We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form a covariance operator field. We show that, in most circumstances, covariance fields are continuous. We also solve the inverse problem: recovering distribution from a covariance field. Surprisingly, this is not possible on Euclidean spaces. On non-Euclidean manifolds however, covariance fields are true distribution representations.
This paper presents a parametric family of compactly-supported positive semidefinite kernels aimed to model the covariance structure of second-order stationary isotropic random fields defined in the $d$-dimensional Euclidean space. Both the covariance and its spectral density have an analytic expression involving the hypergeometric functions ${}_2F_1$ and ${}_1F_2$, respectively, and four real-valued parameters related to the correlation range, smoothness and shape of the covariance. The presented hypergeometric kernel family contains, as special cases, the spherical, cubic, penta, Askey, gene…
We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. However, most of the available solutions use coordinate specific definition that makes them less versatile. %We propose improvements in two directions. We define the distributions of interest in coordinate independent way by utilizing co-variant 2-tensors. Then we study the relat…
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 for most of real data applications. Despite of that, the literature on asymmetric covariance functions has been very sparse. In particular, there is some work related to asymmetric covariances on Euclidean spaces, depending on the Euclidean distance. However, for data collected over large portions of planet Earth, the most natural spatial domain is a sphere…
We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal …
Let X Nv(0, Λ) be a normal vector in v dimensions, where Λ is diagonal. With reference to the truncated distribution of X on the interior of a v-dimensional Euclidean ball, we completely prove a variance inequality and a covariance inequality that were recently discussed by F. Palombi and S. Toti [J. Multivariate Anal. 122 (2013) 355-376]. These inequalities ensure the convergence of an algorithm for the reconstruction of Λ only on the basis of the covariance matrix of X truncated to the Euclidean ball. The concept of monotone likelihood ratio is useful in our proofs. Moreover, we also prove a…
The advent of data science has provided an increasing number of challenges with high data complexity. This paper addresses the challenge of space-time data where the spatial domain is not a planar surface, a sphere, or a linear network, but a generalized network (termed a graph with Euclidean edges). Additionally, data are repeatedly measured over different temporal instants. We provide new classes of nonseparable space-time stationary covariance functions where {\em space} can be a generalized network, a Euclidean tree, or a linear network, and where time can be linear or circular (seasonal).…
How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove that the inner product between eigenvectors of the sample and actual covariance matrices decreases proportionally to the respective eigenvalue distance. Our findings imply non-asymptotic concentration bounds for eigenvectors, eigenspaces, and eigenvalues. They also provide condit…
We characterize completely the Gneiting class of space-time covariance functions and give more relaxed conditions on the involved functions. We then show necessary conditions for the construction of compactly supported functions of the Gneiting type. These conditions are very general since they do not depend on the Euclidean norm. Finally, we discuss a general class of positive definite functions, used for multivariate Gaussian random fields. For this class, we show necessary criteria for its generator to be compactly supported.
We consider the Buhmann class of compactly supported radial basis functions, whih includes a wealth of special cases that have been studied in both numerical analysis and spatial statistics literatures. In particular, the celebrated Wu, Wendland and Missing Wendland functions are notable special cases of this class. We propose a very simple difference operator and show the conditions for which the application of it to Buhmann functions preserves positive definiteness on $m$-dimensional Euclidean spaces. We also show that the application of the difference operator increases smoothness at the or…
Given a probability distribution in R^n with general (non-white) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N = N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose the distribution is supported in a centered Euclidean ball of radius \sqrt{n}. We conjecture that the optimal sample size is N = O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the opti…
Brownian motion on manifolds with non-trivial diffusion coefficient can be constructed by stochastic development of Euclidean Brownian motions using the fiber bundle of linear frames. We provide a comprehensive study of paths for such processes that are most probable in the sense of Onsager-Machlup, however with path probability measured on the driving Euclidean processes. We obtain both a full characterization of the resulting family of most probable paths, reduced equation systems for the path dynamics where the effect of curvature is directly identifiable, and explicit equations in special …
We present arguments in favour of the inequalities $var(X_n^2|X \in B_v(ρ)) \le 2λ_n E[X_n^2|X \in B_v(ρ)]$, where $X \sim N_v(0,Λ)$ is a normal vector in $v\ge 1$ dimensions, with zero mean and covariance matrix $Λ= \diag(λ)$, and $B_v(ρ)$ is a centered $v$-dimensional Euclidean ball of square radius $ρ$. Such relations lie at the heart of an iterative algorithm, proposed in ref. [1] to perform a reconstruction of $Λ$ from the covariance matrix of $X$ conditioned to $B_v(ρ)$. In the regime of strong truncation, i.e. for $ρ\lesssim λ_n$, the above inequality is easily proved, whereas it become…
The link between Gaussian random fields and Markov random fields is well established based on a stochastic partial differential equation in Euclidean spaces, where the Matérn covariance functions are essential. However, the Matérn covariance functions are not always positive definite on circles and spheres. In this manuscript, we focus on the extension of this link to circles, and show that the link between Gaussian random fields and Markov random fields on circles is valid based on the circular Matérn covariance function instead. First, we show that this circular Matérn function is the covari…
