Analysis Tools for Small and Big Data Problems

The dissertation focuses on two separate problems. Each is informed by real-world applications. The first problem involves the assessment of an ordinal measurement system in a manufacturing setting. A random-effects model is proposed that is applicable to this repeatability and reproducibility context, and a Bayesian framework is adopted to facilitate inference. This first problem is an example of an analysis tool to solve a small data problem.;The second problem involves statistical machine learning applied to big data problems. As more and more data become available, a need increases to automate the ability to identify particularly relevant features in a prediction or forecasting context. This often involves expanding features using kernel functions to better facilitate predictive capabilities. Simultaneously, there are often manifolds embedded within big data structures that can be exploited to improve predictive performance on real data sets. Bringing together manifold learning with kernel methods provides a powerful and novel tool developed in this dissertation.;This dissertation has the advantage of contributing to a more-classical problem in statistics involving ordinal data and to cutting edge machine learning techniques for the analysis of big data. It is our contention that statisticians need to understand both problem types. The novel tools developed here are demonstrated on practical applications with strong results