Marco Ferreira

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Associate Professor, Director of Graduate Programs
Virginia Tech


Bayesian Multi-scale Modelling

We introduce two classes of multi-scale models: One for time series and another for more general random fields. The novel framework couples standard Markov models for the time series, or the random field stochastic process, at different levels of aggregation, and links the levels via stochastic links to induce new and rich classes of structured linear models for time series and random fields. The framework allows a reconciliation of models and information processing at different levels of temporal and/or spatial resolution. Jeffrey's rule of conditioning is used to revise the implied distributions and ensure that the probability distributions at different levels are strictly compatible within a formal statistical framework. Our construction has several interesting characteristics: with just a few parameters, our framework produces a great variety of autocorrelation functions for time series and variograms for random fields; the models have the ability to coherently and efficiently combine information from different scales; and the models have the capacity to emulate long memory processes for time series and, maybe even more interesting, for random fields. There are at least three uses for our multi-scale framework: integration of information from data observed at different scales of resolution; induction of long-memory type process when the data is observed only at the finest scale; as priors for underlying multi-scale processes, e.g., a permeability field in the problem of fluid flow through porous media. Bayesian estimation based on MCMC analysis is developed. Issues of prediction through simulation are discussed. Several examples in time series include analysis of flow of a river, log-volatility of exchange rates, potential hydroelectric energy and temperature of the northern hemisphere. Some applications to subsurface hydrology include the estimation of 1-D and 2-D permeability fields.