Space-time Covariance Research Articles

Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg\u2013Landau nabla phi Interface Model

We study a continuous-time random walk on {mathbb {Z}}^d in an environment of random conductances taking values in (0,infty ). For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem is then applied to prove a scaling limit result for the space-time covariances in the Ginzburg–Landau nabla phi model. We also show that the associated Gibbs distribution scales to a Gaussian free field. These results apply to convex potentials for which the second derivative may be unbounded.

From BFV to BV and spacetime covariance

The BFV formulation of a given gauge theory is usually significantly easier to obtain than its BV formulation. Based on foundational work by Fisch and Henneaux, Grigoriev and Damgaard introduced simple formulas for obtaining the latter from the former. Since BFV relies on the Hamiltonian version of the gauge theory, however, it does not come as a surprise that in general the resulting BV theory does not exhibit spacetime covariance. We provide an explicit example of this phenomenon in two spacetime dimensions and show how to restore covariance of the BV data by improving the Fisch-Henneaux-Grigoriev-Damgaard procedure with appropriate adaptations of their formulas.

A Unified View of Space–Time Covariance Functions Through Gelfand Pairs

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 functions with positive definite expansion functions, thereby recovering recent results of the author in collaboration with Peron and Porcu. In the special case when the compact Gelfand pair consists of orthogonal groups, the characterization is important in geostatistics and covers a recent result of Porcu and White.

Predicting Morphodynamics for Beach Intertidal Systems in the North Sea: A Space-Time Stochastic Approach

The ability to accurately predict beach morphodynamics is of primary interest for coastal scientists and managers. With this goal in mind, a stochastic model of a sandy macrotidal barred beach is developed that is based on cross-shore elevation profiles. Intertidal elevation was monitored from monthly to annually for 19 years through Real Time Kinematics-GPS (RTK-GPS) and LiDAR surveys, and monthly during two years with an RTK-GPS. In addition, during two campaigns of about two weeks, intensive surveys on a daily basis were performed with an RTK-GPS on a different set of profiles. Based on the measurements, space and time variograms are constructed in order to assess the spatial and temporal dependencies of these elevations. A separable space-time covariance model is then built from them in order to generate a large number of plausible future profiles at arbitrary time instants t+τ, starting from observed profiles at time instants t. For each simulation, the total displaced sand volume is computed and a distribution is obtained. The mean of this distribution is in good agreement with the total displaced sand volume measured on the profiles, provided that they are lower than 45 m3/m. The time variogram also shows that 90% of maximum variability is reached for a time interval τ of three years. These results demonstrate how the temporal evolution of an integrated property, like the total displaced sand volume, can be estimated over time. This suggests that a similar stochastic approach could be useful for estimating other properties as long as one is able to capture the stochastic space-time variability of the underlying processes.

30 Years of space–time covariance functions

AbstractIn this article, we provide a comprehensive review of space–time covariance functions. As for the spatial domain, we focus on either the d‐dimensional Euclidean space or on the unit d‐dimensional sphere. We start by providing background information about (spatial) covariance functions and their properties along with different types of covariance functions. While we focus primarily on Gaussian processes, many of the results are independent of the underlying distribution, as the covariance only depends on second‐moment relationships. We discuss properties of space–time covariance functions along with the relevant results associated with spectral representations. Special attention is given to the Gneiting class of covariance functions, which has been especially popular in space–time geostatistical modeling. We then discuss some techniques that are useful for constructing new classes of space–time covariance functions. Separate treatment is reserved for spectral models, as well as to what are termed models with special features. We also discuss the problem of estimation of parametric classes of space–time covariance functions. An outlook concludes the paper.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

Soft charges from the geometry of field space

Infinite sets of asymptotic soft-charges were recently shown to be related to new symmetries of the S-matrix, spurring a large amount of research on this and related questions. Notwithstanding, the raison-d’être of these soft-charges rests on less firm ground, insofar as their known derivations through generalized Noether procedures tend to rely on the fixing of (gauge-breaking) boundary conditions rather than on manifestly gauge- invariant computations. In this article, we show that a geometrical framework anchored in the space of field configurations singles out the known leading-order soft charges in gauge theories. Our framework unifies the treatment of finite and infinite regions, and thus it explains why the infinite enhancement of the symmetry group is a property of asymptotic null infinity and should not be expected to hold within finite regions, where at most a finite number of physical charges — corresponding to the reducibility parameters of the quasi-local field configuration — is singled out. As a bonus, our formalism also suggests a simple proposal for the origin of magnetic-type charges at asymptotic infinity based on spacetime (Lorentz) covariance rather than electromagnetic duality.

