NAG Library Function Document
nag_gaps_test (g08edc)
1 Purpose
nag_gaps_test (g08edc) performs a gaps test on a sequence of observations.
2 Specification
#include <nag.h> 
#include <nagg08.h> 
void 
nag_gaps_test (Integer n,
const double x[],
Integer num_gaps,
Integer max_gap,
double lower,
double upper,
double length,
double *chi,
double *df,
double *prob,
NagError *fail) 

3 Description
Gaps tests are used to test for cyclical trend in a sequence of observations. nag_gaps_test (g08edc) computes certain statistics for the gaps test.
The term gap is used to describe the distance between two numbers in the sequence that lie in the interval $\left({r}_{l},{r}_{u}\right)$. That is, a gap ends at ${x}_{j}$ if ${r}_{l}\le {x}_{j}\le {r}_{u}$. The next gap then begins at ${x}_{j+1}$. The interval $\left({r}_{l},{r}_{u}\right)$ should lie within the region of all possible numbers. For example if the test is carried out on a sequence of $\left(0,1\right)$ random numbers then the interval $\left({r}_{l},{r}_{u}\right)$ must be contained in the whole interval $\left(0,1\right)$. Let ${t}_{\text{len}}$ be the length of the interval which specifies all possible numbers.
nag_gaps_test (g08edc) counts the number of gaps of different lengths. Let ${c}_{\mathit{i}}$ denote the number of gaps of length $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k1$. The number of gaps of length $k$ or greater is then denoted by ${c}_{k}$. An unfinished gap at the end of a sequence is not counted. The following is a trivial example.
Suppose we called nag_gaps_test (g08edc) with the following sequence and with
${r}_{l}=0.30$ and
${r}_{u}=0.60$:
 $0.20$ $0.40$ $0.45$ $0.40$ $0.15$ $0.75$ $0.95$ $0.230.27$ $0.40$ $0.25$ $0.10$ $0.34$ $0.39$ $0.61$ $0.12$.
nag_gaps_test (g08edc) would have counted the gaps of the following lengths:
 2, $1$, $1$, $6$, $3$ and $1$.
When the counting of gaps is complete nag_gaps_test (g08edc) computes the expected values of the counts. An approximate
${\chi}^{2}$ statistic with
$k$ degrees of freedom is computed where
where
 ${e}_{i}=\mathit{ngaps}\times p\times {\left(1p\right)}^{i1}$, if $i<k$;
 ${e}_{i}=\mathit{ngaps}\times {\left(1p\right)}^{i1}$, if $i=k$;
 $\mathit{ngaps}=\text{}$ the number of gaps found and
 $p=\left({r}_{u}{r}_{l}\right)/{t}_{\text{len}}$.
The use of the ${\chi}^{2}$distribution as an approximation to the exact distribution of the test statistic improves as the expected values increase.
You may specify the total number of gaps to be found. If the specified number of gaps is found before the end of a sequence nag_gaps_test (g08edc) will exit before counting any further gaps.
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 Arguments
 1:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the length of the current sequence of observations.
Constraint:
${\mathbf{n}}\ge 1$.
 2:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – const doubleInput

On entry: the sequence of observations.
 3:
$\mathbf{num\_gaps}$ – IntegerInput

On entry: the maximum number of gaps to be sought. If ${\mathbf{num\_gaps}}\le 0$ then there is no limit placed on the number of gaps that are found.
Constraint:
${\mathbf{num\_gaps}}\le {\mathbf{n}}$.
 4:
$\mathbf{max\_gap}$ – IntegerInput

On entry: $k$, the length of the longest gap for which tabulation is desired.
Constraint:
$1<{\mathbf{max\_gap}}\le {\mathbf{n}}$.
 5:
$\mathbf{lower}$ – doubleInput

On entry: the lower limit of the interval to be used to define the gaps, ${r}_{l}$.
 6:
$\mathbf{upper}$ – doubleInput

On entry: the upper limit of the interval to be used to define the gaps, ${r}_{u}$.
Constraint:
${\mathbf{upper}}>{\mathbf{lower}}$.
 7:
$\mathbf{length}$ – doubleInput

On entry: the total length of the interval which contains all possible numbers that may arise in the sequence.
Constraint:
${\mathbf{length}}>0.0$ and ${\mathbf{upper}}{\mathbf{lower}}<{\mathbf{length}}$.
 8:
$\mathbf{chi}$ – double *Output

On exit: contains the ${\chi}^{2}$ test statistic, ${X}^{2}$, for testing the null hypothesis of randomness.
 9:
$\mathbf{df}$ – double *Output

On exit: contains the degrees of freedom for the ${\chi}^{2}$ statistic.
 10:
$\mathbf{prob}$ – double *Output

On exit: contains the upper tail probability associated with the ${\chi}^{2}$ test statistic, i.e., the significance level.
 11:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 2.7 in How to Use the NAG Library and its Documentation).
6 Error Indicators and Warnings
 NE_2_INT_ARG_GT

On entry, ${\mathbf{num\_gaps}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{num\_gaps}}\le {\mathbf{n}}$.
 NE_2_REAL_ARG_GE

On entry, ${\mathbf{lower}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{upper}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{upper}}>{\mathbf{lower}}$.
 NE_3_REAL_ARG_CONS

On entry, ${\mathbf{lower}}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{upper}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{length}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{upper}}{\mathbf{lower}}<{\mathbf{length}}$.
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_G08ED_FREQ_LT_ONE

The expected frequency of at least one class is less than one.
This implies that the ${\chi}^{2}$ may not be a very good approximation to the distribution of the test statistics.
All statistics are returned and may still be of use.
 NE_G08ED_FREQ_ZERO

The expected frequency in class
$i=\u2329\mathit{\text{value}}\u232a$ is zero. The value of
$\left({\mathbf{upper}}{\mathbf{lower}}\right)/{\mathbf{length}}$ may be too close to
$0.0\text{ or}1.0$. or
max_gap is too large relative to the number of gaps found.
 NE_G08ED_GAPS

The number of gaps requested were not found, only $\u2329\mathit{\text{value}}\u232a$ out of the requested $\u2329\mathit{\text{value}}\u232a$ where found.
All statistics are returned and may still be of use.
 NE_G08ED_GAPS_ZERO

No gaps were found. Try using a longer sequence, or increase the size of the interval ${\mathbf{upper}}{\mathbf{lower}}$.
 NE_INT_2

On entry, ${\mathbf{max\_gap}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1<{\mathbf{max\_gap}}\le {\mathbf{n}}$.
On entry, ${\mathbf{max\_gap}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{max\_gap}}\le {\mathbf{n}}$.
 NE_INT_ARG_LT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 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.
An unexpected error has been triggered by this function. Please contact
NAG.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_REAL_ARG_LE

On entry, ${\mathbf{length}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{length}}>0.0$.
7 Accuracy
The computations are believed to be stable. The computation of
prob given the values of
chi and
df will obtain a relative accuracy of five significant places for most cases.
8 Parallelism and Performance
nag_gaps_test (g08edc) is not threaded in any implementation.
The time taken by nag_gaps_test (g08edc) increases with the number of observations $n$.
10 Example
The following program performs the gaps test on
$5000$ pseudorandom numbers taken from a uniform distribution
$U\left(0,1\right)$, generated by
nag_rand_uniform (g05sqc). All gaps of length
$10$ or more are counted together.
10.1 Program Text
Program Text (g08edce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (g08edce.r)