Runs tests may be used to investigate for trends in a sequence of observations. g08eac computes statistics for the runs up test. If the runs down test is desired then each observation must be multiplied by

- 1

before g08eac is called with the modified vector of observations.

A run up is a sequence of numbers in increasing order. A run up ends at

x_{k}

when

x_{k} > x_{k + 1}

and the new run then begins at

x_{k + 1}

. g08eac counts the number of runs up of different lengths. Let

c_{i}

denote the number of runs of length

i

, for

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

. The number of runs of length

r

or greater is then denoted by

c_{r}

. An unfinished run at the end of a sequence is not counted. The following is a trivial example.

Suppose we called g08eac with the following sequence:

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

Then g08eac would have counted the runs up of the following lengths:

$3$ , $1$ , $3$ , $3$ , $1$ , and $4$ .

When the counting of runs is complete g08eac computes the expected values and covariances of the counts,

c_{i}

. For the details of the method used see Knuth (1981). An approximate

χ^{2}

statistic with

r

degrees of freedom is computed, where

X^{2} = {(c - μ_{c})}^{T} Σ_{c}^{- 1} (c - μ_{c}),

where

$c$ is the vector of counts, $c_{i}$ , for $i = 1, 2, \dots, r$ ,
$μ_{c}$ is the vector of expected values,
$e_{i}$ , for $i = 1, 2, \dots, r$ , where $e_{i}$ is the expected value for $c_{i}$ under the null hypothesis of randomness, and
$Σ_{c}$ is the covariance matrix of $c$ under the null hypothesis.

The use of the

χ^{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.

You may specify the total number of runs to be found. If the specified number of runs is found before the end of a sequence g08eac will exit before counting any further runs. The number of runs actually counted and used to compute the test statistic is returned via nruns.

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: $n$ – Integer Input: On entry: $n$ , the length of the current sequence of observations.

Constraint: $n \geq 3$ .
2: $x [n]$ – const double Input: On entry: the sequence of observations.
3: $max_run$ – Integer Input: On entry: $r$ , the length of the longest run for which tabulation is desired. That is, all runs with length greater than or equal to $r$ are counted together.

Constraint: $1 \leq max_run < n$ .
4: $nruns$ – Integer * Output: On exit: the number of runs actually found.
5: $chi$ – double * Output: On exit: contains the approximate $χ^{2}$ test statistic, $X^{2}$ .
6: $df$ – double * Output: On exit: contains the degrees of freedom of the $χ^{2}$ statistic.
7: $prob$ – double * Output: On exit: contains the upper tail probability corresponding to the $χ^{2}$ test statistic, i.e., the significance level.
8: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_GE: On entry, $max_run = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $max_run < n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_G08EA_COVAR: Internally computed covariance matrix is not positive definite.
This may be because the value of max_run is too large relative to the full length of the series.
Thus the approximate $χ^{2}$ test statistic cannout be computed.
NE_G08EA_RUNS: The number of runs requested were not found, only $⟨ value ⟩$ out of the requested $⟨ value ⟩$ where found.
All statistics are returned and may still be of use.
NE_G08EA_RUNS_LENGTH: The total length of the runs found is less than max_run.
$max_run = ⟨ value ⟩$ whereas the total length of all runs is $⟨ value ⟩$ .
NE_G08EA_TIE: There is a tie in the sequence of observations.
NE_INT_ARG_LT: On entry, $max_run = ⟨ value ⟩$ .
Constraint: $max_run \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 3$ .
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.

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 figures for most cases.

8 Parallelism and Performance

g08eac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.