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Boltzmann-Gibbs local-interaction models for spatial regression problems

Toganidis Nikolaos

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Summary

Machine learning and Geostatistics are powerful mathematical frameworks for modeling spatial data. Both approaches, however, suffer from poor scaling of the required computational resources for large data applications. In 2015 the Stohastic Local Interaction (SLI) model was presented, which combined machine learning and geostatistics and with its local representation employment improved computational efficiency. Though the model is very stable and performs well, its performance relies on some assumptions that do not always hold. Specifically, due to it is based on a joint probability density function (Boltzmann-Gibbs pdf) defined by an energy functional and is expressed in terms of explicit, typically sparse, precision (inverse covariance) matrix, the curvature term (used to construct this precision matrix) must be semi-positive definite, and depends on the given dataset. In this thesis, to eliminate those assumptions, the Graph Laplacian (GL) version of the model is introduced. Curvature terms now are calculated with the use of the second order of the Graph Laplacian, constructed using kernel functions and its local bandwidths to keep the local representation employment and at the same time the low complexity, and prediction performance of the original model. This version leads to a spatial analysis model, for any dimension with a good performance and low complexity, and now for any kind of spatial dataset.

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