Estimating the Distribution of Parameters in Differential Equations with Repeated Cross-Sectional Data

TL;DR

This paper introduces EPD, a novel method for accurately estimating the distribution of parameters in differential equations using repeated cross-sectional data, overcoming limitations of traditional approaches.

Contribution

The paper presents EPD, a new approach that captures the full parameter distribution from RCS data without data loss, improving modeling accuracy in diverse systems.

Findings

EPD outperforms traditional methods in estimating parameter distributions.

EPD accurately captures diverse distribution shapes in real-world data.

The method demonstrates superiority across multiple biological and ecological models.

Abstract

Differential equations are pivotal in modeling and understanding the dynamics of various systems, offering insights into their future states through parameter estimation fitted to time series data. In fields such as economy, politics, and biology, the observation data points in the time series are often independently obtained (i.e., Repeated Cross-Sectional (RCS) data). With RCS data, we found that traditional methods for parameter estimation in differential equations, such as using mean values of time trajectories or Gaussian Process-based trajectory generation, have limitations in estimating the shape of parameter distributions, often leading to a significant loss of data information. To address this issue, we introduce a novel method, Estimation of Parameter Distribution (EPD), providing accurate distribution of parameters without loss of data information. EPD operates in three main steps: generating synthetic time trajectories by randomly selecting observed values at each time point, estimating parameters of a differential equation that minimize the discrepancy between these trajectories and the true solution of the equation, and selecting the parameters depending on the scale of discrepancy. We then evaluated the performance of EPD across several models, including exponential growth, logistic population models, and target cell-limited models with delayed virus production, demonstrating its superiority in capturing the shape of parameter distributions. Furthermore, we applied EPD to real-world datasets, capturing various shapes of parameter distributions rather than a normal distribution. These results effectively address the heterogeneity within systems, marking a substantial progression in accurately modeling systems using RCS data.