We consider a stationary spatio-temporal random process and assume that we have a sample. By defining a sequence of discrete Fourier transforms at canonical frequencies at each location, and using these complex valued random varables as observed sample, we obtain expressions for the spatio-temporal covariance functions and the spectral density functions of the spatio-temporal random processes. These spectra correspond to non separable class of random processes. The spatio-temporal covariance functions, obtained here are functions of the spatial distance and the temporal frequency and are similar to Matern class. These are in terms of modified Bessel functions of the second kind. and the parameters are in terms of the second order spectral density functions of the random proces and the spatial distances. We consider the estimation of the parameters of the covariance function and also briefly mention their asymptotic properties. The estimation of the entire data at a known location, and also the estimation of a value given the above sample is also considered. The predictors are obtained using the vectors of Discrete Fourier Transforms. We also describe a statistical test for testing the independence of the m spatial time series (testing for spatial independence) using the Finite Fourier Transforms and it is based on the likelihood ratio test of complex valued random variables The methods are illustrated with real data.
Keywords: Discrete Fourier Transforms, Covariance functions, Spectral density functions, Space-Time Processses, Prediction(kriging) Laplacian operators, Frequency Variogram, Tests for independence, Whittle likelihood.
In this paper, we present and compare functional and spatio-temporal (Sp.T.) kriging approaches to predict spatial functional random processes (which can also be viewed as Sp.T. random processes). Comparisons with respect to computational time and prediction performance via functional cross-validation is evaluated, mainly through a simulation study but also on two real data sets. We restrict comparisons to Sp.T. kriging versus ordinary kriging for functional data (OKFD), since the more flexible functional kriging approaches, pointwise functional kriging (PWFK) and functional kriging total mode…
Environmental monitoring is a task that requires to surrogate system-wide information with limited sensor readings. Under the proximity principle, an environmental monitoring system can be based on the virtual sensing logic and then rely on distance-based prediction methods, such as $k$-nearest-neighbors, inverse distance weighted regression and spatio-temporal kriging. The last one is cumbersome with large datasets, but we show that a suitable separability assumption reduces its computational cost to an extent broader than considered insofar. Only spatial interpolation needs to be performed i…
Time series forecasting and spatiotemporal kriging are the two most important tasks in spatiotemporal data analysis. Recent research on graph neural networks has made substantial progress in time series forecasting, while little attention has been paid to the kriging problem -- recovering signals for unsampled locations/sensors. Most existing scalable kriging methods (e.g., matrix/tensor completion) are transductive, and thus full retraining is required when we have a new sensor to interpolate. In this paper, we develop an Inductive Graph Neural Network Kriging (IGNNK) model to recover data fo…
Gaussian random fields (GRF) are a fundamental stochastic model for spatiotemporal data analysis. An essential ingredient of GRF is the covariance function that characterizes the joint Gaussian distribution of the field. Commonly used covariance functions give rise to fully dense and unstructured covariance matrices, for which required calculations are notoriously expensive to carry out for large data. In this work, we propose a construction of covariance functions that result in matrices with a hierarchical structure. Empowered by matrix algorithms that scale linearly with the matrix dimensio…
This paper presents a kriging method for spatial prediction of temporal intensity functions, for situations where a temporal point process is observed at different spatial locations. Assuming that several replications of the processes are available at the spatial sites, this method avoids assumptions like isotropy, which are not valid in many applications. As part of the derivations, new nonparametric estimators for the mean and covariance functions of temporal point processes are introduced, and their properties are studied theoretically and by simulation. The method is applied to the analysi…
There is growing evidence in the epidemiologic literature of the relationship between air pollution and adverse health outcomes. Prediction of individual air pollution exposure in the Environmental Protection Agency (EPA) funded Multi-Ethnic Study of Atheroscelerosis and Air Pollution (MESA Air) study relies on a flexible spatio-temporal prediction model that integrates land-use regression with kriging to account for spatial dependence in pollutant concentrations. Temporal variability is captured using temporal trends estimated via modified singular value decomposition and temporally varying s…
Circular data arise in many areas of application. Recently, there has been interest in looking at circular data collected separately over time and over space. Here, we extend some of this work to the spatio-temporal setting, introducing space-time dependence. We accommodate covariates, implement full kriging and forecasting, and also allow for a nugget which can be time dependent. We work within a Bayesian framework, introducing suitable latent variables to facilitate Markov chain Monte Carlo (MCMC) model fitting. The Bayesian framework enables us to implement full inference, obtaining predict…
This paper considers the surrogate modeling of a complex numerical code in a multifidelity framework when the code output is a time series. Using an experimental design of the low-and high-fidelity code levels, an original Gaussian process regression method is proposed. The code output is expanded on a basis built from the experimental design. The first coefficients of the expansion of the code output are processed by a co-kriging approach. The last coefficients are collectively processed by a kriging approach with covariance tensorization. The resulting surrogate model taking into account the…
This paper deals with several issues related to the pointwise consistency of the kriging predictor when the mean and the covariance functions are known. These questions are of general importance in the context of computer experiments. The analysis is based on the properties of approximations in reproducing kernel Hilbert spaces. We fix an erroneous claim of Yakowitz and Szidarovszky (J. Multivariate Analysis, 1985) that the kriging predictor is pointwise consistent for all continuous sample paths under some assumptions.
