Nonseparable Covariance Functions Research Articles

Response envelopes for linear coregionalization models

Dimension reduction provides a useful tool for statistical data analysis with high-dimensional data. In this paper, we develop a parsimonious multivariate spatial regression model with a non-separable covariance function. The efficacy of this new solution is illustrated through simulation studies and a real data analysis. We show that for cases where the marginal spatial correlations are different from each other, the proposed non-separable model provides better estimation and inference than the related separable model, and provides tighter inference than a non-separable spatial model without dimension reduction when there is immaterial variation in the data.

Wind power scenario generation with non-separable spatio-temporal covariance function and fluctuation-based clustering

Short-term wind power scenarios significantly affect the economic efficiency of the stochastic power system scheduling. In order to better capture the nonlinear spatio-temporal correlations of wind power, this paper proposes a scenario generation method integrated with non-separable spatio-temporal covariance function and fluctuation-based clustering for short-term wind power output. By taking advantage of well-calibrated marginal distribution modeled by Gaussian mixture model, the non-separable covariance function is incorporated in the scenario generation method to capture the complex interactions between spatial and temporal components of wind power. To estimate the covariance matrix more precisely, the historical data is grouped into K clusters with different fluctuations using the K-means clustering algorithm. Two indices are proposed to evaluate the scenarios in capturing the spatial and temporal correlations from the perspective of system operators. The proposed method is applied to a modified IEEE-118 system with four wind farms. Simulation results verify the superiority of the proposed method in capturing spatial and temporal correlations, and validate the economic benefits for the power system operation.

Reconstructing Cloud Contaminated Pixels Using Spatiotemporal Covariance Functions and Multitemporal Hyperspectral Imagery

One of the major challenges in optical-based remote sensing is the presence of clouds, which imposes a hard constraint on the use of multispectral or hyperspectral satellite imagery for earth observation. While some studies have used interpolation models to remove cloud affected data, relatively few aim at restoration via the use of multi-temporal reference images. This paper proposes not only the use of image time-series, but also the implementation of a geostatistical model that considers the spatiotemporal correlation between them to fill the cloud-related gaps. Using Hyperion hyperspectral images, we demonstrate a capacity to reconstruct cloud-affected pixels and predict their underlying surface reflectance values. To do this, cloudy pixels were masked and a parametric family of non-separable covariance functions was automated fitted, using a composite likelihood estimator. A subset of cloud-free pixels per scene was used to perform a kriging interpolation and to predict the spectral reflectance per each cloud-affected pixel. The approach was evaluated using a benchmark dataset of cloud-free pixels, with a synthetic cloud superimposed upon these data. An overall root mean square error (RMSE) of between 0.5% and 16% of the reflectance was achieved, representing a relative root mean square error (rRMSE) of between 0.2% and 7.5%. The spectral similarity between the predicted and reference reflectance signatures was described by a mean spectral angle (MSA) of between 1° and 11°, demonstrating the spatial and spectral coherence of predictions. The approach provides an efficient spatiotemporal interpolation framework for cloud removal, gap-filling, and denoising in remotely sensed datasets.

Nonseparable covariance models on circles cross time: A study of Mexico City ozone

AbstractWe propose a continuous spatiotemporal model for Mexico City ozone levels that account for distinct daily seasonality, as well as variation across the city and over the peak ozone season (April and May) of 2017. To account for these patterns, we use covariance models over space, circles, and time. We review relevant existing covariance models and develop new classes of nonseparable covariance models appropriate for seasonal data collected at many locations. We compare the predictive performance of a variety of models that utilize various nonseparable covariance functions. We use the best model to predict hourly ozone levels at unmonitored locations in April and May to infer compliance with Mexican air quality standards and to estimate the respiratory health risk associated with ozone exposure. We find that predicted compliance with air quality standards and estimated respiratory health risk vary greatly over space and time. In some regions, we predict exceedance of national standards for more than a third of the hours in April and May, and on many days, we predict that nearly all of Mexico City exceeds nationally legislated ozone thresholds at least once. In southern Mexico City, we estimate the respiratory risk for ozone to be 55% higher, on average, than the annual average risk and as much at 170% higher on some days.

