NAG C Library Function Document

nag_runs_test (g08eac)


nag_runs_test (g08eac) performs a runs up (or a runs down) test on a sequence of observations.


#include <nag.h>
#include <nagg08.h>
void  nag_runs_test (Integer n, const double x[], Integer max_run, Integer *nruns, double *chi, double *df, double *prob, NagError *fail)


Runs tests may be used to investigate for trends in a sequence of observations. nag_runs_test (g08eac) computes statistics for the runs up test. If the runs down test is desired then each observation must be multiplied by -1 before nag_runs_test (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 xk when xk>xk+1 and the new run then begins at xk+1. nag_runs_test (g08eac) counts the number of runs up of different lengths. Let ci denote the number of runs of length i, for i=1,2,,r-1. The number of runs of length r or greater is then denoted by cr. An unfinished run at the end of a sequence is not counted. The following is a trivial example.
Suppose we called nag_runs_test (g08eac) with the following sequence:
Then nag_runs_test (g08eac) would have counted the runs up of the following lengths:
When the counting of runs is complete nag_runs_test (g08eac) computes the expected values and covariances of the counts, ci. For the details of the method used see Knuth (1981). An approximate χ2 statistic with r degrees of freedom is computed, where
X2= c-μcT Σ c -1 c-μc ,  
The use of the χ2-distribution as an approximation to the exact distribution of the test statistic, X2, 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 nag_runs_test (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.


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


1:     n IntegerInput
On entry: n, the length of the current sequence of observations.
Constraint: n3.
2:     x[n] const doubleInput
On entry: the sequence of observations.
3:     max_run IntegerInput
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: 1max_run<n.
4:     nruns Integer *Output
On exit: the number of runs actually found.
5:     chi double *Output
On exit: contains the approximate χ2 test statistic, X2.
6:     df double *Output
On exit: contains the degrees of freedom of the χ2 statistic.
7:     prob double *Output
On exit: contains the upper tail probability corresponding to the χ2 test statistic, i.e., the significance level.
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

On entry, max_run=value and n=value.
Constraint: max_run<n.
Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
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 χ2 test statistic cannout be computed.
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.
The total length of the runs found is less than max_run.
max_run=value whereas the total length of all runs is value.
There is a tie in the sequence of observations.
On entry, max_run=value.
Constraint: max_run1.
On entry, n=value.
Constraint: n3.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.


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.

Parallelism and Performance

nag_runs_test (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.

Further Comments

The time taken by nag_runs_test (g08eac) increases with the number of observations n.


The following program performs a runs up test on 10000 pseudorandom numbers taken from a uniform distribution U0,1, generated by nag_rand_uniform (g05sqc). All runs of length 6 or more are counted together.

Program Text

Program Text (g08eace.c)

Program Data


Program Results

Program Results (g08eace.r)