Space-time Covariance Research Articles

Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces

AbstractGaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).

Space-time covariance models on networks

Space-time covariance models on networks

On the frequency domain composite likelihood methods for estimating space-time covariance functions for large datasets

Covariance function plays very important roles in modeling and in prediction of spatial data. For instance, many statistical inferences such as maximum likelihood estimation and best linear unbiased prediction involve covariance matrices which must be inverted and this evaluation is slow when the number of observations is large, and the complexity increases even further in a space-time context. In this paper, we propose approximation methods based on three types of frequency domain composite likelihoods for the estimation of spatial-temporal covariance functions: paired, differenced, and conditional composite likelihoods to overcome this problem. The proposed likelihoods involve computationally efficient and matrix-free Gaussian likelihood approximations that have been successful in fitting parametric space-time covariance functions for massive datasets. We conduct a simulation study to evaluate the performance of the proposed estimation methods from statistical and computational viewpoints. The frequency domain spectral composite likelihood estimates demonstrate better performance compared to the observation domain composite likelihoods. Furthermore, our numerical examples show that paired and conditional frequency domain composite likelihoods work better than differenced frequency domain composite likelihood from statistical point of view.

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.

Stationary nonseparable space-time covariance functions on networks

Abstract 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 generalised network (termed a graph with Euclidean edges). Additionally, data are repeatedly measured over different temporal instants. We provide new classes of stationary nonseparable space-time covariance functions where space can be a generalised network, a Euclidean tree, or a linear network, and where time can be linear or circular (seasonal). Because the construction principles are technical, we focus on illustrations that guide the reader through the construction of statistically interpretable examples. A simulation study demonstrates that the correct model can be recovered when compared to misspecified models. In addition, our simulation studies show that we effectively recover simulation parameters. In our data analysis, we consider a traffic accident dataset that shows improved model performance based on covariance specifications and network-based metrics.

Characterization theorems for pseudo cross-variograms

AbstractPseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.

Extending the Gneiting class for modeling spatially isotropic and temporally symmetric vector random fields

We provide new classes of nonseparable univariate and multivariate space-time covariance functions that extend the Gneiting class. In particular, we prove that the spatial generator of the Gneiting class does not need to be completely monotone and can be replaced with the radial profile of a continuous positive semidefinite function in a finite dimensional Euclidean space, and we weaken the assumptions on the temporal margins so as to include more general behaviors than that of the original Gneiting construction. Our findings allow for covariance functions that are compactly-supported in space and/or nonmonotone in time, i.e., they offer versatility to model complex features commonly observed on data indexed with space-time coordinates.

Speckle Space–Time Covariance in High-contrast Imaging

We introduce a new framework for point-spread function subtraction based on the spatiotemporal variation of speckle noise in high-contrast imaging data where the sampling timescale is faster than the speckle evolution timescale. One way that space–time covariance arises in the pupil is as atmospheric layers translate across the telescope aperture and create small, time-varying perturbations in the phase of the incoming wavefront. The propagation of this field to the focal plane preserves some of that space–time covariance. To utilize this covariance, our new approach uses a Karhunen–Loève transform on an image sequence, as opposed to a set of single reference images as in previous applications of Karhunen–Loève Image Processing (KLIP) for high-contrast imaging. With the recent development of photon-counting detectors, such as microwave kinetic inductance detectors, this technique now has the potential to improve contrast when used as a post-processing step. Preliminary testing on simulated data shows this technique can improve contrast by at least 10%–20% from the original image, with significant potential for further improvement. For certain choices of parameters, this algorithm may provide larger contrast gains than spatial-only KLIP.

Structured Matrix Approximations via Tensor Decompositions

We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric block Toeplitz). The goal is to uncover {\it additional latent structure} that will in turn lead to computationally efficient algorithms when the new structured matrix approximations are employed in the place of the original operator. Our approach has three steps: map the structured matrix to tensors, use tensor compression algorithms, and map the compressed tensors back to obtain two different matrix representations -- sum of Kronecker products and block low-rank format. The use of tensor decompositions enables us to uncover latent structure in the problem and leads to compressed representations of the original matrix that can be used efficiently in applications. The resulting matrix approximations are memory efficient, easy to compute with, and preserve the error that is due to the tensor compression in the Frobenius norm. Our framework is quite general. We illustrate the ability of our method to uncover block-low-rank format on structured matrices from two applications: system identification, space-time covariance matrices. In addition, we demonstrate that our approach can uncover sum of structured Kronecker products structure on several matrices from the SuiteSparse collection. Finally, we show that our framework is broad enough to encompass and improve on other related results from the literature, as we illustrate with the approximation of a three-dimensional blurring operator.