Testing the type of non-separability and some classes of space-time covariance function models

In statistical space-time modeling, the use of non-separable covariance functions is often more realistic than separable models. In the literature, various tests for separability may justify this choice. However, in case of rejection of the separability hypothesis, none of these tests include testing for the type of non-separability of space-time covariance functions. This is an important and further significant step for choosing a class of models. In this paper a method for testing positive and negative non-separability is given; moreover, an approach for testing some well known classes of space-time covariance function models has been proposed. The performance of the tests has been shown using real and simulated data.

Space and circular time log Gaussian Cox processes with application to crime event data

We view the locations and times of a collection of crime events as a space-time point pattern. So, with either a nonhomogeneous Poisson process or with a more general Cox process, we need to specify a space-time intensity. For the latter, we need a \emph{random} intensity which we model as a realization of a spatio-temporal log Gaussian process. Importantly, we view time as circular not linear, necessitating valid separable and nonseparable covariance functions over a bounded spatial region crossed with circular time. In addition, crimes are classified by crime type. Furthermore, each crime event is recorded by day of the year which we convert to day of the week marks. The contribution here is to develop models to accommodate such data. Our specifications take the form of hierarchical models which we fit within a Bayesian framework. In this regard, we consider model comparison between the nonhomogeneous Poisson process and the log Gaussian Cox process. We also compare separable vs. nonseparable covariance specifications. Our motivating dataset is a collection of crime events for the city of San Francisco during the year 2012. We have location, hour, day of the year, and crime type for each event. We investigate models to enhance our understanding of the set of incidences.

Pairwise Meta-Modeling of Multivariate Output Computer Models Using Nonseparable Covariance Function

Gaussian process (GP) is a popular method for emulating deterministic computer simulation models. Its natural extension to computer models with multivariate outputs employs a multivariate Gaussian process (MGP) framework. Nevertheless, with significant increase in the number of design points and the number of model parameters, building an MGP model is a very challenging task. Under a general MGP model framework with nonseparable covariance functions, we propose an efficient meta-modeling approach featuring a pairwise model building scheme. The proposed method has excellent scalability even for a large number of output levels. Some properties of the proposed method have been investigated and its performance has been demonstrated through several numerical examples. Supplementary materials for this article are available online.

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.

A Centered Bivariate Spatial Regression Model for Binary Data with an Application to Presettlement Vegetation Data in the Midwestern United States

Spatially structured discrete data arise in diverse areas of application, such as forestry, epidemiology, or soil sciences. Data from several binary variables are often collected at each location. Variation in distributional properties across the spatial domain is of interest. The specific application that motivates our work involves characterizing historical distributions of two species of Oak in the Driftless Area in the Midwestern United States. Scientists are interested in understanding the patterns of interaction between species, as well as their relationships to spatial covariates. Accounting for spatial dependence is not only of inherent interest but also reduces prediction mean squared error, and is necessary for obtaining appropriate measures of uncertainty (i.e., standard errors and confidence intervals). To address the needs of the application, we introduce a centered bivariate autologistic model, which accounts for the statistical dependence in two response variables simultaneously, for the association between them and for the effect of spatial covariates. The model proposed here offers a relatively stable large-scale model structure, with model parameters which can be interpreted in the usual sense across levels of dependence. Since the model allows for separate dependence parameters for each variable, it offers, in essence, the equivalent of a model with a non-separable covariance function. The flexible model framework permits straightforward generalizations to structures with more than two variables, a temporal component, or an irregular lattice domain. Supplementary materials accompanying this paper appear on-line.

Kriging and Spatial Design Accelerated by Orders of Magnitude: Combining Low-Rank Covariance Approximations with FFT-Techniques

Computational power poses heavy limitations to the achievable problem size for Kriging. In separate research lines, Kriging algorithms based on FFT, the separability of certain covariance functions, and low-rank representations of covariance functions have been investigated, all three leading to drastic speedup factors. The current study combines these ideas, and so combines the individual speedup factors of all ideas. This way, we reduce the mathematics behind Kriging to a computational complexity of only $\mathcal{O}(dL^{*} \log L^{*})$ , where L ∗ is the number of points along the longest edge of the involved lattice of estimation points, and d is the physical dimensionality of the lattice. For separable (factorized) covariance functions, the results are exact, and nonseparable covariance functions can be approximated well through sums of separable components. Only outputting the final estimate as an explicit map causes computational costs of $\mathcal{O}(n)$ , where n is the number of estimation points. In illustrative numerical test cases, we achieve speedup factors up to 108 (eight orders of magnitude), and we can treat problem sizes of up to 15 trillion and two quadrillion estimation points for Kriging and spatial design, respectively, within seconds on a contemporary desktop computer. The current study assumes second-order stationarity and simple Kriging on a regular, equispaced lattice, without working with restricted neighborhoods. Extensions to many other cases are straightforward.

