# Fractals and self-similarity

November 14, 2011 — September 22, 2021

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

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.

TBD

## 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.

## 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 Annals of Probability.
Ball. 2012. Architectural Design.
Barnsley, M F. 1986. Constructive Approximation.
Barnsley, Michael F. 2000. Fractals Everywhere.
Barnsley, Michael Fielding. 2006. SuperFractals.
Barnsley, Michael F., Elton, and Hardin. 1989. Constructive Approximation.
Barnsley, Michael, Hutchinson, and Stenflo. 2005. Fractals.
Barnsley, Michael F., Hutchinson, and Stenflo. 2008. Advances in Mathematics.
Chamorro-Posada. 2016. Chaos, Solitons & Fractals.
Chen. 2011. Chaos, Solitons & Fractals.
Davis. 1998. IEEE Transactions on Image Processing.
Draves, and Reckase. 1992. “The Fractal Flame Algorithm.”
Edwards, Phillips, Watkins, et al. 2007. Nature.
Eliazar, and Klafter. 2009. 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. In Image and Text Compression. The Kluwer International Series in Engineering and Computer Science.
Grassberger. 1988. Physics Letters A.
Gualdi, Yeung, and Zhang. 2011. arXiv:1108.1325 [Physics].
Hayakawa, Sato, and Matsushita. 1987. Phys. Rev. A.
Hutchinson. 1981. Indiana University Mathematics Journal.
Jacquin. 1992. IEEE Transactions on Image Processing.
———. 1993. Proceedings of the IEEE.
Kloeckner. 2021. 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. Nature.
Mandelbrot, Benoit B, and Hudson. 2006. The Misbehavior of Markets: A Fractal View of Financial Turbulence.
Masucci, Stanilov, and Batty. 2013. PLoS ONE.
Ogata, and Katsura. 1991. Biometrika.
Roberts, and Cronin. 1996. Physica A: Statistical and Theoretical Physics.
Smith. 1984. 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. Nature.
Theiler. 1990. Journal of the Optical Society of America A.
Vicsek, Tamás. 1983. Journal of Physics A: Mathematical and General.
Vicsek, Tamas. 1992. Fractal Growth Phenomena.
Vicsek, Tamás, and Szalay. 1987. Physical Review Letters.
West, Brown, and Enquist. 1997. Science.
Zabrodin. 2006. In Applications of Random Matrices in Physics.
Zhang, Yang, and Gao. n.d. The European Physical Journal B - Condensed Matter and Complex Systems.
Zhou. 1993. Journal of Theoretical Probability.