Non-Linear Time Series … 2nd part Léonard Studer IPHE-UNIL leonard.******@ 11/10/2017 1 Refresh on chaotic dynamical systems theory 1/2 Necessary cond to have dyn chaos: Either system of differential equations with 3 or more degrees of freedom Or invertible discrete time maps with 2 or more dof Logistic map is a noninvertible map with 1 dof that can show chaos 11/10/2017 2 Léonard Studer, IPHE-UNIL Refresh from chaotic dynamical systems theory 2/2 BUT chaos has structure in phase space Chaos is irregular in time and slightly predictable 11/10/2017 3 Léonard Studer, IPHE-UNIL Main steps in nonlinear time series proc’g 2/2 Simple nonlinear noise reduction … Control … A lot of new applications are actively studied in academic and private labs (see ref Proceedings of the IEEE) Synchronization of chaotic systems, chaos-based techniques for spread-spectrum sequences generation, modulation based on chaotic signal, Cryptographic coding with chaos etc Not detailed presented today, 11/10/2017 4 Léonard Studer, IPHE-UNIL Simple nonlinear noise reduction n0-2 n0-1 n0 n0+1 Select trajectories that stay close for a few time steps to the segment to be cleaned. Below only trajectory X is OK The filtered-cleaned-new value x’n0 is the average value of scalars sj that are in the ball neighborhood of the xn0 ! X 11/10/2017 5 Léonard Studer, IPHE-UNIL Example of simple noise reduction Time series is air flow through the nose of a human as an indicator of breath activity (example from TISEAN tutorial, see ref) 11/10/2017 6 Léonard Studer, IPHE-UNIL Advanced methods in nonlinear noise reduction Refinements of the 2nd step local projective noise reduction ANN etc nonlinear noise reduction in a data stream Here, only past sj values are known… But it can be done ! See T. Schreiber and M. Richter, Nonlinear projective filtering in a data stream, in Int. J. Bifurcat. Chaos (1999). 11/10/2017 7 Léonard Studer, IPHE-UNIL Unstable Pe
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