Data-Adaptive Modeling using Convex Regression

Thesis (Ph.D.)--University of Washington, 2016-08

The collection and storage of large amounts of data has greatly increased in recent years, which has led to a greater interest in developing methods to explore and model this data. In settings in which little is known about the relationship among variables in advance, exploratory data analysis is particularly important. Furthermore, when the number of covariates is large, traditional exploratory data analysis becomes challenging. In this setting, fitting flexible and interpretable models can be a useful tool for visualizing the conditional relationships between the covariates and outcome. In Chapter 2, we present a method that adaptively selects covariates to include in the model and for those included, models their conditional associations with the outcome as piecewise constant functions with adaptively-chosen knots. In Chapter 3, we present a related method that is useful in settings in which interactions between pairs of covariates are of particular interest. In Chapter 4, we turn our attention to a specific type of data: calcium imaging data, which measures large populations of neurons at cellular resolution in behaving animals. As a first step of analyzing calcium imaging data, two goals must be accomplished: neuron identification and calcium quantification. Through combining image segmentation, clustering, and convex regression, we are able to extract the locations of neurons in the field of view, as well as estimate the neurons' intracellular calcium concentrations over time. This extracted data is useful in downstream analyses, which aim to provide unprecedented insight into neural activity.