Vol. 16, No. 1, 2021

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Stochastic parametrization with VARX processes

Nick Verheul and Daan Crommelin

Vol. 16 (2021), No. 1, 33–57

In this study we investigate a data-driven stochastic methodology to parametrize small-scale features in a prototype multiscale dynamical system, the Lorenz ’96 (L96) model. We propose to model the small-scale features using a vector autoregressive process with exogenous variables (VARX), estimated from given sample data. To reduce the number of parameters of the VARX we impose a diagonal structure on its coefficient matrices. We apply the VARX to two different configurations of the 2-layer L96 model, one with common parameter choices giving unimodal invariant probability distributions for the L96 model variables, and one with nonstandard parameters giving trimodal distributions. We show through various statistical criteria that the proposed VARX performs very well for the unimodal configuration, while keeping the number of parameters linear in the number of model variables. We also show that the parametrization performs accurately for the very challenging trimodal L96 configuration by allowing for a dense (nondiagonal) VARX covariance matrix.

stochastic parametrization, constrained autoregressive models, linear number parameters, multiscale modeling, Lorenz '96
Mathematical Subject Classification 2010
Primary: 62F30, 60H10, 65C20, 68U20, 70K70
Received: 29 July 2019
Revised: 24 July 2020
Accepted: 13 September 2020
Published: 19 January 2021
Nick Verheul
Centrum Wiskunde & Informatica
Daan Crommelin
Centrum Wiskunde & Informatica
Korteweg-de Vries Institute for Mathematics
University of Amsterdam