26 research outputs found
Kryging: Geostatistical analysis of large-scale datasets using Krylov subspace methods
Analyzing massive spatial datasets using Gaussian process model poses
computational challenges. This is a problem prevailing heavily in applications
such as environmental modeling, ecology, forestry and environmental heath. We
present a novel approximate inference methodology that uses profile likelihood
and Krylov subspace methods to estimate the spatial covariance parameters and
makes spatial predictions with uncertainty quantification. The proposed method,
Kryging, applies for both observations on regular grid and irregularly-spaced
observations, and for any Gaussian process with a stationary covariance
function, including the popular \Matern covariance family. We make use of the
block Toeplitz structure with Toeplitz blocks of the covariance matrix and use
fast Fourier transform methods to alleviate the computational and memory
bottlenecks. We perform extensive simulation studies to show the effectiveness
of our model by varying sample sizes, spatial parameter values and sampling
designs. A real data application is also performed on a dataset consisting of
land surface temperature readings taken by the MODIS satellite. Compared to
existing methods, the proposed method performs satisfactorily with much less
computation time and better scalability
Certified and fast computations with shallow covariance kernels
Many techniques for data science and uncertainty quantification demand
efficient tools to handle Gaussian random fields, which are defined in terms of
their mean functions and covariance operators. Recently, parameterized Gaussian
random fields have gained increased attention, due to their higher degree of
flexibility. However, especially if the random field is parameterized through
its covariance operator, classical random field discretization techniques fail
or become inefficient. In this work we introduce and analyze a new and
certified algorithm for the low-rank approximation of a parameterized family of
covariance operators which represents an extension of the adaptive cross
approximation method for symmetric positive definite matrices. The algorithm
relies on an affine linear expansion of the covariance operator with respect to
the parameters, which needs to be computed in a preprocessing step using, e.g.,
the empirical interpolation method. We discuss and test our new approach for
isotropic covariance kernels, such as Mat\'ern kernels. The numerical results
demonstrate the advantages of our approach in terms of computational time and
confirm that the proposed algorithm provides the basis of a fast sampling
procedure for parameter dependent Gaussian random fields