arXiv subjects

Quentin Clairon

Publications and source records attributed to Quentin Clairon.

9 recordsLinked to original sources

arXiv2026

Neural ODE enhanced linear mixed effect models for estimating complex association patterns of time-varying covariates with the marker trajectory

Longitudinal cohort studies produce repeated data that enable the assessment of time-varying association patterns between exposures and health outcomes. Classical linear mixed-effects models (LMMs) can accommodate a large variety of association patterns while accounting for the irregularly spaced, partially observed measurement. But they require the analyst to pre-specify the functional form linking the exposure history to the outcome. We propose the Neural ODE-LMM, which embeds a Neural Ordinary Differential Equation (Neural ODE) within the linear mixed-effects framework: a learned vector field encodes covariate trajectories into a continuous-time latent state that drives both the fixed- and random-effect design, while preserving the standard LMM observation model. This retains classical likelihood-based inference while learning complex, potentially cumulative, covariate effects flexibly. All parameters are estimated by maximising a penalised marginal likelihood. To quantify covariate effects, we introduce contrasts of counterfactual predictions that compare the expected outcome under alternative covariate trajectories with variance estimated via the delta method. In simulations, the model recovers both instantaneous and cumulative-burden effects without prior specification of the functional form. Applied to the Trois-Cit\'{e}s (3C) cohort, a population-based study of 7{,}324 participants, the method reveals trajectory-dependent associations of BMI and fasting glucose with cognitive decline.

stat.ML↗

arXiv2026

Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes

Dynamical modelling is central to many scientific domains, including pharmacometrics, systems biology, physiology, and epidemiology. In these settings, heterogeneity is often intrinsic: different subjects or units follow related but distinct continuous-time dynamics. Classical nonlinear mixed-effects Ordinary Differential Equation (ODE) models address this by combining population-level structure with subject-specific effects, but they rely on a parametric vector field and are therefore vulnerable to structural misspecification and unmodelled mechanisms. This motivates nonparametric approaches that can retain principled uncertainty quantification, yet existing nonparametric ODE methods typically assume a single shared dynamical system rather than an explicit mixed-effect hierarchy over subject-specific dynamics. We propose MEGPODE, a Bayesian nonparametric mixed-effect ODE model in which each subject's vector field is decomposed into a shared population component and a subject-specific deviation, both endowed with Gaussian process (GP) priors. To avoid repeated ODE solves per subject during training, we combine state-space GP trajectory priors with virtual collocation observations, yielding Kalman-smoothing trajectory updates and closed-form regressions for the vector fields. Across controlled heterogeneous ODE benchmarks spanning oscillatory, biomedical systems, MEGPODE improves population-field recovery and subject-level trajectory prediction relative to strong baselines.

cs.LG↗

arXiv2026

Variational autoencoder for inference of nonlinear mixed effect models based on ordinary differential equations

We propose a variational autoencoder (VAE) approach for parameter estimation in nonlinear mixed-effects models based on ordinary differential equations (NLME-ODEs) using longitudinal data from multiple subjects. In moderate dimensions, likelihood-based inference via the stochastic approximation EM algorithm (SAEM) is widely used, but it relies on Markov Chain Monte-Carlo (MCMC) to approximate subject-specific posteriors. As model complexity increases or observations per subject are sparse and irregular, performance often deteriorates due to a complex, multimodal likelihood surface which may lead to MCMC convergence difficulties. We instead estimate parameters by maximizing the evidence lower bound (ELBO), a regularized surrogate for the marginal likelihood. A VAE with a shared encoder amortizes inference of subject-specific random effects by avoiding per-subject optimization and the use of MCMC. Beyond pointwise estimation, we quantify parameter uncertainty using observed-information-based variance estimator and verify that practical identifiability of the model parameters is not compromised by nuisance parameters introduced in the encoder. We evaluate the method in three simulation case studies (pharmacokinetics, humoral response to vaccination, and TGF-$\beta$ activation dynamics in asthmatic airways) and on a real-world antibody kinetics dataset, comparing against SAEM baselines.

stat.ME↗

arXiv2021

Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations

We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by degenerate diffusion coefficients. This feature often causes the failure of contrast estimators based on Euler Maruyama discretization scheme and dramatically impairs classic stochastic filtering methods used to reconstruct the unobserved states. All of theses issues make the estimation problem in hypoelliptic SDEs difficult to solve. To overcome this, we construct a well-defined cost function no matter the elliptic nature of the SDEs. We also bypass the filtering step by considering a control theory perspective. The unobserved states are estimated by solving deterministic optimal control problems using numerical methods which do not need strong assumptions on the diffusion coefficient conditioning. Numerical simulations made on different partially observed hypoelliptic SDEs reveal our method produces accurate estimate while dramatically reducing the computational price comparing to other methods.

math.OC↗

arXiv2021

Parameter estimation in nonlinear mixed effect models based on ordinary differential equations: an optimal control approach

