📂Statistical Analysis

Ensemble Sindy E-SINDy

Algorithm ¹

SINDy algorithm: Let a dynamical system with state space $\mathbb{R}^{n}$ be given by the following smooth function $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$. $$ \dot{\mathbf{x}} = f \left( \mathbf{x} \right) $$ Form a matrix of derived variables by applying some nonlinear functions to the independent variables of $X$, yielding $\Theta \left( X \right) \in \mathbb{R}^{m \times p}$, and perform STLSQ on this together with the following matrix equation to discover the governing equation. This algorithm is called SINDy. $$ \dot{X} = \Theta \left( X \right) \Xi $$

As a variant of SINDy, an algorithm that applies bagging to reduce the library is called Ensemble-SINDy.

Explanation

In practice, applying SINDy to noisy data can produce wildly different equation forms depending on how one selects a subset of the data. Nevertheless, if certain coefficients and terms appear consistently across repeated bootstraps, those components are likely the elements of an equation that well describes the system.

By design, the SINDy library is given ‘sufficiently large’ so that relevant terms are found sparsely via sparse regression. However, as the size (number) of this sufficiently large library grows, beyond mere computational cost it can fail to be full rank or otherwise cause problems for least squares. Ensemble-SINDy can be viewed as a method that searches for smaller libraries and thereby improves both performance and stability.

As illustrated in the figure, E-SINDy has two implementations: bootstrapping the data points themselves, and bootstrapping the library. In either approach, one counts how often a specific library term appears and deletes it if it does not exceed a given tolerance, yielding the final model. Depending on how SINDy is implemented, implementing library-bootstrapped E-SINDy may be more difficult.