Fractal and multifractal geometry: scaling symmetry and statistics
Advanced Review
Michael Frame, William Martino
Published Online: Mar 22 2012
DOI: 10.1002/wics.1207
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Abstract
Abstract Scaling symmetry is explored in several settings, deterministic, stochastic, and natural. Complexity of geometric fractals is quantified by dimension; complexity of a scaling measure is quantified by the multifractal spectrum. Some statistical examples are explored. WIREs Comput Stat 2012, 4:249–274. doi: 10.1002/wics.1207 This article is categorized under: Algorithms and Computational Methods > Computer Graphics Applications of Computational Statistics > Computational Mathematics Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data
Images
Left to right: Two examples of self‐similar fractals, one with some compositions forbidden, one nonlinear, and one random.
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The Mandelbrot set (top left), and a collection of magnifications.
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The transition graph for this Markov process.
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A driven IFS example: is the unoccupied address 323 forbidden or just unlikely?
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Cartoon and increment graphs for Δt _{1} = 4/9 (top left), Δt _{1} = 1/3 (top right), Δt _{1} = 2/9 (bottom left), and Δt _{1} = 1/9 (bottom right).
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Left: the sixth iterate of the process of Figure 21. Right: a randomized version of the left image.
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The initiator (left), generator (middle), and first iterate (right) of the cartoon with generator vertices (24).
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Lévy flight. Left: the path; right, the first difference.
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Top: Fractional Brownian motion simulations with α = 0.25, α = 0.5, and α = 0.75. Bottom: increments X (t + 1) − X (t ) of the graphs above.
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Left: the Koch curve. Center: a randomized Koch curve. Right: a randomized Sierpinski gasket.
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Examples of parametric plots of f (α ) curves.
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The f (α ) curves from Eq. ( 20), with p _{1} = 0.3 (left) and p _{1} = 0.45 (right).
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Log–log plot for computing the dimension of a nonlinear gasket.
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Left to right: IFS with memory, random, nonlinear, and self‐affine IFS, on a grid for computing the dimension.
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Covering the gasket by boxes of side length 1, 1/2, 1/4, and 1/8.
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Left: A small DLA simulation. Right: Natural fractal dendrites, modeled by DLA.
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Left: Steps in measuring the length of the Koch curve. Right: Steps in measuring the area of the Koch curve.
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Top: Time series and driven IFS for the logistic map with equal‐size bins. Bottom: Equal‐weight bins.
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Left: IFS driven by the DNA sequence for amylase. Right: an IFS random except that T _{4} never immediately follows T _{1} .
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Left: 2‐IFS generated by ℱ_{5} . Middle: an incorrect realization of the corresponding second higher block IFS. Right: the correct implementation.
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Left: IFS and vertex transition graph violating conditions (2) (and (3)). Right: violating only condition (3).
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Vertex transition graphs for ℱ_{1} , ℱ_{2} , and ℱ_{3}
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Top: four examples of IFS with memory. Bottom: Empty length 2 and 3 addresses.
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Fractal tree, fractal fern, fractal spiral.
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Some iterations of a process generating the left‐most fractal of Figure 1.
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The fractals generated by the IFS of the numbered tables.
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References Mandelbrot, B. The Fractal Geometry of Nature. San Francisco: W. H. Freeman ; 1982.
Resources
Mandelbrot B. The Fractal Geometry of Nature. San Francisco: W. H. Freeman; 1982.
How to Cite
Frame Michael, Martino William. Fractal and multifractal geometry: scaling symmetry and statistics. WIREs Comp Stat 2012, 4: 249274. doi: 10.1002/wics.1207