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For spatially dependent functional data, a generalized Karhunen-Loève expansion is commonly used to decompose data into an additive form of temporal components and spatially correlated coefficients. This structure provides a convenient model to investigate the space-time interactions, but may not hold for complex spatio-temporal processes. In this work, we introduce the concept of weak separability, and propose a formal test to examine its validity for non-replicated spatially stationary functional field. The asymptotic distribution of the test statistic that adapts to potentially diverging ranks is derived by constructing lag covariance estimation, which is easy to compute for practical implementation. We demonstrate the efficacy of the proposed test via simulations and illustrate its usefulness in two real examples: China PM 2.5 data and Harvard Forest data. for this article are available online. [ABSTRACT FROM AUTHOR]

Regularization has become a primary tool for developing reliable estimators of the covariance matrix in high-dimensional settings. To curb the curse of dimensionality, numerous methods assume that the population covariance (or inverse covariance) matrix is sparse, while making no particular structural assumptions on the desired pattern of sparsity. A highly-related, yet complementary, literature studies the specific setting in which the measured variables have a known ordering, in which case a banded population matrix is often assumed. While the banded approach is conceptually and computationally easier than asking for "patternless sparsity," it is only applicable in very specific situations (such as when data are measured over time or one-dimensional space). This work proposes a generalization of the notion of bandedness that greatly expands the range of problems in which banded estimators apply. We develop convex regularizers occupying the broad middle ground between the former approach of "patternless sparsity" and the latter reliance on having a known ordering. Our framework defines bandedness with respect to a known graph on the measured variables. Such a graph is available in diverse situations, and we provide a theoretical, computational, and applied treatment of two new estimators. An R package, called ggb, implements these new methods. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Numerical climate model simulations run at high spatial and temporal resolutions generate massive quantities of data. As our computing capabilities continue to increase, storing all of the data is not sustainable, and thus it is important to develop methods for representing the full datasets by smaller compressed versions. We propose a statistical compression and decompression algorithm based on storing a set of summary statistics as well as a statistical model describing the conditional distribution of the full dataset given the summary statistics. We decompress the data by computing conditional expectations and conditional simulations from the model given the summary statistics. Conditional expectations represent our best estimate of the original data but are subject to oversmoothing in space and time. Conditional simulations introduce realistic small-scale noise so that the decompressed fields are neither too smooth nor too rough compared with the original data. Considerable attention is paid to accurately modeling the original dataset—1 year of daily mean temperature data—particularly with regard to the inherent spatial nonstationarity in global fields, and to determining the statistics to be stored, so that the variation in the original data can be closely captured, while allowing for fast decompression and conditional emulation on modest computers. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

In this article, we propose two methods for estimating space and space-time covariance functions from a Gaussian random field, based on the composite likelihood idea. The first method relies on the maximization of a weighted version of the composite likelihood function, while the second one is based on the solution of a weighted composite score equation. This last scheme is quite general and could be applied to any kind of composite likelihood. An information criterion for model selection based on the first estimation method is also introduced. The methods are useful for practitioners looking for a good balance between computational complexity and statistical efficiency. The effectiveness of the methods is illustrated through examples, simulation experiments, and by analyzing a dataset on ozone measurements. [ABSTRACT FROM AUTHOR]

The technology to harvest electricity from wind energy is now advanced enough to make entire cities powered by it a reality. High-quality, short-term forecasts of wind speed are vital to making this a more reliable energy source. Gneiting et al. (2006) have introduced a model for the average wind speed two hours ahead based on both spatial and temporal information. The forecasts produced by this model are accurate, and subject to accuracy, the predictive distribution is sharp, that is, highly concentrated around its center. However, this model is split into nonunique regimes based on the wind direction at an offsite location. This paper both generalizes and improves upon this model by treating wind direction as a circular variable and including it in the model. It is robust in many experiments, such as predicting wind at other locations. We compare this with the more common approach of modeling wind speeds and directions in the Cartesian space and use a skew- t distribution for the errors. The quality of the predictions from all of these models can be more realistically assessed with a loss measure that depends upon the power curve relating wind speed to power output. This proposed loss measure yields more insight into the true value of each model’s predictions. [ABSTRACT FROM AUTHOR]

