Browsing by Subject "Shrinkage"

Bayesian latent factor models are key tools for modeling linear structure in data and performing dimension reduction for correlated variables. Recent advances in prior specification allow the estimation of semi- and non-parametric infinite factor mod- els. These models provide significant theoretical and practical advantages at the cost of computationally intensive sampling and non-identifiability of some parameters. We provide a package for the R programming environment that includes functions for sampling from the posterior distributions of several recent latent factor mod- els. These computationally efficient samplers are provided for R with C++ source code to facilitate fast sampling of standard models and provide component sam- pling functions for more complex models. We also present an efficient algorithm to remove the non-identifiability that results from the included shrinkage priors. The infinitefactor package is available in developmental version on GitHub at https://github.com/poworoznek/infinitefactor and in release version on the CRAN package repository.

Knowledge of the drying process of geomaterials is meaningful and helpful in the field of geotechnical and geo-environmental engineering. This experimental study focuses on drying tests on geomaterial samples with monitored surface moisture contents and controlled environmental conditions. Using the digital image correlation (DIC) method to analyze the sample’s displacement, the 3D displacement plots and volumetric strain maps are obtained after calculation. By combining the monitored moisture content with analyzed displacement and volumetric strain plots, the phenomenon and characteristics of a geomaterial’s drying process are discussed and concluded. This study offers a better understanding of the deformation of geomaterials during the drying process in 3D.

The focus of this dissertation is on geometric aspects of point estimation problems. In the first half of this work, we examine the estimation of location parameters for non-Euclidean data that lies in a known manifold or metric space. Ideas from statistical decision theory motivate the construction of new estimators for location parameters. The second half of this work explores information geometric aspects of covariance matrix estimation. In a regular statistical model the Fisher information metric endows the parameter space with a Riemannian manifold structure. Parameter estimation can therefore also be viewed as problem in non-Euclidean data analysis.

Chapter 2 introduces and formalizes the problem of estimating Frechet mean location parameters for metric space valued data. We highlight the importance of the isometry group of distance preserving transformations, and how this group interacts with Frechet means. Pitman's minimum risk equivariant estimator for location models in Euclidean space is generalized to manifold settings, where we discuss aspects of the performance and computation of this minimum risk equivariant estimator.

Turning from equivariant estimation to shrinkage estimation, Chapter 3 introduces a shrinkage estimator for Frechet means that is inspired by Stein's estimator. This estimator utilizes metric space geodesics to shrink an estimate towards a pre-specified, shrinkage point. It is shown that the performance of this geodesic James-Stein estimator depends on the curvature of the underlying metric space, where shrinkage is especially beneficial in non-positively curved spaces.

Chapter 4 discusses shrinkage estimation for covariance matrices that approximately have a Kronecker product structure. The idea of geodesic shrinkage can be applied with respect to alpha-geodesics that arise from the information geometry of a statistical model. These alpha-geodesics also lead to interpretable parameter space decompositions. In a Wishart model we propose an empirical Bayes procedure for estimating approximate Kronecker covariance matrices; a procedure which can be viewed as shrinkage along (-1)-geodesics.

The last chapter of this work further discusses information geometric aspects of covariance estimation, with a view towards asymptotic efficiency. The asymptotic performance of the partial trace estimator for a Kronecker covariance is contrasted with the maximum likelihood estimator. A correction to the partial trace estimator is proposed which is both asymptotically efficient and simple to compute.