Isotropic Covariance Function Research Articles

Revised and wider classes of isotropic space-time covariance functions

AbstractSeveral classes of space-time correlation models have been proposed by various authors in the last years. However, most of these families utilize non negative covariance functions to be adapted to different case studies: indeed, the traditional classes of covariances, such as the Whittle–Matern class and the several families constructed by applying the classical properties are not so flexible to describe covariance functions characterized by negative values. A recent analysis, regarding the difference between two isotropic covariance functions, has underlined that these new families of models are more flexible than the traditional ones because the same models are able to select covariance functions which are always positive in their domain, as well as covariance functions which could be negative in a subset of their field of definition. Moreover, within the same class of models, it is possible to select covariance models which present different behaviours in proximity of the origin. In this paper several families of isotropic space-time covariance functions, among the ones proposed in the literature, have been reviewed in order to enrich the same families including models characterized by negative values in a subset of their domain. Furthermore, the definition of separability has been revised in order to enlarge the classical definition. Apart from the theoretical importance related to the new aspects, these new classes of covariance models are characterized by an extremely simple formalism and can be easily adapted to several case studies.

Estimating the parameters of some common Gaussian random fields with nugget under fixed-domain asymptotics

This article considers parameter estimation for a class of Gaussian random fields on [0,1)d that are observed with measurement error and irregularly spaced design sites. This class comprises Gaussian random fields with suitably smooth mean functions and isotropic powered exponential, Matérn or generalized Wendland covariance functions. Under fixed-domain asymptotics, consistent estimators are proposed for three microergodic parameters, namely the nugget, the smoothness parameter and a parameter related to the coefficient of the principal irregular term of the isotropic covariance function. Upper bounds for the convergence rate of these estimators are established. Simulations are conducted to study the finite sample accuracy of the proposed estimators.

Special classes of isotropic covariance functions

In the literature classical properties concerning the class of covariance functions are well illustrated. A recent analysis has provided the conditions under which the difference between two covariance functions is still a covariance function. In this paper the properties of these new classes of models have been explored; in particular, the present analysis has been given for isotropic covariance functions, because of their importance in many applied areas; moreover, isotropic covariance functions can be considered the starting point to construct anisotropic models. It has been pointed out that these new families of models are more flexible than the traditional ones because the same models, according to the values of their parameters, are able to select covariance functions which are always positive in their domain, as well as covariance functions which could be negative in a subset of their field of definition. Moreover, within the same class of models, it is possible to select covariance models which present a parabolic behaviour near the origin from covariance models which present a linear behaviour in proximity of the origin. Apart from the theoretical importance related to the new aspects presented throughout the paper, it is relevant to underline the practical aspects, since these new classes of isotropic covariance models are characterized by an extremely simple formalism and can be easily adapted to several case studies, hence they result very useful for many practitioners.

Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces

Let p, d be positive integers, with d odd. Let \(\varvec{\phi }:[0,+\infty ) \rightarrow {\mathbb {R}}^{p \times p}\) be the isotropic part of a matrix-valued and isotropic covariance function (a positive semidefinite matrix-valued function) that is defined over the d-dimensional Euclidean space. If \(\varvec{\phi }\) is compactly supported over \([0,\pi ]\), then we show that the restriction of \(\varvec{\phi }\) to \([0,\pi ]\) is the isotropic part of a matrix-valued covariance function defined on a d-dimensional sphere, where isotropy in this case means that the covariance function depends on the geodesic distance. Our result does not need any assumption of continuity for the mapping \(\varvec{\phi }\). Further, when \(\varvec{\phi }\) is continuous, we provide an analytical expression of the d-Schoenberg sequence associated with the compactly-supported covariance on the sphere, which only requires knowledge of the Fourier transform of its isotropic part, and illustrate with the Gauss hypergeometric covariance model, which encompasses the well-known spherical, cubic, Askey and generalized Wendland covariances, and with a hole effect covariance model. Special cases of the results presented in this paper have been provided by other authors in the past decade.

