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The Mat\'ern covariance model is ubiquitous in spatial modelling, but there is no default choice for spatio-temporal modelling. In this paper, we consider the recently proposed ``diffusion-based'' extension of the spatial Mat\'ern…
In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will…
Spatial models for areal data are often constructed such that all pairs of adjacent regions are assumed to have near-identical spatial autocorrelation. In practice, data can exhibit dependence structures more complicated than can be…
The spatial random-effects model is flexible in modeling spatial covariance functions, and is computationally efficient for spatial prediction via fixed rank kriging. However, the success of this model depends on an appropriate set of basis…
Identifying an appropriate covariance function is one of the primary interests in spatial and spatio-temporal statistics because it allows researchers to analyze the dependence structure of the random process. For this purpose, spatial…
We introduce a novel class of non-stationary covariance functions for random fields on linear networks that allows both the variance and the correlation range of the random field to vary spatially. The proposed covariance functions are…
In climate and atmospheric research, many phenomena involve more than one meteorological spatial processes covarying in space. To understand how one process is affected by another, maximum covariance analysis (MCA) is commonly applied.…
Modeling data with non-stationary covariance structure is important to represent heterogeneity in geophysical and other environmental spatial processes. In this work, we investigate a multistage approach to modeling non-stationary…
Covariance parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar…
The Mat\'ern covariance function is a popular choice for modeling dependence in spatial environmental data. Standard Mat\'ern covariance models are, however, often computationally infeasible for large data sets. In this work, recent results…
This paper considers a generalization of Gaussian random field with covariance function of Whittle-Mat$\acute{\text{e}}$rn family. Such a random field can be obtained as the solution to the fractional stochastic differential equation with…
We study how sampling geometry contributes to uncertainty in modeling spatial geophysical observations as sampled random fields characterized by stationary, isotropic, parametric covariance functions. We incorporate the signature of…
We derive explicit expressions for a family of radially symmetric, non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involve the modified Bessel function of the second kind. In addition to the characteristic length and…
Gaussian random fields with Mat\'ern covariance functions are popular models in spatial statistics and machine learning. In this work, we develop a spatio-temporal extension of the Gaussian Mat\'ern fields formulated as solutions to a…
Matrix-valued covariance functions are crucial to geostatistical modeling of multivariate spatial data. The classical assumption of symmetry of a multivariate covariance function is overlay restrictive and has been considered as unrealistic…
The Wendland functions are a class of compactly supported radial basis functions with a user-specified smoothness parameter. We prove that with a linear change of variables, both the original and the "missing" Wendland functions converge…
Smoothing of noisy sample covariances is an important component in functional data analysis. We propose a novel covariance smoothing method based on penalized splines and associated software. The proposed method is a bivariate spline…
The Mat\'ern model has been a cornerstone of spatial statistics for more than half a century. More recently, the Mat\'ern model has been central to disciplines as diverse as numerical analysis, approximation theory, computational…
Geophysical and other natural processes often exhibit non-stationary covariances and this feature is important to take into account for statistical models that attempt to emulate the physical process. A convolution-based model is used to…
Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class…