Spatio-temporal covariances are important for describing the spatio-temporal variability of underlying random processes in geostatistical data. For second-order stationary processes, there exist subclasses of covariance functions that assume a simpler spatio-temporal dependence structure with separability and full symmetry. However, it is challenging to visualize and assess separability and full symmetry from spatio-temporal observations. In this work, we propose a functional data analysis approach that constructs test functions using the cross-covariances from time series observed at each pai…
We discuss the experiences and results of the AppStatUZH team's participation in the comprehensive and unbiased comparison of different spatial approximations conducted in the Competition for Spatial Statistics for Large Datasets. In each of the different sub-competitions, we estimated parameters of the covariance model based on a likelihood function and predicted missing observations with simple kriging. We approximated the covariance model either with covariance tapering or a compactly supported Wendland covariance function.
Understanding and predicting environmental phenomena often requires the construction of spatio-temporal statistical models, which are typically Gaussian processes. A common assumption made on Gaussian processes is that of covariance stationarity, which is unrealistic in many geophysical applications. In this article, we introduce a deep-learning-inspired approach to construct descriptive nonstationary spatio-temporal models by modeling stationary processes on warped spatio-temporal domains. The warping functions we use are constructed using several simple injective warping units which, when co…
We propose a new approach to represent nonparametrically the linear dependence structure of a spatio-temporal process in terms of latent common factors. Though it is formally similar to the existing reduced rank approximation methods (Section 7.1.3 of Cressie and Wikle, 2011), the fundamental difference is that the low-dimensional structure is completely unknown in our setting, which is learned from the data collected irregularly over space but regularly over time. Furthermore a graph Laplacian is incorporated in the learning in order to take the advantage of the continuity over space, and a n…
Determinantal point processes are models for regular spatial point patterns, with appealing probabilistic properties. We present their spatio-temporal counterparts and give examples of these models, based on spatio-temporal covariance functions which are separable and non-separable in space and time.
I present an approach for modeling areal spatial covariance by considering the stationary distribution of a spatio-temporal Markov random walk. In the areal data case, this stationary distribution corresponds to an intrinsic simultaneous autoregressive (SAR) model for spatial correlation, and provides a principled approach to specifying areal spatial models when a spatio-temporal generating process can be assumed. I apply the approach to a study of spatial genetic variation of trout in a stream network in Connecticut, USA, and a study of crime rates in neighborhoods of Columbus, OH, USA.
We describe spatio-temporal random processes using linear mixed models. We show how many commonly used models can be viewed as special cases of this general framework and pay close attention to models with separable or product-sum covariances. The proposed linear mixed model formulation facilitates the implementation of a novel algorithm using Stegle eigendecompositions, a recursive application of the Sherman-Morrison-Woodbury formula, and Helmert-Wolf blocking to efficiently invert separable and product-sum covariance matrices, even when every spatial location is not observed at every time po…
The share of wind power in fuel mixes worldwide has increased considerably. The main ingredient when deriving wind power predictions are wind speed data; the closer to the wind farms, the better they forecast the power supply. The current paper proposes a hybrid model for predicting wind speeds at convenient locations. It is then applied to Southern California power price area. We build random fields with time series of gridded historical forecasts and actual wind speed observations. We estimate with ordinary kriging the spatial variability of the temporal parameters and derive predictions. Th…
Fine particulate matter (PM$_{2.5}$) has become a great concern worldwide due to its adverse health effects. PM$_{2.5}$ concentrations typically exhibit complex spatio-temporal variations. Both the mean and the spatio-temporal dependence evolve with time due to seasonality, which makes the statistical analysis of PM$_{2.5}$ challenging. In geostatistics, Gaussian process is a powerful tool for characterizing and predicting such spatio-temporal dynamics, for which the specification of a spatio-temporal covariance function is the key. While the extant literature offers a wide range of choices fo…
When analyzing the spatio-temporal dependence in most environmental and earth sciences variables such as pollutant concentrations at different levels of the atmosphere, a special property is observed: the covariances and cross-covariances are stronger in certain directions. This property is attributed to the presence of natural forces, such as wind, which cause the transport and dispersion of these variables. This spatio-temporal dynamics prompted the use of the Lagrangian reference frame alongside any Gaussian spatio-temporal geostatistical model. Under this modeling framework, a whole new cl…
Functional spatio-temporal data naturally arise in many environmental and climate applications where data are collected in a three-dimensional space over time. The MATLAB D-STEM v1 software package was first introduced for modelling multivariate space-time data and has been recently extended to D-STEM v2 to handle functional data indexed across space and over time. This paper introduces the new modelling capabilities of D-STEM v2 as well as the complexity reduction techniques required when dealing with large data sets. Model estimation, validation and dynamic kriging are demonstrated in two ca…