Kryging: geostatistical analysis of large-scale datasets using Krylov subspace methods

Analyzing massive spatial datasets using a Gaussian process model poses computational challenges. This is a problem prevailing heavily in applications such as environmental modeling, ecology, forestry and environmental health. 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 for point-referenced spatial data. “Kryging” combines Kriging and Krylov subspace methods and applies for both observations on regular grid and irregularly spaced observations, and for any Gaussian process with a stationary isotropic (and certain geometrically anisotropic) covariance function, including the popular Matérn covariance family. We make use of the block Toeplitz structure with Toeplitz blocks of the covariance matrix and use fast Fourier transform methods to bypass the computational and memory bottlenecks of approximating log-determinant and matrix-vector products. 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.

A deep look into the dagum family of isotropic covariance functions

AbstractThe Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

Criteria and characterizations for spatially isotropic and temporally symmetric matrix-valued covariance functions

We consider spatial matrix-valued isotropic covariance functions in Euclidean spaces and provide a very short proof of a celebrated characterization result proposed by earlier literature. We then provide a characterization theorem to create a bridge between a class of matrix-valued functions and the class of matrix-valued positive semidefinite functions in finite-dimensional Euclidean spaces. We culminate with criteria of the Pólya type for matrix-valued isotropic covariance functions, and with a generalization of Schlather’s class of multivariate spatial covariance functions. We then challenge the problem of matrix-valued space–time covariance functions, and provide a general class that encompasses all the proposals on the Gneiting nonseparable class provided by earlier literature.

Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

Optimal linear prediction (aka. kriging) of a random field {Z(x)}x∈X indexed by a compact metric space (X,dX) can be obtained if the mean value function m:X→R and the covariance function ϱ:X×X→R of Z are known. We consider the problem of predicting the value of Z(x∗) at some location x∗∈X based on observations at locations {xj}j=1n, which accumulate at x∗ as n→∞ (or, more generally, predicting φ(Z) based on {φj(Z)}j=1n for linear functionals φ,φ1,…,φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m˜,ϱ˜), without any restrictive assumptions on ϱ, ϱ˜ such as stationarity. We, for the first time, provide necessary and sufficient conditions on (m˜,ϱ˜) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on X⊂Rd with Matérn or periodic covariance functions, and on the sphere X= S2 for the case of two isotropic covariance functions.

The [formula omitted]-family of covariance functions: A Matérn analogue for modeling random fields on spheres

The Matérn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural distance between any two locations is the great circle distance. In this setting, the Matérn family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset of precipitable water content over a large portion of the Earth, and show that the model gives more precise predictions of the underlying process at unsampled locations than does the Matérn model using chordal distances.

Spatial estimation methods for mapping corn silage and grain yield monitor data

Harvester-mounted yield monitor systems are increasingly used to document corn (Zea mays L.) yield. The three most commonly used spatial estimation methods to convert point data gathered with yield monitors to regular, grid-based, raster maps include nearest neighbor (NN), inverse distance weighting (IDW) and kriging. Seven spatial estimation methods (NN, IDW using 10, 20, 30 and all data points and kriging with exponential and Matern covariance functions) were evaluated to determine the method that most accurately captures intra-field spatial variability of corn silage and corn grain yield in New York. Yield monitor data from two dairy farms and two grain operations were cleaned using Yield Editor prior to spatial analyses. The dataset included 7–10 years of data per farm for a combined 7484 ha (245 fields) of silage and 6971 ha (253 fields) of grain. Data were split into training (80%) and cross-validation datasets (remaining 20% of the data). Normalized root mean squared error (NRMSE) was used to evaluate the accuracy of the spatial estimation methods. Kriging with the Matern covariance function resulted in the most accurate corn silage and grain yield raster maps both at the farm and field level. There were statistically significant differences in NRMSE between kriging with the Matern isotropic covariance function and all other models for both corn silage and grain, regardless of field size, year when data were obtained or farm that supplied the data. These results are beneficial to ensure accurate and precise spatial mapping of yield products toward optimized corn growth management.

