This is a problem I'm encountering in the context of analyzing a data set comprised of all crime locations in a city over a fixed time interval, although it could potentially arise in other types of point processes. The problem has to do with the fact that crime locations are not observed exactly. They are located by whatever street address the police wrote down on their report. In my city of study (and probably almost any city), there are "natural barriers" in the landscape that make these "locations" inherently imprecise.
For example, suppose there is a large nature reserve/park in the middle of a city. Then, all crimes occurring within the nature reserve are mapped to a single address - presumably the address of the main office of the nature reserve. This kind of "censoring" causes artificial clustering in the data set and, most likely, biases the estimate of the intensity function and the associated covariate effects, etc. I'd imagine these sort of "natural barriers" exist in pretty much any city so that this issue has probably been fomented by other researchers so my question is: Are there known methods for handling this type of data?
At the moment, I have analyzed this data set using an ordinary inhomogeneous poisson process model and have gotten some interesting results. I really think these results are "real" based on previous descriptive analysis, etc. and the model fits reasonably well except near these natural barriers where there are huge residuals due to smoothing over "zero density" areas, making the model wildly fail any kind of "goodness of fit" test despite the fact that the empirical density of the observed data agrees fairly closely with the empirical density of data simulated under the fitted model.
Here are the main possibilities I've considered (and why I've decided against them):
Delete these "natural barriers" from the window of observation and view this as a point process on a grid. I don't want to do this since it fundamentally changes the parameter you're estimating and, effectively, sweeps the "censoring" issue under the rug.
Bin the data into areal units (e.g. census-based groups), since this censoring infrequently crosses over census boundaries. This may be a good solution in some cases, but oversmoothing is a concern and, more importantly, the city I'm working with is too small (not enough census units) to do this.
Develop my own model for this. I'm pretty sure this is what I have to do but I wanted to check first that I'm not reinventing the wheel. Based on my own literature search, I'm not, but some specialists here may know something I don't..