Point pattern analysis is concerned with the location of events, and with answering questions about the distribution of those locations, specifically whether they are clustered, randomly or regularly distributed. Point pattern analysis is very sensitive to the definition of the study area, since a regularly distributed pattern can be made to seem clustered by including large margins within the study area. Measures are also subject to boundary corrections, and most often study area boundaries have to be defined as convex polygons over the study area, or in the simplest form as rectangles bounding the points under analysis. It is of course always important to plot the events to detect outliers visually, together with the boundaries being applied (Bailey and Gatrell, 1995, Cressie, 1993).
The simplest way of exploring point pattern data is by examining a two-dimensional frequency distribution of counts within equal-area units imposed on the study area, giving an impression of how the intensity of the point process varies; this can be extended to kernel estimation. Nearest neighbour distances are also used to analyse intensity. Intensity in this sense is a first order property, the mean number of events per unit area at point 
. Spatial dependence is captured by the second order properties of a spatial point process, which involve the relationship between numbers of events in pairs of arbitrary areas within the chosen study area: 
. For a stationary process, this relationship depends on the distance and direction between the pair of areas; when the relationship depends on distance alone, the process is termed isotropic.
Having an empirical data set is not sufficient to test for divergences from randomness. In general, tests are conducted against a standard model for complete spatial randomness following a homogeneous Poisson process over the study area. This implies that any of the events could have occurred anywhere in the study area, and that the locations of the events are mutually independent. This is enough for a start, but quickly encounters difficulties, when the underlying control distribution is not homogeneous across the study area. Further, one may wish to test hypotheses that the incidence of events is raised at or near given locations. Both of these issues have attracted substantial contributions in the past decade, and methods are now available for testing point patterns against hypotheses of non-randomness in relation to a second control variable with a varying spatial distribution (Cuzick and Edwards, 1990, Diggle, 1990, Diggle and Chetwynd, 1991, Diggle and Rowlingson, 1994, Kingham, Gatrell and Rowlingson, 1995, Gatrell et al., 1996).
These developments have led to empirical work using point patterns for cases -- observed events -- and controls -- for the underlying non-homogeneous distribution. In this framework, K functions are defined for a labelled stationary isotropic point process for case-case, control-control, and case-control pairs for distances up to an arbitrary maximum, and the difference is calculated between the case-case and control-control pairs for the chosen distance steps. An confidence interval envelope can be constructed around the null of no difference, permitting the analyst to detect at which distances significant differences occur between the distances between cases and between controls. These methods have been employed by Jones, Langford and Bentham (1996) to explore the outcomes of road accidents, and within the field of location by Sweeney and Feser (1998) to examine small manufacturing business location patterns in North Carolina. They find conclusive evidence of plants from 8-49 employees, with 8-17 employee plants displaying clustered locations at ranges up to 15 kilometres, while the larger 18-49 employee plants clustered at all spatial scales within the bound calculated. Large plants with over 205 employees were found to seek dispersed locations significantly.