Project 3, Part A: An Inhomogeneous Poisson Process

Now, you are going to look at the inhomogeneous Poisson process. This is a spatial point process that generates events based on an underlying intensity that varies from place to place. In other words, it departs from IRP due to a first-order trend across the study area.

To introduce a trend, we have to define a function in place of the fixed intensity value of the standard (homogenous) Poisson process.

> pp <- rpoispp(function(x,y) {100*x + 100*y})

This tells R that the intensity of the Poisson point process varies with the x and y coordinates according to the function specified.

NOTE: if you want to visualize a 3D plot of any equation, you can simply copy and paste the equation into a Google search and an interactive 3D graph of the equation will appear. Google supports equation graphing through their calculator options.

You might also want to plot the density and the contours of points. The first line creates and plots a density plot. The second line overplots the points on top of the density plot. The pch parameter controls which graphical mark is used to plot the points. You can see a full list of different mark symbols here.

> plot(density(pp))
> plot(pp, pch = 1, add = TRUE)

To plot contours, use the following code:

> contour(density(pp))
> plot(pp, pch = 1, add = TRUE)

Make a few patterns with this process and have a look at them. Experiment with this by changing the x and y values (e.g.,: 100*x + 50*y) until you understand how they interact with one another.

We have also provided two additional functions for you to explore using this same procedure.

Vary the intensity with the square of x and y coordinates:

pp <- rpoispp (function(x,y) {100*x*x + 100*y*y})

Vary the intensity with the distance from the center of the unit square:

pp <- rpoispp (function(x,y) {200*sqrt((x-0.5)^2 + (y-0.5)^2)})