NAG Toolbox: nag_nonpar_randtest_pairs (g08eb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{cl}$ – string (length ≥ 1)

Indicates the type of call to nag_nonpar_randtest_pairs (g08eb).

$\mathbf{cl}=\text{'S'}$

This is the one and only call to nag_nonpar_randtest_pairs (g08eb) (single call mode). All data are to be input at once. All test statistics are computed after the counting of pairs is complete.

$\mathbf{cl}=\text{'F'}$

This is the first call to the function. All initializations are carried out and the counting of pairs begins. The final test statistics are not computed since further calls will be made to nag_nonpar_randtest_pairs (g08eb).

$\mathbf{cl}=\text{'I'}$

This is an intermediate call during which the counts of pairs are updated. The final test statistics are not computed since further calls will be made to nag_nonpar_randtest_pairs (g08eb).

$\mathbf{cl}=\text{'L'}$

This is the last call to nag_nonpar_randtest_pairs (g08eb). The test statistics are computed after the final counting of runs is complete.

Constraint: $\mathbf{cl}=\text{'S'}$, $\text{'F'}$, $\text{'I'}$ or $\text{'L'}$.

2:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

The sequence of observations.

Constraint: $0.0\le \mathbf{x}\left(\mathit{i}\right)\le 1.0$, for $\mathit{i}=1,2,\dots ,n$.

3:     $\mathrm{lag}$ – int64int32nag_int scalar

$l$, the lag to be used in choosing pairs.

If $\mathbf{lag}=1$, then we consider the pairs $\left(\mathbf{x}\left(\mathit{i}\right),\mathbf{x}\left(\mathit{i}+1\right)\right)$, for $\mathit{i}=1,3,\dots ,n-1$, where $n$ is the number of observations.

If $\mathbf{lag}>1$, then we consider the pairs $\left(\mathbf{x}\left(i\right),\mathbf{x}\left(i+l\right)\right)$, for $i=1,2,\dots ,l,2l+1,2l+2,\dots ,3l,4l+1,\dots ,n-l$, where $n$ is the number of observations.

lag must not be changed between calls to nag_nonpar_randtest_pairs (g08eb).

Constraints:

• $\mathbf{lag}\ge 1$;
• if $\mathbf{cl}=\text{'S'}$, $\mathbf{lag}<\mathbf{n}$.

4:     $\mathrm{ncount}\left(\mathit{ldc},\mathbf{msize}\right)$ – int64int32nag_int array

ldc, the first dimension of the array, must satisfy the constraint $\mathit{ldc}\ge \mathbf{msize}$.

If $\mathbf{cl}=\text{'S'}$ or $\text{'F'}$, ncount need not be set.

If $\mathbf{cl}=\text{'I'}$ or $\text{'L'}$, ncount must contain the values returned by the previous call to nag_nonpar_randtest_pairs (g08eb).

5:     $\mathrm{wrk}\left(2\times \mathbf{lag}\right)$ – double array

wrk is used to store information between successive calls to nag_nonpar_randtest_pairs (g08eb) and therefore must not be changed.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the array x.

$n$, the number of observations.

Constraints:

• if $\mathbf{cl}=\text{'S'}$, $\mathbf{n}\ge 2$;
• otherwise $\mathbf{n}\ge 1$.

2:     $\mathrm{msize}$ – int64int32nag_int scalar

Default: the first dimension of the array ncount and the second dimension of the array ncount. (An error is raised if these dimensions are not equal.)

$m$, the size of the matrix of counts.

msize must not be changed between calls to nag_nonpar_randtest_pairs (g08eb).

Constraint: $\mathbf{msize}\ge 2$.

Output Parameters

1:     $\mathrm{ncount}\left(\mathit{ldc},\mathbf{msize}\right)$ – int64int32nag_int array

Is an msize by msize matrix containing the counts of the number of pairs in each cell, $c_{ij}$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,m$.

2:     $\mathrm{ex}$ – double scalar

If $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then ex contains the expected number of counts in each cell, $e$.

Otherwise ex is not set.

3:     $\mathrm{chi}$ – double scalar

If $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then chi contains the ${\chi }^2$ test statistic, $X^2$, for testing the null hypothesis of randomness.

Otherwise chi is not set.

4:     $\mathrm{df}$ – double scalar

If $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then df contains the degrees of freedom for the ${\chi }^2$ statistic.

Otherwise df is not set.

5:     $\mathrm{prob}$ – double scalar

If $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then prob contains the upper tail probability associated with the ${\chi }^2$ test statistic, i.e., the significance level.

Otherwise prob is not set.

6:     $\mathrm{wrk}\left(2\times \mathbf{lag}\right)$ – double array

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

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.

   $\mathbf{ifail}=1$

On entry, $\mathbf{cl}=\_$.
Constraint: $\mathbf{cl}=\text{'S'}$, $\text{'F'}$, $\text{'I'}$ or $\text{'L'}$.

   $\mathbf{ifail}=2$

Constraint: if $\mathbf{cl}=\text{'S'}$, $\mathbf{n}\ge 2$, otherwise $\mathbf{n}\ge 1$.

   $\mathbf{ifail}=3$

Constraint: $\mathbf{msize}\ge 2$ 

   $\mathbf{ifail}=4$

Constraint: $\mathbf{lag}>0$ and if $\mathbf{cl}=\text{'S'}$, $\mathbf{lag}<\mathbf{n}$.

   $\mathbf{ifail}=5$

Constraint: $\mathit{ldc}\ge \mathbf{msize}$.

   $\mathbf{ifail}=6$

On entry, at least one element of x is out of range.
Constraint: $0\le \mathbf{x}\left(i\right)\le 1$, for $i=1,2,\dots ,\mathbf{n}$.

   $\mathbf{ifail}=7$

No pairs were found. This will occur if the value of lag is greater than or equal to the total number of observations.

W  $\mathbf{ifail}=8$

msize is too large relative to the number of pairs, therefore the expected value for at least one cell is less than or equal to $5.0$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g08eb_example

function g08eb_example


fprintf('g08eb 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);

lag = int64(1);
wrk = zeros(2*lag, 1);
m   = 5;
ncount = zeros(m, m, 'int64');
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
  [ncount, ex, chi, df, prob, wrk, ifail] = ...
  g08eb( ...
         cl, x, lag, ncount, wrk);

  % Adjust CL
  cl = 'I';
  if i==nsampl-1
     cl = 'L';
  end
end

[ifail] = x04ea( ...
                 'General', ' ', ncount, 'Count matrix');
fprintf('\nExpected value = %8.2f\n', ex);
fprintf('Chisq          = %10.4f\n', chi);
fprintf('DF             = %7.1f\n', df);
fprintf('Probability    = %10.4f\n', prob);

g08eb example results

 Count matrix
     1  2  3  4  5
 1   7 10  5 16  8
 2   9 10  7  6  8
 3  13 15 10 10 12
 4  10 21  7  5 13
 5  13  5 10 12  8

Expected value =    10.00
Chisq          =    34.8000
DF             =    24.0
Probability    =     0.0714