Covatest: An R Package for Selecting a Class of Space-Time Covariance Functions

Although a very rich list of classes of space-time covariance functions exists, specific tools for selecting the appropriate class for a given data set are needed. Thus, the main topic of this paper is to present the new R package, covatest, which can be used for testing some characteristics of a covariance function, such as symmetry, separability and type of non-separability, as well as for testing the adequacy of some classes of space-time covariance models. These last aspects can be relevant for choosing a suitable class of covariance models. The proposed results have been applied to an environmental case study.

Space–Time Kriging of Precipitation: Modeling the Large-Scale Variation with Model GAMLSS

Knowing the dynamics of spatial–temporal precipitation distribution is of vital significance for the management of water resources, in highlight, in the northeast region of Brazil (NEB). Several models of large-scale precipitation variability are based on the normal distribution, not taking into consideration the excess of null observations that are prevalent in the daily or even monthly precipitation information of the region under study. This research proposes a novel way of modeling the trend component by using an inflated gamma distribution of zeros. The residuals of this regression are generally space–time dependent and have been modeled by a space–time covariance function. The findings show that the new techniques have provided reliable and precise precipitation estimates, exceeding the techniques used previously. The modeling provided estimates of precipitation in nonsampled locations and unobserved periods, thus serving as a tool to assist the government in improving water management, anticipating society’s needs and preventing water crises.

Half-spectral analysis of spatial-temporal data: The case study of Iranian daily wind speed data

In this paper, we first study the theory of the spatial-temporal half spectral modeling and describe some properties of recently proposed half spectral models. Next, we propose an estimation method for the estimation of spatial-temporal covariance functions in the half-spectral setting. To assess the performance of the proposed half-spectral models, we conduct two simulations, in which we compare the proposed fitting approach with respect to the other classical estimation methods. The proposed methods have great success in fitting parametric space-time covariance functions specifically for massive data sets. Finally, we apply the proposed methods for a real daily wind speed data in Iran.

Foodwebs based on unreliable foundations: spatiotemporal masting merged with consumer movement, storage, and diet

AbstractMast‐fruiting trees represent a pulsed resource that both supports and destabilizes consumer populations. Whereas a reliable resource is abundant on average and with limited variation in time and space, masting is volatile and localized, and that variability ramifies throughout food‐webs. Theory is developed to evaluate how the space–time structure of masting interacts with consumers who exploit alternative hosts, forage widely in space, and store reserves in time. We derive the space–time–species covariance in resource supply and combine it with the space–time–diet breadth of consumers, or ambit. Direct connection to data is made possible with Mast Inference and Forecasting (MASTIF), a state‐space autoregressive model that fits seed‐trap and canopy observations and predicts resource availability within the canopy and on the forest floor with full uncertainty. A resource score can be assigned to each consumer–habitat combination that integrates the benefits of a high mean supply weighed against the variance cost. As the consumer ambit increases, the volatility of an unreliable resource shifts from a variance cost to a mean benefit. Consumers foraging in the canopy (arboreal arthropods and rodents, song birds) experience space‐time covariance between host trees. Consumers on the forest floor (seed and damping‐off fungi, arthropods, rodents, ground‐nesting birds, mammals) experience instead a redistribution of that covariance by dispersal. For consumers lacking mobility, demographic storage in the form of episodic birth cohorts following mast years is important for population persistence. Consumers additionally compensate volatility with diet breadth. Depending on the dominant masting strategies of host tree species in the diet, habitats differentially limit consumers depending on the misalignment between consumer ambit and spatiotemporal covariance of hosts. The impact of adding or subtracting a diet item can be gauged with the standard error (SE) rule or the benefit of an added diet item balanced against the variance cost, both of which depend on the existing diet, the abundance of the new host, and the consumer's foraging ambit. Results rank habitats by their capacities to support wildlife and other consumers from a resource perspective. Results are connected directly to data, with full uncertainty, by MASTIF.

Space-time dynamics of electricity markets incentivize technology decentralization

We study economic incentives provided by space-time dynamics of day-ahead and real-time electricity markets. Specifically, we seek to analyze to what extent such dynamics promote decentralization of technologies for generation, consumption, and storage (which is essential to obtain a more flexible power grid). Incentives for decentralization are also of relevance given recent interest in the deployment of small-scale modular technologies (e.g., modular ammonia and biogas production systems). Our analysis is based on an asset placement problem that seeks to find optimal locations for generators and loads in the network that minimize profit risk. We show that an unconstrained version of this problem can be cast as an eigenvalue problem. Under this representation, optimal network allocations are eigenvectors of the space-time price covariance matrix while the eigenvalues are the associated profit variances. We also construct a more sophisticated placement formulation that captures different risk metrics and constraints on types of technologies to systematically analyze trade-offs in expected profit and risk. Our analysis reveals that space-time market dynamics provide significant incentives for decentralization and strategic asset placement but that full mitigation of risk is only possible through simultaneous investment in generation and loads (which can be achieved using batteries or microgrids).

