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キーワード:
recurrence networks; persistent homology; time series analysis
要旨:
One of the main goals of analyzing a high-dimensional time series is to identify structures in it. Some of these structures correspond to important dynamical features in the underlying system, like different dynamical states and the transitions between these. In this thesis we introduce two new methodologies for the identification of different dynamical features in a system from the analysis of a real-world time series. We focus in the dynamical features corresponding to the different dynamical metastable states (in a system with multiple and well distinguished time scales, these can be understood as the attractors associated to each of the different time scales) in a system and the transitions between dynamical regimes in a system. Our first method is designed for the identification of different dynamical metastable states, and takes a recurrence analysis approach. The results provided by this method seem to be robust to the introduction of noise and missing points. Our second method is designed for the identification of transitions between different dynamical regimes, and takes an algebraic topological approach. It seems that our second method is, by construction, also robust to the noise and outliers in the data. However, it is still not sensitive enough to identify dynamical transitions where the shape of the attractors in a system suffer small changes. Given that both methods introduced in this thesis rely on the geometrical analysis of the state space, another issue treated in this thesis is the reconstruction of the state space from a complex time series. In this thesis, our criteria for an adequate state space reconstruction are given in terms of the gain or loss of geometrical information. These criteria are specifically developed for each of the approaches taken for every method: recurrence analysis and persistent homology.
