Point Process

A point process on a space $E$ (e.g. $[0,\infty )$ for time, ${\mathbb{R}}^2$ for the plane) is a random countable collection of points $\{x_1,x_2,\dots \}\subset E$, formalized as a random counting measure $N(\cdot )$, where $N(A)$ gives the (random) number of points falling in a measurable set $A\subseteq E$. The process is characterized by its intensity measure $\Lambda (A)=\mathbb{E}[N(A)]$ (the expected number of points in $A$) and, more fully, by its finite-dimensional distributions or its factorial moment measures, which capture how points interact (attraction, repulsion, or independence). The Poisson process on the real line and the Poisson point process on the plane are both specific, simplest-case examples: their defining feature is that points occur completely independently of each other. General point processes relax this independence to model clustering (points more likely to occur near existing points) or inhibition (points less likely to occur near existing points).

• Framing 'random points in time' and 'random points in space' as the same underlying mathematical object, a point process, is what lets techniques developed in one setting (survival analysis's hazard-rate intuition, say) transfer directly to another (spatial ecology's intensity-function intuition), the space $E$ is just a label, the theory doesn't care what it represents.
• Every specific named process in this dictionary category built from 'random events happening,' the Poisson process, compound Poisson process, Cox process, Hawkes process, is a point process with additional structure layered on top, understanding the general point-process framework first makes each specific model a variation on one theme rather than a separate topic.
• A marked point process extends the basic framework one step further by attaching an additional random 'mark' or attribute to every point (an event's severity, a tree's diameter), turning the pure location pattern into a richer object still built from the same counting-measure foundation.