NAG Library Routine Document

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.

2

Specification

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)

3

Description

The data consists of

n

observations

x_{i}

, for

i = 1, 2, \dots, n

from a time series.

The quantities calculated are

(a)

The sample mean

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

(b)

The sample variance (for

n \geq 2

)

s^{2} = \frac{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}{(n - 1)} .

(c)

The sample autocorrelation coefficients of lags

k = 1, 2, \dots, K

, where

K

is a user-specified maximum lag, and

K < n

n > 1

The coefficient of lag

k

is defined as

r_{k} = \frac{\sum_{i = 1}^{n - k} (x_{i} - \bar{x}) (x_{i + k} - \bar{x})}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}} .

See page 496 of Box and Jenkins (1976) for further details.

(d)

A test statistic defined as

stat = n \sum_{k = 1}^{K} r_{k}^{2},

which can be used to test the hypothesis that the true autocorrelation function is identically zero.

n

is large and

K

is much smaller than

n

, stat has a

χ_{K}^{2}

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.

4

References

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

5

Arguments

1: $x (nx)$ – Real (Kind=nag_wp) arrayInput: On entry: the time series, $x_{i}$ , for $i = 1, 2, \dots, n$ .
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: $r (nk)$ – Real (Kind=nag_wp) arrayOutput: On exit: the sample autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .
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).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $nk = 〈value〉$ .
Constraint: $nk > 0$ .

On entry, $nx = 〈value〉$ .
Constraint: $nx > 1$ .

On entry, $nx = 〈value〉$ and $nk = 〈value〉$ .
Constraint: $nx > nk$ .

$ifail = 2$: On entry, all values of x are practically identical. This gives zero variance and means that r and stat are undefined on exit.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computations are believed to be stable.

8

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.

9

Further Comments

n < 100

, or

K < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by g13abf is approximately proportional to

n K

, 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 g13abf internally allocates approximately

4 n

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.

10

Example

In the example below, a set of

50

values of sunspot counts is used as input. The first

10

autocorrelations are computed.

This plot shows the autocorrelations for all possible lag values. Reference lines are given at

\pm z_{0.975} / \sqrt{n}

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

G13 (tsa) Chapter Contents

G13 (tsa) Chapter Introduction

NAG Library Routine Document

g13abf (uni_autocorr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose