Spatio-temporal Covariance Models Research Articles

Constructing large nonstationary spatio-temporal covariance models via compositional warpings

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 combined through composition, can induce complex warpings. A stationary spatio-temporal covariance function on the warped domain induces covariance nonstationarity on the original domain. Sparse linear algebraic methods are used to reduce the computational complexity when fitting the model in a big data setting. We show that our proposed nonstationary spatio-temporal model can capture covariance nonstationarity in both space and time, and provide better probabilistic predictions than conventional stationary models in both simulation studies and on a real-world data set.

Spatio-temporal generalized complex covariance models based on convolution

Modeling covariance functions, with values on a complex domain, is essential for geostatistical interpolation or stochastic simulation of complex-valued random fields in space or space-time. However, little has been done for complex spatio-temporal modeling. For this aim, the construction of new classes of spatio-temporal complex-valued covariance models, based on convolution, is provided. Indeed, starting from the Lajaunie and Béjaoui models extended to a space-time domain, generalized families of complex models are obtained through the integration with respect to a positive measure. A procedure for fitting the two parts of the spatio-temporal complex models and for defining the density function considered for the integration is also illustrated. The computational details of this procedure are discussed through a case study on a spatio-temporal dataset of sea currents and the performance of these classes of models is assessed.

New spatio-temporal complex covariance functions for vectorial data through positive mixtures

In the literature, the theory of complex-valued random fields is usually recalled to describe the evolution of vector data in space, without including the temporal dimension. However, as in the real case, the development of the complex formalism in a spatio-temporal context and the construction of some new classes of spatio-temporal complex covariance models are of sure interest for the scientific community partly due to the ongoing explosion in the availability of vector observations in space–time. In this paper, after presenting the fundamental aspects of the complex formalism of a spatio-temporal random field in a complex domain and the extension of some classes of complex-valued covariance models from a spatial domain to a spatio-temporal one, a new family of spatio-temporal complex-valued models obtained through a positive mixture of an infinite number of terms is proposed and various examples are discussed. A case study on modeling the spatio-temporal complex correlation structure of vectorial data is also provided.

Wind Speed Prediction based on Spatio-Temporal Covariance Model Using Autoregressive Integrated Moving Average Regression Smoothing

It is essential to enhance the ability of wind speeds forecasting for wind energy and wind resource planning. For this purpose, a hybrid strategy has been proposed based on spatio-temporal covariance model which combined the spatio-temporal ordinary kriging (STOK) technology with autoregressive integrated moving average (ARIMA) regression smoothing method. This is because wind speed time series exhibits a long-term dependency. In the case study, both STOK method and ARIMA method are employed and their performances are compared. The ARIMA model can obtain a necessary and sufficient smoothing condition for them to be smoothed. Meanwhile, further theoretical analysis is provided to discuss why the STOK method is potentially more accurate than the ARIMA method for wind speed time series prediction. Results show that the proposed method outperforms the Non-Sep-Gneiting model by 9% and 7.2% in terms of mean absolute error (MAE) and root-mean-square error (RMSE).

Short-Term Time Wind Speed Forecasting Based on Spatio-Temporal Geostatistical Approach and Kriging Method

Short-term wind speed prediction is an essential task for wind resource and wind energy planning. However, most of this literature does not take into account the spatio-termporal correlation of wind data from the geographical field. For this reason, we propose an integrated spatio-temporal kriging and functional kriging strategy to exploit such spatio-temporal correlation into the wind speed prediction. First, the deterministic trend component in wind data is estimated to be removed. The residuals are used for spatio-temporal modeling and prediction. Based on the spatio-temporal kriging framework, four spatio-temporal covariance models (product-sum model, separable exponential product model, separable and nonseparable Gneiting models) are considered which describe the spatio-temporal correlation of wind data. In particular, the flexibility of using the nonseparable Gneiting model is highlighted. More specifically, four spatio-temporal random fields are modeled from the 12 wind monitoring stations over Ireland. We also use an involved weighted least squares method for estimating parameters of the four covariance models involved in the spatio-temporal kriging strategy. We apply the fitted covariance models to generate day-ahead wind speed predictions at both observed and nonobserved locations where wind station already exist but also to nearby locations. Leave-one-out cross-validation is applied to check the significance of the difference among the four models, these spatio-temporal ordinary kriging (STOK), functional ordinary kriging (FOK) and autoregressive integrated moving average (ARIMA) methods are compared for day-ahead wind speed predictions. Forecasting results indicate that the predicting accuracy is improved almost 33.5% using FOK compared with three approaches which confirm the effectiveness of the functional kriging method in the paper.

