Selected Problems for High-Dimensional Data - Quantile and Errors-in-Variables Regressions.

Abstract

This dissertation addresses two problems. First, we study joint quantile regression at multiple quantile levels with high dimensional covariates. Variable selection performed at individual quantile levels may lack stability across neighboring quantiles, making it difficult to understand and to interpret the impact of a given covariate on conditional quantile functions. We propose a Dantzig-type penalization method for sparse model selection at each quantile level which at the same time aims to shrink differences of the selected models across neighboring quantiles. We show model selection consistency, and investigate stability of the selected models across quantiles. In the second part of the thesis, we consider the class of covariance models that can be expressed as a Kronecker sum. Taking advantage of our theoretical analysis on matrix decomposition, we demonstrate that our methodology yields computationally efficient and statistically convergent estimates. We show that this decomposition may correspond to a representation of the data as signal plus additive noise. This may in turn be used in a regression framework to accommodate measurement error. We assess performance using simulations and illustrate the methods using a study of hawkmoth flight control (Sponberg et al. 2015). We find that the decomposition successfully isolates signal and noise, and reveals a stronger neural encoding relationship than otherwise would be obtained.