Empirical evolution equations

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 paper, we address the problem of constructing general yet tractable methods for quantifying such uncertainties, by means of asymptotic theory combined with bootstrap methodology. We demonstrates these procedures in important examples including gradient line estimation, diffusion tensor imaging tractography, and local principal component analysis. The bootstrap perspective is particularly appealing as it circumvents the need to simulate from stochastic (partial) differential equations that depend on (infinite-dimensional) unknowns. We assess the performance of the bootstrap procedure via simulations and find that it demonstrates good finite-sample coverage.