Models for the difference of continuous covariance functions

A linear combination, with negative weights, of two continuous covariance functions has been analyzed by a few authors just for special cases and only in the real domain. However, a covariance is a complex valued function: for this reason, the general problem concerning the difference of two covariance functions in the complex domain needs to be analyzed, while it does not yet seem to have been addressed in the literature; hence, exploring the conditions such that the difference of two covariance functions is again a covariance function can be considered a further property. Therefore, this paper yields a contribution to the theory of correlation, hence the results cannot be restricted to the particular field of application. Starting from the difference of two complex covariance functions defined in one dimensional Euclidean space, wide families of models for the difference of two complex covariance functions can be built in any dimensional space, utilizing some well known properties. In particular, the difference of two real covariance functions has been considered; moreover, the difference between some special isotropic covariance functions has also been analyzed. A detailed analysis of the parameters of the models involved has been proposed; this kind of analysis opens a gate for modeling, in any dimensional space, the correlation structure of a peculiar class of complex valued random fields, as well as the subset of real valued random fields. Some relevant hints about how to construct the subset of real covariance functions characterized by negative values have also been given.

Multivariate isotropic random fields on spheres: Nonparametric Bayesian modeling and Lp fast approximations

We study multivariate Gaussian random fields defined over d-dimensional spheres. First, we provide a nonparametric Bayesian framework for modeling and inference on matrix-valued covariance functions. We determine the support (under the topology of uniform convergence) of the proposed random matrices, which cover the whole class of matrix-valued geodesically isotropic covariance functions on spheres. We provide a thorough inspection of the properties of the proposed model in terms of (a) first moments, (b) posterior distributions, and (c) Lipschitz continuities. We then provide an approximation method for multivariate fields on the sphere for which measures of Lp accuracy are established. Our findings are supported through simulation studies that show the rate of convergence when truncating a spectral expansion of a multivariate random field at a finite order. To illustrate the modeling framework developed in this paper, we consider a bivariate spatial data set of two 2019 NCEP/NCAR Flux Reanalyses.

Isotropic covariance functions on graphs and their edges

We develop parametric classes of covariance functions on linear networks and their extension to graphs with Euclidean edges, that is, graphs with edges viewed as line segments or more general sets with a coordinate system allowing us to consider points on the graph which are vertices or points on an edge. Our covariance functions are defined on the vertices and edge points of these graphs and are isotropic in the sense that they depend only on the geodesic distance or on a new metric called the resistance metric (which extends the classical resistance metric developed in electrical network theory on the vertices of a graph to the continuum of edge points). We discuss the advantages of using the resistance metric in comparison with the geodesic metric as well as the restrictions these metrics impose on the investigated covariance functions. In particular, many of the commonly used isotropic covariance functions in the spatial statistics literature (the power exponential, Matérn, generalized Cauchy and Dagum classes) are shown to be valid with respect to the resistance metric for any graph with Euclidean edges, whilst they are only valid with respect to the geodesic metric in more special cases.

Nonparametric Bayesian Modeling and Estimation of Spatial Correlation Functions for Global Data

We provide a nonparametric spectral approach to the modeling of correlation functions on spheres. The sequence of Schoenberg coefficients and their associated covariance functions are treated as random rather than assuming a parametric form. We propose a stick-breaking representation for the spectrum, and show that such a choice spans the support of the class of geodesically isotropic covariance functions under uniform convergence. Further, we examine the first order properties of such representation, from which geometric properties can be inferred, in terms of Hölder continuity, of the associated Gaussian random field. The properties of the posterior, in terms of existence, uniqueness, and Lipschitz continuity, are then inspected. Our findings are validated with MCMC simulations and illustrated using a global data set on surface temperatures.

Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres

We consider the class Ψd of continuous functions that define isotropic covariance functions in the d-dimensional sphere Sd. We provide a new recurrence formula for the solution of Problem 1 in Gneiting (2013b), solved by Fiedler (2013). In addition, we have improved the current bounds for the curvature at the origin of locally supported covariances (Problem 3 in Gneiting (2013b)), which is of applied interest at least for d=2.