Non-Gaussian spatiotemporal modelling through scale mixing

We construct non-Gaussian processes that vary continuously in space and time with nonseparable covariance functions. Starting from a general and flexible way of constructing valid nonseparable covariance functions through mixing over separable covariance functions, the resulting models are generalized by allowing for outliers as well as regions with larger variances. We induce this through scale mixing with separate positive-valued processes. Smooth mixing processes are applied to the underlying correlated processes in space and in time, thus leading to regions in space and time of increased spread. An uncorrelated mixing process on the nugget effect accommodates outliers. Posterior and predictive Bayesian inference with these models is implemented through a Markov chain Monte Carlo sampler. An application to temperature data in the Basque country illustrates the potential of this model in the identification of outliers and regions with inflated variance, and shows that this improves the predictive performance.

A general class of nonseparable space-time covariance models

The aim of this work is to construct nonseparable, stationary covariance functions for processes that vary continuously in space and time. Stochastic modeling of phenomena over space and time is important in many areas of applications such as environmental sciences, agriculture and meteorology. But choice of an appropriate model can be difficult as one must take care to use valid covariance structures. A common choice for the process is a product of purely spatial and temporal random processes. In this case, the resulting process possesses a separable covariance function. Although these models are guaranteed to be valid, they are severely limited, since they do not allow space-time interactions. In this work we propose a general and flexible way of constructing valid nonseparable covariance functions derived through mixing over separable covariance functions. The proposed model allows for different degrees of smoothness across space and time and long-range dependence in time. Moreover, our proposal has as particular cases several covariance models proposed in the literature such as the Matern and the Cauchy Class. We use a Markov chain Monte Carlo sampler for Bayesian inference and apply our modeling approach to the Irish wind data.

Covariance functions that are stationary or nonstationary in space and stationary in time

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio‐temporal processes. Although several nonseparable space–time covariance models are available in the literature, very few of them can be used for spatially anisotropic data. In this paper, we propose a new class of stationary nonseparable covariance functions that can be used for both geometrically and zonally anistropic data. In addition, we show some desirable mathematical features of this class. Another important aspect, only partially covered by the literature, is that of spatial nonstationarity. We show a very simple criteria allowing for the construction of space–time covariance functions that are nonseparable, nonstationary in space and stationary in time. Part of the theoretical results proposed in the paper will then be used for the analysis of Irish wind speed data as in HASLETT and RAFTERY (Applied Statistics, 38, 1989, 1).

Stationary space-time Gaussian fields and their time autoregressive representation

We compare two different modelling strategies for continuous space discrete time data. The first strategy is in the spirit of Gaussian kriging. The model is a general stationary space-time Gaussian field where the key point is the choice of a parametric form for the covariance function. In the main, covariance functions that are used are separable in space and time. Nonseparable covariance functions are useful in many applications, but construction of these is not easy. The second strategy is to model the time evolution of the process more directly. We consider models of the autoregressive type where the process at time t is obtained by convolving the process at time t − 1 and adding spatially correlated noise. Under specific conditions, the two strategies describe two different formulations of the same stochastic process. We show how the two representations look in different cases. Furthermore, by transforming time-dynamic convolution models to Gaussian fields we can obtain new covariance functions and by writing a Gaussian field as a time-dynamic convolution model, interesting properties are discovered. The computational aspects of the two strategies are discussed through experiments on a dataset of daily UK temperatures. Although algorithms for performing estimation, simulation, and so on are easy to do for the first strategy, more computer-efficient algorithms based on the second strategy can be constructed.

Blur-generated non-separable space–time models

Summary Statistical space–time modelling has traditionally been concerned with separable covariance functions, meaning that the covariance function is a product of a purely temporal function and a purely spatial function. We draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant. We show that this model has a non-separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operates successively in time: the spatial field at time t +1 is obtained by ‘blurring’ the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the length of the time steps goes to 0. This gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussian smoothing kernel. We also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously.