NAG FL Interface
g13bcf (multi_​xcorr)

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1 Purpose

g13bcf calculates cross-correlations between two time series.

2 Specification

Fortran Interface
Subroutine g13bcf ( x, y, nxy, nl, s, r0, r, stat, ifail)
Integer, Intent (In) :: nxy, nl
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(nxy), y(nxy)
Real (Kind=nag_wp), Intent (Out) :: s, r0, r(nl), stat
C Header Interface
#include <nag.h>
void  g13bcf_ (const double x[], const double y[], const Integer *nxy, const Integer *nl, double *s, double *r0, double r[], double *stat, Integer *ifail)
The routine may be called by the names g13bcf or nagf_tsa_multi_xcorr.

3 Description

Given two series x1,x2,,xn and y1,y2,,yn the routine calculates the cross-correlations between xt and lagged values of yt:
rxy(l)=t=1 n-l(xt-x¯)(yt+l-y¯) nsxsy ,  l=0,1,,L  
x¯=t= 1nxtn  
sx2=t=1n (xt-x¯) 2n  
and similarly for y.
The ratio of standard deviations sy/sx is also returned, and a portmanteau statistic is calculated:
stat=nl=1Lrxy (l) 2.  
Provided n is large, L much less than n, and both xt,yt 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 χ2-distribution with L degrees of freedom. Values of stat in the upper tail of this distribution provide evidence against the null hypothesis.

4 References

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

5 Arguments

1: x(nxy) Real (Kind=nag_wp) array Input
On entry: the n values of the x series.
2: y(nxy) Real (Kind=nag_wp) array Input
On entry: the n values of the y series.
3: nxy Integer Input
On entry: n, the length of the time series.
Constraint: nxy2.
4: nl Integer Input
On entry: L, the maximum lag for calculating cross-correlations.
Constraint: 1nl<nxy.
5: s Real (Kind=nag_wp) Output
On exit: the ratio of the standard deviation of the y series to the standard deviation of the x series, sy/sx.
6: r0 Real (Kind=nag_wp) Output
On exit: the cross-correlation between the x and y series at lag zero.
7: r(nl) Real (Kind=nag_wp) array Output
On exit: r(l) contains the cross-correlations between the x and y series at lags L, rxy(l), for l=1,2,,L.
8: stat Real (Kind=nag_wp) Output
On exit: the statistic for testing for absence of cross-correlation.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, nl=value.
Constraint: nl1.
On entry, nl=value and nxy=value.
Constraint: 1nl<nxy.
On entry, nxy=value.
Constraint: nxy>1.
One or both of the x and y series have zero variance.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

All computations are believed to be stable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g13bcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13bcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

If n<100, or L<10log(n) then the autocorrelations are calculated directly and the time taken by g13bcf is approximately proportional to nL, otherwise the autocorrelations are calculated by utilizing fast Fourier transforms (FFTs) and the time taken is approximately proportional to nlog(n). If FFTs are used then g13bcf internally allocates approximately 6n real elements.

10 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.

10.1 Program Text

Program Text (g13bcfe.f90)

10.2 Program Data

Program Data (g13bcfe.d)

10.3 Program Results

Program Results (g13bcfe.r)