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## Lecture series: Data analysis

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**Lecture series: Data analysis**Thomas Kreuz, ISC, CNR thomas.kreuz@cnr.it http://www.fi.isc.cnr.it/users/thomas.kreuz/ Lectures: Each Tuesday at 16:00 (First lecture: May 21, last lecture: June 25)**Schedule**• Lecture 1: Example (Epilepsy & spike train synchrony), • Data acquisition, Dynamical systems • Lecture 2: Linear measures, Introduction to non-linear • dynamics • Lecture 3: Non-linear measures • Lecture 4: Measures of continuous synchronization • Lecture 5: Measures of discrete synchronization • (spike trains) • Lecture 6: Measure comparison & Application to epileptic • seizure prediction**First lecture**• Example: Epileptic seizure prediction • Data acquisition • Introduction to dynamical systems**Second lecture**• Non-linear model systems • Linear measures • Introduction to non-linear dynamics • Non-linear measures • - Introduction to phase space reconstruction • - Lyapunov exponent**Third lecture**Non-linear measures - Dimension [ Excursion: Fractals ] - Entropies - Relationships among non-linear measures**Characterizition of a dynamic in phase space**Stability (sensitivity to initial conditions) (Information / Entropy) Density Determinism / Stochasticity Predictability Linearity / Non-linearity Self-similarity (Dimension)**Dimension (classical)**Number of degrees of freedom necessary to characterize a geometric object Euclidean geometry: Integer dimensions Object Dimension Point 0 Line 1 Square (Area) 2 Cube (Volume) 3 N-cube n Time series analysis: Number of equations necessary to model a physical system**Box-counting**Richardson: Counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used. Fractal dimension of a coastline: How does the number of measuring sticks required to measure the coastline change with the scale of the stick?**Example: Koch-curve**Some properties: - Infinite length - Continuous everywhere - Differentiable nowhere - Fractal dimension D=log4/log3≈ 1.26**Strange attractors are fractals**Logistic map Hénonmap RösslerSystem () 2,01**Generalized dimensions**• - Haussdorff-dimension • for - Fractal dimension • - Correlation dimension • Static measure of system complexity (degrees of freedom): • Regular dynamics integer • Chaotic dynamics fractal • Stochastic dynamics Monotonous decrease with**Generalized entropies**• - Topological entropy • for - Metric entropy • - Correlation entropy • Dynamic measure of system disorder: • Regular dynamics • Chaotic dynamics • Stochastic dynamics Monotonous decrease with**Lyapunov-exponent**Rate of separation of infinitesimally close trajectories (Sensitivity to initial conditions) Largest Lyapunov exponent (LLE) (often ): Regular dynamics Chaotic dynamics Stochastic dynamics Stable fixed point**Summary**• Regular dynamics integer; • Chaotic dynamics fractal; • Stochastic dynamics**Today’s lecture**• Motivation • Measures of synchronization for continuous data • Linear measures: Cross correlation, coherence • Mutual information • Phase synchronization (Hilbert transform) • Non-linear interdependences • Measure comparison on model systems • Measures of directionality • Granger causality • Transfer entropy**Motivation: Bivariate time series analysis**• Three different scenarios: • Repeated measurement from one system (different times) • Stationarity, Reliability • Simultaneous measurement from one system (same time) • Coupling, Correlation, Synchronization, Directionality • Simultaneous measurement from two systems (same time) • Coupling, Correlation, Synchronization, Directionality**Synchronization**Etymology: ‘synchronous’ and ‘synchronicity’ (syn = common) and (chronos = time) “Happening at the same time” But: ‘synchronization’ Implies (active) adjustment of rhythms of different oscillating systems due to some kind of interaction or coupling Huygens, 1673: [Huygens: HorologiumOscillatorium. 