We construct flexible spatio-temporal models through stochastic partial differential equations (SPDEs) where both diffusion and advection can be spatially varying. Computations are done through a Gaussian Markov random field approximation of the solution of the SPDE, which is constructed through a finite volume method. The new flexible non-separable model is compared to a flexible separable model both for reconstruction and forecasting, and evaluated in terms of root mean square errors and continuous rank probability scores. A simulation study demonstrates that the non-separable model performs better when the data is simulated from a non-separable model with diffusion and advection. Further, we estimate surrogate models for emulating the output of a ocean model in Trondheimsfjorden, Norway, and simulate observations of autonomous underwater vehicles. The results show that the flexible non-separable model outperforms the flexible separable model for real-time prediction of unobserved locations.
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…
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Three spatial sampling schemes are considered, including irregular, non-uniform and shifted uniform sampling. The irregular sampling scheme is the general scenario, while computationally efficient solutions are available in the spectral domain for non-uniform and shifted uniform sampling. For…
A class of physics-informed spatio-temporal models has recently been proposed for modeling spatio-temporal processes governed by advection-diffusion equations. The central idea is to approximate the process by a truncated Fourier series and let the governing physics determine the dynamics of the spectral coefficients. However, because many spatio-temporal processes in real applications are non-periodic with boundary discontinuities, the well-known Gibbs phenomenon and ripple artifact almost always exist in the outputs generated by such models due to truncation of the Fourier series. Hence, the…
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.
To analyse interaction in marked spatio-temporal point processes (MSTPPs), we introduce marked (cross) second-order reduced moment measures and K-functions for general inhomogeneous second-order intensity reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark categories of the points, are depending on the specific mark space and mark reference measure chosen. We also look closer at how the summary statistics reduce under assumptions such as the MSTPP being multivariate and/or stationary. A new test for independent marking is devised…
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…
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…
We consider a class of spatio-temporal models which extend popular econometric spatial autoregressive panel data models by allowing the scalar coefficients for each location (or panel) different from each other. To overcome the innate endogeneity, we propose a generalized Yule-Walker estimation method which applies the least squares estimation to a Yule-Walker equation. The asymptotic theory is developed under the setting that both the sample size and the number of locations (or panels) tend to infinity under a general setting for stationary and alpha-mixing processes, which includes spatial a…
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…
Most COVID-19 studies commonly report figures of the overall infection at a state- or county-level. This aggregation tends to miss out on fine details of virus propagation. In this paper, we analyze a high-resolution COVID-19 dataset in Cali, Colombia, that records the precise time and location of every confirmed case. We develop a non-stationary spatio-temporal point process equipped with a neural network-based kernel to capture the heterogeneous correlations among COVID-19 cases. The kernel is carefully crafted to enhance expressiveness while maintaining model interpretability. We also incor…
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…
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
Point process data are becoming ubiquitous in modern applications, such as social networks, health care, and finance. Despite the powerful expressiveness of the popular recurrent neural network (RNN) models for point process data, they may not successfully capture sophisticated non-stationary dependencies in the data due to their recurrent structures. Another popular type of deep model for point process data is based on representing the influence kernel (rather than the intensity function) by neural networks. We take the latter approach and develop a new deep non-stationary influence kernel th…
Focusing on regression based analysis of extremes in a presence of systematically missing covariates, this work presents a data-driven spatio-temporal regression based clustering of threshold excesses. It is shown that in a presence of systematically missing covariates the behavior of threshold excesses becomes nonstationary and nonhomogenous. The presented approach describes this complex behavior by a set of local stationary Generalized Pareto Distribution (GPD) models, where the parameters are expressed as regression models, and a latent spatio-temporal switching process. The spatio-temporal…
DeepMIDE: A Multi-Output Spatio-Temporal Method for Ultra-Scale Offshore Wind Energy Forecasting
Department of Industrial Engineering, Clemson University（克莱姆森大学工业工程系）；Department of Computer Science, Rutgers University（罗杰斯大学计算机科学系）；Department of Statistics, Rutgers University（罗杰斯大学统计学系）；Department of Industrial & Systems Engineering, Rutgers University（罗杰斯大学工业与系统工程系）；University College, Korea University（韩国大学大学学院）