Accounting for spatiotemporal correlations of GNSS coordinate time series to estimate station velocities

It is well known that GNSS permanent station coordinate time series exhibit time-correlated noise. Spatial correlations between coordinate time series of nearby stations are also long-established and generally handled by means of spatial filtering techniques. Accounting for both the temporal and spatial correlations of the noise via a spatiotemporal covariance model is however not yet a common practice. We demonstrate in this paper the interest of using such a spatiotemporal covariance model of the stochastic variations in GNSS time series in order to estimate long-term station coordinates and especially velocities.We provide a methodology to rigorously assess the covariances between horizontal coordinate variations and use it to derive a simple exponential spatiotemporal covariance model for the stochastic variations in the IGS repro2 station coordinate time series. We then use this model to estimate station velocities for two selected datasets of 10 time series in Europe and 11 time series in the USA. We show that coordinate prediction as well as velocity determination from short time series are improved when using this spatiotemporal model, as compared with the case where spatiotemporal correlations are ignored.

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.

Visualization and assessment of spatio-temporal covariance properties

Spatio-temporal covariances are important for describing thespatio-temporal variability of underlying random fields in geostatistical data. For second-order stationary random fields, 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 pair of spatial locations. These test functions of temporal lags summarize the properties of separability or symmetry for the given spatial pairs. We use functional boxplots to visualize the functional median and the variability of the test functions, where the extent of departure from zero at all temporal lags indicates the degree of non-separability or asymmetry. We also develop a rank-based nonparametric testing procedure for assessing the significance of the non-separability or asymmetry. Essentially, the proposed methods only require the analysis of temporal covariance functions. Thus, a major advantage over existing approaches is that there is no need to estimate any covariance matrix for selected spatio-temporal lags. The performances of the proposed methods are examined by simulations with various commonly used spatio-temporal covariance models. To illustrate our methods in practical applications, we apply it to real datasets, including weather station data and climate model outputs.

BME Spatiotemporal Estimation of Annual Precipitation and Detection of Drought Hazard Clusters Using Space–Time Scan Statistics in the Yun-Gui-Guang Region, Mainland China

AbstractDrought disasters cause great economic losses in China every year, especially in its southwest, and they have had a major influence on economic development, lives, and property. In this study, precipitation and drought hazards were examined for a region covering Yunnan, Guizhou, and Guangxi Provinces to assess the spatial and temporal distribution of different drought hazard grades in this region. Annual precipitation data from 90 meteorological stations in or around the study area were collected and organized for the period of 1964–2013. A spatiotemporal covariance model was calculated and fitted. The Bayesian maximum entropy (BME) method, which considers physical knowledge bases to reduce errors, was used to provide an optimal estimation of annual precipitation. Regional annual precipitation distributions were determined. To analyze the spatiotemporal patterns of the drought hazard, the annual standardized precipitation index was used to measure drought severity. A method that involves space–time scan statistics was used to detect the most likely spatiotemporal clusters of the drought hazards. Test-significance p values for all of the calculated clusters were less than 0.001, indicating a high significance level. The results showed that Yunnan Province was a drought-prone area, especially in its northwest and center, followed by Guizhou Province. In addition, Yunnan and Guizhou Provinces were cluster areas of severe and extreme drought. The most likely cluster year was 1966; it was clustered five times during the study period. In this study, the evolutionary process of drought hazards, including spatiotemporal distribution and spatiotemporal clustering characteristics, was considered. The results may be used to provide support for prevention and mitigation of drought in the study area such as optimizing the distribution of drought-resisting resources, drought monitoring, and evaluating potential drought impacts.

