In my early days as a research student I was both impressed and influenced by David Kendall's enthusiasm for trying out ideas using computer-generated simulations. (His particular project at that time culminated in Kendall, 1974.) This can provide a bridge between what John Tukey calls “exploratory data analysis” and the more classical “confirmatory” aspects of statistics, model estimation and testing. Comparison of data with simulations is definitely “confirmatory” in that models are involved, yet it has much of the spirit of the “exploratory” phase, with human judgement replacing formal significance tests. Later I formalized this comparison to give Monte Carlo tests, discovered earlier by Barnard (1963) but apparently popularized by Ripley (1977).

Simulation has enabled progress to be made in the study of spatial patterns and processes which had previously seemed intractable. The starting point for all known simulation algorithms for spatial point patterns and random sets, such as those in Ripley (1981), is an algorithm to give independent uniformly distributed points in a unit square or cube. The purpose of this paper is to discuss the properties of these basic algorithms. Much of the material can be found scattered in the literature, but no one source gives a sufficiently complete picture.

CONGRUENTIAL RANDOM NUMBER GENERATORS

The usual way to produce approximately uniformly distributed random variables on (0, 1) in a computer is to sample integers xi uniformly from {0,1,…,M–1} or {1,2,…,M–1} and set Ui = xi/M. Here M is a large integer, usually of the form 2β. The approximation made is comparable with that made to represent real numbers by a finite set.