Grad Student Seminar: Jack Prothero & Marie Dueker

Jack Prothero – UNC-Chapel Hill
Extracting Signal and Noise from Large Matrices

Discerning a low-rank signal from a large, noisy data matrix is a classic problem in statistics and signal processing. We review a fundamental result in random matrix theory and how it applies to recent results on optimal singular value thresholding and shrinkage for signal extraction. In toy examples we find that these thresholding procedures often fail to capture as much signal as is theoretically possible. Our main contribution is a diagnostic quantile-quantile graphic for evaluating signal extraction quality. Such a diagnostic is reliant on an appropriate estimate of remaining noise after signal is extracted from a matrix. For non-square matrices this noise matrix estimation is surprisingly nontrivial; we discuss the challenges that arise and propose potential solutions to these challenges.

Marie Dueker – Ruhr-Universitaet Bochum
Common deterministic trends in varying means model

For possibly nonstationary multivariate data collected over time, common deterministic trends are linear combinations of different components that are stationary over time. The trend in mean in the underlying model is determined by a vector of deterministic functions. Then, the largest number of linearly independent linear combinations that lead to stationarity in mean is referred to as cotrending dimension, and their spanned space as cotrending space. The cotrending dimension and the cotrending space are related to the eigenstructure and eigenspace, respectively, of suitable matrices. Testing procedures for both dimension and space are presented. The talk will also discuss a possible change in the cotrending dimension over time. The results are illustrated by some simulations and data examples.