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.
Highlights
Given a Banach space X and a vector field v on X, an evolution equation (Walker, 1980) is a model of the form γ = v(γ) and γ(0) = x0 ∈ X, (1.1)specifying a flow γ : R → X by relating its time derivative γto the field v
In order to establish valid general inference procedures based on the empirical integral curve, we investigate how the asymptotic behavior of the estimator vn of v relates that of the estimator γn of γ (Section 2) when the state space X is a Banach space
We show how our results can be applied in the context of several modern statistical and machine learning problems (Section 5). These include finite-dimensional evolution equations arising in gradient line estimation, diffusion tensor tractography, and principal curve estimation; and an infinite dimensional setting describing anisotropic heat flow
Summary
The diversity of applied settings and methodological constructions can lead to a multitude of corresponding problem-specific approaches, but the elegant general specification begs the question whether a broad framework of uncertainty quantification that can be tractably ported to each specific context is feasible This question motivates us to study general evolution equations of the form (1.1) through the lens of sampling variation, when an unknown vector field v is replaced by a (potentially nonparametric) estimate vn depending on sampled data whose “sample size” (in a general sense) is n. We show how our results can be applied in the context of several modern statistical and machine learning problems (Section 5) These include finite-dimensional evolution equations arising in gradient line estimation, diffusion tensor tractography, and principal curve estimation; and an infinite dimensional setting describing anisotropic heat flow. The proofs of all formal statements are collected in Appendix B
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