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Robust SDE Parameter Estimation Under Missing Time Information Setting

Long Van Tran, Truyen Tran, Phuoc Nguyen

TL;DR

The paper tackles the challenge of parameter estimation for SDEs when temporal timestamps are missing or corrupted. It introduces ReTrace, a score-based framework that uses forward–backward drift–score discrepancies to identify the true time direction, reconstruct a global temporal order, and then perform maximum-likelihood estimation of the drift and diffusion parameters on the reordered data. A theoretical identifiability analysis distinguishes when temporal direction is recoverable (irreversible processes) from when it is not (reversible processes), and the method is complemented by a practical sorting algorithm with convergence guarantees. Empirical results on irreversible synthetic SDEs and synthetic pharmacological data demonstrate that ReTrace achieves high ordering accuracy and accurate parameter estimates, enabling reliable counterfactual treatment-effect predictions under unordered data. This approach broadens the applicability of SDE-based inference to privacy-preserving or noisy-time settings and supports principled longitudinal analysis in sensitive domains.

Abstract

Recent advances in stochastic differential equations (SDEs) have enabled robust modeling of real-world dynamical processes across diverse domains, such as finance, health, and systems biology. However, parameter estimation for SDEs typically relies on accurately timestamped observational sequences. When temporal ordering information is corrupted, missing, or deliberately hidden (e.g., for privacy), existing estimation methods often fail. In this paper, we investigate the conditions under which temporal order can be recovered and introduce a novel framework that simultaneously reconstructs temporal information and estimates SDE parameters. Our approach exploits asymmetries between forward and backward processes, deriving a score-matching criterion to infer the correct temporal order between pairs of observations. We then recover the total order via a sorting procedure and estimate SDE parameters from the reconstructed sequence using maximum likelihood. Finally, we conduct extensive experiments on synthetic and real-world datasets to demonstrate the effectiveness of our method, extending parameter estimation to settings with missing temporal order and broadening applicability in sensitive domains.

Robust SDE Parameter Estimation Under Missing Time Information Setting

TL;DR

The paper tackles the challenge of parameter estimation for SDEs when temporal timestamps are missing or corrupted. It introduces ReTrace, a score-based framework that uses forward–backward drift–score discrepancies to identify the true time direction, reconstruct a global temporal order, and then perform maximum-likelihood estimation of the drift and diffusion parameters on the reordered data. A theoretical identifiability analysis distinguishes when temporal direction is recoverable (irreversible processes) from when it is not (reversible processes), and the method is complemented by a practical sorting algorithm with convergence guarantees. Empirical results on irreversible synthetic SDEs and synthetic pharmacological data demonstrate that ReTrace achieves high ordering accuracy and accurate parameter estimates, enabling reliable counterfactual treatment-effect predictions under unordered data. This approach broadens the applicability of SDE-based inference to privacy-preserving or noisy-time settings and supports principled longitudinal analysis in sensitive domains.

Abstract

Recent advances in stochastic differential equations (SDEs) have enabled robust modeling of real-world dynamical processes across diverse domains, such as finance, health, and systems biology. However, parameter estimation for SDEs typically relies on accurately timestamped observational sequences. When temporal ordering information is corrupted, missing, or deliberately hidden (e.g., for privacy), existing estimation methods often fail. In this paper, we investigate the conditions under which temporal order can be recovered and introduce a novel framework that simultaneously reconstructs temporal information and estimates SDE parameters. Our approach exploits asymmetries between forward and backward processes, deriving a score-matching criterion to infer the correct temporal order between pairs of observations. We then recover the total order via a sorting procedure and estimate SDE parameters from the reconstructed sequence using maximum likelihood. Finally, we conduct extensive experiments on synthetic and real-world datasets to demonstrate the effectiveness of our method, extending parameter estimation to settings with missing temporal order and broadening applicability in sensitive domains.
Paper Structure (26 sections, 4 theorems, 25 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 26 sections, 4 theorems, 25 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions are:
  3. Background
  4. Stochastic Differential Equations (SDEs)
  5. Related Work
  6. Methods
  7. Problem Formulation
  8. Conditions for Identifiability of Time Direction
  9. Parameter Learning from Unordered Observations
  10. Experiments
  11. Baseline Methods
  12. Evaluation Metrics
  13. Irreversible SDE Datasets
  14. Results
  15. With Observation Noises
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose the drift satisfies $\mathbf{b}(\mathbf{x}) = \frac{1}{2} \mathbf{H} \nabla \log p_*(\mathbf{x})$, i.e., the process is reversible and satisfies detailed balance with respect to $p_*(\mathbf{x})$. Then, for any finite sequence of observations $(\mathbf{x}_0, \ldots, \mathbf{x}_n)$ drawn from Thus, the direction of time cannot be statistically identified from such data, by any criterion bas

Figures (4)

  • Figure 1: Our sorting procedure leverages drift-score discrepancy to reorder data. (a) Compare errors for each states pair in the Data Reordering stage. (b) Alternating between sorting data and estimating parameters.
  • Figure 2: Performance comparisons between ReTrace and baseline methods on (a) Reordering accuracy, and (b) MAEs of drift parameter A under increasing observation noises. The x-axis is the noise percentage.
  • Figure 3: Sample tumor growth trajectories generated from the stochastic PKPD SDE model. (Top) Treated group with chemotherapy and radiotherapy continuously administered. (Bottom) Untreated group with no interventions. Each curve represents an independent patient sampled from the SDE in Eq. \ref{['eq:pkpd']}.
  • Figure 4: Root Mean Squared Error (RMSE) and Treatment Effect Bias (TEB), computed using \ref{['eq:rmse']} and \ref{['eq:teb']}.

Theorems & Definitions (8)

  • Theorem 1: Non-identifiability of Time Direction
  • proof
  • Theorem 2: Identifiability of Time Direction for Asymmetric Diffusions
  • proof
  • Theorem 3: Identifiability of Drift and Diffusion Parameters
  • proof
  • Theorem 4
  • proof