Supporting Optimal Phase Space Reconstructions Using Neural Network Architecture for Time Series Modeling

Time-series analyses has become a key instrument for the evaluation of continuously collected data in several domains such as Medicine, Physics and Statistics [Firmino et al., 2014, Box and Jenkins, 2015]. Such analysis generally involves the creation of a model (a regression function or a classifier, for instance) that usually leads to inconsistent results when built over raw data, specially if it contains chaotic behavior [Brock et al., 1992]. In order to reach more reliable results, an alternative is to study time-series trajectories in the phase space, as proposed by the area of Dynamical Systems [Ott, 2002, Alligood et al., 1996]. Besides leading to more robust models, the phase space also allows the inference of other important measures, such as the correlation dimension [Grassberger and Procaccia, 1983, Mandelbrot, 1977, Theiler, 1990, Clark, 1990, Ding et al., 1993] and the Lyapunov exponent [Sano and Sawada, 1985, Kantz and Schreiber, 2004], which support further analyses in modeling. In this context, Takens' embedding theorem [Takens, 1981] is one of the most used methods in the literature to reconstruct phase spaces from time series [Ravindra and Hagedorn, 1998]. Such method relies on two parameters known as embedding dimension m and time delay τ (see Figure 1) that, although Takens proved an arbitrary τ can be used given m is sufficiently large, the minimum-but-sufficient (from now on denoted as optimal) set of embedding parameters is desirable either to optimize phase-space computations as to better understand the analyzed phenomenon. In this context, several methods based on entropy [Han et al., 2012], fractal dimensions [Theiler, 1990] and/or nearest neighbors [Kennel et al., 1992] were proposed to guide the estimation of optimal embeddings.