1. Page 0 of 8 Lyapunov Exponents – Theory and Implementation Sanjay Patil Intelligent Electronics Systems Human and Systems Engineering Center for Advanced Vehicular Systems URL: www.cavs.msstate.edu/hse/ies/projects/nsf_nonlinear/doc/www.cavs.msstate.edu/hse/ies/projects/ Lyapunov Exponents : An implementation details

2. Page 1 of 8 Lyapunov Exponents – Theory and Implementation Algorithm flow Embed the data Locate nearest points From neighbourhood Evolve 1 step Locate nearest points From neighbourhood Calculate trajectory matrix Perform QR decomposition Calculate exponents from R

3. Page 2 of 8 Lyapunov Exponents – Theory and Implementation Implementation Details Introduction to Chaos Lyapunov exponents or Lyaponuv spectrum Algorithm details

4. Page 3 of 8 Lyapunov Exponents – Theory and Implementation Chaos Chaos is defined as Lorenz and Poincare` studied Dynamical systems. Takens and Sauer – new signal classification algorithm. Time series of observations sampled from a single state variable of a system Reconstructed space equivalent to the original system Slightly different notations than usually used by other researchers.

5. Page 4 of 8 Lyapunov Exponents – Theory and Implementation Algorithm Details – Time Series Embedding Takens and Sauer – new signal classification algorithm. Time series of observations sampled from a single state variable of a system Reconstructed space equivalent to the original system

6. Page 5 of 8 Lyapunov Exponents – Theory and Implementation The Approach Two methods to tackle the issue: 1.Build global vector reconstructions and differentiate signals in a coefficient space. [Kadtke, 1995] 2.Build GMMs of signal trajectory densities in an RPS and differentiate between signals using Bayesian classifiers. [Authors, 2004] The steps (Algorithm): 1.Data Analysis – normalizing the signals, estimating the time lag and dimension of the RPS. 2.Learning GMMs for each signal class – deciding the number of Gaussian mixtures, parameters learning by Expectation-Maximization (EM) algorithm. 3.Classification – going through the above steps for the SUT (signal under test), using Bayesian maximum likelihood classifiers

7. Page 6 of 8 Lyapunov Exponents – Theory and Implementation Algorithm in details and Issues 1.Data Analysis – 1.normalizing the signals 1.Each signal is normalized to zero mean and unit standard deviation. 2.estimating the time lag  1.Using first minimum of the automutual information function. 2.Overall time lag  is the mode of the histogram of the first minima for all signals. 3.estimating dimension d of the RPS 1.Using global false nearest-neighbor technique. 2.Overall RPS dimension is the mean plus two standard deviations of the distribution of individual signal RPS dimensions. 1.How do you normalize the signal to zero mean and unit standard deviation? 2.What is automutual information function? 3.How do you implement the global false nearest-neighbor technique?

8. Page 7 of 8 Lyapunov Exponents – Theory and Implementation References R. Povinelli, M. Johnson, A. Lindgren, and J. Ye, “Time Series Classification using Gaussian Mixture Models of Reconstructed Phase Spaces,” IEEE Transactions on Knowledge and Data Engineering, Vol 16, no 6, June 2004, pp. 770-783. (the referred paper) F. Takens, “Detecting Strange Attractors in Turbulence,” Proceedings Dynamical Systems and Turbulence, 1980, pp 366-381. (background theory) T. Sauer, J. Yorke, and M. Casdagli, “Embedology,” Journal Statistical Physics, vol 65, 1991, pp 579-616. (background theory) A. Petry, D. Augusto, and C. Barone, “Speaker Identification using Nonlinear Dynamical Features,” Choas, Solitions, and Fractals, vol 13, 2002, pp 221-231. (speech related dynamical system) H. Boshoff, and M. Grotepass, “The fractal dimension of fricative Speech Sounds,” Proceddings South African Symposium Communication and Signal Processing, 1991, pp 12-61. (speech related dynamical system) D. Sciamarella and G. Mindlin, “Topological Structure of Chaotic Flows from Human Speech Chaotic Data,” Physical Review Letters, vol. 82, 1999, pp 1450. (speech related dynamical system) T. Moon, “The Expectation-Maximization algorithm,” IEEE Signal Processing Magazine, 1996, pp 47-59. (expectation-maximization algorithm details) Q. Ding, Z. Zhuang, L. Zhu, and Q. Zhang, “Application of the Chaos, Fractal, and Wavelet Theories to the Feature Extraction of Passive Acoustic Signal,” Acta Acustica, vol 24, 1999, pp 197-203. (frequency based speech dynamical system analysis) J. Garofolo, L. Lamel, W. Fisher, J. Fiscus, D. Pallet, N. Dahlgren, and V. Zue, “TIMIT Acoustic-Phonetic Continuous Speech Corpus,” Linguistic Data Consortium, 1993. (speech data set used for experiments)

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