Fractional-order state space reconstruction: a new frontier in multivariate complex time series

This paper presents a novel approach to the phase space reconstruction technique, fractional-order phase space reconstruction (FOSS), which generalizes the traditional integer-order derivative-based method. By leveraging fractional derivatives, FOSS offers a novel perspective for understanding complex time series, revealing unique properties not captured by conventional methods. We further develop the multi-span transition entropy component method (MTECM-FOSS), an advanced complexity measurement technique that builds upon FOSS. MTECM-FOSS decomposes complexity into intra-sample and inter-sample components, providing a more comprehensive understanding of the dynamics in multivariate data. In simulated data, we observe that lower fractional orders can effectively filter out random noise. Time series with diverse long- and short-term memory patterns exhibit distinct extremities at different fractional orders. In practical applications, MTECM-FOSS exhibits competitive or superior classification performance compared to state-of-the-art algorithms when using fewer features, indicating its potential for engineering tasks.