NAG Library Function Document

nag_tsa_cross_corr (g13bcc)


    1  Purpose
    7  Accuracy


nag_tsa_cross_corr (g13bcc) calculates cross-correlations between two time series.


#include <nag.h>
#include <nagg13.h>
void  nag_tsa_cross_corr (const double x[], const double y[], Integer nxy, Integer nl, double *s, double *r0, double r[], double *stat, NagError *fail)


Given two series x1,x2,,xn and y1,y2,,yn the function calculates the cross-correlations between xt and lagged values of yt:
rxyl=t=1 n-lxt-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.


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


1:     x[nxy] const doubleInput
On entry: the n values of the x series.
2:     y[nxy] const doubleInput
On entry: the n values of the y series.
3:     nxy IntegerInput
On entry: n, the length of the time series.
Constraint: nxy2.
4:     nl IntegerInput
On entry: L, the maximum lag for calculating cross-correlations.
Constraint: 1nl<nxy.
5:     s double *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 double *Output
On exit: the cross-correlation between the x and y series at lag zero.
7:     r[nl] doubleOutput
On exit: r[l-1] contains the cross-correlations between the x and y series at lags L, rxyl, for l=1,2,,L.
8:     stat double *Output
On exit: the statistic for testing for absence of cross-correlation.
9:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, nl=value.
Constraint: nl1.
On entry, nxy=value.
Constraint: nxy>1.
On entry, nlnxy: nl=value and nxy=value.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
One or both of the x and y series have zero variance.


All computations are believed to be stable.

Parallelism and Performance

nag_tsa_cross_corr (g13bcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_cross_corr (g13bcc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

If n<100, or L<10logn then the autocorrelations are calculated directly and the time taken by nag_tsa_cross_corr (g13bcc) is approximately proportional to nL, otherwise the autocorrelations are calculated by utilizing fast Fourier transforms (FFTs) and the time taken is approximately proportional to nlogn. If FFTs are used then nag_tsa_cross_corr (g13bcc) internally allocates approximately 6n real elements.


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.

Program Text

Program Text (g13bcce.c)

Program Data

Program Data (g13bcce.d)

Program Results

Program Results (g13bcce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017