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Lecturer(s)
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Kulish Vladimír, doc. Ing. PhD., DSc.
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Hamidreza Namazi, Ph.D.
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Course content
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1. Introduction: complex interconnected systems; examples of multi-scale phenomena and challenging problems to be pursued; prelude to chaos theory & fractal geometry 2. Randomness in a nutshell: concept of randomness; probability as a quantitative measure of randomness; ergodic principle; unexpectedness & risk; Bernoulli trials; concept of probability distribution (discrete & continuous); simple distributions (from binomial to Poisson to normal); some other simple distributions (e.g., gamma, beta, etc.); Kullback-Leibler divergence 3. Randomness harnessed: predictable outcomes of random processes (e.g., Monte Carlo method, hill climbing method, Brownian motion, etc.); some paradoxes of randomness (e.g., survival of leaders) 4. Information, complexity & organisation: randomness & information; redundancy; entropy of a distribution; complexity & organisation; organised complexity; some examples of complex interconnected systems 5. Chaos: determinism versus predictability; the phenomenon of chaos; chaos vs. randomness; Lyapunov exponent & Lyapunov time; the onset of chaos with examples; Feigenbaum constants; strange attractors 6. Fractals: strange attractors as visual identity of chaos; the concept of fractals; the notion of fractal dimension; self-similar geometrical fractals (e.g., Cantor sets, von Koch & Peano curves, etc.); power law and perception of self-similarity; temporal fractals (signals); Hurst parameter as a measure of persistence 7. Multi-fractals: mono-fractals vs. multi-fractals; Rényi entropy as generalisation of Shannon entropy; generalised fractal dimensions; spectra of fractal dimensions & their approximation by the generalised logistic function; Rényi divergence & improved generalised fractal dimensions 8. Introduction to fractional calculus: Laplace transforms & fractional differ-integrals; fractional differ-integrals of some elementary functions; non-field (memory) solutions of differential equations (convolution & its physical meaning); fractional differ-integrals applied to chaos & fractals 9. Fractal & multi-fractal time series (signals): self-similar stochastic processes (Brownian motion vs. fractional Brownian motion); stable laws and Levy motions (stable distributions & simulation of stable random variables); long memory processes; power laws 10. Modelling & analysis of fractals and multi-fractals: modelling fractals by the Weierstrass-Mandelbrot functions (one- and multi-dimensional); non-field fractional models; fractional signal de-noising; generalised fractal dimensions revisited; signals with varying Hurst parameter; Hurst parameter & predictability; pattern recognition & forecasting 11. Applications (part I): signals generated by inanimate complex interconnected systems (e.g., turbulence, markets, internet traffic, etc.) 12. Applications (part II): biological signals [life as organised complexity] (e.g., DNA walk, cardio and brain signals, bio-photonics, Kirlian images) 13. Conclusion: course overview & summary; discussion on course projects; Q & A session with students
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Learning activities and teaching methods
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- Class attendance
- 52 hours per semester
- Semestral paper
- 26 hours per semester
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Learning outcomes
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Complex interconnected systems, including the Internet, stock markets, human heart or brain, and many others are usually comprised of multiple subsystems that exhibit highly nonlinear deterministic as well as stochastic characteristics and are regulated hierarchically. They generate signals that exhibit complex characteristics such as nonlinearity, sensitive dependence on small disturbances, long memory, extreme variations, and non-stationarity. This introductory course is focused on integrating chaos theory and random fractal theory to analyse these complex time series (signals). Starting with the most fundamental concepts of chaos theory and random fractal theory, students gradually learn how to correctly apply the tools offered by the said theories for analysis of complex multi-fractal signals. Various example of real complex signals are considered. The curriculum of the course is structured to meet the needs of students who are analysing complex time series for their own projects.
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Prerequisites
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Undergraduate calculus; high-school probability theory
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Assessment methods and criteria
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unspecified
Skills acquired through lectures and self-study are assessed via a written mid-term test (0?5 points) conducted midway through the semester. The score from this test is combined with the results of a written final test (0?45 points) administered at the end of the semester. A minimum of 30 points is required to qualify for pre-exam credit. Additionally, students must submit and defend their course project before the final exam. The course project is graded (0?50 points) and contributes to the total grade for the written final exam, which comprises two questions (0?25 points each). To pass the exam, students must achieve a total minimum score of 60 points.
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Recommended literature
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