Multichannel Sea Clutter Measurement and Space-Time Characteristics Analysis with L-Band Shore-Based Radar

In order to study the space-time characteristics of sea clutter, the sea clutter is always measured by the airborne multichannel radar; however, the sea clutter shows the heterogeneity between range gates, which means the space-time covariance matrix’s correspondence to the single range gate cannot be estimated accurately. Meanwhile, the measurement of the sea clutter data by the airborne radar is usually affected by the motion of the platform, which makes the analysis results unrepresentative of the space-time characteristics of the pure sea clutter. In this paper, a sea clutter measurement method based on L-band shore-based multichannel radar is proposed, where the transmit sub-array periodically moves with the pulse repetition period to obtain multiple sets of coherent processing interval pulses for each range gate. This measurement method can exclude the influences of the moving platform. Moreover, a sea clutter space-time signal model of the single range gate is proposed, and the model is used to simulate three-dimensional sea clutter data with space-time coupling characteristics. With verification of the measured and simulated data, it can be seen that the data composed of single range gate and multiple coherent processing interval pulses can accurately estimate the space-time covariance matrix corresponding to this single range gate. Furthermore, the space-time characteristics are analyzed based on the measured data. The results show that the eigenvalue spectrum and the spread width of space-time power spectrum are influenced by the backscattering coefficient of sea clutter and the speed of sea surface motion. In comparison, the decorrelation effect caused by the backscattering coefficient of sea clutter is stronger than that caused by the speed of the surface motion. The proposed method is helpful for guiding multichannel sea clutter measurement and the analysis results are of great significance to the clutter suppression algorithms of the marine multichannel radar.

Fully nonseparable Gneiting covariance functions for multivariate space–time data

We broaden the well-known Gneiting class of space–time covariance functions by introducing a very general parametric class of fully nonseparable direct and cross-covariance functions for multivariate random fields, where each component has a spatial covariance function from the Matérn family with its own smoothness and scale parameters and, unlike most of currently available models, its own correlation function in time. We present sufficient conditions that result in valid models with varying degrees of complexity and we discuss the parameterization of those. Continuous-in-space and discrete-in-time simulation algorithms are also given, which are not limited by the number of target spatial coordinates and allow tens of thousands of time coordinates. The application of the proposed model is illustrated on a weather trivariate dataset over France. Our new model yields better fitting and better predictive scores in time compared to a more parsimonious model with a common temporal correlation function.

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.

A Parametric Approach to Space-Time Adaptive Processing in Bistatic Radar Systems

Space-time adaptive processing (STAP) is an important airborne radar technique used to improve target detection in environments of clutter and jammers. Effective STAP implementations are dependent on an accurate estimate of the space-time covariance matrix, which characterizes noise and interference in the radar signal. Inside-looking monostatic radar systems, the estimate based on secondary radar observations is rather straight forward as all the samples in secondary data can be argued to be from a single distribution, and the sample covariance can be used as an estimate of the space-time covariance matrix. However, in many other radar configurations, the vital underlying STAP training assumption that secondary data are identically distributed is violated, which implies that detection performance can be significantly degraded. This article develops a new method that can be used when secondary data do not share a common distribution due to geometry-induced range dependencies. This phenomenon is of particular concern in bistatic radar systems. We propose a model-based approach, where the distribution of noise and clutter for each range bin is parameterized by a set of scenario parameters. Using secondary data, the scenario parameters are estimated by maximizing the likelihood function. Based on the estimated scenario parameters, the STAP covariance estimate is formed for the cell under test. The presented method is compared with other state-of-the-art methods for bistatic radar STAP via numerical simulations. The simulations indicate that the presented method, with a proper initialization, yields an estimate of the STAP covariance matrix that significantly increases the signal-to-interference-plus-noise ratio compared to the other investigated methods.