The prevalence of multivariate space-time data collected from monitoring networks and satellites, or generated from numerical models, has brought much attention to multivariate spatio-temporal statistical models, where the covariance function plays a key role in modeling, inference, and prediction. For multivariate space-time data, understanding the spatio-temporal variability, within and across variables, is essential in employing a realistic covariance model. Meanwhile, the complexity of generic covariances often makes model fitting very challenging, and simplified covariance structures, inc…
The covariance function and the variogram play very important roles in modelling and in prediction of spatial and spatio-temporal data. The assumption of second order stationarity, in space and time, is often made in the analysis of spatial data and the spatio-temporal data. Several times the assumption of stationarity is considered to be very restrictive, and therefore, a weaker assumption that the data is Intrinsically stationary both in space and time is often made and used, mainly by the geo-statisticians and other environmental scientists. In this paper we consider the data to be intrinsi…
This article concerns the predictive modeling for spatio-temporal data as well as model interpretation using data information in space and time. We develop a novel approach based on supervised dimension reduction for such data in order to capture nonlinear mean structures without requiring a prespecified parametric model. In addition to prediction as a common interest, this approach emphasizes the exploration of geometric information from the data. The method of Pairwise Directions Estimation (PDE; Lue, 2019) is implemented in our approach as a data-driven function searching for spatial patter…
Covariance functions and variograms play a fundamental role in exploratory analysis and statistical modelling of spatial and spatio-temporal datasets. In this paper, we construct a new class of spatial covariance functions using the Fourier transform of some higher-order kernels. Further, we extend this class of the spatial covariance functions to the spatio-temporal setting by using the idea used in Ma (2003).
Technological developments and open data policies have made large, global environmental datasets accessible to everyone. For analysing such datasets, including spatiotemporal correlations using traditional models based on Gaussian processes does not scale with data volume and requires strong assumptions about stationarity, separability, and distance measures of covariance functions that are often unrealistic for global data. Only very few modeling approaches suitably model spatiotemporal correlations while addressing both computational scalability as well as flexible covariance models. In this…
In this work we consider the problem of detecting anomalous spatio-temporal behavior in videos. Our approach is to learn the normative multiframe pixel joint distribution and detect deviations from it using a likelihood based approach. Due to the extreme lack of available training samples relative to the dimension of the distribution, we use a mean and covariance approach and consider methods of learning the spatio-temporal covariance in the low-sample regime. Our approach is to estimate the covariance using parameter reduction and sparse models. The first method considered is the representati…
Large spatial datasets are becoming ubiquitous in environmental sciences with the explosion in the amount of data produced by sensors that monitor and measure the Earth system. Consequently, the geostatistical analysis of these data requires adequate methods. Richer datasets lead to more complex modeling but may also prevent from using classical techniques. Indeed, the kriging predictor is not straightforwarldly available as it requires the inversion of the covariance matrix of the data. The challenge of handling such datasets is therefore to extract the maximum of information they contain whi…
In geostatistics, it is common to model spatially distributed phenomena through an underlying stationary and isotropic spatial process. However, these assumptions are often untenable in practice because of the influence of local effects in the correlation structure. Therefore, it has been of prolonged interest in the literature to provide flexible and effective ways to model nonstationarity in the spatial effects. Arguably, due to the local nature of the problem, we might envision that the correlation structure would be highly dependent on local characteristics of the domain of study, namely, …
Design and operation of a utility scale photovoltaic (PV) power plant depends on accurate modeling of the power generated, which is highly correlated with aggregate solar irradiance on the plant's PV modules. At present, aggregate solar irradiance over the area of a typical PV power plant cannot be measured directly. Rather, irradiance measurements are typically available from a few, relatively small sensors and thus aggregate solar irradiance must be estimated from these data. As a step towards finding more accurate methods for estimating aggregate irradiance from avaialble measurements, we e…
Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are computationally simple. However, in many applications one would like to predict for a range of untried points simultaneously. In this work we obtain some error bounds for the (simple) kriging predictor under the uniform metric. It works for a scattered set of input points in an arbitrary dimension, and also covers the case where the covariance function of the Gaussian process is misspecified. These results lead to a better understanding…
