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

TL;DR

This work tackles estimating distributions of differential-equation parameters from Repeated Cross-Sectional data, where each time point provides independent observations. The core innovation, Estimation of Parameter Distribution (EPD), builds $N$ artificial trajectories by sampling observed values at each time point, then uses LSODA-based ODE solves to fit parameters by minimizing a loss $L(\mathbf{p})$ and accepting parameter sets via a logistic-transformed probability $a_n$; varying the scaling factor $C$ controls acceptance and permits either selective sampling or exhaustive exploration. Across exponential growth, logistic, and target cell–limited models, EPD robustly recovers unimodal, bimodal, and trimodal parameter distributions, even under substantial noise, and reveals heterogeneity in real-world data (e.g., amyloid-beta dynamics and influenza infection) that standard unimodal assumptions would miss. The approach advances understanding of system heterogeneity from RCS data and supports more nuanced, personalized interpretations of dynamical processes in biology and beyond.

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.

Figures (10)

• Figure 1: Parameter estimation in the exponential growth model with Repeated Cross-Sectional (RCS) data. An exponential growth model $y’(t)=ay(t)$ represents the amount of population, y(t), changes over time, $t$. We then estimated parameter $a$ that can fit the model to a given RCS data (a-b). When the true parameter distribution of $a$ is unimodal (a, top-panel), corresponding RCS data is generated by parameters $a$, and populations per time do not diverge (a, top).In this case, previous methods, such as Gaussian Process (GP) or All Possible combinations (AP), can estimate true parameter distributions (bottom) (a, bottom). When the true parameter distribution of $a$ is bimodal (b, top-penal), populations per time diverge (b, top). In this case, previous methods fail to estimate the shape of true parameters (b, bottom).
• Figure 1: Parameter estimation in the exponential growth model with Repeated Cross-Sectional (RCS) data. An exponential growth model $y’(t)=ay(t)$ represents the amount of population, y(t), changing over time, $t$. We then estimated parameter $a$ that can fit the model to a given RCS data (a-b). When the true parameter distribution of $a$ is unimodal (a, top-panel), corresponding RCS data is generated by parameters $a$, and populations per time do not diverge (a, top). In this case, previous methods, such as Gaussian Process (GP) or All Possible combinations (AP), can estimate true parameter distributions (bottom) (a, bottom). When the true parameter distribution of $a$ is bimodal (b, top-penal), populations per time diverge (b, top). In this case, previous methods fail to estimate the shape of true parameters (b, bottom).
• Figure 1: Accurate estimation of true distributions by EPD in datasets exhibiting unimodal, bimodal, and trimodal parameter distributions within an Exponential Growth Model. (a) When the true parameter distribution is unimodal, we applied EPD on the observed data (left) and estimated the parameters (right). Notably, EPD remained accurate even when we added 3% or 10% multiplicative noise to the data (b, c). Likewise, EPD was confirmed to estimate Bimodal and Trimodal parameter distributions effectively.
• Figure 1: Estimation results for real experimental datasets on amyloid beta accumulation using a logistic model with EPD. (a) amyloid beta 40, (b) amyloid beta 42. The left plot shows the accumulation of amyloid beta at 4, 8, 12, and 18 months. We present the estimation results for this dataset using EPD on the right. The left plot also includes some trajectories corresponding to the parameters estimated on the right.
• Figure 1: Accept probability $a_n$ for each scaling factor $C$ in EPD Three images represent the accept probability of each estimated parameter by AP method within an exponential growth model across three scaling factors: $C=1$ (left), $C=100$ (Middle), $C=10000$ (Right). In all images, estimated parameters from each artificial trajectory are marked by blue dots. The grey lines represent the true value of the growth rate parameter $a$ in the model.