g08ea:: Nonparametric Statistics (NAG Toolbox)

Runs tests may be used to investigate for trends in a sequence of observations. nag_nonpar_randtest_runs (g08ea) computes statistics for the runs up test. If the runs down test is desired then each observation must be multiplied by

- 1

before nag_nonpar_randtest_runs (g08ea) is called with the modified vector of observations. nag_nonpar_randtest_runs (g08ea) may be used in two different modes:

(i)	a single call to nag_nonpar_randtest_runs (g08ea) which computes all test statistics after counting the runs;
(ii)	multiple calls to nag_nonpar_randtest_runs (g08ea) with the final test statistics only being computed in the last call.

A run up is a sequence of numbers in increasing order. A run up ends at

x_{k}

when

x_{k} > x_{k + 1}

and the new run then begins at

x_{k + 1}

. nag_nonpar_randtest_runs (g08ea) counts the number of runs up of different lengths. Let

c_{i}

denote the number of runs of length

i

, for

i = 1, 2, \dots, r - 1

. The number of runs of length

r

or greater is then denoted by

c_{r}

An unfinished run at the end of a sequence is not counted unless the sequence is part of an initial or intermediate call to nag_nonpar_randtest_runs (g08ea) (i.e., unless there is another call to nag_nonpar_randtest_runs (g08ea) to follow) in which case the unfinished run is used together with the beginning of the next sequence of numbers input to nag_nonpar_randtest_runs (g08ea) in the next call. The following is a trivial example.

When the counting of runs is complete nag_nonpar_randtest_runs (g08ea) computes the expected values and covariances of the counts,

c_{i}

. For the details of the method used see Knuth (1981). An approximate

χ^{2}

statistic with

r

degrees of freedom is computed, where

X^{2} = {(c - μ_{c})}^{T} Σ_{c}^{- 1} (c - μ_{c}),

where

$c$ is the vector of counts, $c_{i}$ , for $i = 1, 2, \dots, r$ ,
$μ_{c}$ is the vector of expected values,
$e_{i}$ , for $i = 1, 2, \dots, r$ , where $e_{i}$ is the expected value for $c_{i}$ under the null hypothesis of randomness, and
$Σ_{c}$ is the covariance matrix of $c$ under the null hypothesis.

The use of the

χ^{2}

-distribution as an approximation to the exact distribution of the test statistic,

X^{2}

, improves as the length of the sequence relative to

m

increases and hence the expected value,

e

, increases.

You may specify the total number of runs to be found. If the specified number of runs is found before the end of a sequence nag_nonpar_randtest_runs (g08ea) will exit before counting any further runs. The number of runs actually counted and used to compute the test statistic is returned via nruns.

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

The time taken by nag_nonpar_randtest_runs (g08ea) increases with the number of observations

n

, and also depends to some extent on whether the call to nag_nonpar_randtest_runs (g08ea) is an only, first, intermediate or last call.

Example

The following program performs a runs up test on

500

pseudorandom numbers. nag_nonpar_randtest_runs (g08ea) is called

5

times with

100

observations each time. No limit is placed on the number of runs to be counted. All runs of length

6

or more are counted together.

function g08ea_example


fprintf('g08ea example results\n\n');

% Initialize the base generator to a repeatable sequence
seed = [int64(324213)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

m      = int64(0);
nruns  = int64(0);
ncount = [int64(0);0;0;0;0;0];
n      = int64(100);
nsampl = 5;
cl     = 'F';

for i=1:nsampl
  % Generate a sample from U(0,1)
  [state, x, ifail] = g05sq( ...
                             n, 0, 1, state);
  % Process the sample
  [nruns, ncount, ex, covar, chi, df, prob, ifail] = ...
  g08ea( ...
         cl, x, m, nruns, ncount);
  % Adjust CL
  cl = 'I';
  if i==nsampl-1
     cl = 'L';
  end
end

fprintf('Total number of runs found = %d\n', nruns);
fprintf('\n%33s\n', 'Count');
head = '        1        2        3        4        5       >5';
fprintf('%s\n', head);
fprintf('%9d', ncount);

fprintf('\n\n%34s\n', 'Expect');
fprintf('%s\n', head);
fprintf('%9.1f', ex);
fprintf('\n\n');

[ifail] = x04ca( ...
                 'General', ' ', covar, 'Covariance matrix');

fprintf('\n\nChisq = %10.4f\n', chi);
fprintf('DF    = %7.1f\n', df);
fprintf('Prob  = %10.4f\n', prob);

g08ea example results

Total number of runs found = 251

                            Count
        1        2        3        4        5       >5
       77      120       39       12        1        2

                            Expect
        1        2        3        4        5       >5
     83.8    104.0     45.6     13.1      2.9      0.6

 Covariance matrix
             1          2          3          4          5          6
 1     64.2222    -9.8639    -7.4780    -3.5759    -1.1406    -0.3305
 2     -9.8639    70.2942   -24.4639    -9.8092    -2.7386    -0.7103
 3     -7.4780   -24.4639    29.9473    -5.8284    -1.5474    -0.3852
 4     -3.5759    -9.8092    -5.8284    11.0343    -0.5319    -0.1289
 5     -1.1406    -2.7386    -1.5474    -0.5319     2.7169    -0.0318
 6     -0.3305    -0.7103    -0.3852    -0.1289    -0.0318     0.5809


Chisq =     9.7559
DF    =     6.0
Prob  =     0.1353

NAG Toolbox: nag_nonpar_randtest_runs (g08ea)

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Purpose

Syntax

Description