Spatio-Temporal Model

Modeling
9

Going back to your original model specification, assuming that you can convert those element-wise univariate normal priors to multivariate normal priors over vectors (over all spatial units for each time step of phi, and over all time steps for delta) the Stan code might look something like:

data {
  ...
  int K; // number of spatial units
  int T; // number of time steps
}
parameters {
  ...
  vector[K] phi[T];
  vector[T] delta;
}
model {
  ...
  for (t in 1:T) 
    // phi[t] is a vector with one element per spatial unit
    phi[t] ~ multi_normal...

  delta ~ multi_normal...
}

There are ways to implement CAR priors more efficiently that do not require using any of the multi_normal* functions, and these are described in the case studies mentioned above.

To use a temporal AR 1 prior on delta, you could use the covariance matrix for an AR 1 process (https://stats.stackexchange.com/questions/295102/how-to-write-variance-covariance-matrix-of-ar1-process-in-r), or you could specify the AR 1 prior using the transformed parameters block (Migrated from Google Group: AR(1) Logistic Regression Funnel).

I’d recommend starting with the simplest version of this model that you can, and then slowly add complexity and verify that you can recover parameters for a sequence of models that are increasing in complexity (essentially following Stan best practices), e.g.,

  1. Ridiculously simple Poisson model
  2. Model with a non-time varying spatial effect
  3. Model with a time varying spatial effect
  4. Model with a time varying spatial effect + a temporal effect
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