Expects one column containing times of events (e.g. earthquakes or clade divergences) or positions along a line (e.g. a transect). The times do not have to be in increasing order.
Density trend (Laplace test)
The 'Laplace' test for a trend in density (intensity) is described by Cox & Lewis (1978). It is based on the test statistic U. U is approximately normally distributed with zero mean and unit variance under the null hypothesis of stationary intensity. This is the basis for the p value.
If p<0.05, a positive U indicates an increasing trend in intensity (decreasing waiting times), while a negative U indicates a decreasing trend. Note that if a trend is detected by this test, the sequence is not stationary and the assumptions of the exp test below are violated.
Exp test for Poisson process
The exp test (Prahl 1999) for a stationary Poisson process (random, independent events) gives a test statistic M. M will tend to zero for a regularly spaced (overdispersed) sequence, and to 1 for a highly clustered sequence. For the null hypothesis of a Poisson process, M is asymptotically normally distributed. This is the basis for the given z test.
In summary, if p<0.05 the sequence is not Poisson. You can then inspect the M statistic; if smaller than the expected value this indicates regularity, if higher it indicates clustering.
For both tests, p values are also estimated by Monte Carlo simulation with 9999 random data sets.
For mathematical details, see the Past manual.
References
Cox, D. R. & P. A. W. Lewis. 1978. The Statistical Analysis of Series of Events. Chapman and Hall, London.
Prahl, J. 1999. A fast unbinned test on event clustering in Poisson processes. Arxiv, Astronomy and Astrophysics September 1999.