g13bc calculates cross-correlations between two time series.

Syntax

C#
public static void g13bc( double[] x, double[] y, int nxy, int nl, out double s, out double r0, double[] r, out double stat, out int ifail )

Visual Basic
Public Shared Sub g13bc ( _ x As Double(), _ y As Double(), _ nxy As Integer, _ nl As Integer, _ <OutAttribute> ByRef s As Double, _ <OutAttribute> ByRef r0 As Double, _ r As Double(), _ <OutAttribute> ByRef stat As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13bc ( _
	x As Double(), _
	y As Double(), _
	nxy As Integer, _
	nl As Integer, _
	<OutAttribute> ByRef s As Double, _
	<OutAttribute> ByRef r0 As Double, _
	r As Double(), _
	<OutAttribute> ByRef stat As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13bc( array<double>^ x, array<double>^ y, int nxy, int nl, [OutAttribute] double% s, [OutAttribute] double% r0, array<double>^ r, [OutAttribute] double% stat, [OutAttribute] int% ifail )

Visual C++

public:
static void g13bc(
	array<double>^ x, 
	array<double>^ y, 
	int nxy, 
	int nl, 
	[OutAttribute] double% s, 
	[OutAttribute] double% r0, 
	array<double>^ r, 
	[OutAttribute] double% stat, 
	[OutAttribute] int% ifail
)

F#
static member g13bc : x : float[] * y : float[] * nxy : int * nl : int * s : float byref * r0 : float byref * r : float[] * stat : float byref * ifail : int byref -> unit

Parameters

x: Type: array<System..::..Double>[]()[][]
An array of size [nxy]
On entry: the $n$ values of the $x$ series.

y: Type: array<System..::..Double>[]()[][]
An array of size [nxy]
On entry: the $n$ values of the $y$ series.

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

Constraint: $nxy \geq 2$ .

nl: Type: System..::..Int32
On entry: $L$ , the maximum lag for calculating cross-correlations.

Constraint: $1 \leq nl < nxy$ .

s: Type: System..::..Double%
On exit: the ratio of the standard deviation of the $y$ series to the standard deviation of the $x$ series, $s_{y} / s_{x}$ .

r0: Type: System..::..Double%
On exit: the cross-correlation between the $x$ and $y$ series at lag zero.

r: Type: array<System..::..Double>[]()[][]
An array of size [nl]
On exit: $r [l - 1]$ contains the cross-correlations between the $x$ and $y$ series at lags $L$ , $r_{x y} (l)$ , for $l = 1, 2, \dots, L$ .

stat: Type: System..::..Double%
On exit: the statistic for testing for absence of cross-correlation.

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

Given two series

x_{1}, x_{2}, \dots, x_{n}

and

y_{1}, y_{2}, \dots, y_{n}

the method calculates the cross-correlations between

x_{t}

and lagged values of

y_{t}

r_{x y} (l) = \frac{\sum_{t = 1}^{n - l} (x_{t} - \bar{x}) (y_{t + l} - \bar{y})}{n s_{x} s_{y}}, l = 0, 1, \dots, L

where

\bar{x} = \frac{\sum_{t = 1}^{n} x_{t}}{n}

s_{x}^{2} = \frac{\sum_{t = 1}^{n} {(x_{t} - \bar{x})}^{2}}{n}

and similarly for

y

The ratio of standard deviations

s_{y} / s_{x}

is also returned, and a portmanteau statistic is calculated:

stat = n \sum_{l = 1}^{L} r_{x y} {(l)}^{2} .

Provided

n

is large,

L

much less than

n

, and both

x_{t}, y_{t}

are samples of series whose true autocorrelation functions are zero, then, under the null hypothesis that the true cross-correlations between the series are zero, stat has a

χ^{2}

-distribution with

L

degrees of freedom. Values of stat in the upper tail of this distribution provide evidence against the null hypothesis.

References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$nxy \leq 1$ ,
or	$nl < 1$ ,
or	$nl \geq nxy$ .

$ifail = 2$: One or both of the $x$ and $y$ series have zero variance and hence cross-correlations cannot be calculated.

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

Accuracy

All computations are believed to be stable.

Parallelism and Performance

None.

Further Comments

n < 100

, or

L < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by g13bc is approximately proportional to

n L

, otherwise the autocorrelations are calculated by utilizing fast Fourier transforms (FFTs) and the time taken is approximately proportional to

n \log (n)

. If FFTs are used then g13bc internally allocates approximately

6 n

real elements.

Example

This example reads two time series of length

20

. It calculates and prints the cross-correlations up to lag

15

for the first series leading the second series and then for the second series leading the first series.

Example program (C#): g13bce.cs

Example program data: g13bce.d

Example program results: g13bce.r