Geostatistics for point processes : predicting the intensity of partially observed point process data
In many studies (e.g. in ecology) mapping the positions of objects (e.g. plant species) is cumbersome as soon as these objects are not accessible by automated methods, as image analysis for example. The knowledge at large scale of the underlying process variability can then only be obtained through sampling and spatial prediction. Here, we aim to estimate the local intensity of a point process at locations where it has not been observed using the best linear unbiased combination of the point process realization on sampled windows. We show that the weight function, $a(x)$, associated to the estimator is the solution of a Fredholm equation of second kind, $a(x) - \int k(x,y) a(y) dy = f(x)$. The kernel of the Fredholm equation, $k(x,y)$, is defined by the pair correlation function between points in observed windows. The function $f(x)$ depends of the pair correlation function between the observed point locations and the location of the prediction. Thus, both $k(x,y)$ and $f(x)$ are related to the second order characteristics of the point process.
Several approximations can be used to solve the Fredholm equation in order to obtain practical solutions. In particular, for $a(x)$ defined as a step function, we get the classical kriging equations where one counts the number of points in a given area. In this case the kriging weights depend on the local structure of the point process (as proposed in Gabriel et al, 2014).
We can also use $a(x) = \sum_i \alpha_i h_i (x)$, where $h_i (x)$ are known basis functions.
Gabriel E., Bonneu F., Monestiez P., Chadoeuf J. (2014) Predicting the local intensity of partially observed data from a revisited kriging for point processes. arXiv:1409.6441