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A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and uncertainty quantification of the map estimation, while resulting in a closed-form and invertible posterior map. We then focus on inferring the distribution of a nonstationary spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible and are motivated by the behavior of a large class of stochastic processes. Our approach is scalable to high-dimensional distributions due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for flexible marginals. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output. for this article are available online. [ABSTRACT FROM AUTHOR]

Spatio-temporal datasets are rapidly growing in size. For example, environmental variables are measured with increasing resolution by increasing numbers of automated sensors mounted on satellites and aircraft. Using such data, which are typically noisy and incomplete, the goal is to obtain complete maps of the spatio-temporal process, together with uncertainty quantification. We focus here on real-time filtering inference in linear Gaussian state-space models. At each time point, the state is a spatial field evaluated on a very large spatial grid, making exact inference using the Kalman filter computationally infeasible. Instead, we propose a multi-resolution filter (MRF), a highly scalable and fully probabilistic filtering method that resolves spatial features at all scales. We prove that the MRF matrices exhibit a particular block-sparse multi-resolution structure that is preserved under filtering operations through time. We describe connections to existing methods, including hierarchical matrices from numerical mathematics. We also discuss inference on time-varying parameters using an approximate Rao-Blackwellized particle filter, in which the integrated likelihood is computed using the MRF. Using a simulation study and a real satellite-data application, we show that the MRF strongly outperforms competing approaches. include Python code for reproducing the simulations, some detailed properties of the MRF and auxiliary theoretical results. [ABSTRACT FROM AUTHOR]

People are increasingly concerned with understanding their personal environment, including possible exposure to harmful air pollutants. To make informed decisions on their day-to-day activities, they are interested in real-time information on a localized scale. Publicly available, fine-scale, high-quality air pollution measurements acquired using mobile monitors represent a paradigm shift in measurement technologies. A methodological framework utilizing these increasingly fine-scale measurements to provide real-time air pollution maps and short-term air quality forecasts on a fine-resolution spatial scale could prove to be instrumental in increasing public awareness and understanding. The Google Street View study provides a unique source of data with spatial and temporal complexities, with the potential to provide information about commuter exposure and hot spots within city streets with high traffic. We develop a computationally efficient spatiotemporal model for these data and use the model to make short-term forecasts and high-resolution maps of current air pollution levels. We also show via an experiment that mobile networks can provide more nuanced information than an equally sized fixed-location network. This modeling framework has important real-world implications in understanding citizens' personal environments, as data production and real-time availability continue to be driven by the ongoing development and improvement of mobile measurement technologies. for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement. [ABSTRACT FROM AUTHOR]

We propose a new class of filtering and smoothing methods for inference in high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models. The main idea is to combine the ensemble Kalman filter and smoother, developed in the geophysics literature, with state-space algorithms from the statistics literature. Our algorithms address a variety of estimation scenarios, including online and off-line state and parameter estimation. We take a Bayesian perspective, for which the goal is to generate samples from the joint posterior distribution of states and parameters. The key benefit of our approach is the use of ensemble Kalman methods for dimension reduction, which allows inference for high-dimensional state vectors. We compare our methods to existing ones, including ensemble Kalman filters, particle filters, and particle MCMC. Using a real data example of cloud motion and data simulated under a number of nonlinear and non-Gaussian scenarios, we show that our approaches outperform these existing methods. for this article are available online. [ABSTRACT FROM AUTHOR]

Automated sensing instruments on satellites and aircraft have enabled the collection of massive amounts of high-resolution observations of spatial fields over large spatial regions. If these datasets can be efficiently exploited, they can provide new insights on a wide variety of issues. However, traditional spatial-statistical techniques such as kriging are not computationally feasible for big datasets. We propose a multi-resolution approximation (M-RA) of Gaussian processes observed at irregular locations in space. TheM-RA process is specified as a linear combination of basis functions at multiple levels of spatial resolution, which can capture spatial structure from very fine to very large scales. The basis functions are automatically chosen to approximate a given covariance function, which can be nonstationary. All computations involving theM-RA, including parameter inference and prediction, are highly scalable for massive datasets. Crucially, the inference algorithms can also be parallelized to take full advantage of large distributed-memory computing environments. In comparisons using simulated data and a large satellite dataset, theM-RA outperforms a related state-of-the-art method. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The ensemble Kalman filter (EnKF) is a computational technique for approximate inference in state-space models. In typical applications, the state vectors are large spatial fields that are observed sequentially over time. The EnKF approximates the Kalman filter by representing the distribution of the state with an ensemble of draws from that distribution. The ensemble members are updated based on newly available data by shifting instead of reweighting, which allows the EnKF to avoid the degeneracy problems of reweighting-based algorithms. Taken together, the ensemble representation and shifting-based updates make the EnKF computationally feasible even for extremely high-dimensional state spaces. The EnKF is successfully used in data-assimilation applications with tens of millions of dimensions. While it implicitly assumes a linear Gaussian state-space model, it has also turned out to be remarkably robust to deviations from these assumptions in many applications. Despite its successes, the EnKF is largely unknown in the statistics community. We aim to change that with the present article, and to entice more statisticians to work on this topic. [ABSTRACT FROM PUBLISHER]

Developing global maps of carbon dioxide (CO2) mole fraction (in units of parts per million) near the Earth’s surface can help identify locations where major amounts of CO2are entering and exiting the atmosphere, thus providing valuable insights into the carbon cycle and mitigating the greenhouse effect of atmospheric CO2. Existing satellite remote sensing data do not provide measurements of the CO2mole fraction near the surface. Japan’s Greenhouse gases Observing SATellite (GOSAT) is sensitive to average CO2over the entire column, and NASA’s Atmospheric InfraRed Sounder (AIRS) is sensitive to CO2in the middle troposphere. One might expect that lower-atmospheric CO2could be inferred by differencing GOSAT column-average and AIRS mid-tropospheric data. However, the two instruments have different footprints, measurement-error characteristics, and data coverages. In addition, the spatio-temporal domains are large, and the AIRS dataset is massive. In this article, we describe a spatio-temporal data-fusion (STDF) methodology based on reduced-dimensional Kalman smoothing. Our STDF is able to combine the complementary GOSAT and AIRS datasets to optimally estimate lower-atmospheric CO2mole fraction over the whole globe. Further, it is designed for massive remote sensing datasets and accounts for differences in instrument footprint, measurement-error characteristics, and data coverages. This article has supplementary material online. [ABSTRACT FROM AUTHOR]