NAG Library Routine Document
G08ECF
1 Purpose
G08ECF performs the triplets test on a sequence of observations from the interval $\left[0,1\right]$.
2 Specification
SUBROUTINE G08ECF ( 
CL, N, X, MSIZE, NCOUNT, LDC, EX, CHI, DF, PROB, IFAIL) 
INTEGER 
N, MSIZE, NCOUNT(LDC,LDC,MSIZE), LDC, IFAIL 
REAL (KIND=nag_wp) 
X(N), EX, CHI, DF, PROB 
CHARACTER(1) 
CL 

3 Description
G08ECF 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
$i=1,4,7,\dots ,n2$, such that
Note that all triplets formed are nonoverlapping 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
where
$e={\displaystyle \sum _{j,k,l=1}^{m}}{c}_{\mathrm{jkl}}/{m}^{3}=\text{}$ 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.
G08ECF may be used in two different modes:
(i) 
a single call to G08ECF which computes all test statistics after counting the triplets; 
(ii) 
multiple calls to G08ECF 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 parameter
CL in
Section 5 for details on how to invoke each mode.
4 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
5 Parameters
 1: CL – CHARACTER(1)Input
On entry: indicates the type of call to G08ECF.
 ${\mathbf{CL}}=\text{'S'}$
 This is the one and only call to G08ECF (single call mode). All data are to be input at once. All test statistics are computed after counting of the triplets is complete.
 ${\mathbf{CL}}=\text{'F'}$
 This is the first call to the routine. 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 G08ECF.
 ${\mathbf{CL}}=\text{'I'}$
 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 G08ECF.
 ${\mathbf{CL}}=\text{'L'}$
 This is the last call to G08ECF. The test statistics are computed after the final counting of the triplets is complete.
Constraint:
${\mathbf{CL}}=\text{'S'}$, $\text{'F'}$, $\text{'I'}$ or $\text{'L'}$.
 2: N – INTEGERInput
On entry: $n$, the number of observations.
Constraints:
 if ${\mathbf{CL}}=\text{'S'}$, ${\mathbf{N}}\ge 3$;
 otherwise ${\mathbf{N}}\ge 1$.
 3: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the sequence of observations.
Constraint:
$0.0\le {\mathbf{X}}\left(\mathit{i}\right)\le 1.0$, for $\mathit{i}=1,2,\dots ,n$.
 4: MSIZE – INTEGERInput
On entry:
$m$, the size of the count matrix to be formed.
MSIZE must not be changed between calls to G08ECF.
Constraint:
${\mathbf{MSIZE}}\ge 2$.
 5: NCOUNT(LDC,LDC,MSIZE) – INTEGER arrayInput/Output
On entry: 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 G08ECF.
On exit: is an
MSIZE by
MSIZE by
MSIZE matrix containing the counts of the number of triplets,
${c}_{jkl}$, for
$\mathit{j}=1,2,\dots ,m$,
$\mathit{k}=1,2,\dots ,m$ and
$\mathit{l}=1,2,\dots ,m$.
 6: LDC – INTEGERInput
On entry: the first dimension of the array
NCOUNT and the second dimension of the array
NCOUNT as declared in the (sub)program from which G08ECF is called.
Constraint:
${\mathbf{LDC}}\ge {\mathbf{MSIZE}}$.
 7: EX – REAL (KIND=nag_wp)Output
On exit: if
${\mathbf{CL}}=\text{'S'}$ 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.
 8: CHI – REAL (KIND=nag_wp)Output

On exit: 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.
 9: DF – REAL (KIND=nag_wp)Output

On exit: 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.
 10: PROB – REAL (KIND=nag_wp)Output

On exit: 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.
 11: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: G08ECF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{CL}}\ne \text{'S'}$, $\text{'F'}$, $\text{'I'}$ or $\text{'L'}$. 
 ${\mathbf{IFAIL}}=2$
On entry,  ${\mathbf{N}}<1$, 
or  ${\mathbf{CL}}=\text{'S'}$ and ${\mathbf{N}}<3$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{MSIZE}}\le 1$. 
 ${\mathbf{IFAIL}}=4$

On entry,  ${\mathbf{LDC}}<{\mathbf{MSIZE}}$. 
 ${\mathbf{IFAIL}}=5$
On entry,  ${\mathbf{X}}\left(i\right)<0.0$, 
or  ${\mathbf{X}}\left(i\right)>1.0$, for some $i=1,2,\dots ,n$. 
 ${\mathbf{IFAIL}}=6$
No triplets were found because less than $3$ observations were provided in total.
 ${\mathbf{IFAIL}}=7$
The expected value for the counts in each element of the count matrix 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.
7 Accuracy
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.
If the call to G08ECF is an initial call or intermediate call with further calls to follow then any unused observations are saved for use at the beginning of the new sequence provided in the following call. Clearly any observations left over from an only or final call to G08ECF are ignored.
The time taken by the routine increases with the number of observations $n$, and also depends to some extent whether the call to G08ECF is an only, first, intermediate or last call.
9 Example
The following program performs the triplets test on $500$ pseudorandom numbers. G08ECF is called $5$ times with $100$ observations on each call. The triplets are tallied into a $2$ by $2$ by $2$ matrix.
9.1 Program Text
Program Text (g08ecfe.f90)
9.2 Program Data
Program Data (g08ecfe.d)
9.3 Program Results
Program Results (g08ecfe.r)