Non-Linear Analysis od Biological Time-Series

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Information and Communication Technology in Medicine

Synopsis:

Summary of practical applications of fractal an multifractal analysis, applied

to biological time-series. Introduction to deterministic chaos. Takens theorem,

practical computation of selected invariant parameters from experimental data

(correlation dimension, Lyapunov exponents etc.). Tests for determinism and

nonlinearity. High-dimensional chaos. Multifractal formalism, estimators of

Hurst exponents, self-similarity of time series.

Requirements:

Syllabus of lectures:

1. Description of the linear systems, linear models, stability of the linear

system,

2. Description of non-linear systems, comparison with the linear systems,

stability of non-linear systems.

3. Deterministic chaos - introduction, State-space and phase diagram,

bifurcation diagram, strange attractor.

4. Fractals, non-integer dimension.Examples of geometrical fractal structures.

5. One-dimensional discrete systems, bifurcation, doubling of the periods, non-

stabil orbits, Henon maps

6. Information and entropy. Stationarity of time-series. Influence of

discretization. Time-frequency analysis and wavelet transform.

7. Metods of state space, reconstruction of the chaotic attractor, delayed

coordinates, Takens-theorem, embedded dimension and reconstruction delay.

8. Average mutual information, false nearest neighbours, practical computation

of the correlation dimension from the experimental data

9. Largest Lyapunov exponent, spectrum of Lyapunov exponents - methods of

computation.

10. Self-similarity in time series, fluctuation analysis, Hurst exponent, 1/f

noise.

11. Estimators of the global Hurst exponent, Detrended Flucuation Analysis.

12. From the fractal to the multifractal: generalized fractal dimension,

spectrum of the local Hurst exponents, utilization of the Wavelet transform,

WTMM-estimator.

13. Comparsion and utilization of classical and non-linear methods, methods of

biological data visualisaton, descriptors and invariant descriptors.

14. Exapmles of successful application of the non-linear analysis in medicine,

biology and engineering. Contemporary research centers and Information sources.

Syllabus of tutorials:

Study Objective:

Summary of practical applications of fractal an multifractal analysis, applied

to biological time-series. Introduction to deterministic chaos. Takens theorem,

practical computation of selected invariant parameters from experimental data

(correlation dimension, Lyapunov exponents etc.). Tests for determinism and

nonlinearity. High-dimensional chaos. Multifractal formalism, estimators of

Hurst exponents, self-similarity of time series.

Study materials:

[1] Harte: Multifractals, Theory and Applications, Chapman & Hall, 2001

[2] Hilborn: Chaos and Nonlinear Dynamics, Oxford University Press, 2003

[3] Kantz, Schreiber: Nonlinear Time Series Analysis, Cambridge University

Press, 2002

[4] Davies: Exploring Chaos, Perseus Publushing, 1999

[5] Sprott: Chaos and Time-Series Analysis, Oxford University Press, 2003

[6] Abarbanel et al.: Introduction to Nonlinear Dynamics for Physicists, World

Scientific, vol 53., 1996

[7] Kantz, Schreiber: Nonlinear Time Series Analysis, Cambridge University

Press, 2002

[8] Strogatz: Nonlinear Dynamics and Chaos, Westview Press, 2000

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: