Fisher Information and Dynamical Sampling I
Mattia Carrino, Stefan Hohenegger
TLDR
This paper calculates the bias of Fisher information for dynamical systems reconstructed from sampled data, showing clustering improves accuracy.
Key contributions
- Calculates the bias of Fisher information for large $n$ when reconstructing dynamics from sampled data.
- Demonstrates that clustering degrees of freedom reduces bias, enhancing reconstruction accuracy.
- Provides a quantitative assessment of information loss due to clustering, estimating extractable information.
- Illustrates the findings using a simple compartmental model, applicable to general dynamical systems.
Why it matters
This work provides a method to quantify the accuracy of reconstructing dynamical systems from sampled data using Fisher information. It demonstrates how clustering degrees of freedom can improve this accuracy, offering insights into reliable information extraction. These findings are broadly applicable to various multi-degree-of-freedom models.
Original 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.
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