1. Spatial Point Pattern Analysis
GRAD6104/8104 INES 8090
Spatial Statistic- Spring 2017
Spatial Point Pattern Analysis

2. Point Process Models

3. Point Process Models Homogeneous Poisson process Cluster process
Regular process

4. Point Process Models
Superpositioning
Thinning
Clustering

5. Point Process Models
Superpositioning
HPP1+HPP2+…HPPn -> HPP + = +=

6. Point Process Models Thinning
Events in one process are eliminated based on some probability p
p-thinning
Related probability: p
p(s)-thinning
Retention probability is given by the deterministic function p(s), where 0 ≤ p(s) ≤ 1
E.g., x-location of an event
π-thinning
Thinning function is a stochastic
Thinning obtained by drawing a realization p(s) of the random function π(s) and apply p(s)-thinning

7. Point Process Models Clustering Poisson Cluster Process (PCP)
Based on three rules
Cluster centers form a HPP with intensity ρ
The number of events in each cluster are iid variates with mean µ
Positions of events within a cluster, relative to its center, are iid ~ pdf f(.) (e.g., bivariate normal or uniform)

8. Point Process Models Clustering Poisson Cluster Process (PCP)
Stationary
Isotropic
Simulation: in spatstat,
rPoissonCluster(kappa, rmax, rcluster, win = owin(c(0,1),c(0,1)),..., lmax=NULL, nsim=1)

9. Point Process Models Regular Process
Sequential Inhibition Process (SIP)
First event is uniformly distributed
The distribution of each subsequent event, conditional on all previously realized events, is uniform on that portion of D that lies no closer than δ to any previously realized event.

10. Point Process Models Regular Process
Sequential Inhibition Process (SIP)
Attributes
Stationary and isotropic
Distance threshold δ cannot be too large
Computer Simulation
Generate events as for a HPP, but retain only those events that are no closer than δ to all previously generated events.
In spatstat
rSSI(r, n=Inf, win = square(1), giveup = 1000, x.init=NULL, ..., nsim=1)

11. Point Process Models Inhomogeneous Poisson Process (IPP)
Nonstationary process with non-constant intensity function
For every B D, N(B) ~ Poisson with mean
For any two disjoint regions B1 and B2, N(B1) and N(B2) are independent

12. Point Process Models Inhomogeneous Poisson Process (IPP)
Nonstationary process with non-constant intensity function
A possible framework for the incorporation of covariates via an intensity function
A Cox process is obtained by first generating a realization λ(s) from a nonnegative-valued random field and then generating events from an IPP with intensity function λ(s)

13. Point Process Models Inhomogeneous Poisson Process (IPP)
Nonstationary process with non-constant intensity function
Computer Simulation
Generate an event from the uniform distribution on D. Call its coordinate vector s
Retain the event at s with probability λ(s) / λ0, where λ0=max λ(s) (over the entire study region)
Repeat step 1 and 2 until n events have been retained.

14. Point Process Models Inhomogeneous Poisson Process (IPP) In spatstat,
rpoispp(lambda, lmax=NULL, win=owin(), ..., nsim=1)
rpoispp(function(x,y) {2000 * exp(-3*x)}, 2000)

15. Spatstat