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Bias and Coverage Properties of the WENDy-IRLS Algorithm

Abhi Chawla, David M. Bortz, Vanja Dukic

TL;DR

This paper evaluates the bias and coverage properties of the original WENDy-IRLS algorithm across five benchmark ODE models under four noise distributions and varying data resolutions. By formulating dynamics in a weak form and applying iteratively reweighted least squares, the study demonstrates robust parameter and state estimation under substantial noise, with coverage generally near nominal for many models but notable challenges for FHN and Hindmarsh-Rose in high-noise or low-resolution regimes. Key findings show that higher data resolution improves coverage and reduces bias, while certain noise structures (e.g., MLN, ACN, ATN) induce parameter-specific biases that may require correction or more data. The results support WENDy as a fast, noise-tolerant tool for parameter/state inference in dynamical systems, and offer practical data-density guidance (e.g., minimum data points per state) to achieve reliable inference in noisy settings.

Abstract

The Weak form Estimation of Nonlinear Dynamics (WENDy) method is a recently proposed class of parameter estimation algorithms that exhibits notable noise robustness and computational efficiency. This work examines the coverage and bias properties of the original WENDy-IRLS algorithm's parameter and state estimators in the context of the following differential equations: Logistic, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark. The estimators' performance was studied in simulated data examples, under four different noise distributions (normal, log-normal, additive censored normal, and additive truncated normal), and a wide range of noise, reaching levels much higher than previously tested for this algorithm.

Bias and Coverage Properties of the WENDy-IRLS Algorithm

TL;DR

This paper evaluates the bias and coverage properties of the original WENDy-IRLS algorithm across five benchmark ODE models under four noise distributions and varying data resolutions. By formulating dynamics in a weak form and applying iteratively reweighted least squares, the study demonstrates robust parameter and state estimation under substantial noise, with coverage generally near nominal for many models but notable challenges for FHN and Hindmarsh-Rose in high-noise or low-resolution regimes. Key findings show that higher data resolution improves coverage and reduces bias, while certain noise structures (e.g., MLN, ACN, ATN) induce parameter-specific biases that may require correction or more data. The results support WENDy as a fast, noise-tolerant tool for parameter/state inference in dynamical systems, and offer practical data-density guidance (e.g., minimum data points per state) to achieve reliable inference in noisy settings.

Abstract

The Weak form Estimation of Nonlinear Dynamics (WENDy) method is a recently proposed class of parameter estimation algorithms that exhibits notable noise robustness and computational efficiency. This work examines the coverage and bias properties of the original WENDy-IRLS algorithm's parameter and state estimators in the context of the following differential equations: Logistic, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark. The estimators' performance was studied in simulated data examples, under four different noise distributions (normal, log-normal, additive censored normal, and additive truncated normal), and a wide range of noise, reaching levels much higher than previously tested for this algorithm.

Paper Structure

This paper contains 39 sections, 24 equations, 82 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Output Error-based Non-linear Least Squares
  3. Weak-form Estimation of Non-linear Dynamics
  4. Coverage
  5. Bias
  6. Outline
  7. Methods
  8. Benchmark Models
  9. Logistic Growth Model
  10. Lotka-Volterra Model
  11. FitzHugh-Nagumo Model
  12. Hindmarsh-Rose Model
  13. Protein Transduction Benchmark Model
  14. Data Generation
  15. Coverage and Bias
  16. ...and 24 more sections

Figures (82)

  • Figure 1: Logistic growth curve with 103 data points, and multiplicative log-normally distributed noise: left 1% noise; right: 5% noise. The following initial conditions and true parameter values were used: $u_0 = 0.01$; $\mathbf{w}^* = (1, -1)$
  • Figure 2: Lotka-Volterra model with varying levels of multiplicative log-normal noise. The following initial conditions and parameter values where used: $\mathbf{u}_0 = (1, 1)$; $\mathbf{w}^* = (3, -1, -6, 1)$
  • Figure 3: FitzHugh-Nagumo model with varying levels of log-normal noise. The following initial conditions and parameter values where used: $\mathbf{u}_0 = (0, 0.1)$; $\mathbf{w}^* = (3, -3, 3, -1/3, 17/150, 1/15)$
  • Figure 4: Hindmarsh-Rose model with varying levels of log-normal noise. The following initial conditions and parameter values were used: $\mathbf{u}_0 = (-1.31, -7.6, -0.2)$; $\mathbf{w}^* = (10, -10, 30, -10, 10, -50, -10, 0.04, 0.0319, -0.01)$
  • Figure 5: Protein Transduction Benchmark Model with varying levels of log-normal noise. The following initial conditions and parameter values were used: $\mathbf{u}_0 = (1, 0, 1, 0, 1)$; $\mathbf{w}^* = (-0.07, -0.6, 0.35, 0.07, -0.6, 0.05, 0.17, 0.6, -0.35, 0.3, -0.017)$
  • ...and 77 more figures