Bayesian non-parametric modeling for integro-difference equations
- 217 Downloads
Integro-difference equations (IDEs) provide a flexible framework for dynamic modeling of spatio-temporal data. The choice of kernel in an IDE model relates directly to the underlying physical process modeled, and it can affect model fit and predictive accuracy. We introduce Bayesian non-parametric methods to the IDE literature as a means to allow flexibility in modeling the kernel. We propose a mixture of normal distributions for the IDE kernel, built from a spatial Dirichlet process for the mixing distribution, which can model kernels with shapes that change with location. This allows the IDE model to capture non-stationarity with respect to location and to reflect a changing physical process across the domain. We address computational concerns for inference that leverage the use of Hermite polynomials as a basis for the representation of the process and the IDE kernel, and incorporate Hamiltonian Markov chain Monte Carlo steps in the posterior simulation method. An example with synthetic data demonstrates that the model can successfully capture location-dependent dynamics. Moreover, using a data set of ozone pressure, we show that the spatial Dirichlet process mixture model outperforms several alternative models for the IDE kernel, including the state of the art in the IDE literature, that is, a Gaussian kernel with location-dependent parameters.
KeywordsDirichlet process mixtures Hamiltonian Markov chain Monte Carlo Hermite polynomials Spatial Dirichlet process
This research is part of the first author’s Ph.D. dissertation completed at University of California, Santa Cruz. A. Kottas was supported in part by the National Science Foundation under award DMS 1310438. B. Sansó was supported in part by the National Science Foundation under award DMS 1513076. The authors wish to thank an Associate Editor and two reviewers for constructive feedback and for comments that improved the presentation of the material in the paper.
- Chen, T., Fox, E.B., Guestrin, C.: Stochastic gradient Hamiltonian Monte Carlo. In: ICML, pp. 1683–1691 (2014)Google Scholar
- Geweke, J. et al.: Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments, vol. 196. Federal Reserve Bank of Minneapolis, Research Department Minneapolis(1991)Google Scholar
- Jones, R.H., Zhang, Y.: Models for continuous stationary space-time processes. In: Gregoire, T.G., Brillinger, D.R., Diggle, P.J., Russek-Cohen, E., Warren, W.G., Wolfinger, R.D. (eds.) Modelling Longitudinal and Spatially Correlated Data, pp. 289–298. Springer, New York (1997)CrossRefGoogle Scholar
- Nolan, J.: Stable Distributions: models for Heavy-Tailed Data. Birkhauser, New York (2003)Google Scholar
- Olver, F.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)Google Scholar
- Richardson, R., Kottas, A., Sansó, B.: Flexible Integro-Difference Equation Modeling for Spatio-Temporal Data, To appear in Computational Statistics and Data Analysis (2017)Google Scholar
- Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient Langevin dynamics. In: Proceedings of the 28th International Conference on Machine Learning (ICML-11), pp. 681–688 (2011)Google Scholar