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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_uni_autocorr (g13ab)

## Purpose

nag_tsa_uni_autocorr (g13ab) 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.

## Syntax

[xm, xv, r, stat, ifail] = g13ab(x, nk, 'nx', nx)
[xm, xv, r, stat, ifail] = nag_tsa_uni_autocorr(x, nk, 'nx', nx)

## Description

The data consists of $n$ observations ${x}_{i}$, for $\mathit{i}=1,2,\dots ,n$ from a time series.
The quantities calculated are
(a) The sample mean
 $x-=∑i=1nxin.$
(b) The sample variance (for $n\ge 2$)
 $s2=∑i=1n xi-x- 2 n-1 .$
(c) The sample autocorrelation coefficients of lags $k=1,2,\dots ,K$, where $K$ is a user-specified maximum lag, and $K, $n>1$.
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
 $stat=n∑k= 1Krk2,$
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 ${\chi }_{K}^{2}$ distribution under the hypothesis of a zero autocorrelation function. Values of stat in the upper tail of the distribution provide evidence against the hypothesis; nag_stat_prob_chisq (g01ec) 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.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{nx}}\right)$ – double array
The time series, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
2:     $\mathrm{nk}$int64int32nag_int scalar
$K$, the number of lags for which the autocorrelations are required. The lags range from $1$ to $K$ and do not include zero.
Constraint: $0<{\mathbf{nk}}<{\mathbf{nx}}$.

### Optional Input Parameters

1:     $\mathrm{nx}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the number of values in the time series.
Constraint: ${\mathbf{nx}}>1$.

### Output Parameters

1:     $\mathrm{xm}$ – double scalar
The sample mean of the input time series.
2:     $\mathrm{xv}$ – double scalar
The sample variance of the input time series.
3:     $\mathrm{r}\left({\mathbf{nk}}\right)$ – double array
The sample autocorrelation coefficient relating to lag $\mathit{k}$, for $\mathit{k}=1,2,\dots ,K$.
4:     $\mathrm{stat}$ – double scalar
The statistic used to test the hypothesis that the true autocorrelation function of the time series is identically zero.
5:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{nx}}\le {\mathbf{nk}}$, or ${\mathbf{nx}}\le 1$, or ${\mathbf{nk}}\le 0$.
W  ${\mathbf{ifail}}=2$
On entry, all values of x are practically identical, giving zero variance. In this case r and stat are undefined on exit.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The computations are believed to be stable.

If $n<100$, or $K<10\mathrm{log}\left(n\right)$ then the autocorrelations are calculated directly and the time taken by nag_tsa_uni_autocorr (g13ab) is approximately proportional to $nK$, otherwise the autocorrelations are calculated by utilizing fast fourier transforms (FFTs) and the time taken is approximately proportional to $n\mathrm{log}\left(n\right)$. If FFTs are used then nag_tsa_uni_autocorr (g13ab) internally allocates approximately $4n$ real elements.
If the input series for nag_tsa_uni_autocorr (g13ab) was generated by differencing using nag_tsa_uni_diff (g13aa), ensure that only the differenced values are input to nag_tsa_uni_autocorr (g13ab), and not the reconstituting information.

## Example

In the example below, a set of $50$ values of sunspot counts is used as input. The first $10$ autocorrelations are computed.
```function g13ab_example

fprintf('g13ab example results\n\n');

x = [ 5;  11;  16;  23;  36;  58;  29;  20;  10;   8;
3;   0;   0;   2;  11;  27;  47;  63;  60;  39;
28;  26;  22;  11;  21;  40;  78; 122; 103;  73;
47;  35;  11;   5;  16;  34;  70;  81; 111; 101;
73;  40;  20;  16;   5;  11;  22;  40;  60;  80.9];

nk = int64(10);
% Compute autocorrelation
[xm, xv, r, stat, ifail] = g13ab( ...
x, nk);

% Display results
fprintf('The first %4d coefficients are required\n',nk);
fprintf('The input array has sample mean     %12.4f\n', xm);
fprintf('The input array has sample variance %12.4f\n', xv);
fprintf('The sample autocorrelation coefficients are\n\n');
fprintf('   Lag    Coeff      Lag    Coeff\n');
ivar = double([1:nk]);
fprintf('%6d%10.4f%8d%10.4f\n',[ivar; r(1:nk)']);
fprintf('\nThe value of stat is %12.4f\n', stat);

% For plotting use all posible lags
nk = int64(numel(x)-1);
[xm, xv, r, stat, ifail] = g13ab( ...
x, nk);

fig1 = figure;
refline = 1.96/sqrt(numel(x));
hold on
h = bar(r,0.1);
h.FaceColor = [1 0 0];
h.EdgeColor = [1 0 0];
h.ShowBaseLine = 'off';
plot([0 nk+1],[refline refline],'green');
plot([0 nk+1],[-refline -refline],'green');
axis([0 50 -0.6 1]);
ax = gca;
ax.YTick = [-0.6:0.2:1];
ax.XTick = [0:10:50];
xlabel('Lag');
ylabel('ACF');
title('Sample autocorrelation coefficients');
hold off

```
```g13ab example results

The first   10 coefficients are required
The input array has sample mean          37.4180
The input array has sample variance    1002.0301
The sample autocorrelation coefficients are

Lag    Coeff      Lag    Coeff
1    0.8004       2    0.4355
3    0.0328       4   -0.2835
5   -0.4505       6   -0.4242
7   -0.2419       8    0.0550
9    0.3783      10    0.5857

The value of stat is      92.1231
``` This plot shows the autocorrelations for all possible lag values. Reference lines are given at $±{z}_{0.975}/\sqrt{n}$.