Ecologists use the relative abundance of fossil pollen in sediments to estimate how tree species abundances change over space and time. To predict historical forest composition and quantify the available information, we build a Bayesian hierarchical model of forest composition in central New England, USA, based on pollen in a network of ponds. The critical relationships between abundances of taxa in the pollen record and abundances as actual vegetation are estimated for the modern and colonial periods, for which both pollen and direct vegetation data are available, based on a latent multivariate spatial process representing forest composition. For time periods in the past with only pollen data, we use the estimated model parameters to constrain predictions about the latent spatio-temporal process conditional on the pollen data. We develop an innovative graphical assessment of feature significance to help to infer which spatial patterns are reliably estimated. The model allows us to estimate the spatial distribution and relative abundances of tree species over the last 2,500 years, with an assessment of uncertainty, and to draw inference about how these patterns have changed over time. Cross-validation suggests that our feature significance approach can reliably indicate certain large-scale spatial features for many taxa, but that features on scales smaller than 50 km are difficult to distinguish, as are large-scale features for some taxa. We also use the model to quantitatively investigate ecological hypotheses, including covariate effects on taxa abundances and questions about pollen dispersal characteristics. The critical advantages of our modeling approach over current ecological analyses are the explicit spatio-temporal representation, quantification of abundance on the scale of trees rather than pollen, and uncertainty characterization. [ABSTRACT FROM AUTHOR]

Maximum likelihood estimation of branching point process models via numerical optimization procedures can be unstable and computationally intensive. We explore an alternative estimation method based on the expectation-maximization algorithm. The method involves viewing the estimation of such branching processes as analogous to incomplete data problems. Using an application from seismology, we show how the epidemic-type aftershock sequence (ETAS) model can, in fact, be estimated this way, and we propose a computationally efficient procedure to maximize the expected complete data log-likelihood function. Using a space-time ETAS model, we demonstrate that this method is extremely robust and accurate and use it to estimate declustered background seismicity rates of geologically distinct regions in Southern California. All regions show similar declustered background intensity estimates except for the one covering the southern section of the San Andreas fault system to the east of San Diego in which a substantially higher intensity is observed. [ABSTRACT FROM AUTHOR]

We propose a unified framework for testing various assumptions commonly made for covariance functions of stationary spatio-temporal random fields. The methodology is based on the asymptotic normality of space-time covariance estimators. We focus on tests for full symmetry and separability in this article, but our framework naturally covers testing for isotropy and Taylor's hypothesis. Our test successfully detects the asymmetric and nonseparable features in two sets of wind speed data. We perform simulation experiments to evaluate our test and conclude that our method is reliable and powerful for assessing common assumptions on space-time covariance functions. [ABSTRACT FROM AUTHOR]

This work considers a number of properties of space-time covariance functions and how these relate to the spatial-temporal interactions of the process. First, it examines how the smoothness away from the origin of a space-time covariance function affects, for example, temporal correlations of spatial differences. Models that are not smoother away from the origin than they are at the origin, such as separable models, have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances. Smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments. These results are used to obtain a parametric class of spectral densities whose corresponding space-time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time. Second, this work considers models that are asymmetric in space-time; the covariance between site x at time t and site y at time s is different than the covariance between site x at time s and site y at time t. A general approach is described for generating asymmetric models from symmetric models by taking derivatives. Finally, the implications of a Markov assumption in time on space-time covariance functions for Gaussian processes are examined, and an explicit characterization of all such continuous covariance functions is given. Several of the new models described in this work are applied to wind data from Ireland. [ABSTRACT FROM AUTHOR]

This article is concerted with objective estimation of the spatial intensity function of the background earthquake occurrences from an earthquake catalog that includes numerous clustered events in space and time, and also with an algorithm for producing declustered catalogs from the original catalog. A space-time branching process model (the ETAS model) is used for describing how each event generates offspring events. It is shown that the background intensity function can be evaluated if the total spatial seismicity intensity and the branching structure can be estimated, In fact the whole space-time process is split into two subprocesses, the background events and the clustered events. The proposed algorithm combines a parametric maximum likelihood estimate of the clustering structures using the space time ETAS model and a nonparametric estimate of the background seismicity that we call a variable weighted kernel estimate. To demonstrate the present methods, we estimate the background seismic activities in the central region of New Zealand and in the central and western regions of Japan, then use these estimates to produce catalogs of background events. [ABSTRACT FROM AUTHOR]