Tensor-based methods for blind spatial signature estimation under arbitrary and unknown source covariance structure

Spatial signature estimation is a problem encountered in several applications in signal processing such as mobile communications, sonar, radar, astronomy and seismology. In this paper, we propose higher-order tensor methods to solve the blind spatial signature estimation problem using planar arrays. By assuming that sources' powers vary between successive time blocks, we recast the spatial and spatiotemporal covariance models for the received data as third-order PARATUCK2 and fourth-order Tucker4 tensor decompositions, respectively. Firstly, by exploiting the multilinear algebraic structure of the proposed tensor models, new iterative algorithms are formulated to blindly estimate the spatial signatures. Secondly, in order to achieve a better spatial resolution, we propose an expanded form of spatial smoothing that returns extra spatial dimensions in comparison with the traditional approaches. Additionally, by exploiting the higher-order structure of the resulting expanded tensor model, a multilinear noise reduction preprocessing step is proposed via higher-order singular value decomposition. We show that the increase on the tensor order provides a more efficient denoising, and consequently a better performance compared to existing spatial smoothing techniques. Finally, a solution based on a multi-stage Khatri–Rao factorization procedure is incorporated as the final stage of our proposed estimators. Our results demonstrate that the proposed tensor methods yield more accurate spatial signature estimates than competing approaches while operating in a challenging scenario where the source covariance structure is unknown and arbitrary (non-diagonal), which is actually the case when sample covariances are computed from a limited number of snapshots.

Spatio-Temporal Interpolation using gstat

We present new spatio-temporal geostatistical modelling and interpolation capabilities of the R package gstat. Various spatio-temporal covariance models have been implemented, such as the separable, product-sum, metric and sum-metric models. In a real-world application we compare spatiotemporal interpolations using these models with a purely spatial kriging approach. The target variable of the application is the daily mean PM 10 concentration measured at rural air quality monitoring stations across Germany in 2005. R code for variogram fitting and interpolation is presented in this paper to illustrate the workflow of spatio-temporal interpolation using gstat. We conclude that the system works properly and that the extension of gstat facilitates and eases spatio-temporal geostatistical modelling and prediction for R users

New compactly supported spatiotemporal covariance functions from SPDEs

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.

Merging daily sea surface temperature data from multiple satellites using a Bayesian maximum entropy method

Sea surface temperature (SST) is an important variable for understanding interactions between the ocean and the atmosphere. SST fusion is crucial for acquiring SST products of high spatial resolution and coverage. This study introduces a Bayesian maximum entropy (BME) method for blending daily SSTs from multiple satellite sensors. A new spatiotemporal covariance model of an SST field is built to integrate not only single-day SSTs but also time-adjacent SSTs. In addition, AVHRR 30-year SST climatology data are introduced as soft data at the estimation points to improve the accuracy of blended results within the BME framework. The merged SSTs, with a spatial resolution of 4 km and a temporal resolution of 24 hours, are produced in the Western Pacific Ocean region to demonstrate and evaluate the proposed methodology. Comparisons with in situ drifting buoy observations show that the merged SSTs are accurate and the bias and root-mean-square errors for the comparison are 0.15°C and 0.72°C, respectively.

The role of visualisation in the choice of stationary non-separable space–time covariance functions: an application to air pollution data

Modelling spatio-temporal dependencies resulting from dynamic processes that evolve in both space and time is essential in many scientific fields. Spatio-temporal Kriging is one of the space–time procedures, which has progressed the most over the last few years. Kriging predictions strongly depend on the covariance function associated with the stochastic process under study. Therefore, the choice of such a covariance function, which is usually based on empirical covariance, is a core aspect in the prediction procedure. As the empirical covariance is not necessarily a permissible covariance function, it is necessary to fit a valid covariance model. Due to the complexity of these valid models in the spatio-temporal case, visualising them is of great help, at least when selecting the set of candidate models to represent the spatio-temporal dependencies suggested by the empirical covariogram. We focus on the visualisation of the most interesting stationary non-separable covariance functions and how they change as their main parameters take different values. We wrote a specialised code for visualisation purposes. In order to illustrate the usefulness of visualisation when choosing the appropriate non-separable spatio-temporal covariance model, we focus on an important pollution problem, namely the levels of carbon monoxide, in the city of Madrid, Spain.