1673]**Synchronization**• Complete / identical synchronization • Only possible for identical systems • Phase synchronization • mn-phase locking, amplitudes uncorrelated • Generalized synchronization • Unidirectionally coupled systems: • Largest Lyapunov exponent of the responder negative • [Necessary (but not sufficient) condition] [Pecora & Carroll. Synchronization in chaotic systems. Phys Rev Lett1990]**Synchronization**In-phase synchronization [Pikovsky& Rosenblum: Synchronization. Scholarpedia (2007)]**Synchronization**Anti-phase synchronization [Pikovsky& Rosenblum: Synchronization. Scholarpedia (2007)]**Synchronization**Synchronization with phase shift [Pikovsky& Rosenblum: Synchronization. Scholarpedia (2007)]**Synchronization**No synchronization [Pikovsky& Rosenblum: Synchronization. Scholarpedia (2007)]**Synchronization**In-phase synchronization Anti-phase synchronization Synchronization with phase shift No synchronization [Pikovsky& Rosenblum: Synchronization. Scholarpedia (2007)]**Measures of synchronization**Synchronization Directionality Cross correlation / Coherence Mutual Information Index of phase synchronization - based on Hilbert transform - based on Wavelet transform Non-linear interdependence Non-linear interdependence Event synchronization Delay asymmetry Transfer entropy Granger causality**Linear**correlation**Static linear correlation: Pearson’s r**Two sets of data points: -1 - completely anti-correlated r = 0 - uncorrelated (linearly!) 1 - completely correlated**Examples: Pearson’s r**Undefined [An example of the correlation of x and y for various distributions of (x,y) pairs; Denis Boigelot 2011]**Cross correlation**Two signals and with (Normalized to zero mean and unit variance) Time domain, dependence on time lag : Maximum cross correlation:**Coherence**Linear correlation in the frequency domain Cross spectrum: – Fourier transform, – discrete frequencies, * - Complex conjugation Complex number Phase Coherence = Normalized power in the cross spectrum Welch’s method: average over estimated periodograms of subintervals of equal length**Mutual**information**Shannon entropy**Shannon entropy ~ ‘Uncertainty’ Binary probabilities: In general:**Mutual Information**Marginal Shannon entropy: Joint Shannon entropy: Mutual Information: Kullback-Leibler entropy compares to probability distributions Mutual Information = KL-Entropy with respect to independence Estimation based on k-nearest neighbor distances: - digamma function [Kraskov, Stögbauer, Grassberger: Estimating Mutual Information. Phys Rev E 2004]**Mutual Information**Properties: Non-negativity: Symmetry: Minimum: Independent time series Maximum: for identical systems Venn diagram (Set theory)**Cmax**Cmax Cmax I I I 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 Cross correlation & Mutual Information**Phase**synchronization**Phase synchronization**• Definition of a phase • - Rice phase • - Hilbert phase • - Wavelet phase • Index of phase synchronization • - Index based on circular variance • - [Index based on Shannon entropy] • - [Index based on conditional entropy] [Tass et al. PRL 1998]**Rice phase**Linear interpolation between ‘marker events’ - threshold crossings (mostly zero, sometimes after demeaning) - discrete events (begin of a new cycle) Problem: Can be very sensitive to noise**Hilbert phase**Analytic signal: ‘Artificial’ imaginary part: - Cauchy principal value Frequency domain: Phase shift of original signal by Instantaneous Hilbert phase: [Rosenblum et al., Phys. Rev. Lett. 1996]**Wavelet phase**Basis functions with finite support Example: complex Morlet wavelet Wavelet phase: Wavelet = Hilbert + filter [QuianQuiroga, Kraskov, Kreuz, Grassberger. Phys. Rev. E 2002]**Index of phase synchronization:**Circular variance (CV)**Non-linear**interdependence**Taken’s embedding theorem**Trajectory of a dynamical system in - dimensional phase space . One observable measured via some measurement function : ; M: It is possible to reconstruct a topologically equivalent attractor via time delay embedding: - time lag, delay; – embedding dimension [F. Takens. Detecting strange attractors in turbulence. Springer, Berlin, 1980]**Non-linear interdependences**Nonlinear interdependence S Nonlinear interdependence H And equivalent for and Synchronization Directionality [Arnhold, Lehnertz, Grassberger, Elger. Physica D 1999]