Fisher Information and Dynamical Sampling I

TL;DR

This paper analyzes how accurately the Fisher information can be reconstructed from sampled data of dynamical systems and shows that clustering degrees of freedom improves this accuracy.

Contribution

It provides a quantitative bias estimate for Fisher information from sampled data and assesses how clustering reduces bias and information loss in dynamical system reconstruction.

Findings

Bias of Fisher information decreases with clustering of degrees of freedom.

Clustering improves the accuracy of dynamical system reconstruction from data.

The results are demonstrated on a simple epidemiological model.

Abstract

Information theory is a powerful framework to capture aspects of dynamical systems with multiple degrees of freedom. Mathematically, the dynamics can be represented as a continuous curve $\mathcal{C}$ on a suitable hyperplane in flat space and the Fisher information provides the norm of an infinitesimal displacement along this curve. In many applications, however, we do not have direct access to $\mathcal{C}$. Instead, we have to reconstruct the latter from a time-series of measurements (obtained as samples of size $n$), which are represented by an ordered set of points $\widehat{\mathcal{C}}$ on the same hyperplane. In this work, we calculate the bias of the Fisher information for large $n$, which provides a quantitative estimation for how accurately the dynamics of a system can be reconstructed from a given set of sampled data. Based on this result, we show that a clustering of the degrees of freedom reduces the bias and thus improves the accuracy with which the new system can be described with the same data. Inspired by a recent proposal for such a clustering, we provide a quantitive assessment of the loss of information, which allows to estimate how much information about the dynamics of a system can reliably be extracted based on a given set of data. We illustrate our findings in the case of a simple compartmental model. Although the latter is inspired by epidemiology, the results of this work are applicable to very general dynamical models with multiple degrees of freedom.