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

(r_{l}, r_{u})

. That is, a gap ends at

x_{j}

r_{l} \leq x_{j} \leq r_{u}

. The next gap then begins at

x_{j + 1}

. The interval

(r_{l}, r_{u})

should lie within the region of all possible numbers. For example if the test is carried out on a sequence of

(0, 1)

random numbers then the interval

(r_{l}, r_{u})

must be contained in the whole interval

(0, 1)

. Let

t_{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_{i}

denote the number of gaps of length

i

, for

i = 1, 2, \dots, k - 1

. 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

$0.20$ $0.40$ $0.45$ $0.40$ $0.15$ $0.75$ $0.95$ $0.23 0.27$ $0.40$ $0.25$ $0.10$ $0.34$ $0.39$ $0.61$ $0.12$ .

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

χ^{2}

statistic with

k

degrees of freedom is computed where

X^{2} = \frac{\sum_{i = 1}^{k} {(c_{i} - e_{i})}^{2}}{e_{i}},

where

$e_{i} = ngaps \times p \times {(1 - p)}^{i - 1}$ , if $i < k$ ;
$e_{i} = ngaps \times {(1 - p)}^{i - 1}$ , if $i = k$ ;
$ngaps =$ the number of gaps found and
$p = (r_{u} - r_{l}) / t_{len}$ .

The use of the

χ^{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: $n$ – IntegerInput: On entry: $n$ , the length of the current sequence of observations.

Constraint: $n \geq 1$ .
2: $x [n]$ – const doubleInput: On entry: the sequence of observations.
3: $num_gaps$ – IntegerInput: On entry: the maximum number of gaps to be sought. If $num_gaps \leq 0$ then there is no limit placed on the number of gaps that are found.

Constraint: $num_gaps \leq n$ .
4: $max_gap$ – IntegerInput: On entry: $k$ , the length of the longest gap for which tabulation is desired.

Constraint: $1 < max_gap \leq n$ .
5: $lower$ – doubleInput: On entry: the lower limit of the interval to be used to define the gaps, $r_{l}$ .
6: $upper$ – doubleInput: On entry: the upper limit of the interval to be used to define the gaps, $r_{u}$ .

Constraint: $upper > lower$ .
7: $length$ – doubleInput: On entry: the total length of the interval which contains all possible numbers that may arise in the sequence.

Constraint: $length > 0.0$ and $upper - lower < length$ .
8: $chi$ – double *Output: On exit: contains the $χ^{2}$ test statistic, $X^{2}$ , for testing the null hypothesis of randomness.
9: $df$ – double *Output: On exit: contains the degrees of freedom for the $χ^{2}$ statistic.
10: $prob$ – double *Output: On exit: contains the upper tail probability associated with the $χ^{2}$ test statistic, i.e., the significance level.
11: $fail$ – 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, $num_gaps = 〈value〉$ and $n = 〈value〉$ .
Constraint: $num_gaps \leq n$ .
NE_2_REAL_ARG_GE: On entry, $lower = 〈value〉$ and $upper = 〈value〉$ .
Constraint: $upper > lower$ .
NE_3_REAL_ARG_CONS: On entry, $lower = 〈value〉$ , $upper = 〈value〉$ and $length = 〈value〉$ .
Constraint: $upper - lower < length$ .
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 $〈value〉$ 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 $χ^{2}$ 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 $i = 〈value〉$ is zero. The value of $(upper - lower) / length$ may be too close to $0.0 or 1.0$ . 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 $〈value〉$ out of the requested $〈value〉$ 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 $upper - lower$ .
NE_INT_2: On entry, $max_gap = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 < max_gap \leq n$ .

On entry, $max_gap = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq max_gap \leq n$ .
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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, $length = 〈value〉$ .
Constraint: $length > 0.0$ .

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.