Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data

Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. More recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications

Spatio-temporal statistical models with application to atmospheric processes

This dissertation is concerned with spatio-temporal processes in the Atmospheric Sciences;In the first chapter, a comprehensive overview of spatio-temporal methods from the atmospheric science literature is presented. Focus is on Empirical Orthogonal Function (EOF), Principal Interaction Pattern (PIP), Principal Oscillation Pattern (POP), and spatio-temporal Canonical Correlation Analysis (CCA) methods. Previously unexamined issues related to measurement error, continuous space, and Bayesian ideas are considered;In the second chapter, harmonic analysis is used to make diagnostic inference about the spatial variation of the semiannual oscillation (SAO) in the Northern Hemisphere (NH) 500-hPa height field. The SAO is explained by the spatial and temporal asymmetries in the annual variation of stationary eddies. The SAO in the NH extratropics is a result of east-west land-sea contrasts, analogous to the well-known Southern Hemisphere (SH) SAO, which is explained by north-south land-sea contrasts;The third chapter examines the seasonal variability of mixed Rossby-gravity waves (MRGWs) in the lower stratosphere over the tropical western Pacific. Thirty-one years of lower stratospheric wind observations from four tropical Pacific stations are examined with seasonally varying cross-spectral analysis, which suggests significant twice-yearly peaks in the v-wind power and the mean squared coherence between the u- and v-winds, with peaks occurring in the winter-early spring and in summer-early fall. Horizontal momentum flux convergence is found with these waves, with the sign of the convergence opposite during the two seasonal maxima. Cyclic spectral analyses show that the frequency of the maximum v-wind power in the MRGW frequency band shifts seasonally;In the fourth chapter, a spatio-temporal statistical model is proposed that assumes a first-order Markov dynamic process combined with a spatially descriptive colored noise process. With a measurement error equation, a spatio-temporal Kalman filter gives predictions in time and at any spatial location. The model prediction equation includes a simple kriging analog as a special case. The model predicts well with simulated spatio-temporal data, and is superior to simple kriging applied independently at each time. Predictions of precipitation over the data-sparse South China Sea captures the dynamic variation of the spatial precipitation

Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models

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 on-line 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

Hierarchical Bayesian Models for Predicting The Spread of Ecological Processes

This is the pre-print version of the article found in Ecology (http://www.esajournals.org/loi/ecol).There is increasing interest in predicting ecological processes. Methods to accomplish such predictions must account for uncertainties in observation, sampling, models, and parameters. Statistical methods for spatio-temporal processes are powerful, yet difficult to implement in complicated, high-dimensional settings. However, recent advances in hierarchical formulations for such processes can be utilized for ecological prediction. These formulations are able to account for the various sources of uncertainty, and can incorporate scientific judgment in a probabilistically consistent manner. In particular, analytical diffusion models can serve as motivation for the hierarchical model for invasive species. We demonstrate by example that such a framework can be utilized to predict spatially and temporally, the house finch relative population abundance over the eastern United States.This research has been supported by a grant from the U.S. Environmental Protection Agency's Science to Achieve Results (STAR) program, Assistance Agreement No. R827257-01-0

Bayesian Semiparametric Hierarchical Empirical Likelihood Spatial Models

We introduce a general hierarchical Bayesian framework that incorporates a flexible nonparametric data model specification through the use of empirical likelihood methodology, which we term semiparametric hierarchical empirical likelihood (SHEL) models. Although general dependence structures can be readily accommodated, we focus on spatial modeling, a relatively underdeveloped area in the empirical likelihood literature. Importantly, the models we develop naturally accommodate spatial association on irregular lattices and irregularly spaced point-referenced data. We illustrate our proposed framework by means of a simulation study and through three real data examples. First, we develop a spatial Fay-Herriot model in the SHEL framework and apply it to the problem of small area estimation in the American Community Survey. Next, we illustrate the SHEL model in the context of areal data (on an irregular lattice) through the North Carolina sudden infant death syndrome (SIDS) dataset. Finally, we analyze a point-referenced dataset from the North American Breeding Bird survey that considers dove counts for the state of Missouri. In all cases, we demonstrate superior performance of our model, in terms of mean squared prediction error, over standard parametric analyses.Comment: 29 pages, 3 figue