Using information to generate derivative coordinates from noisy time series

Abstract

This paper describes an approach for recovering a signal, along with the derivatives of the signal, from a noisy time series. To mimic an experimental setting, noise was superimposed onto a deterministic time series. Data smoothing was then used to successfully recover the derivative coordinates; however, the appropriate level of data smoothing must be determined. To investigate the level of smoothing, an information theoretic is applied to show a loss of information occurs for increased levels of noise; conversely, we have shown data smoothing can recover information by removing noise. An approximate criterion is then developed to balance the notion of information recovery through data smoothing with the observation that nearly negligible information changes occur for a sufficiently smoothed time series.

Introduction

It is rarely practical to measure each state variable in an experimental setting. To circumvent this issue, methods exist for reconstructing a pseudo-state space from a small number of measured observables [1], [2], [3]. A primary benefit of the reconstruction is that it produces a pseudo-state space with dynamics equivalent to those of the original state space. As a consequence of the equivalence, an attractor in the reconstructed state space has the same invariants, such as Lyapunov exponents and dimension, as the original attractor [4].

While delayed embedding is the predominant choice for attractor reconstruction [3], [5], [6], [7], [8], the use of derivative coordinates is appealing, given their obvious physical meaning. For instance, many physical systems can be described by a set of equations with states that are related through their derivatives. While the numerical derivatives of a signal can be used for noise-free data, the presence of noise renders the signal derivatives to be poor approximates – owing to noise amplification in the signal derivatives.

The inherent goal of signal analysis is to extract useful information about a system from the observed data. Consider a continuous and deterministic system that has produced the scalar time series q(t). To mimic realistic data from an experiment, noise is superimposed onto the noise-free data as follows: η(t) = q(t) + σα(t), where α(t) is a time series with normally distributed random noise, a zero mean, and a standard deviation equal to σ. The underlying goal of this investigation is to extract q(t) and the derivatives of q(t) from the noisy time series η(t).

This work investigates smoothing η(t) to recover q(t) and its derivatives. While the process of data smoothing is well known, the question of how much smoothing yields accurate derivative coordinates is unclear. To answer this question, we explored the use of an information theoretic, known as the average mutual information, to develop a criterion for the appropriate amount of data smoothing.

The work of this paper is organized as follows. The next section describes the data smoothing technique and average mutual information tools used in our analyses. These discussions are followed by a series of example results that apply the average mutual information to investigate the influence of noise and an approximate criterion for the level of data smoothing.