naginterfaces.library.nonpar.randtest_​triplets

naginterfaces.library.nonpar.randtest_triplets(cl, x, ncount, comm)

randtest_triplets performs the triplets test on a sequence of observations from the interval $[0,1]$.

For full information please refer to the NAG Library document for g08ec

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g08/g08ecf.html

Parameters

clstr, length 1

Indicates the type of call to randtest_triplets.

$cl=\text{'S'}$

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

$cl=\text{'F'}$

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

$cl=\text{'I'}$

This is an intermediate call during which counts of the triplets are updated. The final test statistics are not computed since further calls will be made to randtest_triplets.

$cl=\text{'L'}$

This is the last call to randtest_triplets. The test statistics are computed after the final counting of the triplets is complete.

xfloat, array-like, shape $\left(n\right)$

The sequence of observations.

ncountint, array-like, shape $(\text{msize},\text{msize},\text{msize})$

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

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

commdict, communication object, modified in place

Communication structure.

On initial entry: need not be set.

Returns

ncountint, ndarray, shape $(\text{msize},\text{msize},\text{msize})$

Is an $\text{msize}$ by $\text{msize}$ by $\text{msize}$ matrix containing the counts of the number of triplets, $c_{jkl}$, for $\text{l}=1,2,\dots ,m$, for $\text{k}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,m$.

exfloat

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

Otherwise $ex$ is not set.

chifloat

If $cl=\text{'S'}$ 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.

dffloat

If $cl=\text{'S'}$ 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.

probfloat

If $cl=\text{'S'}$ 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.

Raises

NagValueError

(errno $1$)

On entry, $cl=⟨value⟩$.

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

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: if $cl=\text{'S'}$, $n\ge 3$, otherwise $n\ge 1$.

(errno $3$)

On entry, $\text{msize}=⟨value⟩$.

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

(errno $5$)

On entry, at least one element of $x$ is out of range.

Constraint: $0\le x[i-1]\le 1$, for $i=1,2,\dots ,n$.

(errno $6$)

No triplets were found because less than $3$ observations were provided in total.

Warns

NagAlgorithmicWarning

(errno $7$)

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

This implies that the ${\chi }^2$ distribution may not be a very good approximation to the distribution of the test statistic.

$\text{msize}=⟨value⟩$, number of triplets $=⟨value⟩$ and $\text{expected value}=⟨value⟩$.

All statistics are returned and may still be of use.

Notes

randtest_triplets computes the statistics for performing a triplets test which may be used to investigate deviations from randomness in a sequence, $x=\{x_i:i=1,2,\dots ,n\}$, of $[0,1]$ observations.

An $m\times m$ matrix, $C$, of counts is formed as follows. The element $c_{jkl}$ of $C$ is the number of triplets $(x_i,x_{i+1},x_{i+2})$ for $i=1,4,7,\dots ,n-2$, such that

$$\frac{j-1}{m}\le x_i<\frac{j}{m}$$

$$\frac{k-1}{m}\le x_{i+1}<\frac{k}{m}$$

$$\frac{l-1}{m}\le x_{i+2}<\frac{l}{m}\text{.}$$

Note that all triplets formed are non-overlapping and are thus independent under the assumption of randomness.

Under the assumption that the sequence is random, the expected number of triplets for each class (i.e., each element of the count matrix) is the same; that is, the triplets should be uniformly distributed over the unit cube ${[0,1]}^3$. Thus the expected number of triplets for each class is just the total number of triplets, ${\sum }_{j,k,l=1}^m{c}_{jkl}$, divided by the number of classes, $m^3$.

The ${\chi }^2$ test statistic used to test the hypothesis of randomness is defined as

$$X^2=\sum _{j,k,l=1}^m\frac{{(c_{jkl}-e)}^2}{e}\text{,}$$

where $e={\sum }_{j,k,l=1}^m{c}_{jkl}/m^3=$ expected number of triplets in each class.

The use of the ${\chi }^2$-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.

randtest_triplets may be used in two different modes:

1. a single call to randtest_triplets which computes all test statistics after counting the triplets;

2. multiple calls to randtest_triplets with the final test statistics only being computed in the last call.

The second mode is necessary if all the data do not fit into the memory. See argument $cl$ in Parameters for details on how to invoke each mode.

References

Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press

Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison–Wesley

Morgan, B J T, 1984, Elements of Simulation, Chapman and Hall

Ripley, B D, 1987, Stochastic Simulation, Wiley