NAG Library Function Document

nag_tsa_auto_corr (g13abc)

+− 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

nag_tsa_auto_corr (g13abc) 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

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_auto_corr (const double x[], Integer nx, Integer nk, double mean, double var, double r[], double stat, NagError fail)

3 Description

The data consist 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

(d)

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}}

(e)

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

(f)

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.

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] – const doubleInput: On entry: the time series, $x_{i}$ , for $i = 1, 2, \dots, n$ .
2: nx – IntegerInput: On entry: the number of values, $n$ , in the time series.
Constraint: $nx > 1$ .
3: nk – IntegerInput: On entry: the number of lags, $K$ , for which the autocorrelations are required. The lags range from 1 to $K$ and do not include zero.
Constraint: $0 < nk < nx$ .
4: mean – double *Output: On exit: the sample mean of the input time series.
5: var – double *Output: On exit: the sample variance of the input time series.
6: r[nk] – doubleOutput: On exit: the sample autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .
7: stat – double *Output: On exit: the statistic used to test the hypothesis that the true autocorrelation function of the time series is identically zero.
8: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LE: On entry, $nx = ⟨value⟩$ while $nk = ⟨value⟩$ . These arguments must satisfy $nx > nk$ .
NE_INT_ARG_LE: On entry, $nk = ⟨value⟩$ .
Constraint: $nk > 0$ .
On entry, $nx = ⟨value⟩$ .
Constraint: $nx > 1$ .
NE_TIME_SERIES_IDEN: On entry, all values of x are practically identical, giving zero variance. In this case r and stat are undefined on exit.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_tsa_auto_corr (g13abc) is approximately proportional to

nx \times nk

10 Example

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

NAG Library Function Documentnag_tsa_auto_corr (g13abc)