Dynamical and statistical properties of estimated high-dimensional ODE models: The case of the Lorenz '05 type II model

The performance of estimated models is often evaluated in terms of their predictive capability. In this study, we investigate another important aspect of estimated model evaluation: the disparity between the statistical and dynamical properties of estimated models and their source system. Specifically, we focus on estimated models obtained via the regression method, sparse identification of nonlinear dynamics (SINDy), one of the promising algorithms for determining equations of motion from time series of dynamical systems. We chose our data source dynamical system to be a higher-dimensional instance of the Lorenz 2005 type II model, an important meteorological toy model. We examine how the dynamical and statistical properties of the estimated models are affected by the standard deviation of white Gaussian noise added to the numerical data on which the estimated models were fitted. Our results show that the dynamical properties of the estimated models match those of the source system reasonably well within a range of data-added noise levels, where the estimated models do not generate divergent (unbounded) trajectories. Additionally, we find that the dynamics of the estimated models become increasingly less chaotic as the data-added noise level increases. We also perform a variance analysis of the (SINDy) estimated model's free parameters, revealing strong correlations between parameters belonging to the same component of the estimated model's ordinary differential equation.