Stochastic Parameterization of Subgrid-Scale Velocity Enhancement of Sea Surface Fluxes

Abstract Subgrid-scale (SGS) velocity variations result in gridscale sea surface flux enhancements that must be parameterized in weather and climate models. Traditional parameterizations are deterministic in that they assign a unique value of the SGS velocity flux enhancement to any given configuration of the resolved state. In this study, we assess the statistics of SGS velocity flux enhancement over a range of averaging scales (as a proxy for varying model resolution) through systematic coarse-graining of a convection-permitting atmospheric model simulation over the Indian Ocean and west Pacific warm pool. Conditioning the statistics of the SGS velocity flux enhancement on 1) the fluxes associated with the resolved winds and 2) the precipitation rate, we find that the lack of a separation between “resolved” and “unresolved” scales results in a distribution of flux enhancements for each configuration of the resolved state. That is, the SGS velocity flux enhancement should be represented stochastically rather than deterministically. The spatial and temporal statistics of the SGS velocity flux enhancement are investigated by using basic descriptive statistics and through a fit to an anisotropic space–time covariance structure. Potential spatial inhomogeneities of the statistics of the SGS velocity flux enhancement are investigated through regional analysis, although because of the relatively short duration of the simulation (9 days) distinguishing true inhomogeneity from sampling variability is difficult. Perspectives for the implementation of such a stochastic parameterization in weather and climate models are discussed.

Markov Chain Monte Carlo estimation of spatial dynamic panel models for large samples

Focus is on efficient estimation of a dynamic space–time panel data model that incorporates spatial dependence, temporal dependence, as well as space–time covariance and can be implemented where there are a large number of spatial units and time periods. Quasi-maximum likelihood (QML) estimation in cases involving large samples poses computational challenges because optimizing the (log) likelihood requires: (1) evaluating the log-determinant of a large matrix that appears in the likelihood, (2) imposing stability restrictions on parameters reflecting space–time dynamics, and (3) simulations to produce an empirical distribution of the partial derivatives used to interpret model estimates that require numerous inversions of large matrices.A Markov Chain Monte Carlo (MCMC) estimation procedure is set forth that produces estimates equivalent to those from QML along with a Monte Carlo integrated estimate of the log-marginal likelihood, useful for model comparison. An applied illustration is based on over six million observations. The MCMC estimation procedure uses: (1) a Taylor series approximation to the log-determinant based on traces of matrix products calculated prior to MCMC sampling, (2) block sampling of the spatiotemporal parameters, which allows imposition of the stability restrictions, and (3) a Metropolis–Hastings guided Monte Carlo integration of the log-marginal likelihood. In addition, an efficient approach to simulations needed to produce the empirical distribution of the partial derivatives for model interpretation is set forth.

Covariance functions on spheres cross time: Beyond spatial isotropy and temporal stationarity

Spectral representations uniquely define the covariance functions associated to random fields defined over spheres or spheres cross time. Covariance functions on spheres cross time are usually modeled under the assumptions of either spatial isotropy or axial symmetry, and the assumption of temporal stationarity. This paper goes beyond these assumptions. In particular, we consider the problem of spatially anisotropic covariance functions on spheres. The crux of our criterion is to escape from the addition theorem for spherical harmonics. We also challenge the problem of temporal nonstationarity in nonseparable space–time covariance functions, where space is the n-dimensional sphere.

Geostatistical stationary space-time covariance functions modeling of Yellow Sigatoka progress in banana crop

Banana production is affected by Yellow Sigatoka, one of the causes of leaf lesions, which causes the reduction of the photosynthetic area of the plant and, consequently, the quality of the fruit and the production. The objective of this study was to analyze using geostatistics and comparing separable and non-separable spatio-temporal covariance models with different adjustment methods. The experiment was carried out in a banana plantation of the Prata-Ana variety, which presented high severity of the disease, without any control measures, which allowed the study of behavior under natural conditions. The Separable Doubly Exponential and the non-separable model of Gneiting were tested with the Weight Least Squares (WLS), Restricted Maximum Likelihood (REML) and Likelihood Pairwise estimation methods. The Gneiting model, WLS curve-fitting methods for estimatives space-time covariance structure, allowed to reduce the uncertainties of the spatial and temporal prediction of the disease, as well as to characterize the spatio-temporal pattern of the disease.

