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Models for Space-Time Data Inspired from Statistical Physics

Date:
Location:
MDS 220
Speaker(s) / Presenter(s):
Dr. Dionisis Hristopulos

Abstract: This presentation will focus on statistical models for space-time data which are motivated by ideas from statistical physics. The latter provides a general framework for developing space-time models based on Boltzmann-Gibbs probability density functions and stochastic partial differential equations (SPDEs). In geostatistics, on the other hand, spatial models are typically defined in terms of an explicit covariance function (or a family of covariance functions). In contrast, in the Boltzmann-Gibbs approach the covariance function is intrinsically generated from the underlying joint probability density model. The latter is determined from the respective energy function model which incorporates interactions between different sites. In the SPDE formulation, the covariance function is determined from the “driving equation” of the random field, which leads to a respective partial differential equation for the covariance function.

 

I will briefly discuss the connection between the Boltzmann-Gibbs and SPDE formulations for Gaussian random fields. I will then review some results which are based on Boltzmann-Gibbs densities equipped with an energy function comprising short-range interactions. These results include: (1) A class of flexible spatial covariance functions; (2) a non-separable covariance function with a composite spacetime metric; (3) a family of non-separable covariance functions that are based on linear response theory combined with the space transform; and (4) ongoing efforts to generalize Boltzmann-Gibbs models from continuum and lattice spaces to irregular sampling geometries. The space-time models generated by means of the Boltzmann-Gibbs formulation with short-range energy functions involve sparse precision matrices by construction. This is a significant asset for the processing of big spatial or space-time datasets, since the computationally demanding inversion of large covariance matrices (common in geostatistics and Gaussian process regression) is avoided. I will illustrate these concepts with applications to environmental and energy resources datasets.

 

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