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Abstract Geostatistical modeling for continuous point‐referenced data has extensively been applied to neuroimaging because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging technique characterizing the brain's anatomical structure, produces a positive‐definite (p.d.) matrix for each voxel. Currently, only a few geostatistical models for p.d. matrices have been proposed because introducing spatial dependence among p.d. matrices properly is challenging. In this paper, we use the spatial Wishart process, a spatial stochastic process (random field), where each p.d. matrix‐variate random variable marginally follows a Wishart distribution, and spatial dependence between random matrices is induced by latent Gaussian processes. This process is valid on an uncountable collection of spatial locations and is almost‐surely continuous, leading to a reasonable way of modeling spatial dependence. Motivated by a DTI data set of cocaine users, we propose a spatial matrix‐variate regression model based on the spatial Wishart process. A problematic issue is that the spatial Wishart process has no closed‐form density function. Hence, we propose an approximation method to obtain a feasible Cholesky decomposition model, which we show to be asymptotically equivalent to the spatial Wishart process model. A local likelihood approximation method is also applied to achieve fast computation. The simulation studies and real data application demonstrate that the Cholesky decomposition process model produces reliable inference and improved performance, compared to other methods. 

Wind is a critical component of the Earth system and has unmistakable impacts on everyday life.The CYGNSS satellite mission improves observational coverage of ocean windsviaa fleet of eightmicro-satellites that use reflected GNSS signals to infer surface wind speed. We present analysescharacterizing variability in wind speed measurements among the eight CYGNSS satellites andbetween antennas, using a Gaussian process model that leverages comparisons between CYGNSSand Jason-3 during a one-year period from September 2019 to September 2020. The CYGNSS sen-sors exhibit a range of biases, mostly between1.0 m/s andþ0.2 m/s with respect to Jason-3,indicating that some CYGNSS sensors are biased with respect to one another and with respect toJason-3. The biases between the starboard and port antennas within a CYGNSS satellite aresmaller. Our results are consistent with, yet sharper than, a more traditional paired comparisonanalysis. We also explore the possibility that the bias depends on wind speed, finding some evi-dence that CYGNSS satellites have positive biases with respect to Jason-3 at low wind speeds.However, we argue that there are subtle issues associated with estimating wind speed-dependentbiases, so additional careful statistical modeling and analysis is warranted. 

Lightning is a destructive and highly visible product of severe storms, yet there is still much to be learned about the conditions under which lightning is most likely to occur. The GOES-16 and GOES-17 satellites, launched in 2016 and 2018 by NOAA and NASA, collect a wealth of data regarding individual lightning strike occurrence and potentially related atmospheric variables. The acute nature and inherent spatial correlation in lightning data renders standard regression analyses inappropriate. Further, computational considerations are foregrounded by the desire to analyze the immense and rapidly increasing volume of lightning data. We present a new computationally feasible method that combines spectral and Laplace approximations in an EM algorithm, denoted SLEM, to fit the widely popular log-Gaussian Cox process model to large spatial point pattern datasets. In simulations we find SLEM is competitive with contemporary techniques in terms of speed and accuracy. When applied to two lightning datasets, SLEM provides better out-of-sample prediction scores and quicker runtimes, suggesting its particular usefulness for analyzing lightning data which tend to have sparse signals. 

We introduce computational methods that allow for effective estimation of a flexible nonstationary spatial model when the field size is too large to compute the multivariate normal likelihood directly. In this method, the field is defined as a weighted spatially varying linear combination of a globally stationary process and locally stationary processes. Often in such a model, the difficulty in its practical use is in the definition of the boundaries for the local processes, and therefore, we describe one such selection procedure that generally captures complex nonstationary relationships. We generalize the use of a stochastic approximation to the score equations in this nonstationary case and provide tools for evaluating the approximate score in O(n log n ) operations and O(n) storage for data on a subset of a grid. We perform various simulations to explore the effectiveness and speed of the proposed methods and conclude by predicting average daily temperature. Supplementary materials for this article are available online. 

We propose computationally efficient methods for estimating stationary multivariate spatial and spatial–temporal spectra from incomplete gridded data. The methods are iterative and rely on successive imputation of data and updating of model estimates. Imputations are done according to a periodic model on an expanded domain. The periodicity of the imputations is a key feature that reduces edge effects in the periodogram and is facilitated by efficient circulant embedding techniques. In addition, we describe efficient methods for decomposing the estimated cross spectral density function into a linear model of coregionalization plus a residual process. The methods are applied to two storm datasets, one of which is from Hurricane Florence, which struck the southeastern United States in September 2018. The application demonstrates how fitted models from different datasets can be compared, and how the methods are computationally feasible on datasets with more than 200,000 total observations. 

We study the problem of sparse signal detection on a spatial domain. We propose a novel approach to model continuous signals that are sparse and piecewise-smooth as the product of independent Gaussian (PING) processes with a smooth covariance kernel. The smoothness of the PING process is ensured by the smoothness of the covariance kernels of the Gaussian components in the product, and sparsity is controlled by the number of components. The bivariate kurtosis of the PING process implies that more components in the product results in the thicker tail and sharper peak at zero. We develop an efficient computation algorithm based on spectral methods. The simulation results demonstrate superior estimation using the PING prior over Gaussian process prior for different image regressions. We apply our method to a longitudinal magnetic resonance imaging dataset to detect the regions that are affected by multiple sclerosis computation in this domain. Supplementary materials for this article are available online. 

We derive a single-pass algorithm for computing the gradient and Fisher information of Vecchia’s Gaussian process loglikelihood approximation, which provides a computationally efficient means for applying the Fisher scoring algorithm for maximizing the loglikelihood. The advantages of the optimization techniques are demonstrated in numerical examples and in an application to Argo ocean temperature data. The new methods find the maximum likelihood estimates much faster and more reliably than an optimization method that uses only function evaluations, especially when the covariance function has many parameters. This allows practitioners to fit nonstationary models to large spatial and spatial–temporal datasets. 

Abstract We conduct a study of the aliased spectral densities of Matérn covariance functions on a regular grid of points, providing clarity on the properties of a popular approximation based on stochastic partial differential equations. While others have shown that it can approximate the covariance function well, we find that it assigns too much power at high frequencies and does not provide increasingly accurate approximations to the inverse as the grid spacing goes to zero, except in the one-dimensional exponential covariance case.