267,246 research outputs found
Linear theory for filtering nonlinear multiscale systems with model error
We study filtering of multiscale dynamical systems with model error arising
from unresolved smaller scale processes. The analysis assumes continuous-time
noisy observations of all components of the slow variables alone. For a linear
model with Gaussian noise, we prove existence of a unique choice of parameters
in a linear reduced model for the slow variables. The linear theory extends to
to a non-Gaussian, nonlinear test problem, where we assume we know the optimal
stochastic parameterization and the correct observation model. We show that
when the parameterization is inappropriate, parameters chosen for good filter
performance may give poor equilibrium statistical estimates and vice versa.
Given the correct parameterization, it is imperative to estimate the parameters
simultaneously and to account for the nonlinear feedback of the stochastic
parameters into the reduced filter estimates. In numerical experiments on the
two-layer Lorenz-96 model, we find that parameters estimated online, as part of
a filtering procedure, produce accurate filtering and equilibrium statistical
prediction. In contrast, a linear regression based offline method, which fits
the parameters to a given training data set independently from the filter,
yields filter estimates which are worse than the observations or even divergent
when the slow variables are not fully observed
Parameter estimation of ODE's via nonparametric estimators
Ordinary differential equations (ODE's) are widespread models in physics,
chemistry and biology. In particular, this mathematical formalism is used for
describing the evolution of complex systems and it might consist of
high-dimensional sets of coupled nonlinear differential equations. In this
setting, we propose a general method for estimating the parameters indexing
ODE's from times series. Our method is able to alleviate the computational
difficulties encountered by the classical parametric methods. These
difficulties are due to the implicit definition of the model. We propose the
use of a nonparametric estimator of regression functions as a first-step in the
construction of an M-estimator, and we show the consistency of the derived
estimator under general conditions. In the case of spline estimators, we prove
asymptotic normality, and that the rate of convergence is the usual
-rate for parametric estimators. Some perspectives of refinements of
this new family of parametric estimators are given.Comment: Published in at http://dx.doi.org/10.1214/07-EJS132 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Identification and estimation of continuous time dynamic systems with exogenous variables using panel data
This paper deals with the identification and maximum likelihood estimation of the parameters of a stochastic differential equation from discrete time sampling. Score function and maximum likelihood equations are derived explicitly. The stochastic differential equation system is extended to allow for random effects and the analysis of panel data. In addition, we investigate the identifiability of the continuous time parameters, in particular the impact of the inclusion of exogenous variables
Learning Model-Based Sparsity via Projected Gradient Descent
Several convex formulation methods have been proposed previously for
statistical estimation with structured sparsity as the prior. These methods
often require a carefully tuned regularization parameter, often a cumbersome or
heuristic exercise. Furthermore, the estimate that these methods produce might
not belong to the desired sparsity model, albeit accurately approximating the
true parameter. Therefore, greedy-type algorithms could often be more desirable
in estimating structured-sparse parameters. So far, these greedy methods have
mostly focused on linear statistical models. In this paper we study the
projected gradient descent with non-convex structured-sparse parameter model as
the constraint set. Should the cost function have a Stable Model-Restricted
Hessian the algorithm produces an approximation for the desired minimizer. As
an example we elaborate on application of the main results to estimation in
Generalized Linear Model
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