# NAG CL Interfaceg08ecc (randtest_​triplets)

## 1Purpose

g08ecc performs the triplets test on a sequence of observations from the interval $\left[0,1\right]$.

## 2Specification

 #include
 void g08ecc (Integer n, const double x[], Integer max_count, double *chi, double *df, double *prob, NagError *fail)
The function may be called by the names: g08ecc, nag_nonpar_randtest_triplets or nag_triplets_test.

## 3Description

g08ecc computes the statistics for performing a triplets test which may be used to investigate deviations from randomness in a sequence, $x=\left\{{x}_{i}:i=1,2,\dots ,n\right\}$, of $\left[0,1\right]$ observations.
An $m$ by $m$ matrix, $C$, of counts is formed as follows. The element ${c}_{\mathrm{jkl}}$ of $C$ is the number of triplets $\left({x}_{i},{x}_{i+1},{x}_{i+2}\right)$ for $i=1,4,7,\dots ,n-2$, such that
 $j-1m≤xi
 $k- 1m≤xi+ 1< km$
 $l-1m≤xi+2< lm.$
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 ${\left[0,1\right]}^{3}$. Thus the expected number of triplets for each class is just the total number of triplets, $\sum _{j,k,l=1}^{m}{c}_{\mathrm{jkl}}$, divided by the number of classes, ${m}^{3}$.
The ${\chi }^{2}$ test statistic used to test the hypothesis of randomness is defined as
 $X2=∑j,k,l=1m cjkl-e 2e,$
where $e=\sum _{j,k,l=1}^{m}{c}_{\mathrm{jkl}}/{m}^{3}=\text{}$ 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.

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 3$.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the sequence of observations.
Constraint: $0.0\le {\mathbf{x}}\left[\mathit{i}-1\right]\le 1.0$, for $\mathit{i}=1,2,\dots ,n$.
3: $\mathbf{max_count}$Integer Input
On entry: $m$, the size of the count matrix to be formed.
Constraint: ${\mathbf{max_count}}\ge 2$.
4: $\mathbf{chi}$double * Output
On exit: contains the ${\chi }^{2}$ test statistic, ${X}^{2}$, for testing the null hypothesis of randomness.
5: $\mathbf{df}$double * Output
On exit: contains the degrees of freedom for the ${\chi }^{2}$ statistic.
6: $\mathbf{prob}$double * Output
On exit: contains the upper tail probability associated with the ${\chi }^{2}$ test statistic, i.e., the significance level.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_G08EC_CELL
max_count 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.
${\mathbf{max_count}}=〈\mathit{\text{value}}〉$, number of triplets $\text{}=〈\mathit{\text{value}}〉$ and $\text{expected value}=〈\mathit{\text{value}}〉$.
All statistics are returned and may still be of use.
NE_G08EC_TRIPLETS
No triplets were found because less than $3$ observations were provided in total.
NE_INT_ARG_LE
On entry, ${\mathbf{max_count}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{max_count}}\ge 2$.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 3$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARRAY_CONS
On entry, at least one element of x is out of range.
Constraint: $0\le {\mathbf{x}}\left[i-1\right]\le 1$, for $i=1,2,\dots ,{\mathbf{n}}$.

## 7Accuracy

The computations are believed to be stable. The computations of prob given the values of chi and df will obtain a relative accuracy of five significant figures for most cases.

## 8Parallelism and Performance

g08ecc is not threaded in any implementation.

The time taken by the function increases with the number of observations $n$.

## 10Example

The following program performs the triplets test on $10000$ pseudorandom numbers taken from a uniform distribution $U\left(0,1\right)$, generated by g05sac. g08ecc is called with ${\mathbf{max_count}}=5$.

### 10.1Program Text

Program Text (g08ecce.c)

None.

### 10.3Program Results

Program Results (g08ecce.r)