Nested covariance functions on graphs with Euclidean edges cross time

Covariance functions over generalized networks have been explored to a very limited extent. We consider nested spatial or space-time covariance models, where space is a generalized network, and where time can be linear (the real line) or circular. We show sufficient conditions allowing preservation of positive semidefiniteness when at least one of the weights involved in the linear combination is negative. Several examples illustrate our findings. In particular, we show nested constructions for Euclidean trees with a finite number of leaves involving basic covariance functions with different scale parameters or different compact supports. We also provide criteria that allow one to build space-time models through half spectral modeling on graphs cross linear or circular time.

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

We introduce a method to construct general multivariate positive definite kernels on a nonempty set XX that employs a prescribed bounded completely monotone function and special multivariate functions on XX. The method is consistent with a generalized version of Aitken's integral formula for Gaussians. In the case in which XX is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate version of the well-established Gneiting's model for constructing space-time covariances commonly highly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.

Moment estimates for some renormalized parabolic Anderson models

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: ∂tut(x)=12Δut(x)+ut(x)W˙t(x), for t∈R+ and x∈Rd, where W˙t(x) is a Gaussian noise whose space time covariance function is singular. In this rough context we shall give some information about the moments of ut(x) when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension 1+1, where the disorder is given by the occupation field of a Poisson system of independent random walks on Z. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel.In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establish an L1-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments.

Space-Time Covariance Structures and Models

In recent years, interest has grown in modeling spatio-temporal data generated from monitoring networks, satellite imaging, and climate models. Under Gaussianity, the covariance function is core to spatio-temporal modeling, inference, and prediction. In this article, we review the various space-time covariance structures in which simplified assumptions, such as separability and full symmetry, are made to facilitate computation, and associated tests intended to validate these structures. We also review recent developments on constructing space-time covariance models, which can be separable or nonseparable, fully symmetric or asymmetric, stationary or nonstationary, univariate or multivariate, and in Euclidean spaces or on the sphere. We visualize some of the structures and models with visuanimations. Finally, we discuss inference for fitting space-time covariance models and describe a case study based on a new wind-speed data set.

Nonstationary space–time covariance functions induced by dynamical systems

AbstractThis article provides a novel approach to nonstationarity by considering a bridge between differential equations and spatial fields. We consider the dynamical transformation of a given spatial process undergoing the action of a temporal flow of space diffeomorphisms. Such dynamical deformations are shown to be connected to certain classes of ordinary and partial differential equations. The natural question arises of how such dynamical diffeomorphisms convert the original spatial covariance function, specifically if the original covariance is spatially stationary or isotropic. We first challenge this question from a general perspective, and then turn into the special cases of both d‐dimensional Euclidean spaces, and hyperspheres. Several examples of dynamical diffeomorphisms defined in these spaces are given and some emphasis has been put on the stationary reducibility problem. We provide a simple illustration to show the performance of the maximum likelihood estimation of the parameters of a family of dynamically deformed covariance functions.

Statistical Modeling for Spatio-Temporal Data From Stochastic Convection-Diffusion Processes

This article proposes a physical-statistical modeling approach for spatio-temporal data arising from a class of stochastic convection-diffusion processes. Such processes are widely found in scientific and engineering applications where fundamental physics imposes critical constraints on how data can be modeled and how models should be interpreted. The idea of spectrum decomposition is employed to approximate a physical spatio-temporal process by the linear combination of spatial basis functions and a multivariate random process of spectral coefficients. Unlike existing approaches assuming spatially and temporally invariant convection-diffusion, this article considers a more general scenario with spatially varying convection-diffusion and nonzero-mean source-sink. As a result, the temporal dynamics of spectral coefficients is coupled with each other, which can be interpreted as the nonlinear energy redistribution across multiple scales from the perspective of physics. Because of the spatially varying convection-diffusion, the space-time covariance is nonstationary in space. The theoretical results are integrated into a hierarchical dynamical spatio-temporal model. The connection is established between the proposed model and the existing models based on integro-difference equations. Computational efficiency and scalability are also investigated to make the proposed approach practical. The advantages of the proposed methodology are demonstrated by numerical examples, a case study, and comprehensive comparison studies. Computer code is available on GitHub. Supplementary materials for this article are available online.