The strong dependence between samples in large spatial data sets is the primary challenge of designing statistically consistent and computationally efficient inference algorithms. Gaussian processes provide a powerful tool for modelling the spatial dependence patterns and play a crucial role in numerous tractable inference algorithms.This thesis addresses two important problems on high-dimensional Gaussian spatial processes. We first focus on scalable estimation of covariance parameters. Evaluating the log-likelihood function of Gaussian process data can be computationally intractable, particularly for large and irregularly spaced observations. We build a broad family of surrogate loss functions based on local moment-matching and a block diagonal approximation of the covariance matrix. This class of algorithms provides a versatile balance between the estimation accuracy and the computational cost. The fixed domain asymptotic behavior of the proposed method is thoroughly studied for the isotropic Matern processes observed on amulti-dimensional irregular lattice.In the second part, the main emphasis is on minimax optimal detection of abrupt changes in the mean of a one-dimensional Gaussian process. Our main contribution is to show that in the fixed-domain asymptotic regime, neglecting the dependence structures leads to suboptimal performance. We first show that plugging the estimated covariance matrix into the Generalized Likelihood Ratio Test (GLRT) provides a test with near minimax asymptotic optimality. On the other hand, the suboptimality of the cumulative sum test, which ignores the dependence structure of data, is substantiated for a vast range of covariance functions.
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Detection and Estimation in Gaussian Random Fields: Minimax Theory and Efficient Algorithms
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