Is it reasonable to use AIC to rank point process models that are based on different likelihood functions (Poisson and Gibbs in this case)?

Question

I am using spatstat version 3.0.2 to explore settlement patterns in archaeological data in five different periods throughout prehistory. I have formulated several point process models with different covariates including elevation, slope, etc.
In terms of model formalisation, each environmental variable was examined on its own to delineate its relationship with archaeological sites during each chronological phase. Several point process models were created (as realisations of the Nonstationary Poisson Process). Subsequently, a model including all the environmental variables was formulated for each period. Finally, a Gibbs model including both environmental variables and the social interaction between sites was created. All of the created models were compared per chronological period. AIC was used to measure the models’ quality, not in terms of the goodness of fit, but rather in terms of how well they act as approximations of the processes that generated the observed point patterns. This was done to compare competing hypotheses regarding site location, reaching a “best model” for each chronological period-that is a model that could best predict the observed point pattern given a trade-off between the complexity of the data and the model’s fit.

My question has to do with the use of AIC, which in my case is used to rank Poisson and Gibbs point process models. Is it reasonable to use a likelihood-based statistic like AIC to compare point process models that are based on different likelihood functions (Poisson and Gibbs in this case)? Any insights would be very much appreciated.

Answer 1

Yes, you can do this, but you need to be careful. You need to use the same "edge correction" in each model --- otherwise the likelihoods and pseudolikelihoods are not comparable. In a Poisson model fitted by ppm, the default behaviour is not to do any edge correction, because it is not necessary for Poisson models. In a Gibbs model fitted by ppm the default is to use the "border correction" which effectively restricts the spatial domain by cutting off a margin of width r at the edges, where r is the range of spatial interaction in the model. So if you are comparing different Gibbs models, or Gibbs and Poisson models, you can either

• find the largest interaction range R used in any of the models, and re-fit all the models using ppm with argument rbord=R
• re-fit all the models using ppm with correction="isotropic" or correction="translate"

This is done automatically if you use anova.ppm instead of comparing models using AIC.

Answer 2

I have to double check to be absolutely sure, but I think the answer is yes. For Gibbs point processes we use the pseudo-likelihood to estimate parameters, and it is also underlying the calculation in functions like loglik() and AIC(). For Poisson models pseduo-likelihood and likelihood are equivalent, so you could say that you are comparing pseudo-likelihood based quantities for all models, which is perfectly valid. My only concern is whether there is some normalization factor that is implemented differently in the two cases, but I don't think so. Really, (pseudo-)likelihoods are only defined up to a constant of proportionality (so an additive constant for a log-(pseudo-)likelihood), so if this constant is not compatible across Poisson and Gibbs models it would be problematic, but I'm fairly certain that there is no problem with the spatstat implementation.