I am currently trying to redo plots which could be found on p. 120 in the textbook "Statistical Analysis and Modelling of Spatial Point Patterns". The following information should be sufficient to help me without having a look into the mentioned textbook. Using the fantastic spatstat package, I try to simulate point patterns in the unit square resulting from inhomgenous Poisson point processes (IPP) with the intensity functions (a) $\lambda(x,y)=a*(x+y)$ (linear trend) and (b) $/lambda(r)=c*exp(-dr^2)$, with $r$ being the distance from the origin.
For (a) I did the following:
library(spatstat)
linear <- function(x,y,a) {a*(x+y)}
plot(rpoispp(lambda = linear, a=150))
The resulting plot is not too bad from my understanding. I am unable figuring out how to implement (b) and would appreciate any help.
Hopefully, understanding how the implemantation of (b) works helps me to fit a model to an observed point pattern, with only a few clusters, probably one, which is likely to stem from an IPP using ppm(pattern, function describing the simple model) or kppm.
Note. The reason I am asking this question is self-interest. I could easily retrieve the plots from the source, but this does not help me understanding how to implement intensities, or create and fit simple models to observed point patterns.
If my question is answered elsewhere I would appreciate the provision of links. Thank you!
If you want to code an intensity function as a function in the R language, then it should be a function of the spatial location (x,y).
In (b) the intensity function is $\lambda(x,y) = c exp(-d (x^2 + y^2))$ where we use the fact that the distance from the origin (0,0) to the point (x, y) is $r = sqrt(x^2 + y^2)$. The code is
lam <- function(x,y,c,d) { c * exp(- d * (x^2 + y^2))
In this example the value of lambda(x,y) depends only on the distance r, so we say loosely that "the intensity is a function of r", which may be the source of your confusion.