Complete Spatial Randomness

As the first step to analyze spatial point pattern data, we need to check CSR, the complete spatial randomness.

Preliminaries 1

Consider a point process $N$, as a random counting measure on a space $\mathcal{S}\subseteq\mathbb{R}^d$, usually $d=2, 3$ at spatial context.

• $N$ takes non-negative integer values, is finite on bounded sets, and is countably additive.

• That is, if $A=\cup_{i=1}^\infty A_i$ then $N(A)=\sum_i N(A_i)$

Definition 1 (The homogeneous Poisson process).

The most fundamental point process is the homogeneous Poisson process. For this, if $A$, $A_i(i=1,\ldots,k)$ are bounded Borel subsets of $\mathcal{S}$, and are disjoint then the following hold.

1. $N(A)$ follows a Poisson distribution as $N(A) \sim \mathrm{Poisson}(\lambda|A|)$ where $|A|$ is the volume(Lebesgue measure) of $A$, and the constant $\lambda$ is called intensity which indicates the mean number of events per unit area.

2. $N(A_1),\ldots,N(A_k)$ are independent random variables.

Why interested in CSR?

• Property 2 at the definition above is known as the property of complete spatial randomness.

• If a test of CSR is not rejected, any further formal analysis may not be necessary(except estimating the intensity).

• Distinguish whether the point pattern has some aggregated patterns or not.

References

• 서울대학교 공간자료분석 강의노트
• Handbook of Spatial Statistics