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Intrinsic Gaussian Processes on Complex Constrained Domains
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81(3), 2019.
Abstract
We propose a class of intrinsic Gaussian processes (GPs) for
interpolation, regression and classification on manifolds with a
primary focus on complex constrained domains or irregularly shaped
spaces arising as subsets or submanifolds of $\Re$, $\Re^2$, $\Re^3$
and beyond. For example, intrinsic GPs can accommodate spatial
domains arising as complex subsets of Euclidean space. Intrinsic GPs
respect the potentially complex boundary or interior conditions as
well as the intrinsic geometry of the spaces. The key novelty of the
approach proposed is to utilize the relationship between heat
kernels and the transition density of Brownian motion on manifolds
for constructing and approximating valid and computationally
feasible covariance kernels. This enables intrinsic GPs to be
practically applied in great generality, whereas existing approaches
for smoothing on constrained domains are limited to simple special
cases. The broad utilities of the intrinsic GP approach are
illustrated through simulation studies and data examples.