Dynamical Invariants from a Time Series

1. Dynamical Invariants from a Time Series Saurabh Prasad Intelligent Electronic Systems Human and Systems Engineering Department of Electrical and Computer Engineering

2. Estimating the correlation integral from a time series Correlation Integral of an attractor’s trajectory : Correlation sum of a system’s attractor is a (probabilistic) measure quantifying the average number of neighbors in a neighborhood of radius along the trajectory. where represents the i’th point on the trajectory, is a valid norm and is the Heaviside’s unit step function (serving as a count function here) Correlation Dimension : In the limit that we have an infinitely large data set and a very small neighborhood For small epsilons, the correlation integral is expected to grow exponentially with the true dimension of the attractor Hence,

3. Correlation Integral and dimension estimation: Practical considerations • Temporal correlations vs. geometric correlations : Choosing neighbors in a small neighborhood about a point forces the inclusion of temporally correlated points •  This results in biasing the estimator, yielding a lower dimension estimate • Theiler’s correction : The solution is simple – exclude temporally correlated points from analysis. • ‘w’ – Theiler’s correction factor • An optimal value for w may be found by a space-time separation plot of the data-set.

4. Some results on the Lorentz attractor

5. Fitting the current discussion into the scope of Pattern Recognition Input Time Series (scalar / vector) RPS Higher Order Statistics Lyapunov Spectra K2 Entropy Correlation Dimension Measure discriminability in a space comprised of all possible combinations of these “features”

6. A note on Vector-Embedding • Vector Embedding: x(t) = [s(t) s(t - t) s(t - 2t) … s(t - (d - 1)t)] x(t) = [s1(t) s2(t)….sm(t) s1(t - t) s2(t - t) ….sm(t - t) … s1(t - (d - 1)t) s2(t - (d - 1)t)…sm(t - (d -1)t)] • It is typically assumed that the delay coordinates chosen are such that components of the embedded vectors are uncorrelated. • For scalar embedding, an optimal choice of t ensures this. • For vector embedding, a strong correlation between components of the (observed) vector stream may hurt the embedding procedure. • PCA/SVD based de-correlation may help remove correlations in second order statistics. • If correlations in higher order statistics of the data stream are removed, it is hoped that it will provide a more meaningful reconstruction.