11,433 research outputs found
A Variational Approach to Parameter Estimation in Ordinary Differential Equations
Ordinary differential equations are widely-used in the field of systems
biology and chemical engineering to model chemical reaction networks. Numerous
techniques have been developed to estimate parameters like rate constants,
initial conditions or steady state concentrations from time-resolved data. In
contrast to this countable set of parameters, the estimation of entire courses
of network components corresponds to an innumerable set of parameters. The
approach presented in this work is able to deal with course estimation for
extrinsic system inputs or intrinsic reactants, both not being constrained by
the reaction network itself. Our method is based on variational calculus which
is carried out analytically to derive an augmented system of differential
equations including the unconstrained components as ordinary state variables.
Finally, conventional parameter estimation is applied to the augmented system
resulting in a combined estimation of courses and parameters. The combined
estimation approach takes the uncertainty in input courses correctly into
account. This leads to precise parameter estimates and correct confidence
intervals. In particular this implies that small motifs of large reaction
networks can be analysed independently of the rest. By the use of variational
methods, elements from control theory and statistics are combined allowing for
future transfer of methods between the two fields
Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations
This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies
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- 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).Comment: 35 pages, 5 figure
AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs
Stochastic differential equations are an important modeling class in many
disciplines. Consequently, there exist many methods relying on various
discretization and numerical integration schemes. In this paper, we propose a
novel, probabilistic model for estimating the drift and diffusion given noisy
observations of the underlying stochastic system. Using state-of-the-art
adversarial and moment matching inference techniques, we avoid the
discretization schemes of classical approaches. This leads to significant
improvements in parameter accuracy and robustness given random initial guesses.
On four established benchmark systems, we compare the performance of our
algorithms to state-of-the-art solutions based on extended Kalman filtering and
Gaussian processes.Comment: Published at the Thirty-sixth International Conference on Machine
Learning (ICML 2019
A variational approach to path estimation and parameter inference of hidden diffusion processes
We consider a hidden Markov model, where the signal process, given by a
diffusion, is only indirectly observed through some noisy measurements. The
article develops a variational method for approximating the hidden states of
the signal process given the full set of observations. This, in particular,
leads to systematic approximations of the smoothing densities of the signal
process. The paper then demonstrates how an efficient inference scheme, based
on this variational approach to the approximation of the hidden states, can be
designed to estimate the unknown parameters of stochastic differential
equations. Two examples at the end illustrate the efficacy and the accuracy of
the presented method.Comment: 37 pages, 2 figures, revise
Methods for the identification of material parameters in distributed models for flexible structures
Theoretical and numerical results are presented for inverse problems involving estimation of spatially varying parameters such as stiffness and damping in distributed models for elastic structures such as Euler-Bernoulli beams. An outline of algorithms used and a summary of computational experiences are presented
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