BMTRY 763. Space-time (ST) Modeling (BDM13, ch 12) Some notation Assume counts within fixed spatial and temporal periods: map evolutions Both space and.

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BMTRY 763

Space-time (ST) Modeling (BDM13, ch 12) Some notation Assume counts within fixed spatial and temporal periods: map evolutions Both space and time are subscripts in the analysis Consider separable models (with spatial and separate temporal terms) Also interaction effects ©Andrew Lawson 2014

Notation ©Andrew Lawson 2014

Expected Counts Computation (simplest - overall average): ©Andrew Lawson 2014

Basic retrospective model Infinite population; small disease probability Poisson assumption ©Andrew Lawson 2014

Full data set: 21 years of Ohio lung cancer 10 years of SMRs standardized with the statewide rate: Frequently analyzed Row wise from 1979 ©Andrew Lawson 2014

Some Random Effect models ©Andrew Lawson 2014

Random Walk Prior distribution Model 1 b: we assume a random effect for the time element and this has a random walk prior distribution: More generally an AR1 prior could be used: ©Andrew Lawson 2014

Interaction priors A variety of priors for the interaction can be assumed (both correlated and non-separable) Knorr-Held (2000) first suggested dependent priors (see Lawson (2013) ch12) Two simple separable examples of possible priors are: ©Andrew Lawson 2014

Model fitting Results ModelDICpD 1a b ©Andrew Lawson 2014

Interpretation The temporal trend model does not provide a better fit than the random walk (1a, 1b) The extra RE in model 2 is not needed The inclusion of the interaction in model 3 is significant but model 4 is not good Model 5 with the random walk interaction seems best as it has lowest DIC and smaller pD than model 3 ©Andrew Lawson 2014

Space-time Kalman Filter The Kalman Filter consists of a coupled set of equations describing the behavior of a system and also the measurement made on the system In a dynamic setting it is an appropriate model for an evolving system observed with error ©Andrew Lawson 2014

System structure evolution e.g. risk evolving over time ©Andrew Lawson 2014

Measurement model Poisson (independent) error In the classic Kalman filter the errors would be Gaussian and the measurement model could also have correlated errors ©Andrew Lawson 2014

Gaussian approximation We can proceed to generalize this to allow correlation if we assume a log Gaussian form This leads to a hidden Markov model ©Andrew Lawson 2014

Full Model Structural model Measurement model ©Andrew Lawson 2014

WinBUGS Code ©Andrew Lawson 2014 DIC: 1657 (767)

Clustering in ST data ©Andrew Lawson 2014

Clustering in ST data Clustering is a different issue. Earlier we examined exceedence probabilities These can also be used with ST data ©Andrew Lawson 2014

Ohio SMR 21 years ©Andrew Lawson 2014

Added simulated clusters Exceedence C=1 ©Andrew Lawson 2014

Exceedence C=2 ©Andrew Lawson 2014