Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems

We compare two approaches to the predictive modeling of dynamical systems from partial observations at discrete times. The first is continuous in time, where one uses data to infer a model in the form of stochastic differential equations, which are then discretized for numerical solution. The second is discrete in time: the model one infers is a parametric representation of a time series that can be directly used for computation. The comparison is performed in a special case where the observations are known to have been obtained from a hypoelliptic stochastic differential equation. We show that the discrete-time approach has better predictive skills, especially when the data are relatively sparse in time. We discuss open questions as well as the broader significance of the results.

[1]  Georg A. Gottwald,et al.  Stochastic Climate Theory , 2016, 1612.07474.

[2]  Richard A. Davis,et al.  Continuous-time Gaussian autoregression , 2007 .

[3]  Michael Ghil,et al.  Data-driven non-Markovian closure models , 2014, 1411.4700.

[4]  Michael Sørensen,et al.  Estimating functions for diffusion-type processes , 2012 .

[5]  Peter J. Brockwell,et al.  Recent results in the theory and applications of CARMA processes , 2014 .

[6]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[7]  Michael Sørensen,et al.  Estimating functions for diffusion-type processes , 2012 .

[8]  Alexandre J. Chorin,et al.  Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics , 2015, Proceedings of the National Academy of Sciences.

[9]  H. Herzel,et al.  Positive Lyapunov exponents in the Kramers oscillator , 1993 .

[10]  D. Talay.  Stochastic Hamiltonian Systems : Exponential Convergence to the Invariant Measure , and Discretization by the Implicit Euler Scheme , 2002 .

[11]  H. Sørensen.  Parametric Inference for Diffusion Processes Observed at Discrete Points in Time: a Survey , 2004 .

[12]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[13]  Richard H. Jones.  FITTING A CONTINUOUS TIME AUTOREGRESSION TO DISCRETE DATA , 1981 .

[14]  Andrew J. Majda,et al.  Physics constrained nonlinear regression models for time series , 2012 .

[15]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[16]  Alexandre J. Chorin,et al.  Data-based stochastic model reduction for the Kuramoto--Sivashinsky equation , 2015, 1509.09279.

[17]  A. Samson,et al.  Contrast estimator for completely or partially observed hypoelliptic diffusion , 2011 .

[18]  H. Kramers.  Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .

[19]  Jonathan C. Mattingly,et al.  Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , 2002 .

[20]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[21]  A. W. Phillips,et al.  THE ESTIMATION OF PARAMETERS IN SYSTEMS OF STOCHASTIC DIFFERENTIAL EQUATIONS , 1959 .

[22]  T. Faniran.  Numerical Solution of Stochastic Differential Equations , 2015 .

[23]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[24]  P. Brockwell.  Continuous-time ARMA processes , 2001 .

[25]  A. Stuart,et al.  Parameter estimation for partially observed hypoelliptic diffusions , 2007, 0710.5442.

[26]  Physikalische Gesellschaft.  Position-dependent diffusion coefficients and free energies from Bayesian analysis of equilibrium and replica molecular dynamics simulations , 2005 .

[27]  Arnaud Gloter,et al.  Parameter Estimation for a Discretely Observed Integrated Diffusion Process , 2006 .

[28]  P. Imkeller,et al.  The Kramers Oscillator Revisited , 2000 .

[29]  M. V. Tretyakov,et al.  Computing ergodic limits for Langevin equations , 2007 .

[30]  E. B. Andersen,et al.  Asymptotic Properties of Conditional Maximum‐Likelihood Estimators , 1970 .

[31]  Jonathan C. Mattingly,et al.  A weak trapezoidal method for a class of stochastic differential equations , 2009, 0906.3475.

[32]  Andrew M. Stuart,et al.  Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations , 2009, SIAM J. Numer. Anal..

[33]  Prakasa Rao.  Statistical inference for diffusion type processes , 1999 .

[34]  Yaozhong Hu.  Strong and weak order of time discretization schemes of stochastic differential equations , 1996 .