Fractals and self-similarity

November 13, 2011 — September 21, 2021

model selection
photon choreography
regression
self similar
signal processing
statistics
stochastic processes
time series

Suspiciously similar content

Figure 1: From the original patent application of the original “fractal” vise

Objects with a fractional Hausdorff dimension, AFAICT. For some equivocation on this theme, see wikipedia.

1 Iterated function systems

Connection to design grammars.

Fun fact: there is a fractal renderer built into GIMP.

2 Mandelbrot sets I guess

TBD

3 Non-local derivatives

See fractional derivatives.

4 Long memory time series

See, for example, fractional brownian motion, long memory time series.

5 In nature

Pattern formation in nature often looks fractal to some approximation, or at some range of scales.

Figure 2: Horse lung
Figure 3: Starfish of infinite surface area.

6 Relationship to dimensions

David S. Richeson, A Mathematician’s Guided Tour Through High Dimensions does a nice job of relating fractals to measure-theoretic notions of dimension.

7 Estimating fractal dimension

Question: How closely related is this to estimating a Hurst exponent? How close to grammatical induction? Various classic methods based on naïve plug-in versions of mathematical definitions are given in Theiler (1990). A new one, which I am curious about, is Chamorro-Posada (2016) based on some kind of compression argument, basically, gzipping copies of the image that have been downsampled by various ratios and watching how the file size changes as a kind of entropy estimate.

I am reminded of Cosma Shalizi’s cautionary note on estimating Entropies and Information using Lempel-Ziv, and the caveat:

Jose M. Amigo, Janusz Szczepanski, Elek Wajnryb and Maria V. Sanchez-Vives, “Estimating the Entropy Rate of Spike Trains via Lempel-Ziv Complexity”, Neural Computation 16 (2004): 717–736: Normally, I have strong views on using Lempel-Ziv to measure entropy rates, but here they are using the 1976 Lempel-Ziv definitions, not the 1978 ones. The difference is subtle, but important; 1978 leads to gzip and practical compression algorithms, but very bad entropy estimates; 1976 leads, as they show numerically, to reasonable entropy rate estimates, at least for some processes. Thanks to Dr. Szczepanski for correspondence about this paper.]

