Centre-Satellite models

"The compound and generalized Poisson models contain one particularly unrealistic constraint, namely that the `offspring' are coincident with the `parent' - in other words, they suppose that the entire cluster is situated at a single point."

Thus we add displacement (so-called centre-satellite models - note cool British spelling, since I stole this from those brits):

• Parent events form a Poisson forest
• the number of offspring generated (via some other distribution, e.g. logarithmic)
• the locations of each offspring displaced relative to parent according to e.g. bivariate normal. Here's an example.

We show here a similar situation, although the pattern was created in a slightly different way.

This example was created as follows:

1. a set of "susceptible" points was simulated randomly on the unit square;
2. another set of "infected" points was added;
3. each infected individual infects a group of susceptible neighbors, the number infected distributed as a Poisson variable - but only neighbors within a radius of .1 units of the infected point. (If there were not enough neighbors within that radius, all available were infected, but no more).

Below is the isotropic sample variogram, plotted with a model:

It is a "nested" model (that is, a positive linear combination of valid models). The model is composed of

• a nugget of about .071
• a Gaussian model with a sill of about .063 and a range of .14, and
• an exponential model with a tiny sill (.002) and range of .07.

This is an example of "indicator kriging": rather than rates, or some other continuous variable, we are actually trying to estimate the probability that someone is ill at a given location by using the binary variable of "disease status". Here is the resulting kriged surface: