Spatial point patterns

• Instead of some attributes over areas or measuring continuous surfaces, we are interested in the locations of individual events
• Countable sets of points as realisations of stochastic spatial point processes
  • $\{x_1,x_2,...x_{N(A)}\}$ in the plane
  • $N(A)$ for the number of events in a planar region $A$
• Collection of random variables $X_i$ (values in the sample place) where $i$ belongs to an indexing set $I$
  • $\{X_1,X_2,...,X_{N(A)}\}$ taking values in $A\subset\R^2$
• Examples
  • Locations of trees of a certain species in a forest
  • Cancer cells in the tissue

Intensity

![Tree distribution can be influenced by 1st order effects such as elevation gradient or spatial distribution of soil characteristics; this, in turn, changes the tree density distribution across the study area. Tree distribution can also be influenced by 2nd order effects such as seed dispersal processes where the process is independent of location and, instead, dependent on the presence of other trees.

](attachment:d35b6dec-7f15-495e-9d25-90061433afe3:a30092e4-d857-413e-b83d-2999517dde42.png)

Tree distribution can be influenced by 1st order effects such as elevation gradient or spatial distribution of soil characteristics; this, in turn, changes the tree density distribution across the study area. Tree distribution can also be influenced by 2nd order effects such as seed dispersal processes where the process is independent of location and, instead, dependent on the presence of other trees.

• Let $dx$ and $dy$ be small regions containing the points $x$ and $y$ respectively
  • When $dx\rightarrow 0$ and $dy\rightarrow0$, we can define density
• First-order intensity
  • Overall density of points across a space
  • Mean number of events per unit area
  • Example: number of malaria cases might be higher in urban areas due to higher population density

$$ \lambda(x)=\lim_{|dx|\rightarrow0}\frac{E[N(dx)]}{|dx|} $$

• Second-order intensity
  • Interaction between pairs of points
  • Clustering: do points like to be near each other?
  • Inhibition: do points avoid each other?
  • Example: aftershocks tend to occur close to main earthquakes

$$ \lambda_2(x)=\lim_{|dx|\rightarrow0,|dy|\rightarrow0}\frac{E[N(dx)N(dy)]}{|dx||dy|} $$

• If a process is stationary
  • Statistical properties are unchanged by shifting the pattern in space
  • The average number of points per unit area is the same everywhere
  • $\frac{E[N(A)]}{|A|}$ remains constant $\rightarrow$ constant $\lambda$
• If a process is isotropic
  • Statistical properties are unchanged by rotation about the origin
  • Relationship between any two parts only depends on the distance $h$ but not angle
  • $\lambda_2(x,y)=\lambda_2(||x-y||)=\lambda_2(h)$

Homogenous Poisson processes

• Let $\lambda(\cdot)$ be a intensity function of the spatial point process, which is a non-negative valued function