# Stationarity assumption for a point process

as I understand stationarity of a point process implies that it is invariant under translation. Imagine one has numerous realizations of the same point process. Due to the nature of this process, it can be observed only in a bounded region (i.e. islands, cell), the are can differ from observation to observation and so the number of points (i.e. it is random variable). Would it be correct to envision this process as stationary (since an infinite number of realization of this process is possible and nothing is in stone, but the concept of boundary) or it is just a finite process? Would it be true that non-homogenous process (clustering) is non-stationary? I am a bit confused and would appreciate hearing your opinion. Thanks!

Stationarity implies that the statistical parameters of an underlying process do not vary over space (or time), and is often formally described as "invariance under translation" (not to be confused with "invariance under rotation": isotropy). Homogeneity has a more specific meaning that pertains to the assumption that the average "intensity" $(\lambda)$ of a point process (i.e, the average number of points per unit area) is constant. In other words, it is akin to the concept of first order stationarity in the context of a point process.

For a homogeneous Poisson point process $(\mathbf{X})$, the expected number of points falling in some sub-region $B$ is $\mathbb{E} [n(\mathbf{X}{\bigcap}B]) =\lambda\cdot|B|$. The observed counts in different regions may vary because they represent random samples, but each sample is drawn from the same underlying distribution (with the same mean intensity)...in other words, a point process that is first-order stationary.

In practice, clustering and inhomogeneity can be hard to differentiate, but clustering by itself does not necessarily imply non-stationarity. That said, a pattern that is clustered cannot be modeled well by a homogeneous Poisson point process.

Common methods of assessing dispersion in point pattern analysis, such as Ripley's K function, assume that the process is homogeneous...and that the point process extends beyond the study area boundary. If you truly have a bounded region, you are dealing with something called a "small world model"....which complicates matters because the pattern around the edges will be inhomogeneous.

There are many excellent books on point pattern analysis. My personal recommendation is Baddeley, Rubak, and Turner: "Spatial Point Patterns: Methodology and Applications with R". The associated R package (spatstat) is super powerful and the authors are really responsive. If you can't afford the book in the short term, go to the spatstat website and download the pdf of the workshop notes (by Baddeley).

• +1. But please don't equate isotropy with stationarity: they are distinctly different. Think of stationarity as being translation invariance and isotropy as being rotation invariance.
– whuber
Aug 24, 2017 at 15:21
• Thank you for clarifying this whuber...edited accordingly. Aug 24, 2017 at 15:27