Converting intensities to probabilities in ppp

Question

Apologies for the overlap with existing questions; mine is at a more basic skill level. I am working with very sparse occurrences spanning very large areas, so I would like to calculate probability at pixels using the density.ppp function (as opposed to relrisk.ppp, where specifying presences+absences would be computationally intractable). Is there a straightforward way to convert density (intensity) to probabilities at each point?

Maxdist=50
dtruncauchy=function(x,L=60) L/(diff(atan(c(-1,1)*Maxdist/L)) * (L^2 + x^2)) 
dispersfun=function(x,y) dtruncauchy(sqrt(x^2+y^2))
n=1e3; PPP=ppp(1:n,1:n, c(1,n),c(1,n), marks=rep(1,n));
density.ppp(PPP,cutoff=Maxdist,kernel=dispersfun,at="points",leaveoneout=FALSE) #convert to probabilies?

Thank you!!

Answer 1

I think there is a misunderstanding about fundamentals. The spatstat package is designed mainly for analysing "mapped point patterns", datasets which record the locations where events occurred or things were located. It is designed for "presence-only" data, not "presence/absence" data (with some exceptions).

The relrisk function expects input data about the presence of two different types of events, such as the mapped locations of trees belonging to two different species, and then estimates the spatially-varying probability that a tree will belong to each species.

If you have 'presence-only' data stored in a point pattern object X of class "ppp", then density(X, ....) will produce a pixel image of the spatially-varying intensity (expected number of points per unit area). For example if the spatial coordinates were expressed in metres, then the intensity values are "points per square metre". If you want to calculate the probability of presence in each pixel (i.e. for each pixel, the probability that there is at least one presence point in the pixel), you just need to multiply the intensity value by the area of one pixel, which gives the expected number of points in the pixel. If pixels are small (the usual case) then the presence probability is just equal to this value. For physically larger pixels the probability is 1 - exp(-m) where m is the expected number of points.

Example:

X <- redwood
D <- density(X, 0.2)
pixarea <- with(D, xstep * ystep)
M <- pixarea * D
p <- 1 - exp(-M)

then M and p are images which should be almost equal, and can both be interpreted as probability of presence.

For more information see Chapter 6 of the spatstat book.

If, instead, you had a pixel image of presence/absence data, with pixel values equal to 1 or 0 for presence or absence respectively, then you can just use the function blur in the spatstat package to perform kernel smoothing of the image, and the resulting pixel values are presence probabilities.