# Complete Spatial Randomness

Tags :: Spatial Statistics

## Overview

- An event is equally likely to occur at any location in the region of interest
- Events follow a uniform distribution across the are and are independent of each other
**Uniform does not mean evenly dispersed but rather the event locations follow a uniform probability distribution**

## CSR

A point process that produces CSR is the homogenous Poisson Point Process There are several equivalent definitions - but the simplist is as follows:

- The number of events occuring within a finite region \(A\), written \(N(A)\) is a random variable following a Poisson Distribution with mean \(\lambda |A|\) for some positive constant \(\lambda\) and \(|A|\) denoting the area of A.
- Given \(N(A)\), the total number of events occuring within an area \(A\), the locations of the events represent an independent random sample of \(N(A)\) locations, where each point is equally likely to be chosen as an event.

## Testing for CSR

Consider a set of event locations $\{s_1, s_2, \ldots, s_n\}$in some spatial region \(\mathcal{D}\).
Assume this set of events is generated by some underlying process (point process). Further we define a point to be any location in the spatial area of interest where an
*event could occur*, and an event location to be the point at which the *event did occur*.

We want to Is there evidence of spatial clustering, randomness, or regularity. To do so we use the K-function and the L-function.