8 References

Abraham. 1992. Dynamics–the geometry of behavior.
Aldous. 1991. “The Continuum Random Tree. I.” The Annals of Probability.
Ball. 2012. “Pattern Formation in Nature: Physical Constraints and Self-Organising Characteristics.” Architectural Design.
Barnsley, M F. 1986. “Fractal Functions and Interpolation.” Constructive Approximation.
Barnsley, Michael F. 2000. Fractals Everywhere.
Barnsley, Michael Fielding. 2006. SuperFractals.
Barnsley, Michael F., Elton, and Hardin. 1989. “Recurrent Iterated Function Systems.” Constructive Approximation.
Barnsley, Michael, Hutchinson, and Stenflo. 2005. “A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy.” Fractals.
Barnsley, Michael F., Hutchinson, and Stenflo. 2008. “V-Variable Fractals: Fractals with Partial Self Similarity.” Advances in Mathematics.
Chamorro-Posada. 2016. “A Simple Method for Estimating the Fractal Dimension from Digital Images: The Compression Dimension.” Chaos, Solitons & Fractals.
Chen. 2011. “Zipf’s Law, 1/f Noise, and Fractal Hierarchy.” Chaos, Solitons & Fractals.
Davis. 1998. “A Wavelet-Based Analysis of Fractal Image Compression.” IEEE Transactions on Image Processing.
Draves, and Reckase. 1992. “The Fractal Flame Algorithm.”
Edwards, Phillips, Watkins, et al. 2007. “Revisiting Lévy Flight Search Patterns of Wandering Albatrosses, Bumblebees and Deer.” Nature.
Eliazar, and Klafter. 2009. “Universal Generation of Statistical Self-Similarity: A Randomized Central Limit Theorem.” Physical Review Letters.
Falconer. 2014. Fractal Geometry: Mathematical Foundations and Applications.
Fisher, Yuval. 1994. Fractal Image Compression: Theory and Application.
Fisher, Y., Jacobs, and Boss. 1992. “Fractal Image Compression Using Iterated Transforms.” In Image and Text Compression. The Kluwer International Series in Engineering and Computer Science.
Grassberger. 1988. “Finite Sample Corrections to Entropy and Dimension Estimates.” Physics Letters A.
Gualdi, Yeung, and Zhang. 2011. “Tracing the Evolution of Physics on the Backbone of Citation Networks.” arXiv:1108.1325 [Physics].
Hayakawa, Sato, and Matsushita. 1987. “Scaling Structure of the Growth-Probability Distribution in Diffusion-Limited Aggregation Processes.” Phys. Rev. A.
Hutchinson. 1981. “Fractals and Self-Similarity.” Indiana University Mathematics Journal.
Jacquin. 1992. “Image Coding Based on a Fractal Theory of Iterated Contractive Image Transformations.” IEEE Transactions on Image Processing.
———. 1993. “Fractal Image Coding: A Review.” Proceedings of the IEEE.
Kloeckner. 2021. “Optimal Transportation and Stationary Measures for Iterated Function Systems.” Mathematical Proceedings of the Cambridge Philosophical Society.
Kuffner, and LaValle. 2009. “Space-Filling Trees.”
Mandelbrot, Benoit B. 1999. Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976).
Mandelbrot, Benoit B. 2010. Fractals and Scaling In Finance: Discontinuity, Concentration, Risk.
Mandelbrot, Benoit B, and Evertsz. 1990. “The Potential Distribution Around Growing Fractal Clusters.” Nature.
Mandelbrot, Benoit B, and Hudson. 2006. The Misbehavior of Markets: A Fractal View of Financial Turbulence.
Masucci, Stanilov, and Batty. 2013. “Limited Urban Growth: London’s Street Network Dynamics Since the 18th Century.” PLoS ONE.
Ogata, and Katsura. 1991. “Maximum Likelihood Estimates of the Fractal Dimension for Random Spatial Patterns.” Biometrika.
Roberts, and Cronin. 1996. “Unbiased Estimation of Multi-Fractal Dimensions of Finite Data Sets.” Physica A: Statistical and Theoretical Physics.
Smith. 1984. “Plants, Fractals, and Formal Languages.” In SIGGRAPH Comput. Graph.
Stanley, Amaral, Goldberger, et al. 1999. “Statistical Physics and Physiology: Monofractal and Multifractal Approaches.” Physica A: Statistical and Theoretical Physics.
Stanley, and Meakin. 1988. “Multifractal Phenomena in Physics and Chemistry.” Nature.
Theiler. 1990. “Estimating Fractal Dimension.” Journal of the Optical Society of America A.
Vicsek, Tamás. 1983. “Fractal Models for Diffusion Controlled Aggregation.” Journal of Physics A: Mathematical and General.
Vicsek, Tamas. 1992. Fractal Growth Phenomena.
Vicsek, Tamás, and Szalay. 1987. “Fractal Distribution of Galaxies Modeled by a Cellular-Automaton-Type Stochastic Process.” Physical Review Letters.
West, Brown, and Enquist. 1997. “A General Model for the Origin of Allometric Scaling Laws in Biology.” Science.
Zabrodin. 2006. “Matrix Models and Growth Processes: From Viscous Flows to the Quantum Hall Effect.” In Applications of Random Matrices in Physics.
Zhang, Yang, and Gao. n.d. “Role of Fractal Dimension in Random Walks on Scale-Free Networks.” The European Physical Journal B - Condensed Matter and Complex Systems.
Zhou. 1993. “Resistance Dimension, Random Walk Dimension and Fractal Dimension.” Journal of Theoretical Probability.