Learning stochasticity: a nonparametric framework for intrinsic noise estimation

TL;DR

This work tackles the challenge of recovering state-dependent intrinsic noise from time-series data of stochastic dynamical systems where parametric models are often inadequate. It introduces TRINE, a nonparametric, kernel-based three-phase regression framework that jointly estimates the drift $f$ and diffusion $g$, and recovers latent noise realizations via Stage 2, with the noise variance profile learned in Stage 3. By designing custom kernels and leveraging Gaussian Process Regression, TRINE achieves near-oracle performance and outperforms existing heteroscedastic approaches on biological and ecological benchmarks. The approach provides interpretable insights into how intrinsic noise shapes system dynamics, enabling noise-aware modeling and potential control strategies in gene regulation and other stochastic networks.

Abstract

Understanding the principles that govern dynamical systems is a central challenge across many scientific domains, including biology and ecology. Incomplete knowledge of nonlinear interactions and stochastic effects often renders bottom-up modeling approaches ineffective, motivating the development of methods that can discover governing equations directly from data. In such contexts, parametric models often struggle without strong prior knowledge, especially when estimating intrinsic noise. Nonetheless, incorporating stochastic effects is often essential for understanding the dynamic behavior of complex systems such as gene regulatory networks and signaling pathways. To address these challenges, we introduce Trine (Three-phase Regression for INtrinsic noisE), a nonparametric, kernel-based framework that infers state-dependent intrinsic noise from time-series data. Trine features a three-stage algorithm that com- bines analytically solvable subproblems with a structured kernel architecture that captures both abrupt noise-driven fluctuations and smooth, state-dependent changes in variance. We validate Trine on biological and ecological systems, demonstrating its ability to uncover hidden dynamics without relying on predefined parametric assumptions. Across several benchmark problems, Trine achieves performance comparable to that of an oracle. Biologically, this oracle can be viewed as an idealized observer capable of directly tracking the random fluctuations in molecular concentrations or reaction events within a cell. The Trine framework thus opens new avenues for understanding how intrinsic noise affects the behavior of complex systems.

Figures (7)

• Figure 1: Structured Intrinsic Noise Estimation via Trine. The proposed method decomposes the estimation of structured intrinsic noise into three sequential phases using Gaussian Process Regression (GPR) with customized kernels: Step 1---Sign Estimation: A GPR model with a smooth kernel is used to describe the deterministic component $f$ of the system dynamics. The residuals from this model are then used to estimate the signs of the intrinsic noise realizations, capturing directional information. Step 2---Intrinsic Noise Realization: Based on the estimated signs, a second GPR model is employed, featuring a structured kernel that captures both the discontinuities in the intrinsic noise realizations and the smooth variation of the variance profile with respect to the system state. This model is used to recover the realizations of the intrinsic noise. Step 3---Noise Variance Profiling: The absolute residuals from Step 2, appropriately scaled, are modeled using a third GPR with a smooth kernel to estimate the state-dependent noise standard deviation profile. Overall, this modular approach allows for flexible modeling of nonhomogeneous, state-dependent noise in complex dynamical systems by decoupling directionality, structure, and variance.
• Figure 2: Results for Ricker model. The top panel show the simulations of the system using $r=2.5$ and intrinsic noise SD $g(x)=\sqrt{0.3^2+0.05^2x}$Bashkirtseva2014. The bottom left panel shows the estimated absolute values of the intrinsic noise realizations returned by the second stage of Trine (grey dots), the true $g$ function (red) and the estimated one returned by Trine (blue). The bottom right panel displays the true deterministic part $f$ (red) and the estimate by Trine (blue).
• Figure 3: Results for FitzHugh--Nagumo model. The top panel shows a representative simulated trajectory in the 2D plane $(V,W)$ obtained using parameters $\epsilon = 0.08$, $a = 0.7$, $b = 0.8$, $I_{\rm ext} = 0.5$, $\sigma_V = 0.1$, $\sigma_W = 0.05$ and $\alpha = \beta = 0.8$. Stochastic dynamics qualitatively show two basins of attraction. The left-bottom panel shows the true state-dependent standard deviation driving $\dot{V}$, while the right-bottom panel displays its Trine estimate.
• Figure 4: Results for Self-Promoter model. The top left panel shows the self-promoter gene regulatory network. The top right panel displays a representative simulated trajectory, whose dynamics exhibit pronounced burst-like stochastic behavior. Data are generated using the parameters $a_0 = 0.05$ (basal activity), $b = 10$ (feedback strength), $m_0 = 25$ (copy-number scale), $\kappa = 1$ (promoter-switching noise) Elston2001. The bottom panel reports the true SD profile (red) and the Trine estimate (blue).
• Figure 5: Results for Toggle switch model. The top left panel shows the gene regulatory network, while the top right panel shows a representative simulated trajectory of the two proteins in the 2D plane. Their stochastic dynamics show two basins of attraction, typical of bistable systems. The left-bottom panel shows the true intrinsic noise and state-dependent standard deviation while Trine estimates are in the right-bottom panel. Data are generated using the parameters $b = 0.28$, $m_0 = 1000$, $\kappa = 2.37$ (see Appendix for further details).

Theorems & Definitions (1)

• Proposition 1