Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference
arxiv(2024)
摘要
This study presents a Bayesian maximum a posteriori (MAP) framework
for dynamical system identification from time-series data. This is shown to be
equivalent to a generalized zeroth-order Tikhonov regularization, providing a
rational justification for the choice of the residual and regularization terms,
respectively, from the negative logarithms of the likelihood and prior
distributions. In addition to the estimation of model coefficients, the
Bayesian interpretation gives access to the full apparatus for Bayesian
inference, including the ranking of models, the quantification of model
uncertainties and the estimation of unknown (nuisance) hyperparameters. Two
Bayesian algorithms, joint maximum a posteriori (JMAP) and variational
Bayesian approximation (VBA), are compared to the popular SINDy algorithm for
thresholded least-squares regression, by application to several dynamical
systems with added noise. For multivariate Gaussian likelihood and prior
distributions, the Bayesian formulation gives Gaussian posterior and evidence
distributions, in which the numerator terms can be expressed in terms of the
Mahalanobis distance or “Gaussian norm” ||-||^2_M^-1 =
(-)^⊤M^-1 (-), where is a vector
variable, is its estimator and M is the covariance matrix. The
posterior Gaussian norm is shown to provide a robust metric for quantitative
model selection.
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