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Title: ESTIMATION OF LYAPUNOV SPECTRA FROM A TIME SERIES


1
ESTIMATION OF LYAPUNOV SPECTRAFROM A TIME SERIES
S. Srinivasan, S. Prasad, S. Patil, G. Lazarou
and J. Picone Intelligent Electronic
Systems Center for Advanced Vehicular
Systems Mississippi State University URL
http//www.cavs.msstate.edu/hse/ies/publications/c
onferences/ieee_secon/2006/lyapunov_exponents/
2
  • Motivation
  • Analysis of chaotic signals
  • Reconstruction a phase-space from a scalar
    observable
  • Lyapunov exponents as a tool to analyze chaos
  • Lyapunov spectra of chaotic and non-chaotic time
    series
  • Optimize parameters of Lyapunov spectra
    estimation

3
  • Definitions
  • A deterministic signal or system every event is
    the result of preceding events and actions hence
    predictable completely
  • Stochastic noise signal that is not
    deterministic, i.e., inherently unpredictable
  • A chaotic signal or system sensitive to initial
    conditions (Butterfly Effect)
  • Chaos says predictability holds only in
    principle, hence chaotic signals are also called
    deterministic noise.
  • Dimension of a system number degrees of freedom
    possessed by the system
  • Deterministic Chaos or Stochastic Noise?
  • Both have continuous power spectra (and not
    easily distinguishable)
  • Noise is infinite-dimensional.
  • Chaotic signals are finite dimensional, but
    dimension no longer associated with number of
    independent frequencies, but a statistical
    feature related to both temporal evolution and
    geometric aspect (self-similar structure of the
    attractor)

4
  • Power Spectrum of a Lorentz Signal
  • Power spectra of chaotic signals are continuous,
    though the system is finite dimensional. For
    example, the power spectrum of a 3-dimensional
    chaotic Lorentz signal is shown below. Stochastic
    systems have similar spectra even though they are
    infinite dimensional.

5
  • Attractors for Dynamical Systems
  • System Attractor Trajectories approach a limit
    with increasing time, irrespective of the initial
    conditions within a region
  • Basin of Attraction Set of initial conditions
    converging to a particular attractor
  • Attractors Non-chaotic (point, limit cycle or
    torus),or chaotic (strange attactors)
  • Example point and limit cycle attractors of a
    logistic map (a discrete nonlinear chaotic map)

6
  • Strange Attractors
  • Strange Attractors attractors whose shapes are
    neither points nor limit cycles. They typically
    have a fractal structure (i.e., they have
    dimensions that are not integers but fractional)
  • Example a Lorentz system with parameters

7
  • Characterizing Chaos
  • Exploit geometrical (self-similar structure)
    aspects of an attractor or the temporal evolution
    for system characterization
  • Geometry of a Strange Attractor
  • Most strange attractors show a similar structure
    at various scales, i.e., parts are similar to the
    whole.
  • Fractal dimensions can be used to quantify this
    self-similarity.
  • e.g., Hausdorff, correlation dimensions.
  • Temporal Aspect of Chaos
  • Characteristic exponents or Lyapunov Exponents
    (LEs) - captures rate of divergence (or
    convergence) of nearby trajectories
  • Also Correlation Entropy captures similar
    information.
  • Any characterization presupposes that phase-space
    is available.
  • What if only one scalar time series measurement
    of the system (and not its actual phase space) is
    available?

8
  • Reconstructed Phase Space (RPS) Embedding
  • Embedding A mapping from an one-dimensional
    signal to an m-dimensional signal
  • Takens Theorem
  • Can reconstruct a phase space equivalent to the
    original phase space by embedding with m  2d1
    (d is the system dimension)
  • Embedding Dimension a theoretically sufficient
    bound in practice, embedding with a smaller
    dimension is adequate.
  • Equivalence
  • means the system invariants characterizing the
    attractor are the same
  • does not mean reconstructed phase space (RPS) is
    exactly the same as original phase space
  • RPS Construction techniques include differential
    embedding, integral embedding, time delay
    embedding, and SVD embedding

9
  • Reconstructed Phase Space (RPS) Time Delay
    Embedding
  • Uses delayed copies of the original time series
    as components of RPS to form a matrix
  • m embedding dimension, delay parameter
  • Each row of the matrix is a point in the RPS

10
  • Reconstructed Phase Space (RPS)

Time Delay Embedding of a Lorentz time series
11
  • Reconstructed Phase Space (RPS) Time Delay
    Embedding
  • Setting very small delay value leads to highly
    correlated vector elements, concentrated around
    the diagonal in embedding space. Structure
    perpendicular to the diagonal not captured
    adequately.
  • Setting very large delay value leads elements of
    the vector to behave as if they are independent.
    Evolutionary information in the system is lost.
  • Quantitative tools for fixing delay plots of
    autocorrelation and auto-mutual information are
    useful guides.
  • Advantages easy to compute the attractor
    structure is not distorted since no extra
    processing is done on it.
  • Disadvantages choice of delay parameter value is
    not obvious leads to poor RPS in presence of
    noise.

