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# Point Sets and Distance Statistics

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# Point Sets and Distance Statistics

The aggregation of point-based observations into quadrats results in a substantial loss of information regarding the underlying distribution of events. The procedure is also highly susceptible to issues relating to the shape, size, orientation and contiguity of the quadrats. By retaining the specific locations of the original points, assuming that this information is available, more detailed analysis can be undertaken. Typically this form of spatial analysis examines the variations in density or intensity of observed points over the study region, and the distances between point pairs — either between events, or between events and random or control points. Distances are typically computed as Euclidean, but spherical distances may also be provided by some packages. Network-based analyses are rarely if ever provided (the SANET software being an important exception), other than indirectly through the use of the L1 metric or other Lp‑type measure (e.g. such metrics are supported in Crimestat and the new defunct LOLA project).

For analysis of this type of data, events should be distinct objects (possibly weighted) and not the result of computations (e.g. zone or point set centroids). All events in a meaningful delimited study region (incorporating areas in which events are observed and also are not observed) are normally included in an analysis — analysis of subsets of the data (samples) should be undertaken with caution. If point events include associated attribute data, such as the type of tree, or the time of an event, they are said to be marked. Several attributes may be associated with each point, in which case the marking is multivariate. Analyses of the relationship between a given point pattern and a separate, usually continuous explanatory variable defined over the same region, may yield an improved understanding of the pattern and processes at work. Such explanatory variables are usually described as covariates.

For more complete discussions of basic point pattern analysis see O’Sullivan and Unwin (2010), Mitchell (2005, Ch.3) and the free and extensive Crimestat Manual (most chapters). More specialized books covering this subject area include Cressie (1993), Diggle (1983) and Bailey and Gatrell (1995). With the increasing importance of the R-Spatial programme, many tools and techniques for spatial point pattern analysis have been incorporated into the R-Spatial library, notably within spatstat (see also, www.spatstat.org and the reference article by the authors, Baddeley and Turner, 2005). Spatsat supports five main functional areas:

Creation, manipulation and plotting of point patterns

Exploratory data analysis, including density mapping

Simulation of point processes

Parametric model fitting, and

Hypothesis tests, residual plots and diagnostics

In addition to the 500-page manual on spatstat, an excellent 200 page course notes document (Analysing spatial point patterns in 'R') based on spatstat has been developed by the authors which is downloadable from the CSIRO website: research.csiro.au/software/r-workshop-notes/. In the sections that follow, we can only touch upon some of the more common problems and issues involved in point-pattern analysis − interested researchers are strongly recommended to work through the literature cited, particularly the Crimestat and spatstat documents, for a broader understanding of the topic and the tools and techniques now available.