This article presents a multivariate spatial prediction methodology in a Bayesian framework. The method is especially suited for use in environmetrics, where vector-valued responses are observed at a small set of ambient monitoring staions "(gauged sites)" at successive time points. However, the stations may have varying start-up times so that the data have a "staircase" pattern("monotone" pattern in the terminology of Rubin and Shaffer). The lowest step corresponds to the newest station in the monitoring network. We base our approach on a hierarchical Bayes prior involving a Gaussian generalized inverted Wishart model. For given hyperparameters, we derive the predictive distribution for currently gauged sites at times before their start-up when no measurements were taken. The resulting predictive distribution is a matric / distribution with appropriate covariance parameters and degrees of freedom. We estimate the hyperparameters using the method of moments (MOM) as an easy-to-implement alternative tot he more complex EM algorithm. The MOM in particular gives exact parameter estimates and involves less cumbersome calculation than the EM algorithm. Finally, we obtain the predictive distribution for unmeasured responses at "ungauged" sites. The results obtained here allow us to pool the data from different sites that measure different pollutants and also to treat cases where the observed data monitoring stations have a monotonic "staircase" structure. We demonstrate the use of this methodology by mapping PM3.5 fields for Philadelphia during the period of May 1992 to September 1993. Large amounts of data missing by design make this application challenging. We give empirical evidence that the method performs well. [ABSTRACT FROM AUTHOR]

Opinion. Presents the authors' view on dataset for studying the spatial variation of ozone levels. Information about pseudoreplicates; Advantages of space-time datasets.

Starting in the early 1970s, the decline in column ozone over much of the earth has received much attention. Satellite ozone data, with the advantage of global coverage, now play an important role in assessing long-term trends in ozone distributions. We consider a class of space-time regression models for the analysis of satellite data on a fixed latitude, which take into account temporal and longitudinal dependence of the observations. The models can be used to test the uniformity of long-term trends in different longitudinal ozone series. Using the property of circular matrices, explicit expressions of the likelihood functions are obtained. Asymptotic properties of the parameter estimates are briefly discussed. A diagnostic method is proposed to tentatively select the orders in the noise terms of the models. The space-time regression models are applied to the total ozone mapping spectrometer (TOMS) data for trend assessment. [ABSTRACT FROM AUTHOR]

A marked spatial point pattern of trees and their diameters is the result of a dynamic biological process that takes place over time as well as space. Such patterns can be modeled as realizations of marked space-time survival point processes, where trees are born at some random location and time and then live, grow, and produce offspring in a random fashion A model for a marked space-time survival point process is fit to data from a longleaf pine (Pinus palustris) forest in southern Georgia The space-time survival point process is divided into three components a birth process. a growth process, and a survival process Each of the component processes is analyzed individually, from which conclusions regarding the dynamic ecological processes can be made By using this reductionist approach. questions concerning each individual process can be addressed that might not have been answerable otherwise. [ABSTRACT FROM AUTHOR]

Real data experiments show that, for large areas, the area average rain rate and the fraction of the area covered by rain rate above a fixed threshold are highly correlated. For the right choice of the threshold level the correlation can easily exceed 95% and may even reach 99%. This remarkable fact observed in nature is the basis for the so-called threshold method for measuring rainfall from space via satellite. The method depends critically on the threshold level, showing significant improvement in performance for optimal and nearly optimal thresholds. Under the assumption that the continuous part of the distribution of rain rate is lognormal, two theoretically derived optimal levels agree closely with the best level obtained empirically from a well-known data set of rain rate. [ABSTRACT FROM AUTHOR]

Opinion. Presents the author's view on Carroll and colleagues' paper that considers ambient ozone pollution in Harris County, Texas between 1980 and 1993. Some features of the space-time model that require discussion; Comments about styles of space-time modelling.