g13ca calculates the smoothed sample spectrum of a univariate time series using one of four lag windows – rectangular, Bartlett, Tukey or Parzen window.

Syntax

C#
public static void g13ca( int nx, int mtx, double px, int iw, int mw, int ic, int nc, double[] c, int kc, int l, int lg, double[] xg, out int ng, double[] stats, out int ifail )

Visual Basic
Public Shared Sub g13ca ( _ nx As Integer, _ mtx As Integer, _ px As Double, _ iw As Integer, _ mw As Integer, _ ic As Integer, _ nc As Integer, _ c As Double(), _ kc As Integer, _ l As Integer, _ lg As Integer, _ xg As Double(), _ <OutAttribute> ByRef ng As Integer, _ stats As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13ca ( _
	nx As Integer, _
	mtx As Integer, _
	px As Double, _
	iw As Integer, _
	mw As Integer, _
	ic As Integer, _
	nc As Integer, _
	c As Double(), _
	kc As Integer, _
	l As Integer, _
	lg As Integer, _
	xg As Double(), _
	<OutAttribute> ByRef ng As Integer, _
	stats As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13ca( int nx, int mtx, double px, int iw, int mw, int ic, int nc, array<double>^ c, int kc, int l, int lg, array<double>^ xg, [OutAttribute] int% ng, array<double>^ stats, [OutAttribute] int% ifail )

Visual C++

public:
static void g13ca(
	int nx, 
	int mtx, 
	double px, 
	int iw, 
	int mw, 
	int ic, 
	int nc, 
	array<double>^ c, 
	int kc, 
	int l, 
	int lg, 
	array<double>^ xg, 
	[OutAttribute] int% ng, 
	array<double>^ stats, 
	[OutAttribute] int% ifail
)

F#
static member g13ca : nx : int * mtx : int * px : float * iw : int * mw : int * ic : int * nc : int * c : float[] * kc : int * l : int * lg : int * xg : float[] * ng : int byref * stats : float[] * ifail : int byref -> unit

static member g13ca : 
        nx : int * 
        mtx : int * 
        px : float * 
        iw : int * 
        mw : int * 
        ic : int * 
        nc : int * 
        c : float[] * 
        kc : int * 
        l : int * 
        lg : int * 
        xg : float[] * 
        ng : int byref * 
        stats : float[] * 
        ifail : int byref -> unit

Parameters

nx: Type: System..::..Int32
On entry: $n$ , the length of the time series.

Constraint: $nx \geq 1$ .

mtx

Type: System..::..Int32

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

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.

px: Type: System..::..Double
On entry: if covariances are to be calculated by the method ( $ic = 0$ ), 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: $0.0 \leq px \leq 1.0$ .

iw

Type: System..::..Int32

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

mw: Type: System..::..Int32
On entry: $M$ , the ‘cut-off’ point of the lag window. Windowed covariances at lag $M$ or greater are zero.

Constraint: $1 \leq mw \leq nx$ .

ic

Type: System..::..Int32

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

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

nc: Type: System..::..Int32
On entry: the number of covariances to be calculated in the method or supplied in the call to the method.

Constraint: $mw \leq nc \leq nx$ .

c: Type: array<System..::..Double>[]()[][]
An array of size [nc]
On entry: if $ic \neq 0$ , c must contain the nc covariances for lags from $0$ to $(nc - 1)$ , otherwise c need not be set.
On exit: if $ic = 0$ , c will contain the nc calculated covariances.
If $ic \neq 0$ , the contents of c will be unchanged.

kc: Type: System..::..Int32
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 $m \leq 20$ .
If $ic \neq 0$ , 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.

l: Type: System..::..Int32
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.

lg

Type: System..::..Int32

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

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

xg: Type: array<System..::..Double>[]()[][]
An array of size [nxg]
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 $i = 0, 1, \dots, [L / 2]$ in $xg [0]$ to $xg [ng - 1]$ respectively (logged if $\lg = 1$ ). The elements $xg [i - 1]$ , for $i = ng + 1, \dots, nxg$ contain $0.0$ .

ng: Type: System..::..Int32%
On exit: the number of spectral estimates, $[L / 2] + 1$ , in xg.

stats: Type: array<System..::..Double>[]()[][]
An array of size [ $4$ ]
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 $\lg = 1$ ), and the bandwidth in $stats [3]$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

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 method, 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.

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

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.

Parallelism and Performance

None.

Further Comments

g13ca 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 method for an FFT of length

n

is approximately proportional to

n \log (n)

(but see [Further Comments] in c06pa for further details).

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 g13ca to calculate the univariate spectrum and statistics and prints the autocovariances and the spectrum together with its

95 %

confidence multiplying limits.

Example program (C#): g13cae.cs

Example program data: g13cae.d

Example program results: g13cae.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also