11/5/18
Subhadip Pal, Assistant Professor, Department of Bioinformatics and Biostatistics, School of Public Health and Information Science, University of Louisville
A Bayesian Framework for Modeling Data on the Stiefel Manifold
Directional data emerges in a wide array of applications, ranging from atmospheric sciences to medical imaging. Modeling such data, however, poses unique challenges by virtue of their being constrained to non-Euclidean spaces like manifolds. Here, we present a unified Bayesian framework for inference on the Stiefel manifold using the Matrix Langevin distribution. Specifically, we propose a novel family of conjugate priors and establish a number of theoretical properties relevant to statistical inference.
Conjugacy enables translation of these properties to their corresponding posteriors, which we exploit to develop the posterior inference scheme. For the implementation of the posterior computation, including the posterior sampling, we adopt a novel computational procedure for evaluating the hypergeometric function of matrix arguments that appears as normalization constants in the relevant densities.
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11/2/18
Daniel Simpson, Assistant Professor, Department of Statistical Sciences, University of Toronto
Esther Williams in the Harold Holt Memorial Swimming Pools: Practical Problems with Complex Statistical Models
Thirty years of computational research have lead us to a place where statisticians and quantitative scientists can routinely fit extremely complex models to very messy data. In this talk, I'm going to talk about some ways in which we can keep track on that complexity when building Bayesian models and find problems with our model fits.
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10/22/18
Professor Yimin Xiao, Department of Statistics and Probability, Michigan State University
Gaussian Random Fields on the Sphere: Some Probabilistic and Statistical Properties
Let $T= \{T(x), x \in {\mathbb{S}}^{2}\}$ be a real-valued Gaussian random field, where ${\mathbb S}^2$ is the unit sphere in $\mathbb R^3$. This talk is concerned with probabilistic and statistical properties of $T$.
When $T$ is isotropic, we show that the regularity, geometric properties, and optimal bounds for the prediction errors of the spherical Gaussian field $T$ are explicitly determined by the high-frequency behavior of its angular power spectrum. It can be shown that similar results also hold for random intrinsic functions of order $\kappa=1$.
If time permits, we will discuss extensions of these results to anisotropic and/or multivariate Gaussian random fields on the sphere ${\mathbb S}^{d-1}$ or on $\R_+\times {\mathbb S}^{2}$.
This talk is based on joint works with Dan Cheng, Xiaohong Lan, and Domenico Marinucci.
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10/15/18
D. Andrew Brown, Assistant Professor, Department of Mathematical Sciences, Clemson University
Some Statistical Problems in Uncertainty Quantification
Scientists and engineers are becoming increasingly reliant on computer models for simulating the behavior of complex physical processes that often are only partially understood. Studying how faithful these models are to reality involves collecting physical data, which can be resource-intensive so that limited data are available. Further, evaluating the computer code can be computationally expensive. The field of uncertainty quantification (UQ), or computer experiments, sits at the intersection of modeling complex systems and the data collected to study those models. The goal is to quantify the uncertainties associated with using such models. In this talk, I will discuss a couple of UQ projects that I, as a statistician, have done in cooperation with engineers and applied mathematicians. The talk will be in two parts. In the first part, I will present some recent work on the so-called computer model calibration problem in which some calibration parameters are thought to be functions of the control inputs. This problem is approached from the perspective of treating the computer code a black box, wherein a statistician is only concerned with its inputs and outputs. In the second part of the talk, I will discuss an approach to accelerating the Bayesian computations necessary for solving a so-called hierarchical Bayesian inverse problem. This latter approach “breaks open the black box,” allowing us to use a low-rank approximation to the covariance matrix to more efficiently access the posterior distribution of plausible solutions.
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10/8/18
Christopher K. Wikle, Curators’ Distinguished Professor and Chair, Department of Statistics, University of Missouri
Using parsimonious “deep” models from machine learning to facilitate efficient implementation of multiscale spatio-temporal statistical models
Spatio-temporal data are ubiquitous in engineering and the sciences, and their study is important for understanding and predicting a wide variety of processes. One of the chief difficulties in modeling spatial processes that change with time is the complexity of the dependence structures that must describe how such a process varies, and the presence of high-dimensional complex datasets and large prediction domains. It is particularly challenging to specify parameterizations for nonlinear dynamical spatio-temporal models that are simultaneously useful scientifically and efficient computationally. Current statistical methodologies for modeling these processes are often highly parameterized and thus, challenging to implement from a computational perspective. One potential parsimonious solution to this problem is a method from the dynamical systems and engineering literature referred to as an echo state network (ESN). ESN models use so-called reservoir computing to efficiently compute recurrent neural network (RNN) forecasts. Moreover, so-called “deep” models have recently been shown to be successful at predicting high-dimensional complex nonlinear processes, particularly those with multiple spatial and temporal scales of variability (such as we often find in spatio-temporal environmental data). Here we introduce a deep ensemble ESN (D-EESN) model in a hierarchical Bayesian framework that naturally accommodates non-Gaussian data types and multiple levels of uncertainties. The methodology is first applied to a data set simulated from a novel non-Gaussian multiscale Lorenz-96 dynamical system simulation model and then to a long-lead United States (U.S.) soil moisture forecasting application.
