This video is a guide on how to implement time-series reconstruction algorithms in Python with example of the Lorenz attractor. The Lorenz attractor is a mathematical chaotic system and, thus, fully disclosed. The reconstruction algorithms applied are the Takens delay-time embedding or simply Takens approach and the spectral embedding algorithm based on Laplacian Eigenmaps in combination with principal component analysis (PCA) and nearest neighbour algorithms (k-NN) taken out of machine learning. Strange (fractal) attractor reconstruction of time-series is an important methodology in nonlinear dynamics, nonlinear time-series analysis and chaos analysis.
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Content:
00:00 - 00:25 Animated Introduction
00:25 - 01:11 Content Presentation
01:11 - 03:31 Lorenz Attractor Theory
03:32 - 06:02 Python Code for Lorenz Attractor
06:03 - 08:38 Python Lorenz Attractor
08:39 - 11:05 Takens Delay Time Embedding Algorithm Theory
11:05 - 12:25 Python Code for Takens Approach
12:25 - 16:43 Python Takens Delay Time Embedding
16:43 - 19:00 Spectral Embedding Theory
19:00 - 20:45 Python Code Display for Spectral Embedding
20:45 - 22:00 Python Spectral Embedding Lorenz (Total)
22:00 - 27:06 Python Spectral Embedding Lorenz (Single Variables)
27:06 - End References and Final Comments
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Scientific Reference:
Belkin, M., & Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6), 1373-1396.
Çoban, G., & Büyüklü, A. H. (2009). Deterministic flow in phase space of exchange rates: evidence of chaos in filtered series of Turkish Lira-Dollar daily growth rates. Chaos, Solitons and Fractals, 42(2), 1062-1067.
Harikrishnan, K., Misra, R., & Ambika, G. (2017). Is a hyperchaotic attractor superposition of two multifractals? Chaos, Solitons and Fractals, 103, 450-459.
Hirsch, M. (1997). Differential Topology. Berlin, New York: Springer.
Lewandowski, M., Makris, D., Velastin, S., & Nebel, J.-C. (2014). Structural Laplacian Eigenmaps for modeling sets of multivariate sequences. IEEE Transactions on Cybernetics, 44(6), 936-949.
Lorenz, E. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130-141.
Song, X., Niu, D., & Zhang, Y. (2016). The Chaotic Attractor Analysis of DJIA Based on Manifold Embedding and Laplacian Eigenmaps. Mathematical Problems in Engineering, 4, 1-10.
Strogatz, S. (2014). Nonlinear Dynamics and Chaos. Colorado: Westview Press.
Takens, F. (1981). Detecting strange attractors in fluid turbulence. in: D. Rand. L.-S. Young (Eds.). Dynamical Systems and Turbulence. Springer Berlin, 366-381.
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Hastags
#chaos #nonlinear #nonlineardynamics #Lorenz #butterfly #strangeattractor #attractor #Takens #spectraltheory #embedding #complexity #dynamics #mathematics #python #pythonprogramming #data #datascientist #datscience #3danimation #science #reconstruction #analysis