Exploring geometric anisotropy for point-referenced spatial data

Isotropic covariance functions are routinely adopted in specifying models for point-referenced spatial data but carry the limitation that spatial dependence is not directional. Geometric anisotropic covariance functions offer a class of stationary specifications which incorporate directional dependence through spatial range. They have received modest attention in the literature but can provide flexible and readily interpretable directional dependence. Within a Bayesian framework, this paper attempts to illuminate when and how much such models for random effects in geostatistical settings improve out-of-sample predictive performance. We show that geometric anisotropy yields better predictive performance when the residuals strongly depart from isotropy (anisotropy ratio is much greater than one) and sample size is fairly large. Also, improvement is more prominent when the spatial variance is much greater than the pure error variance. The investigation of predictive performance is illustrated through simulation as well as modeling of real data on PM2.5 monitoring stations.

Fast and exact simulation of Gaussian random fields defined on the sphere cross time

We provide a method for fast and exact simulation of Gaussian random fields on the sphere having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over the sphere cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude and latitude.

On the Choice of Momentum Control Variables and Covariance Modeling for Mesoscale Data Assimilation

Abstract For mesoscale variational data assimilation with high-resolution observations, there has been an issue concerning the choice of momentum control variables and related covariance modeling. This paper addresses the theoretical aspect of this issue. First, relationships between background error covariance functions for differently chosen momentum control variables are derived, and different choices of momentum control variables are proven to be theoretically equivalent in the sense that they lead to the same optimally analyzed incremental wind field in the limit of infinitely high spatial resolution provided their error covariance functions satisfy the derived relationships. It is then shown that when the velocity potential χ and streamfunction ψ are used as momentum control variables with their background error autocovariance functions modeled by single-Gaussian functions, the derived velocity autocovariance functions contain significant negative sidelobes. These negative sidelobes can represent background wind error structures associated with baroclinic waves on the synoptic scale but become unrepresentative on the mesoscale. To reduce or remove these negative sidelobes for mesoscale variational data assimilation, Gaussian functions are used with two types of modifications to model the velocity covariance functions in consistency with the assumed homogeneity and isotropy in variational data assimilation. In this case, the random (χ, ψ) background error fields have no classically valid homogeneous and isotropic covariance functions, but generalized (χ, ψ) covariance functions can be derived from the modified velocity covariance functions for choosing (χ, ψ) as momentum control variables. Mathematical properties of generalized covariance functions are explored with physical interpretations. Their important implications are discussed for mesoscale data assimilation.

Robust Gaussian stochastic process emulation

We consider estimation of the parameters of a Gaussian Stochastic Process (GaSP), in the context of emulation (approximation) of computer models for which the outcomes are real-valued scalars. The main focus is on estimation of the GaSP parameters through various generalized maximum likelihood methods, mostly involving finding posterior modes; this is because full Bayesian analysis in computer model emulation is typically prohibitively expensive. The posterior modes that are studied arise from objective priors, such as the reference prior. These priors have been studied in the literature for the situation of an isotropic covariance function or under the assumption of separability in the design of inputs for model runs used in the GaSP construction. In this paper, we consider more general designs (e.g., a Latin Hypercube Design) with a class of commonly used anisotropic correlation functions, which can be written as a product of isotropic correlation functions, each having an unknown range parameter and a fixed roughness parameter. We discuss properties of the objective priors and marginal likelihoods for the parameters of the GaSP and establish the posterior propriety of the GaSP parameters, but our main focus is to demonstrate that certain parameterizations result in more robust estimation of the GaSP parameters than others, and that some parameterizations that are in common use should clearly be avoided. These results are applicable to many frequently used covariance functions, for example, power exponential, Matern, rational quadratic and spherical covariance. We also generalize the results to the GaSP model with a nugget parameter. Both theoretical and numerical evidence is presented concerning the performance of the studied procedures.

Determinantal point process models on the sphere

We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb{S}^{d}$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb{S}^{d}\times\mathbb{S}^{d}$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^{d}$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.