An Analytic Approach to Practical State Space Reconstruction

Abstract:

We study the three standard methods for reconstructing a state space from a time series: delays, derivatives, and principal components. We derive a closed-form solution to principal component analysis in the limit of small window widths. This solution explains the relationship between delays, derivatives, and principal components, it shows how the singular spectrum scales with dimension and delay time, and it explains why the eigenvectors resemble the Legendre polynomials. Most importantly, the solution allows us to derive a guideline for choosing a good window width. Unlike previous suggestions, this guideline is based on first principles and simple quantities. We argue that discrete Legendre polynomials provide a quick and not-so-dirty substitute for principal component analysis, and that they are a good practical method for state space reconstruction.