G13ABF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

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.

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.

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

On entry: the time series,

x_{i}

, for

i = 1, 2, \dots, n

On entry:

n

, the number of values in the time series.

Constraint:

NX > 1

On entry:

K

, the number of lags for which the autocorrelations are required. The lags range from

1

K

and do not include zero.

Constraint:

0 < NK < NX

On exit: the sample mean of the input time series.

On exit: the sample variance of the input time series.

On exit: the sample autocorrelation coefficient relating to lag

k

, for

k = 1, 2, \dots, K

On exit: the statistic used to test the hypothesis that the true autocorrelation function of the time series is identically zero.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

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:

On entry, all values of X are practically identical, giving zero variance. In this case R and STAT are undefined on exit.

The computations are believed to be stable.

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.

In the example below, a set of

50

values of sunspot counts is used as input. The first

10

autocorrelations are computed.

NAG Library Routine Document

G13ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

On entry,	$NX \leq NK$ ,
or	$NX \leq 1$ ,
or	$NK \leq 0$ .

NAG Library Routine DocumentG13ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G13ABF