[R-sig-Geo] how to assess spatial randomness of an spp given the distribution of another spp

[Adrian Baddeley]

sylvain willart <sylvain.willart at gmail.com> writes:

  > I am working with spatstat package to analyse two
 > spatial points data frames:
 > - the locations of hospitals in France
 >  - the location of the population (each 'point' here
 > represents about 3,000 people)
 > I used quadrat.test and kstest to test for spatial randomness of
 > hospitals, which, hopefully is not random,
 > I would like to know if there's a (simple non-bayesian) way of
 > assessing spatial randomness for hospitals given the distribution of
 > the population,
 > and ultimately if it is possible to do so using MarkedPointProcess
 > package (the hospitals being of different sizes, and populations
 > having different needs for healthcare)

Suppose X is the point pattern of hospitals and Y the point pattern of 
the reference population. First smooth the reference population to get a 
population density e.g.
        Yden <- density(Y)
(see 'density.ppp' for additional arguments to control the smoothing)
Next rescale this intensity to have the same total as the hospitals

        nn <- npoints(X)/npoints(Y)
        Href <- eval.im(Yden * nn)

Thus Href is (a smooth estimate of) the predicted intensity of hospitals 
under the assumption that hospitals are randomly allocated in proportion 
to population density, independent of one another.

Finally compute the inhomogeneous K-function

        Ki <- Kinhom(X, Href)
        plot(Ki)

The result has an interpretation similar to the usual K-function except 
that it is adjusted for the population density. [In the latest versions 
of spatstat, Href can be replaced by Yden in the last command, because 
Kinhom automatically renormalises]

To generate simulation envelopes,

         e <- expression(rpoispp(Href))
         plot(envelope(X, Kinhom, lambda=Yden, simulate=e, nsim=39))

Adrian Baddeley