feature/spdep

I'm just starting an issue to track the feature/spdep branch which incorporates (some) spatial models from spdep in rstanarm.

Currently it looks something like this:

       ┌─ stan_lagsarlm.R ───┐
user ──┤                     ├─ stan_sp.fit.R ── spatial.stan
       └─ stan_errorsarlm.R ─┘

spdep doesn't use .fit.R files for doing the heavy lifting for their modeling functions so currently I deal with all of the models in stan_sp.fit.R (to stay somewhat consistent with the other rstanarm models). As for spatial weights, the user can include a a listw object (as required in spdep) or a symmetric matrix (usually row normalized) in the call.

There are two other models that we might also want to include,

• spdep::sacsarlm() (Spatial simultaneous autoregressive SAC model estimation)
• spdep::lagmess() (matrix exponential spatial lag model)

Stuff to implement for stan_lagsarlm.R and stan_errorsarlm.R:

• check spatial weights (row normalized, convert from listw, etc.)
• center data
• QR decomposition
• prior for spatial autocorrelation (beta)
• prior for intercept
• prior summary
• optimize/vb
• print/summary stuff (in stanreg.R)
• predict/posterior_predict
• loo
• examples
• documentation
• vignette(s)
• tests

The response variable is basically modeled as multivariate normal with an NxN covariance matrix,

Sigma = solve( (I - rho * W) %*% t(I - rho * W) )

(where I is a diagonal matrix, W is a spatial weight matrix, and rho is spatial correlation term) and linear predictor,

mu = X * beta (times other stuff, depending on the model)