We present a parameter estimation method for nonlinear mixed effect models based on ordinary differential equations (NLME-ODEs). The method presented here aims at regularizing the estimation problem in presence of model misspecifications, practical identifiability issues and unknown initial conditions. For doing so, we define our estimator as the minimizer of a cost function which incorporates a possible gap between the assumed model at the population level and the specific individual dynamic. The cost function computation leads to formulate and solve optimal control problems at the subject level. This control theory approach allows to bypass the need to know or estimate initial conditions for each subject and it regularizes the estimation problem in presence of poorly identifiable parameters. Comparing to maximum likelihood, we show on simulation examples that our method improves estimation accuracy in possibly partially observed systems with unknown initial conditions or poorly identifiable parameters with or without model error. We conclude this work with a real application on antibody concentration data after vaccination against Ebola virus coming from phase 1 trials. We use the estimated model discrepancy at the subject level to analyze the presence of model misspecification.

stat.ME↗

arXiv2018

A regularization method for the parameter estimation problem in ordinary differential equations via discrete optimal control theory

We present a parameter estimation method in Ordinary Differential Equation (ODE) models. Due to complex relationships between parameters and states the use of standard techniques such as nonlinear least squares can lead to the presence of poorly identifiable parameters. Moreover, ODEs are generally approximations of the true process and the influence of misspecification on inference is often neglected. Methods based on control theory have emerged to regularize the ill posed problem of parameter estimation in this context. However, they are computationally intensive and rely on a nonparametric state estimator known to be biased in the sparse sample case. In this paper, we construct criteria based on discrete control theory which are computationally efficient and bypass the presmoothing step of signal estimation while retaining the benefits of control theory for estimation. We describe how the estimation problem can be turned into a control one and present the numerical methods used to solve it. We show convergence of our estimator in the parametric and well-specified case. For small sample sizes, numerical experiments with models containing poorly identifiable parameters and with various sources of model misspecification demonstrate the acurracy of our method. We finally test our approach on a real data example.

stat.ME↗

arXiv2014

A Tracking Approach to Parameter Estimation in Linear Ordinary Differential Equations

Ordinary Differential Equations are widespread tools to model chemical, physical, biological process but they usually rely on parameters which are of critical importance in terms of dynamic and need to be estimated directly from the data. Classical statistical approaches (nonlinear least squares, maximum likelihood estimator) can give unsatisfactory results because of computational difficulties and ill-posedness of the statistical problem. New estimation methods that use some nonparametric devices have been proposed to circumvent these issues. We present a new estimator that shares properties with Two-Step estimator and Generalized Smoothing (introduced by Ramsay et al, 2007). We introduce a perturbed model and we use optimal control theory for constructing a criterion that aims at minimizing the discrepancy with data and the model. Here, we focus on the case of linear Ordinary Differential Equations as our criterion has a closed-form expression that permits a detailed analysis. Our approach avoids the use of a nonparametric estimator of the derivative, which is one of the main cause of inaccuracy in Two-Step estimators. Moreover, we take into account model discrepancy and our estimator is more robust to model misspecification than classical methods. The discrepancy with the parametric ODE model correspond to the minimum perturbation (or control) to apply to the initial model. Its qualitative analysis can be informative for misspecification diagnosis. In the case of well-specified model, we show the consistency of our estimator and that we reach the parametric root-n rate when regression splines are used in the first step.

stat.ME↗

arXiv2014

State and Parameter Estimation of Partially Observed Linear Ordinary Differential Equations with Deterministic Optimal Control

Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give unsatisfactory results because the inverse problem can be ill-posed, even when the differential equation is linear. Following recent approaches that use approximate solutions of the ODE model, we propose a new method that converts parameter estimation into an optimal control problem: our objective is to determine a control and a parameter that are as close as possible to the data. We derive then a criterion that makes a balance between discrepancy with data and with the model, and we minimize it by using optimization in functions spaces: our approach is related to the so-called Deterministic Kalman Filtering, but different from the usual statistical Kalman filtering. e show the root-$n$ consistency and asymptotic normality of the estimators for the parameter and for the states. Experiments in a toy model and in a real case shows that our approach is generally more accurate and more reliable than Nonlinear Least Squares and Generalized Smoothing, even in misspecified cases.

stat.ME↗

arXiv2014

Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions

Differential equations are commonly used to model dynamical deterministic systems in applications. When statistical parameter estimation is required to calibrate theoretical models to data, classical statistical estimators are often confronted to complex and potentially ill-posed optimization problem. As a consequence, alternative estimators to classical parametric estimators are needed for obtaining reliable estimates. We propose a gradient matching approach for the estimation of parametric Ordinary Differential Equations observed with noise. Starting from a nonparametric proxy of a true solution of the ODE, we build a parametric estimator based on a variational characterization of the solution. As a Generalized Moment Estimator, our estimator must satisfy a set of orthogonal conditions that are solved in the least squares sense. Despite the use of a nonparametric estimator, we prove the root-$n$ consistency and asymptotic normality of the Orthogonal Conditions estimator. We can derive confidence sets thanks to a closed-form expression for the asymptotic variance. Finally, the OC estimator is compared to classical estimators in several (simulated and real) experiments and ODE models in order to show its versatility and relevance with respect to classical Gradient Matching and Nonlinear Least Squares estimators. In particular, we show on a real dataset of influenza infection that the approach gives reliable estimates. Moreover, we show that our approach can deal directly with more elaborated models such as Delay Differential Equation (DDE).

stat.ME↗