Revisiting linear regression of dynamical systems within the context of
Zwanzig-Mori theory: tests on a simple system
Abstract
Linear regression can be applied to time series data to extract model
parameters such as the effective force and friction constant matrices of
the system. Even highly nonlinear systems can be analyzed by linear
regression, if the total amount of data is broken up into shorter “time
windows”, so that the dynamics is considered to be piece-wise linear.
Traditionally, linear regression has been performed on the equation of
motion itself (which approach we refer to as LRX). There has been
surprisingly little published on the accuracy and reliability of LRX as
applied to time series data. Here we show that linear regression can
also be applied to the time correlation function of the dynamical
observables (which approach we refer to as LRC), and that this approach
is better justified within the context of statistical physics, namely,
Zwanzig-Mori theory. We test LRC against LRX on a simple system of two
damped harmonic oscillators driven by Gaussian random noise. We find
that LRC allows one to improve the signal to noise ratio in a way that
is not possible within LRX. Linear regression using time correlation
functions (LRC) thus appears to be not only better justified
theoretically, but it is more accurate and more versatile than LRX.