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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..

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  • $\begingroup$ I am somewhat familiar with this literature and haven't seen this problem addressed in any reasonable way. The datasets I have had access to have been large, yet had so few values of this sort that it was easy to downweight them when fitting models and easy to overlook any high residuals. $\endgroup$ – whuber Mar 11 '13 at 22:30
  • $\begingroup$ Hi @whuber - can you please clarify what you mean by "downweight them when fitting models"? In regression-type models the meaning is clear but I'm a little confused what you mean in the context of a point process model. $\endgroup$ – Macro Mar 11 '13 at 22:57
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    $\begingroup$ (+1) Are these regions known and annotated as such in your data or do you have to "identify" them after the fact? Is it reasonable/sensible to estimate via a resampling approach, e.g., redistributing the points at random within the censored region? I can imagine how that might not be satisfactory either. Presumably the nature reserve would have well-travelled trails/paths. $\endgroup$ – cardinal Mar 11 '13 at 23:16
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    $\begingroup$ Are you interested in covariate effects, or merely interested in estimating the intensity of the process (i.e. forecasting or fitting a causal model)? This is a fundamental limitation of the data, it doesn't occur just in parks, it occurs everywhere. The only paper I could think of related to where you might want to go is Zimmerman (2008). Looking at the papers that have cited Zimmerman brings up this article that might be of interest, although it appears more aimed at edge effects. $\endgroup$ – Andy W Mar 12 '13 at 1:14
  • $\begingroup$ Hi @cardinal, in a perfect world, I'd like to do this without knowing where these "regions" are, but I'm pretty sure that's not feasible. Perhaps they could be detected by observing many crimes in exactly the same location. Hypothetically, data would exist to characterize these regions, such as street networks. Then, this could be viewed as a point process that was "pushed" onto the grid defined by the street networks. I like your suggestion and I may do something like that if I can't find an alternative that avoids the computational burden (these models take a good ~30 sec to fit, each). $\endgroup$ – Macro Mar 12 '13 at 14:18
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To start, this change in support problem is an active area of research, and so although it is usual to treat units of analysis in criminology as discrete, you can certainly make a case for treating it as a continuous field. Although I don't hold as such a negative view of using discrete units as you, I look forward to any further research and advances on the topic. I can see the hesitation to use larger areal units, like census block groups, but I don't have as much problem with smaller units like street segments or parcel units, which are becoming more popular in criminology all the time.

On its face, it seems the problem is related to edge effects (as you labeled the question as a censored point process), but it is a little different. Edge effects are typically handled by creating a weight for observed point process, and points near the edge of the study space are weighted higher. You have a process you observe though, but it is binned at a precision that doesn't suit your fancy.

This is very similar to the change of support problem, and attempts to solve it can be characterized as attempting to allocate the points (the suggestion) or by utilizing different techniques to take into the uncertainty in the aggregation. The dasymetric techniques listed are synonymous with the permutation/reallocation cardinal suggests in the comments. See the previous (two) answers of mine to a question of simulating similar outcomes. You can see examples of EM both for calculating ecological regressions, and for allocating estimates (see Tsutsumi & Murakami, 2012 - which is admittedly quite rough around the edges - as you can imaging by the title of the paper). The simulation approach you suggest is very similar to what Goovaerts (2008) suggests in his area-to-point kriging models.

I would note though, that there are areas that can be legitimately considered places that should be cut out of the window. For instance, although technically possible, you can assume the lake in the park is an area where no crime occurs. It also changes depending on the type of crime; burglary (as usually defined) can only occur in non-public areas, a public park can not be the victim of a burglary. Similarly if you wanted to fit a model of traffic accidents or DUI they are pretty much limited to actual roads. Most contemporary examples of crime forecasting though ignore this (and still produce estimates in areas where it is not possible to have a crime committed). Taking such places into account is what intelligent dasymetric mapping is all about.

Sorry, I believe there is no easy answer (and the best I can do is just provide a long list of literature that won't directly answer your question anyway).


As a sidebar, I'm not diametrically opposed to representing reality like this, but the examples you in the comments are obviously not measures that are continuous fields. The only continuous field I can think of that criminologists have been interested in is (unless you are a really persnickety physicist) weather related.

But, as perhaps always, how you go about representing the variables should be partly guided by what substantive questions you want answered. The continuous field approach to smoothing in the end confounds local and spatial spillover effects, and so is not reasonable to answer particular questions.

For example, if a concerned citizen came to me and said,

A parcel a block away, in a separate neighborhood group, was recently proposed to be rezoned to allow the sale of alcohol. Although it isn't in my neighborhood, I think my neighborhood should be allowed to vote on whether it is rezoned. Can you provide evidence if bars have an effect on crime not just where they are on the street, but a block away?

This answer can not be addressed with bars as a continuous density formulation in the model. But this can be by treating street blocks as units of analysis and fitting a spatial model to take into account bars not only on the local street, but on the neighboring street.

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  • $\begingroup$ Thanks for this answer and for the references (+1) and for confirming my suspicion that I'll have to "make my own" to handle this problem. I agree that barriers like lakes - where crimes literally cannot happen - may be sensibly removed from the "space". Regarding your sidebar - obviously there are issues of interpretation but in a continuous space model like this (which I have good reasons for needing to do in this project), you have the practical consideration of needing to represent covariates continuously, so... Aside from that, there is literature regarding how alcohol outlet density, $\endgroup$ – Macro Mar 15 '13 at 22:48
  • $\begingroup$ not necessarily the location of a specific alcohol outlet, or the nearest outlet, or the number within a certain radius (etc...) affects the community atmosphere and the dynamics of the process you're studying. See, e.g. Gruenewald PJ. The spatial ecology of alcohol problems: niche theory and assortative drinking. Addiction. 2007;102(6):870–878. So, I think there is some validity to representing reality this way. Of course, you are right that for some particular research questions, this would not be useful or appropriate. $\endgroup$ – Macro Mar 15 '13 at 22:54
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Censored spatial data is very common in the field of Cosmology. The issue is typically dealt with by creating a sample of random points with a flat distribution but that receives the same censoring as the data you are trying to model. Then, your analysis is adjusted by comparing effects seen in your data relative to effects seen in the random (but censored) data.

The classic estimator is one by Landy & Szalay, for the analysis of spatial clustering in real space (as opposed to Fourier space). Go here for the article http://adsabs.harvard.edu/abs/1993ApJ...412...64L

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