State Space Reconstruction Parameters in the Analysis of Chaotic Time Series - the Role of the Time Window Length

Abstract

The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay $τ$ and the embedding dimension $m$. We discuss the applicability of these techniques with a critical view as to their validity, and point out the necessity of determining the overall time window length, $τ_w$, for successful embedding. Emphasis is put on the relation between $τ_w$ and the dynamics of the underlying chaotic system, and we suggest to set $τ_w \geq τ_p$, the mean orbital period; $τ_p$ is approximated from the oscillations of the time series. The procedure is assessed using the correlation dimension for both synthetic and real data. For clean synthetic data, values of $τ_w$ larger than $τ_p$ always give good results given enough data and thus $τ_p$ can be considered as a lower limit ($τ_w \geq τ_p$). For noisy synthetic data and real data, an upper limit is reached for $τ_w$ which approaches $τ_p$ for increasing noise amplitude.