g13ab:: Time Series Analysis (NAG Toolbox)

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; 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

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

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

Accuracy

Further Comments

n < 100

, or

K < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by nag_tsa_uni_autocorr (g13ab) 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 nag_tsa_uni_autocorr (g13ab) internally allocates approximately

4 n

real elements.

Example

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

On entry,	$nx \leq nk$ ,
or	$nx \leq 1$ ,
or	$nk \leq 0$ .

NAG Toolbox: nag_tsa_uni_autocorr (g13ab)

▸▿ Contents

Purpose

Syntax