In nonparametric regression, it is common for the inputs to fall in a restricted subset of Euclidean space. Typical kernel-based methods that do not take into account the intrinsic geometry of the domain across which observations are collected may produce sub-optimal results. In this article, we focus on solving this problem in the context of Gaussian process (GP) models, proposing a new class of Graph Laplacian based GPs (GL-GPs), which learn a covariance that respects the geometry of the input domain. As the heat kernel is intractable computationally, we approximate the covariance using fini…
Space-time covariance modeling under the Lagrangian framework has been especially popular to study atmospheric phenomena in the presence of transport effects, such as prevailing winds or ocean currents, which are incompatible with the assumption of full symmetry. In this work, we assess the dimple problem (Kent et al., 2011) for covariance functions coming from transport phenomena. We work under two important cases: the spatial domain can be either the $d$-dimensional Euclidean space $\mathbb{R}^d$ or the spherical shell of $\mathbb{R}^d$. The choice is relevant for the type of metric chosen t…
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data landscape at all spatial scales, unlike the usual covariance tensor that only quantifies global variation about the mean. Empirical forms of localized covariance previously have been used in data analysis and visualization, but we develop a framework for the systematic treatment of theoretical questions and computational models based on localized covariance. …
This paper addresses the problem of finding parametric constraints that ensure the validity of the multivariate Mat{é}rn covariance for modeling the spatial correlation structure of coregionalized variables defined in an Euclidean space. To date, much attention has been given to the bivariate setting, while the multivariate setting has been explored to a limited extent only. The existing conditions often imply severe restrictions on the upper bounds for the collocated correlation coefficients, which makes the multivariate Mat{é}rn model appealing for the case of weak spatial cross-dependence o…
We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.
The asymptotic concentration of the Fr{é}chet mean of IID random variables on a Rieman-nian manifold was established with a central limit theorem by Bhattacharya \& Patrangenaru (BP-CLT) [6]. This asymptotic result shows that the Fr{é}chet mean behaves almost as the usual Euclidean case for sufficiently concentrated distributions. However, the asymptotic covariance matrix of the empirical mean is modified by the expected Hessian of the squared distance. This Hessian matrix was explicitly computed in [5] for constant curvature spaces in order to relate it to the sectional curvature. Although ex…
We give a characterization of positive definite integrable functions on a product of two Gelfand pairs as an integral of positive definite functions on one of the Gelfand pairs with respect to the Plancherel measure on the dual of the other Gelfand pair. In the very special case where the Gelfand pairs are Euclidean groups and the compact subgroups are reduced to the identity, the characterization is a much cited result in spatio-temporal statistics due to Cressie, Huang and Gneiting. When one of the Gelfand pairs is compact the characterization leads to results about expansions in spherical f…
In this paper we relate the matrix $S_B$ of the second moments of a spherically truncated normal multivariate to its full covariance matrix $Σ$ and present an algorithm to invert the relation and reconstruct $Σ$ from $S_B$. While the eigenvectors of $Σ$ are left invariant by the truncation, its eigenvalues are non-uniformly damped. We show that the eigenvalues of $Σ$ can be reconstructed from their truncated counterparts via a fixed point iteration, whose convergence we prove analytically. The procedure requires the computation of multidimensional Gaussian integrals over a Euclidean ball, for …
Centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be ide…
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 the amplitude coefficient, the Spartan covariance parameters include the rigidity coefficient $η_{1}$ which determines the shape of the covariance function. If $ η_{1} >> 1$ Spartan covariance functions exhibit multiscaling. We also derive a family of radially symmetric, infinitely differentiable Bessel-Lommel covariance functions valid in $\mathbb{R}^{d}, …
So far, the pseudo cross-variogram is primarily used as a tool for the structural analysis of multivariate random fields. Mainly applying recent theoretical results on the pseudo cross-variogram, we use it as a cornerstone in the construction of valid covariance models for multivariate random fields. In particular, we extend known univariate constructions to the multivariate case, and generalize existing multivariate models. Furthermore, we provide a general construction principle for conditionally negative definite matrix-valued kernels, which we use to reinterpret previous modeling proposals…
Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Having covariance functions that allow to specify both local and global properties is certainly on demand. This paper provides a method to find new classes of covariance functions having such properties. We term these models hybrid as they are obtained as scale mixtures of piecewise covariance ker…
The paper proves the Strong Law of Large Numbers for integral functionals of random fields with unboundedly increasing covariances. The case of functional data and increasing domain asymptotics is studied. Conditions to guarantee that the Strong Law of Large Numbers holds true are provided. The considered scenarios include wide classes of non-stationary random fields. The discussion about application to weak and long-range dependent random fields and numerical examples are given.
The construction of valid and flexible cross-covariance functions is a fundamental task for modeling multivariate space-time data arising from climatological and oceanographical phenomena. Indeed, a suitable specification of the covariance structure allows to capture both the space-time dependencies between the observations and the development of accurate predictions. For data observed over large portions of planet Earth it is necessary to take into account the curvature of the planet. Hence the need for random field models defined over spheres across time. In particular, the associated covari…
