NAG Library Routine Document

g13abf (uni_autocorr)


g13abf computes the sample autocorrelation function of a time series. It also computes the sample mean, the sample variance and a statistic which may be used to test the hypothesis that the true autocorrelation function is zero.


Fortran Interface
Subroutine g13abf ( x, nx, nk, xm, xv, r, stat, ifail)
Integer, Intent (In):: nx, nk
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x(nx)
Real (Kind=nag_wp), Intent (Out):: xm, xv, r(nk), stat
C Header Interface
#include <nagmk26.h>
void  g13abf_ (const double x[], const Integer *nx, const Integer *nk, double *xm, double *xv, double r[], double *stat, Integer *ifail)


The data consists of n observations xi, for i=1,2,,n from a time series.
The quantities calculated are
(a) The sample mean
(b) The sample variance (for n2)
s2=i=1n xi-x- 2 n-1 .  
(c) The sample autocorrelation coefficients of lags k=1,2,,K, where K is a user-specified maximum lag, and K<n, n>1.
The coefficient of lag k is defined as
rk=i=1 n-kxi-x-xi+k-x- i=1n xi-x- 2 .  
See page 496 of Box and Jenkins (1976) for further details.
(d) A test statistic defined as
stat=nk= 1Krk2,  
which can be used to test the hypothesis that the true autocorrelation function is identically zero.
If n is large and K is much smaller than n, stat has a χK2 distribution under the hypothesis of a zero autocorrelation function. Values of stat in the upper tail of the distribution provide evidence against the hypothesis; g01ecf can be used to compute the tail probability.
Section 8.2.2 of Box and Jenkins (1976) provides further details of the use of stat.


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


1:     xnx – Real (Kind=nag_wp) arrayInput
On entry: the time series, xi, for i=1,2,,n.
2:     nx – IntegerInput
On entry: n, the number of values in the time series.
Constraint: nx>1.
3:     nk – IntegerInput
On entry: K, the number of lags for which the autocorrelations are required. The lags range from 1 to K and do not include zero.
Constraint: 0<nk<nx.
4:     xm – Real (Kind=nag_wp)Output
On exit: the sample mean of the input time series.
5:     xv – Real (Kind=nag_wp)Output
On exit: the sample variance of the input time series.
6:     rnk – Real (Kind=nag_wp) arrayOutput
On exit: the sample autocorrelation coefficient relating to lag k, for k=1,2,,K.
7:     stat – Real (Kind=nag_wp)Output
On exit: the statistic used to test the hypothesis that the true autocorrelation function of the time series is identically zero.
8:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

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, nk=value.
Constraint: nk>0.
On entry, nx=value.
Constraint: nx>1.
On entry, nx=value and nk=value.
Constraint: nx>nk.
On entry, all values of x are practically identical. This gives zero variance and means that r and stat are undefined on exit.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 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 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


The computations are believed to be stable.

Parallelism and Performance

g13abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13abf 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.

Further Comments

If n<100, or K<10logn then the autocorrelations are calculated directly and the time taken by g13abf is approximately proportional to nK, 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 g13abf internally allocates approximately 4n real elements.
If the input series for g13abf was generated by differencing using g13aaf, ensure that only the differenced values are input to g13abf, and not the reconstituting information.


In the example below, a set of 50 values of sunspot counts is used as input. The first 10 autocorrelations are computed.

Program Text

Program Text (g13abfe.f90)

Program Data

Program Data (g13abfe.d)

Program Results

Program Results (g13abfe.r)

This plot shows the autocorrelations for all possible lag values. Reference lines are given at ±z0.975/n.
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 ACF Lag Example Program Sample Autocorrelation Coefficients gnuplot_plot_1 gnuplot_plot_2 gnuplot_plot_3