Global fluctuations for 1D log-gas dynamics. Covariance kernel and support

We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, \[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \] with $\beta >1$, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta $-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty $ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [39] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho _t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.

Towards a complete picture of stationary covariance functions on spheres cross time

With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides results for stationary covariance functions of random fields defined over $d$-dimensional spheres cross time. Specifically, we provide a bridge between the characterization in Berg and Porcu (2017) for covariance functions on spheres cross time and Gneiting’s lemma (Gneiting, 2002) that deals with planar surfaces. We then prove that there is a valid class of covariance functions similar in form to the Gneiting class of space-time covariance functions (Gneiting, 2002) that replaces the squared Euclidean distance with the great circle distance. Notably, the provided class is shown to be positive definite on every $d$-dimensional sphere cross time, while the Gneiting class is positive definite over $\mathbb{R} ^{d}\times \mathbb{R} $ for fixed $d$ only. In this context, we illustrate the value of our adapted Gneiting class by comparing examples from this class to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. For these datasets, we show that examples from our covariance class have better predictive performance than competing models.

Covariant information theory and emergent gravity

Informational dependence between statistical or quantum subsystems can be described with Fisher information matrix or Fubini-Study metric obtained from variations/shifts of the sample/configuration space coordinates. Using these (noncovariant) objects as macroscopic constraints, we consider statistical ensembles over the space of classical probability distributions (i.e. in statistical space) or quantum wave functions (i.e. in Hilbert space). The ensembles are covariantized using dual field theories with either complex scalar field (identified with complex wave functions) or real scalar field (identified with square roots of probabilities). We construct space–time ensembles for which an approximate Schrodinger dynamics is satisfied by the dual field (which we call infoton due to its informational origin) and argue that a full space–time covariance on the field theory side is dual to local computations on the information theory side. We define a fully covariant information-computation tensor and show that it must satisfy certain conservation equations. Then we switch to a thermodynamic description of the quantum/statistical systems and argue that the (inverse of) space–time metric tensor is a conjugate thermodynamic variable to the ensemble-averaged information-computation tensor. In (local) equilibrium, the entropy production vanishes, and the metric is not dynamical, but away from the equilibrium the entropy production gives rise to an emergent dynamics of the metric. This dynamics can be described approximately by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium, these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry, the Onsager tensor is expected to be symmetric. We show that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein–Hilbert term. This proves that general relativity is equivalent to a theory of nonequilibrium (thermo)dynamics of the metric, but the theory is expected to break down far away from equilibrium where the symmetries of the Onsager tensor are to be broken.

Modeling Temporally Evolving and Spatially Globally Dependent Data

SummaryThe last decades have seen an unprecedented increase in the availability of data sets that are inherently global and temporally evolving, from remotely sensed networks to climate model ensembles. This paper provides an overview of statistical modeling techniques for space–time processes, where space is the sphere representing our planet. In particular, we make a distintion between (a) second order‐based approaches and (b) practical approaches to modeling temporally evolving global processes. The former approaches are based on the specification of a class of space–time covariance functions, with space being the two‐dimensional sphere. The latter are based on explicit description of the dynamics of the space–time process, that is, by specifying its evolution as a function of its past history with added spatially dependent noise.We focus primarily on approach (a), for which the literature has been sparse. We provide new models of space–time covariance functions for random fields defined on spheres cross time. Practical approaches (b) are also discussed, with special emphasis on models built directly on the sphere, without projecting spherical coordinates onto the plane.We present a case study focused on the analysis of air pollution from the 2015 wildfires in Equatorial Asia, an event that was classified as the year's worst environmental disaster. The paper finishes with a list of the main theoretical and applied research problems in the area, where we expect the statistical community to engage over the next decade.

A statistical modeling approach for spatio-temporal degradation data

ABSTRACTThis article investigates the modeling of a new type of degradation data: spatio-temporal degradation data collected from a spatial domain over time. Like existing stochastic degradation models, a random field is constructed to describe the spatio-temporal degradation process. We model the degradation process as an additive superposition of two stochastic components: a dynamic spatial degradation generation process and a spatio-temporal propagation process. Some common challenges are addressed, including the spatial heterogeneity of the degradation process, spatial propagation of degradation to neighboring areas, anisotropic and space-time non-separable covariance structures associated with a complex spatio-temporal degradation process, and the computational issues related to parameter estimation and simulation. When spatial dependence is ignored, we show that the proposed spatio-temporal degradation model incorporates some existing purely time-dependent degradation models as its special cases. We also show the connection, under special conditions, between the proposed statistical model and a class of physical-degradation processes given by stochastic partial differential equations. Numerical examples are presented to illustrate modeling approach, parameter estimation, model validation, and applications.