Goals of spatial smoothing in environmental health research

• estimate exposure as a spatial surface based on observations
  • possibly combine with other predictors
    • $\log y_{i}\sim N\left(X_{i}\beta+g(s_{1,i},s_{2,i}),\sigma^{2}\right)$
  • possibly combine observations with deterministic models
• estimate health risk using spatial location as a smooth covariate in a regression model
  • models for non-normal data
    • e.g., spatial logistic regression: $y_{i}\sim\mbox{Bern}\left(\mbox{logit}^{-1}(X_{i}\beta+g(s_{1,i},s_{2,i})\right)$

  • spatial location not necessarily fixed, may be uncertain