A unified and automated approach to attractor reconstruction

Abstract

We present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a symbiotic fashion. Using non-uniform delays allows the successful reconstruction of systems inheriting different time scales. In contrast to the established methods, the minimization of an appropriate cost function determines the embedding dimension without using a threshold parameter. Moreover, the method is capable of detecting stochastic time series and, thus, can handle noise contaminated input without adjusting parameters. The superiority of the proposed method is shown on some paradigmatic models and experimental data from chaotic chemical oscillators.

Localizing category-related information in speech with multi-scale analyses

“…In general, entropy estimation methods are most effective when time series are relatively long or when numerous observation sequences are available. Nonetheless, there are many novel approaches to analysis of time series which are designed to be suitable for nonstationary signals [12,15,16] or which do not require large datasets [14,17,18]. Although we do not apply such methods here, we suspect that it would be valuable to investigate their utility in relation to the goal of identifying speech categories.…”

Optimal State Space Reconstruction via Monte Carlo Decision Tree Search

“…We focused on two scenarios: (1) only the time series of the 2 nd node (univariate embedding) and (2) three time series of nodes 2, 4, and 7 have been used to mimic a uni-and a multivariate embedding case. For each of these time series we performed an embedding, using three classic time delay approaches proposed by Kennel et al [55] (5%-threshold), Cao [13] (slope threshold of 0.2) and Hegger and Kantz [39] (5%-threshold) with a uniform delay value estimated as the first minimum of the auto mutualinformation (only applicable to the univariate case) and the recently proposed PECUZAL algorithm [58]. For Fig.…”

“…We have recently proposed an algorithm [58], which minimizes the L-statistic [93] (the objective function) in each embedding cycle D d over possible delay values in this embedding cycle determined by a continuity statistic [76]. Nichkawde [71] minimizes the FNN-statistic in each embedding cycle over time delays given by a statistic, which maximizes directional derivatives of the actual reconstruction vectors.…”

“…Picking optimal embedding parameters τ j and m comes down to make the resulting components of the reconstruction vectors v(t) as independent as possible [72,81], but at the same time not too independent, in order to keep some information of the correlation structure of the data [16,80,59,22,93]. Besides some unified approaches [11,61,9,10,53,27,84,30,29,92,58], which tackle the estimation of the delays τ j and the embedding dimension m simultaneously, most researchers use two different methods to perform the reconstruction.…”

“…Fig.2Visualization of the expand step of the MCDTS algorithm. Here we exemplary use the continuity statistic ε ⋆ (τ) as the delay preselection statistic Λ τ and the ∆ L-statistic[93] as the objective function Γ , as it has been utilized in the recently proposed PECUZAL algorithm[58].…”