UC Santa Cruz

Large spatial datasets often exhibit fine scale features that only occur in sub-

domains of the space, coupled with large scale features at much larger ranges.

The most commonly used model used for spatial datasets is the Gaussian Process,

but evaluation of likelihood is computationally expensive. Additionally,

traditional Gaussian Processes models make very strong assumptions regarding

the symmetry of the Gaussian field. In particular they assume

stationarity, namely that covariance functions depend only on the displacement

vector between two points, not their locations. This assumption prevents stationary

Gaussian Processes from accounting for multi-scale features that only exist in parts

of the spatial domain. In this work, we develop multi-resolution kernel convolution methods that explicitly account for local multi-scale features through spatially varying resolution. These methods define an increasingly refined set of nested kernels, and induce sparsity on these grids.

We first introduce modifications to existing multi-resolution kernel convolution models

that result in spatially varying resolution through a sparsity inducing prior.

We cast spatially varying resolution as a model selection problem, and develop a

Shotgun Stochastic Search algorithm that considers an infinite number of resolutions,

and permits uncertainty quantification without resorting to MCMC.

We propose a LASSO like prior that achieves spatially varying resolution at its

maximum a posteriori, and develop a proximal gradient descent algorithm to find this

optimum considering an infinite number of resolutions. We then develop a Bayesian

model averaging approach to perform uncertainty quantification in this setting.

We implement these methods in an efficient and reproducible manner via the R package

MSSS. We discuss in detail the computational efficiency achieved by

leveraging parallel computation, compactly supported kernels, add one column

regression updates, and modern optimization methods in {\tt MSSS}. We demonstrate the local feature identification properties of spatially varying resolution and demonstrate the computational performance by considering a land surface temperature dataset from the Ozarks, and a large sea surface temperature dataset collected by a satellite off the coast of California