Full-Scale Approximations of Spatio-Temporal Covariance Models for Large Datasets

This work was partially supported by NSF grant DMS-1007618, NSF grant EARS-1343155, and Award Number KUS-CI-016-04 made by King Abdullah University of Science and Technology (KAUST). Huang's work was also partially supported by NSF grant DMS-1208952.

Positive and negative non-separability for space–time covariance models

Separable spatio-temporal covariance models, defined as the product of purely spatial and purely temporal covariance functions, are often used in practice, but frequently they only represent a convenient assumption. On the other hand, non-separable models are receiving a lot of attention, since they are more flexible to handle empirical covariances showed up in applications. Different forms of non-separability for space–time covariance functions have been recently defined in the literature. In this paper, the notion of positive and negative non-separability is further formalized in order to distinguish between pointwise and uniform non-separability. Various well-known non-separable space–time stationary covariance models are analyzed and classified by using the new definition of non-separability. In particular, wide classes of non-separable spatio-temporal covariance functions, able to capture positive and negative non-separability, are proposed and some examples of these classes are given. General results concerning the non-separability of spatial–temporal covariance functions obtained by a linear combination of spatial–temporal covariance functions and some stability properties are also presented. These results can be helpful to generate as well as to select appropriate covariance models for describing space–time data.

Mathematical Genesis of the Spatio-Temporal Covariance Functions

Problem statement: Obtaining new and flexible classes of nonseparable spatio-temporal covariances have resulted in a key point of research in the last years within the context of spatiotemporal Geostatistics. Approach: In general, the literature has focused on the problem of full symmetry and the problem of anisotropy has been overcome. Results: By exploring mathematical properties of positive definite functions and their close connection to covariance functions we are able to develop new spatio-temporal covariance models taking into account the problem of spatial anisotropy. Conclusion/Recommendations: The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability.

Recent advances to model anisotropic space–time data

Building new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to present an up-to-date overview of recent spatio-temporal covariance models taking into account the problem of spatial anisotropy. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability. In particular, we focus on the problem of modelling anisotropy through isotropy within components. We present the Bernstein class, and a generalisation of Gneiting’s approach (2002a) to obtain new classes of space–time covariance functions which are spatially anisotropic. We also discuss some methods for building covariance functions that attain negative values. We finally present several differentiation and integration operators acting on particular space–time covariance classes.

Interactive spatiotemporal modelling of health systems: the SEKS–GUI framework

This paper describes the spatiotemporal epistematics knowledge synthesis and graphical user interface (SEKS–GUI) framework and its application in medical geography problems. Based on sound theoretical reasoning, the interactive software library of SEKS–GUI explores heterogeneous (spatially non-homogeneous and temporally non-stationary) health attribute distributions (disease incidence, mortality, human exposure, epidemic propagation etc.); expresses the health system’s dependence structure using (ordinary and generalized) spatiotemporal covariance models; synthesizes core knowledge bases, empirical evidence and multi-sourced system uncertainty; and generates a meaningful picture of the real-world system using space–time dependent probability functions and associated maps of health attributes. The implementation stages of the SEKS–GUI library are described in considerable detail using appropriate screens. The wide applicability of SEKS–GUI is demonstrated by reviewing a selection of real-world case studies.

Recent developments on the construction of spatio-temporal covariance models

This paper briefly surveys some recent advances on how to construct spatio-temporal covariance functions, with the emphasis on the methods which can be used to derive covariance functions but not on a summary list of particular closed-form covariance functions. The advantages and shortcomings of some methods are discussed, and a number of proposals for future research are also suggested.