In many applications, a signal is deformed by well-understood dynamics before it can be measured. For example, when a pollutant enters a river, it immediately begins dispersing, flowing, settling, and reacting. If the pollutant enters at a single point, its concentration can be measured before it enters the complex dynamics of the river system. However, in the case of a non-point source pollutant, it is not clear how to efficiently measure its source. One possibility is to record concentration measurements in the river, but this signal is masked by the fluid dynamics of the river. Specifically…
Precipitation is a complex physical process that varies in space and time. Predictions and interpolations at unobserved times and/or locations help to solve important problems in many areas. In this paper, we present a hierarchical Bayesian model for spatio-temporal data and apply it to obtain short term predictions of rainfall. The model incorporates physical knowledge about the underlying processes that determine rainfall, such as advection, diffusion and convection. It is based on a temporal autoregressive convolution with spatially colored and temporally white innovations. By linking the a…
Natural spatiotemporal processes can be highly non-stationary in many ways, e.g. the low-level non-stationarity such as spatial correlations or temporal dependencies of local pixel values; and the high-level variations such as the accumulation, deformation or dissipation of radar echoes in precipitation forecasting. From Cramer's Decomposition, any non-stationary process can be decomposed into deterministic, time-variant polynomials, plus a zero-mean stochastic term. By applying differencing operations appropriately, we may turn time-variant polynomials into a constant, making the deterministi…
The modeling of spatio-temporal trends in temperature extremes can help better understand the structure and frequency of heatwaves in a changing climate. Here, we study annual temperature maxima over Southern Europe using a century-spanning dataset observed at 44 monitoring stations. Extending the spectral representation of max-stable processes, our modeling framework relies on a novel construction of max-infinitely divisible processes, which include covariates to capture spatio-temporal non-stationarities. Our new model keeps a popular max-stable process on the boundary of the parameter space…
We restrict our attention to space-time point pattern data for which we have a single realisation within a finite region. Second-order characteristics are used to analyse the spatio-temporal structure of the underlying point process. In particular, the space-time inhomogeneous pair correlation function and $K$-function measure the spatio-temporal clustering/regularity and the spatio-temporal interaction, and thus help with model choice. Non-parametric estimators of the second-order characteristics require information from outside the study region, resulting to the so-called edge effects which …
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…
We introduce WECS (Wavelet Energies Correlation Sreening), an unsupervised sparse procedure to detect spatio-temporal change points on multi-temporal SAR (POLSAR) images or even on sequences of very high resolution images. The procedure is based on wavelet approximation for the multi-temporal images, wavelet energy apportionment, and ultra-high dimensional correlation screening for the wavelet coefficients. We present two complimentary wavelet measures in order to detect sudden and/or cumulative changes, as well as for the case of stationary or non-stationary multi-temporal images. We show WEC…
This paper describes the methodology used by the team RedSea in the data competition organized for EVA 2021 conference. We develop a novel two-part model to jointly describe the wildfire count data and burnt area data provided by the competition organizers with covariates. Our proposed methodology relies on the integrated nested Laplace approximation combined with the stochastic partial differential equation (INLA-SPDE) approach. In the first part, a binary non-stationary spatio-temporal model is used to describe the underlying process that determines whether or not there is wildfire at a spec…
University of Warwick（华威大学）；The Alan Turing Institute for Data Science & AI（艾伦·图灵数据科学与人工智能研究所）
Bayesian On-line Changepoint Detection is extended to on-line model selection and non-stationary spatio-temporal processes. We propose spatially structured Vector Autoregressions (VARs) for modelling the process between changepoints (CPs) and give an upper bound on the approximation error of such models. The resulting algorithm performs prediction, model selection and CP detection on-line. Its time complexity is linear and its space complexity constant, and thus it is two orders of magnitudes faster than its closest competitor. In addition, it outperforms the state of the art for multivariate …
We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationary spatio-temporal point processes. The statistic is defined through the n-point correlation functions of the point process and it generalises the J-function when stationarity is assumed. We show that our statistic can be represented in terms of the generating functional and that it is related to the inhomogeneous K-function. We further discuss its explicit form under some specific model assumptions and derive a ratio-unbiased estimator. We finally illustrate the use of our statistic on simulated data.
