Fractals and self-similarity

November 14, 2011 — September 22, 2021

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.]

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