naginterfaces.library.nonpar.randtest_​gaps

naginterfaces.library.nonpar.randtest_gaps(cl, x, m, rlo, rup, totlen, ngaps, ncount, comm)

randtest_gaps performs a gaps test on a sequence of observations.

For full information please refer to the NAG Library document for g08ed

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g08/g08edf.html

Parameters

clstr, length 1

Indicates the type of call to randtest_gaps.

$cl=\text{'S'}$

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

$cl=\text{'F'}$

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

$cl=\text{'I'}$

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

$cl=\text{'L'}$

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

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

The sequence of observations.

mint

The maximum number of gaps to be sought. If $m\le 0$ then there is no limit placed on the number of gaps that are found.

$m$ should not be changed between calls to randtest_gaps.

rlofloat

The lower limit of the interval to be used to define the gaps, $r_l$.

$rlo$ must not be changed between calls to randtest_gaps.

rupfloat

The upper limit of the interval to be used to define the gaps, $r_u$.

$rup$ must not be changed between calls to randtest_gaps.

totlenfloat

The total length of the interval which contains all possible numbers that may arise in the sequence.

ngapsint

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

If $cl=\text{'I'}$ or $\text{'L'}$, $ngaps$ must contain the value returned by the previous call to randtest_gaps.

ncountint, array-like, shape $\left(\text{maxg}\right)$

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_gaps.

commdict, communication object, modified in place

Communication structure.

On initial entry: need not be set.

Returns

ngapsint

The number of gaps actually found, $\text{ngaps}$.

ncountint, ndarray, shape $\left(\text{maxg}\right)$

The counts of gaps of the different lengths, $c_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

exfloat, ndarray, shape $\left(\text{maxg}\right)$

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

Otherwise the elements of $ex$ are 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: $n\ge 1$.

(errno $3$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: if $cl=\text{'S'}$, $m\le n$.

(errno $4$)

On entry, $\text{maxg}=⟨value⟩$ and $n=⟨value⟩$.

Constraint: if $cl=\text{'S'}$, $\text{maxg}\le n$.

(errno $4$)

On entry, $\text{maxg}=⟨value⟩$.

Constraint: $\text{maxg}>1$.

(errno $5$)

On entry, $rlo=⟨value⟩$, $rup=⟨value⟩$ and $totlen=⟨value⟩$.

Constraint: $rup-rlo<totlen$.

(errno $5$)

On entry, $totlen=⟨value⟩$.

Constraint: $totlen>0.0$.

(errno $5$)

On entry, $rlo=⟨value⟩$ and $rup=⟨value⟩$.

Constraint: $rup>rlo$.

(errno $6$)

No gaps were found. Try using a longer sequence, or increase the size of the interval $rup-rlo$.

(errno $7$)

The expected frequency in class $i=⟨value⟩$ is zero. The value of $(rup-rlo)/totlen$ may be too close to $0.0$ or $1.0$. or $\text{maxg}$ is too large relative to the number of gaps found.

Warns

NagAlgorithmicWarning

(errno $8$)

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.

(errno $9$)

The expected frequency of at least one class is less than $1$.

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.

Notes

Gaps tests are used to test for cyclical trend in a sequence of observations. randtest_gaps computes certain statistics for the gaps test.

randtest_gaps may be used in two different modes:

1. a single call to randtest_gaps which computes all test statistics after counting the gaps;

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

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

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 $x_j$ if $r_l\le x_j\le r_u$. 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 $t_{\text{len}}$ be the length of the interval which specifies all possible numbers.

randtest_gaps counts the number of gaps of different lengths. Let $c_{\text{i}}$ denote the number of gaps of length $\text{i}$, for $\text{i}=1,2,\dots ,k-1$. 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 unless the sequence is part of an initial or intermediate call to randtest_gaps (i.e., unless there is another call to randtest_gaps to follow) in which case the unfinished gap is used. The following is a trivial example.

Suppose we called randtest_gaps twice (i.e., with $cl=\text{'F'}$ and then with $cl=\text{'L'}$) with the following two sequences and with $rlo=0.30$ and $rup=0.60$:

($0.20$ $0.40$ $0.45$ $0.40$ $0.15$ $0.75$ $0.95$ $0.23$) and

($0.27$ $0.40$ $0.25$ $0.10$ $0.34$ $0.39$ $0.61$ $0.12$).

Then after the second call randtest_gaps would have counted the gaps of the following lengths:

$2$, $1$, $1$, $6$, $3$ and $1$.

When the counting of gaps is complete randtest_gaps computes the expected values of the counts. An approximate ${\chi }^2$ statistic with $k$ degrees of freedom is computed where

$$X^2=\frac{{\sum }_{i=1}^k{(c_i-e_i)}^2}{e_i}\text{,}$$

where

$e_i=\text{ngaps}\times p\times {(1-p)}^{i-1}$, if $i<k$;

$e_i=\text{ngaps}\times {(1-p)}^{i-1}$, if $i=k$;

$\text{ngaps}=$ the number of gaps found and

$p=(r_u-r_l)/t_{\text{len}}$.

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 randtest_gaps will exit before counting any further gaps.

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