Parameter Estimation for Differential Equations: a Generalized Smoothing Approach

TLDR

Differential equations model system outputs by linking derivatives to the process itself, yet current parameter‑estimation methods are computationally intensive and poorly suited for statistical inference and interval estimation. The study proposes a new method for estimating parameters in non‑linear differential equation models using noisy measurements on a subset of variables. The method modifies data‑smoothing techniques and generalizes profiled estimation to estimate the model parameters. It produces low‑bias estimates with good coverage in simulated chemical‑engineering and neurobiology data, and performs well on real chemistry and lupus datasets.

Abstract

Summary We propose a new method for estimating parameters in models that are defined by a system of non-linear differential equations. Such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself. Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation. The paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non-linear differential equations. The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation. We derive estimates and confidence intervals, and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology. The performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus.