The Annals of Applied Statistics

Modeling temporal gradients in regionally aggregated California asthma hospitalization data

Harrison Quick, Sudipto Banerjee, and Bradley P. Carlin

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Advances in Geographical Information Systems (GIS) have led to the enormous recent burgeoning of spatial-temporal databases and associated statistical modeling. Here we depart from the rather rich literature in space–time modeling by considering the setting where space is discrete (e.g., aggregated data over regions), but time is continuous. Our major objective in this application is to carry out inference on gradients of a temporal process in our data set of monthly county level asthma hospitalization rates in the state of California, while at the same time accounting for spatial similarities of the temporal process across neighboring counties. Use of continuous time models here allows inference at a finer resolution than at which the data are sampled. Rather than use parametric forms to model time, we opt for a more flexible stochastic process embedded within a dynamic Markov random field framework. Through the matrix-valued covariance function we can ensure that the temporal process realizations are mean square differentiable, and may thus carry out inference on temporal gradients in a posterior predictive fashion. We use this approach to evaluate temporal gradients where we are concerned with temporal changes in the residual and fitted rate curves after accounting for seasonality, spatiotemporal ozone levels and several spatially-resolved important sociodemographic covariates.

Article information

Ann. Appl. Stat., Volume 7, Number 1 (2013), 154-176.

First available in Project Euclid: 9 April 2013

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Zentralblatt MATH identifier

Gaussian process gradients Markov chain Monte Carlo spatial process models spatially associated functional data


Quick, Harrison; Banerjee, Sudipto; Carlin, Bradley P. Modeling temporal gradients in regionally aggregated California asthma hospitalization data. Ann. Appl. Stat. 7 (2013), no. 1, 154--176. doi:10.1214/12-AOAS600.

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Supplemental materials

  • Supplementary material: Imputation of missing daily hospitalization counts, MCMC details, alternative models and comparison with discrete-time models. As data for days with between one and four asthma hospitalizations are missing, we impute county-specific values for these days using a method similar to Besag’s iterated conditional modes method [Besag (1986)] but with means. We also lay out the details for the MCMC implementation, discuss more general versions of our model and compare our gradient estimates to finite differences from a simple discrete-time model.