How to define the likelihood for a log-Gaussian Cox Process in TMB with SPDE approach implemented from INLA

[李大一]

Hi all,

I am very new to TMB, and I am looking at some examples of using TMB for fitting spatial model.

For example, this example is for the Leukimea survival rate from Lindgren et al. 2011:

https://github.com/kaskr/adcomp/blob/master/tmb_examples/spde.cpp

This example shows how to use the SPDE approach in INLA proposed by Lindgren et al. 2011. But I am not sure how to use TMB for a log-Gaussian Cox process to model some point pattern. Specifically, I do not know how should on write the likelihood of a Poisson process based on the SPDE approach. 

Traditionally, one write the likelihood based on a discretizing grids of the entire study region, but the SPDE approach does not have a discretizing grids. So how would one specify the likelihood in TMB in such a case?

A simple example code would be extremely helpful and much appreciated!

Best,

David

[Hans Skaug]

Hi,

In principle that is just the likelihood of a non-homogeneous Poisson process.

However, this involves the integral of the density exp(Xs) over the spatial region, where Xs is the GMRF,

which will break the sparsity of the Hessian in the Laplace approximation. 

TMB is not efficient for such models. 

Instead one has to discretize the spatial region in some way. I assume this is the basis

for implementation of  log-Gaussian Cox Processes in INLA:

Janine B. Illian. Sigrunn H. Sørbye. Håvard Rue. "A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA)." Ann. Appl. Stat. 6 (4) 1499 - 1530, December 2012. https://doi.org/10.1214/11-AOAS530

 

I am not aware of any TMB code for this.

Hans

[李大一]

Hi Hans,

Thanks for the clarification. So in this case, does that mean I have to manually construct the response first (which is the point count in each discretized grid), and then fit the latent GMRF using SPDE in a similar way as shown in the example for the Leukimia data?

On another note, I went back to the original paper for fitting LGCP using SPDE by Simpson et al.:  https://doi.org/10.1093/biomet/asv064

In this paper, the author suggest to approximate the likelihood of the Poisson process using the basis function expansion for Xs, is this possible to do in TMB somehow?

David

[Hans Skaug]

Hi,

What you say in the first sentence is correct. The trick is to do the discretization so that 

1) you do not loose to much of the Poisson process aspect, and 2) the model

is computationally feasible. This is where I suspect that the reference I pointed to will be useful.

Hans