g13cd calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

Syntax

C#
public static void g13cd( int nxy, int mtxy, double pxy, int mw, int ish, double pw, int l, int kc, double[] xg, double[] yg, out int ng, out int ifail )

Visual Basic
Public Shared Sub g13cd ( _ nxy As Integer, _ mtxy As Integer, _ pxy As Double, _ mw As Integer, _ ish As Integer, _ pw As Double, _ l As Integer, _ kc As Integer, _ xg As Double(), _ yg As Double(), _ <OutAttribute> ByRef ng As Integer, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13cd ( _
	nxy As Integer, _
	mtxy As Integer, _
	pxy As Double, _
	mw As Integer, _
	ish As Integer, _
	pw As Double, _
	l As Integer, _
	kc As Integer, _
	xg As Double(), _
	yg As Double(), _
	<OutAttribute> ByRef ng As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13cd( int nxy, int mtxy, double pxy, int mw, int ish, double pw, int l, int kc, array<double>^ xg, array<double>^ yg, [OutAttribute] int% ng, [OutAttribute] int% ifail )

F#
static member g13cd : nxy : int * mtxy : int * pxy : float * mw : int * ish : int * pw : float * l : int * kc : int * xg : float[] * yg : float[] * ng : int byref * ifail : int byref -> unit

Parameters

nxy: Type: System..::..Int32
On entry: $n$ , the length of the time series $x$ and $y$ .

Constraint: $nxy \geq 1$ .

mtxy

Type: System..::..Int32

On entry: whether the data is to be initially mean or trend corrected.

$mtxy = 0$: For no correction.
$mtxy = 1$: For mean correction.
$mtxy = 2$: For trend correction.

Constraint:

0 \leq mtxy \leq 2

pxy: Type: System..::..Double
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: $0.0 \leq pxy \leq 1.0$ .

mw: Type: System..::..Int32
On entry: $M$ , the frequency width of the smoothing window as $\frac{2 π}{M}$ .
A value of $n$ implies that no smoothing is to be carried out.

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

ish: Type: System..::..Int32
On entry: $S$ , the alignment shift between the $x$ and $y$ series. If $x$ leads $y$ , the shift is positive.

Constraint: $- l < ish < l$ .

pw: Type: System..::..Double
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.

If $mw = nxy$ (i.e., no smoothing is carried out) then pw is not used.

Constraint: if $mw \neq nxy$ , $0.0 \leq pw \leq 1.0$ .

l

Type: System..::..Int32

On entry:

L

, the frequency division of smoothed cross spectral estimates as

\frac{2 π}{L}

Constraints:

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

kc

Type: System..::..Int32

On entry: the order of the fast Fourier transform (FFT) used to calculate the spectral estimates. kc should be a product 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:

$kc \geq 2 \times nxy$ ;
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.

xg: Type: array<System..::..Double>[]()[][]
An array of size [kc]
On entry: the nxy data points of the $x$ series.
On exit: the real parts of the ng cross spectral estimates in elements $xg [0]$ to $xg [ng - 1]$ , and $xg [ng]$ to $xg [kc - 1]$ contain $0.0$ . The $y$ series leads the $x$ series.

yg: Type: array<System..::..Double>[]()[][]
An array of size [kc]
On entry: the nxy data points of the $y$ series.
On exit: the imaginary parts of the ng cross spectral estimates in elements $yg [0]$ to $yg [ng - 1]$ , and $yg [ng]$ to $yg [kc - 1]$ contain $0.0$ . The $y$ series leads the $x$ series.

ng: Type: System..::..Int32%
On exit: the number of spectral estimates, $[L / 2] + 1$ , whose separate parts are held in xg and yg.

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 supplied time series may be mean and trend corrected and tapered as in the description of g13cb before calculation of the unsmoothed sample cross-spectrum

f_{x y}^{*} (ω) = \frac{1}{2 π n} \{\sum_{t = 1}^{n} y_{t} \exp (i ω t)\} \times \{\sum_{t = 1}^{n} x_{t} \exp (- i ω t)\}

for frequency values

ω_{j} = \frac{2 π j}{K}

0 \leq ω_{j} \leq π

A correction is made for bias due to any tapering.

As in the description of g13cb for univariate frequency window smoothing, the smoothed spectrum is returned as a subset of these frequencies,

ν_{l} = \frac{2 π l}{L}, l = 0, 1, \dots, [L / 2]

where [ ] denotes the integer part.

Its real part or co-spectrum

c f (ν_{l})

, and imaginary part or quadrature spectrum

q f (ν_{l})

are defined by

f_{x y} (ν_{l}) = c f (ν_{l}) + i q f (ν_{l}) = \sum_{|ω_{k}| < \frac{π}{M}} {\tilde{w}}_{k} f_{x y}^{*} (ν_{l} + ω_{k})

where the weights

{\tilde{w}}_{k}

are similar to the weights

w_{k}

defined for g13cb, but allow for an implicit alignment shift

S

between the series:

{\tilde{w}}_{k} = w_{k} \exp (- 2 π i S k / L) .

It is recommended that

S

is chosen as the lag

k

at which the cross-covariances

c_{x y} (k)

peak, so as to minimize bias.

If no smoothing is required, the integer

M

, which determines the frequency window width

\frac{2 π}{M}

, should be set to

n

The bandwidth of the estimates will normally have been calculated in a previous call of g13cb for estimating the univariate spectra of

y_{t}

and

x_{t}

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,	$nxy < 1$ ,
or	$mtxy < 0$ ,
or	$mtxy > 2$ ,
or	$pxy < 0.0$ ,
or	$pxy > 1.0$ ,
or	$mw < 1$ ,
or	$mw > nxy$ ,
or	$pw < 0.0$ and $mw \neq nxy$ ,
or	$pw > 1.0$ and $mw \neq nxy$ ,
or	$l < 1$ ,
or	$\|ish\| \geq l$ .

$ifail = 2$

On entry,	$kc < 2 \times nxy$ ,
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 in calls to g13cd. Seek expert help.

$ifail = -9000$: 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

g13cd carries out an FFT of length kc to calculate the sample cross spectrum. The time taken by the method for this is approximately proportional to

kc \times \log (kc)

(but see method document (C06PAF not in this release) for further details).

Example

This example reads two time series of length

296

. It selects mean correction and a 10% tapering proportion. It selects a

2 π / 16

frequency width of smoothing window, a window shape parameter of

0.5

and an alignment shift of

3

. It then calls g13cd to calculate the smoothed sample cross spectrum and prints the results.

Example program (C#): g13cde.cs

Example program data: g13cde.d

Example program results: g13cde.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also