Dissertation Defence: Differential Equation-Constrained Local Polynomial Regression Models
July 23 at 9:00 am - 1:00 pm
Chunlei Ge, supervised by Dr. John Braun, will defend their dissertation titled “Differential Equation-Constrained Local Polynomial Regression Models” in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics.
An abstract for Chunlei Ge’s dissertation is included below.
Examinations are open to all members of the campus community as well as the general public. This examination will be offered in hybrid format. Registration is not required to attend in person, but please email john.braun@ubc.ca to receive the Zoom link for this exam.
Abstract
This thesis develops a unified framework of differential equation constrained local polynomial regression (DE-constrained LPR) for nonparametric estimation of a smooth mean function g(x) whose derivative structure is governed by an ordinary differential equation (ODE). The central motivation is the prevalence of sparse, irregularly spaced data in the life and environmental sciences, where physical or biological knowledge encoded in an ODE can compensate for the scarcity of observations that would otherwise render conventional smoothers unreliable.
The framework is developed progressively across five model classes. The DE1-k estimator is first introduced for the exponential growth model g′(x) = λg(x) (Chapter 3), then extended to the nonlinear quasi-exponential model g′(x) = λgα(x) (Chapter 4), the general first-order linear ODE g′(x) = a(x)g(x) + b(x) with arbitrary coefficient functions (Chapter 5), and the fully general nonlinear constraint g′(x) = F(x,g(x)) (Chapter 6). Chapter 7 removes the first-order restriction, developing DE2-k and DE3-k estimators for second- and third-order ODE constraints, respectively.
For each model class, conditional asymptotic bias and variance expressions are derived, and data-driven bandwidth selection procedures are established. A key finding, consistent across all chapters, is that ODE constraints reduce bias without inflating variance relative to unconstrained local polynomial regression, with the largest gains occurring in sparse-design settings where physical structure is most needed to guide estimation. Corrected bias theorems for the DE2-k and DE3-k estimators, which account for the Jacobian structure of the higher-dimensional nonlinear least squares problem, are among the principal theoretical contributions of the thesis.
Three real-data applications demonstrate the breadth of the framework: mouse tumour growth data (exponential and quasi-exponential model), fire brand burning rate data with as few as three observations per group (general linear model), and Korean influenza-like illness surveillance data analysed via the SIR epidemiological model (higher-order model). In each setting, DE-constrained LPR recovers smooth, scientifically interpretable curves that unconstrained methods cannot produce at the available sample sizes.