12
  • Reconstructed Phase Space (RPS) SVD-based
    Embedding
  • Works in two stages
  • Delay embed, with one sample delay, to a
    dimension larger than twice the actual embedding
    dimension
  • Reduce this matrix using SVD to finally have
    number of columns equal to embedding dimension.
  • (SVD-based matrix reduction is done by
    projecting each row onto only the first few
    eigenvectors and then reconstructing it to a
    lower-dimensional space)
  • SVD window size dimension of time delayed
    embedded matrix over which SVD operates
  • Advantages No delay parameter value to be set
    more robust to noise due to SVD stage
  • Disadvantages Noise reducing property of SVD may
    also distort the attractor properties

13
Reconstructed Phase Space (RPS) Reconstruction
Attractor reconstruction using SVD embedding
(for a Lorentz system)

14
  • Lyapunov Exponents
  • Quantifies separation in time between
    trajectories, assuming rate of growth (or decay)
    is exponential in time, as
  • where J is the Jacobian matrix at point p.
  • Captures sensitivity to initial conditions.
  • Analyzes separation in time of two trajectories
    with close initial points
  • where is the systems evolution function.

15
  • Lyapunov Exponents Some Properties
  • m-dimensional system has m LEs
  • LE is a measure averaged over the whole attractor
  • Sum of first k LEs rate of growth of
    k-dimensional Euclidean volume element
  • Bounded attractor Sum of all LEs equals zero
    (conservative) or negative (dissipative)
  • Zero exponents indicate periodic attractor (limit
    cycle) or a flow
  • Negative exponents pull points in the basin of
    attraction to the attractor
  • Positive exponents indicate divergence signature
    for existence of chaos

16
  • Lyapunov Exponents Computation
  • Embed time series to form RPS matrix. Rows
    represent points in phase space
  • Take first point as center
  • Form neighborhood matrix, each row obtained by
    subtracting a neighbor from the centre
  • Find evolution of each neighbor and form the
    evolved neighborhood matrix by subtracting each
    evolved neighbor from the evolved centre
  • Compute trajectory matrix at the center by
    multiplying pseudo-inverse of neighborhood matrix
    with evolved neighborhood matrix
  • Advance center to a new point and go to step 3,
    averaging the trajectory matrix in each iteration
  • The LEs are given by the average of the
    eigenvalues from each R matrix. Direct averaging
    has numerical issues, hence an iterative QR
    decomposition method (treppen-iteration) is used.

17
  • Lyapunov Exponents Computation Flowchart


18
  • Experimental Design
  • Three systems tested two chaotic (Lorentz and
    Rossler) and one non-chaotic (sine signal)
  • Two test conditions clean and noisy (10 dB white
    noise)
  • Lorentz system
  • Parameters
  • Expected LEs (1.37, 0, -22.37)
  • Rossler system
  • Parameters a 0.15, b 0.2, c 10
  • Expected LEs (0.090, 0.00, -9.8)
  • Sine Signal
  • Parameters Freq1Hz, Samp freq16Hz, Amp1
  • Expected LEs (0.00, 0.00, -1.85)

19
  • Experimental Design
  • Experiments performed to optimize parameters of
    estimation algorithm
  • 30,000 points were generated for each series in
    both the conditions
  • 5,000 iterations (or the number of evolution
    steps) were used for averaging using QR
    treppen-iteration
  • Variation of LEs with SVD window size and number
    of nearest neighbors
  • Varied number of neighbors with SVD window size
    15 for clean data 50 for noisy data
  • Varied SVD window size with number of neighbors
    15 for clean data 50 for noisy data

20
  • Experimental Results

Lyapunov Exponents (LEs) for a Lorentz System
  • For clean data Positive and zero exponents are
    almost constant at the expected values
  • For noisy data Positive and zero exponents
    converge to the expected value for SVD window
    size about 50 and number of neighbors also about
    50
  • Negative LE estimation not reliable

21
  • Experimental Results

Lyapunov Exponents (LEs) for a Rossler System
  • For clean data Positive and zero exponents are
    almost constant at the expected values
  • For noisy data Positive and zero exponents
    converge to the expected value for SVD window
    size about 60 and number of neighbors also about
    50
  • Negative LE estimation not reliable

22
  • Experimental Results

Lyapunov Exponents (LEs) for Sine Signal
  • For clean data Positive and zero exponents are
    almost constant at the expected values
  • For clean data Positive and zero exponents
    converge to the expected value for SVD window
    size about 40 and number of neighbors also about
    30
  • Negative LE estimation not reliable

23
  • Summary and Future Work
  • LEs are useful in quantifying how chaotic a
    system is.
  • SVD embedding helps reconstructing phase spaces
    in noisy conditions.
  • Parameters of the LE computation algorithm are
    optimized experimentally to get reliable
    estimates.
  • Both the positive and zero LEs are estimated
    near the actual values using optimized
    parameters.
  • Negative LE estimation is unreliable (but this is
    of little concern in chaotic systems).
  • The code for LE estimation is publicly available.
  • Our future work will be to apply Lyapunov
    exponents to model nonlinearities in speech for
    better automatic speech recognition.

24
  • Resources

25
  • References
  • J. P. Eckmann and D. Ruelle, Ergodic Theory of
    Chaos and Strange Attractors, Reviews of Modern
    Physics, vol. 57, pp. 617-656, July 1985.
  • M. Banbrook, Nonlinear analysis of speech from
    a synthesis perspective, PhD Thesis, The
    University of Edinburgh, Edinburgh, UK, 1996.
  • E. Ott, T. Sauer, J. A. Yorke, Coping with
    chaos, Wiley Interscience, New York, New York,
    USA, 1994.
  • M. Sano and Y. Sawada, Measurement of the
    Lyapunov Spectrum from a Chaotic Time Series,
    Physical Review Letters, vol. 55, pp. 1082-1085,
    1985.
  • G. Ushaw, Sigma delta modulation of a chaotic
    signal, PhD Thesis, The University of Edinburgh,
    Edinburgh, UK, 1996.
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