Empirical Evolution Equations

January 22, 2018 - 11:15am to 12:15pm
Biostatistics, UMN
Mechanical Engineering 212
ABSTRACT: Evolution equations comprise a broad framework for describing the dynamics of a system in a general state space: when the state space is finite-dimensional, they give rise to systems of ordinary differential equations; for infinite-dimensional state spaces, they give rise to partial differential equations. Several modern statistical and machine learning methods concern the estimation of objects that can be formalized as solutions to evolution equations, in some appropriate state space, even if not stated as such. The corresponding equations, however, are seldom known exactly, and are empirically derived from data, often by means of non-parametric estimation. This induces uncertainties on the equations and their solutions that are challenging to quantify, and moreover the diversity and the specifics of each particular setting may obscure the path for a general approach. In this work, we address the problem of constructing general yet tractable methods for quantifying such uncertainties, by means of asymptotic theory combined with bootstrap methodology. We develop valid nonparametric bootstrap procedures for empirical evolution equations, and show how these apply in important examples including gradient line estimation, diffusion tensor imaging tractography, and local principal curve estimation.

BIO: Susan Wei is an Assistant Professor in the Division of Biostatistics at the University of Minnesota. Her research interests lie at the intersection of statistics and machine learning and focuses on tackling the demands of modern data analysis. In particular her methodology works have been inspired by applications in medical imaging and medical devices.