

On entry: if

IC = 0

, KC must specify the order of the fast Fourier transform (FFT) used to calculate the covariances. KC should be a product of small primes such as

2^{m}

where

m

is the smallest integer such that

2^{m} \geq NX + NC

, provided

m \leq 20

IC \neq 0

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

Constraint:

KC \geq NX + NC

. 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.

10: L – INTEGERInput

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. L should be a product of small primes such as

2^{m}

where

m

is the smallest integer such that

2^{m} \geq 2 M - 1

, provided

m \leq 20

Constraint:

L \geq 2 \times MW - 1

. The largest prime factor of L must not exceed

19

, and the total number of prime factors of L, counting repetitions, must not exceed

20

. These two restrictions are imposed by the internal FFT algorithm used.

11: LG – INTEGERInput

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

$LG = 0$: Unlogged.
$LG \neq 0$: Logged.

12: NXG – INTEGERInput

On entry: the dimension of the array XG as declared in the (sub)program from which G13CAF is called.

Constraints:

if $IC = 0$ , $NXG \geq \max (KC, L)$ ;
if $IC \neq 0$ , $NXG \geq L$ .

13: XG(NXG) – REAL (KIND=nag_wp) arrayInput/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 (1)

XG (NG)

respectively (logged if

LG = 1

). The elements

XG (i)

, for

i = NG + 1, \dots, NXG

contain

0.0

14: NG – INTEGEROutput

On exit: the number of spectral estimates,

[L / 2] + 1

, in XG.

15: 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 = 1

), and the bandwidth in

STATS (4)

16: 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, if you are not familiar with this parameter, the recommended value is

0

. 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).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NX < 1$ ,
or	$MTX < 0$ and $IC = 0$ ,
or	$MTX > 2$ and $IC = 0$ ,
or	$PX < 0.0$ ,
or	$PX > 1.0$ ,
or	$IW < 1$ ,
or	$IW > 4$ ,
or	$MW < 1$ ,
or	$MW > NX$ ,
or	$NC < MW$ ,
or	$NC > NX$ ,
or	$NXG < \max (KC, L)$ and $IC = 0$ ,
or	$NXG < L$ and $IC \neq 0$ .

$IFAIL = 2$

On entry,	$KC < NX + NC$ ,
or	KC has a prime factor exceeding $19$ ,
or	KC has more than $20$ prime factors, counting repetitions.

This error only occurs when

IC = 0

$IFAIL = 3$

On entry,	$L < 2 \times MW - 1$ ,
or	L has a prime factor exceeding $19$ ,
or	L has more than $20$ prime factors, counting repetitions.

$IFAIL = 4$: 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

G13CAF 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 routine for an FFT of length

n

is approximately proportional to

n \log (n)

(but see Section 8 in C06PAF for further details).

9 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 G13CAF to calculate the univariate spectrum and statistics and prints the autocovariances and the spectrum together with its

95 %

confidence multiplying limits.

NAG Library Routine DocumentG13CAF

+− 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

G13CAF