University of Cambridge（剑桥大学）；Invenia Labs（Invenia实验室）；Microsoft Research（微软研究院）
Stationary stochastic processes (SPs) are a key component of many probabilistic models, such as those for off-the-grid spatio-temporal data. They enable the statistical symmetry of underlying physical phenomena to be leveraged, thereby aiding generalization. Prediction in such models can be viewed as a translation equivariant map from observed data sets to predictive SPs, emphasizing the intimate relationship between stationarity and equivariance. Building on this, we propose the Convolutional Neural Process (ConvNP), which endows Neural Processes (NPs) with translation equivariance and extend…
Large, non-stationary spatio-temporal data are ubiquitous in modern statistical applications, and the modeling of spatio-temporal extremes is crucial for assessing risks in environmental sciences among others. While the modeling of extremes is challenging in itself, the prediction of rare events at unobserved spatial locations and time points is even more difficult. In this editorial, we describe the data competition that was organized for the 11th international conference on Extreme-Value Analysis (EVA 2019), for which several teams modeled and predicted Red Sea surface temperature extremes o…
This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models related to a wide class of linear SPDEs, with applications to spatio-temporal models having non-trivial properties. Within the framework of Generalized Random Fields, a criterion for existence and uniqueness of stationary solutions for this class of SPDEs is proposed and proven. Their covariance are then obtained through their spectral measure. We present a result relating the covariance in the ca…
Climate is known for being characterised by strong non-linearity and chaotic behaviour. Nevertheless, few studies in climate science adopt statistical methods specifically designed for non-stationary or non-linear systems. Here we show how the use of statistical methods from Information Theory can describe the non-stationary behaviour of climate fields, unveiling spatial and temporal patterns that may otherwise be difficult to recognize. We study the maximum temperature at two meters above ground using the NCEP CDAS1 daily reanalysis data, with a spatial resolution of 2.5 by 2.5 degree and cov…
The importance of modelling temperature fields goes beyond the need to understand a region's climate and serves too as a starting point for understanding their socioeconomic, and health consequences. The topography of the study region contributes much to the complexity of modelling these fields and demands flexible spatio-temporal models that are able to handle nonstationarity and changes in trend. In this paper, we develop a flexible stochastic spatio-temporal model for daily temperatures in the Pacific Northwest, and describe a methodology for performing Bayesian spatial prediction. A novel …
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…
A large number of scientific studies and engineering problems involve high-dimensional spatiotemporal data with complicated relationships. In this paper, we focus on a type of space-time interaction named \emph{temporal evolution of spatial dependence (TESD)}, which is a zero time-lag spatiotemporal covariance. For this purpose, we propose a novel Bayesian nonparametric method based on non-stationary spatiotemporal Gaussian process (STGP). The classic STGP has a covariance kernel separable in space and time, failed to characterize TESD. More recent works on non-separable STGP treat location an…
With extreme weather events becoming more common, the risk posed by surface water flooding is ever increasing. In this work we propose a model, and associated Bayesian inference scheme, for generating probabilistic (high-resolution short-term) forecasts of localised precipitation. The parametrisation of our underlying hierarchical dynamic spatio-temporal model is motivated by a forward-time, centred-space finite difference solution to a collection of stochastic partial differential equations, where the main driving forces are advection and diffusion. Observations from both weather radar and gr…