Neural encoding studies explore the relationships between measurements of neural activity and measurements of a behavior that is viewed as a response to that activity. The coupling between neural and behavioral measurements is typically imperfect and difficult to measure.To enhance our ability to understand neural encoding relationships, we propose that a behavioral measurement may be decomposable as a sum of two latent components, such that the direct neural influence and prediction is primarily localized to the component which encodes temporal dependence. For this purpose, we propose to use …
Periodic monitoring of groundwater quality at industrial and commercial sites generates large volumes of spatiotemporal concentration data. Data modelling is typically restricted to either the analysis of monotonic trends in individual wells, or independent fitting of spatial concentration distributions (e.g. Kriging) to separate monitoring time periods. Neither of these techniques satisfactorily elucidate the interaction between spatial and temporal components of the data. Potential negative consequences include an incomplete understanding of groundwater plume dynamics, which can lead to the …
We address the problem of predicting spatio-temporal processes with temporal patterns that vary across spatial regions, when data is obtained as a stream. That is, when the training dataset is augmented sequentially. Specifically, we develop a localized spatio-temporal covariance model of the process that can capture spatially varying temporal periodicities in the data. We then apply a covariance-fitting methodology to learn the model parameters which yields a predictor that can be updated sequentially with each new data point. The proposed method is evaluated using both synthetic and real cli…
Many physical datasets are generated by collections of instruments that make measurements at regular time intervals. For such regular monitoring data, we extend the framework of half-spectral covariance functions to the case of nonstationarity in space and time and demonstrate that this method provides a natural and tractable way to incorporate complex behaviors into a covariance model. Further, we use this method with fully time-domain computations to obtain bona fide maximum likelihood estimators---as opposed to using Whittle-type likelihood approximations, for example---that can still be co…
For a spatiotemporal process $\{X_j(s,t) | ~s \in S~,~t \in T \}_{j =1, \ldots , n} $, where $S$ denotes the set of spatial locations and $T$ the time domain, we consider the problem of testing for a change in the sequence of mean functions. In contrast to most of the literature we are not interested in arbitrarily small changes, but only in changes with a norm exceeding a given threshold. Asymptotically distribution free tests are proposed, which do not require the estimation of the long-run spatiotemporal covariance structure. In particular we consider a fully functional approach and a test …
Spatial statistics often involves Cholesky decomposition of covariance matrices. To ensure scalability to high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vecchia approximation, whose conditional-independence assumptions imply sparsity in the Cholesky factors of both the precision and the covariance matrix. This remarkable property is crucial for applications to high-dimensional spatio-temporal filtering. We present a fast and simple algorithm to compute our hierarchical Vecchia approximation, and we provide…
We consider the application of KronPCA spatio-temporal modeling techniques [Greenewald et al 2013, Tsiligkaridis et al 2013] to the extraction of spatiotemporal features for video dismount classification. KronPCA performs a low-rank type of dimensionality reduction that is adapted to spatio-temporal data and is characterized by the T frame multiframe mean and covariance of p spatial features. For further regularization and improved inverse estimation, we also use the diagonally corrected KronPCA shrinkage methods we presented in [Greenewald et al 2013]. We apply this very general method to the…
Air pollution is a great concern because of its impact on human health and on the environment. Statistical models play an important role in improving knowledge of this complex spatio-temporal phenomenon and in supporting public agencies and policy makers. We focus on the class of hierarchical models that provides a flexible framework for incorporating spatio-temporal interactions at different hierarchical levels. The challenge is to choose a model that is satisfactory in terms of goodness of fit, interpretability, parsimoniousness, prediction capability and computational costs. In order to sup…
Argo floats measure seawater temperature and salinity in the upper 2,000 m of the global ocean. Statistical analysis of the resulting spatio-temporal dataset is challenging due to its nonstationary structure and large size. We propose mapping these data using locally stationary Gaussian process regression where covariance parameter estimation and spatio-temporal prediction are carried out in a moving-window fashion. This yields computationally tractable nonstationary anomaly fields without the need to explicitly model the nonstationary covariance structure. We also investigate Student-$t$ dist…
Atmospheric inverse modelling is a method for reconstructing historical fluxes of green-house gas between land and atmosphere, using observed atmospheric concentrations and an atmospheric tracer transport model. The small number of observed atmospheric concentrations in relation to the number of unknown flux components makes the inverse problem ill-conditioned, and assumptions on the fluxes are needed to constrain the solution. A common practise is to model the fluxes using latent Gaussian fields with a mean structure based on estimated fluxes from combinations of process modelling (natural fl…
