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Fractal and multifractal geometry: scaling symmetry and statistics

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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

Figure 1.

Left to right: Two examples of self‐similar fractals, one with some compositions forbidden, one nonlinear, and one random.

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Figure 2.

Some iterations of a process generating the left‐most fractal of Figure 1.

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Figure 3.

The fractals generated by the IFS of the numbered tables.

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Figure 4.

Fractal tree, fractal fern, fractal spiral.

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Figure 5.

Top: four examples of IFS with memory. Bottom: Empty length 2 and 3 addresses.

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Figure 6.

Vertex transition graphs for ℱ1, ℱ2, and ℱ3

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Figure 7.

Left: IFS and vertex transition graph violating conditions (2) (and (3)). Right: violating only condition (3).

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Figure 8.

Left: 2‐IFS generated by ℱ5. Middle: an incorrect realization of the corresponding second higher block IFS. Right: the correct implementation.

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Figure 9.

Left: IFS driven by the DNA sequence for amylase. Right: an IFS random except that T4 never immediately follows T1.

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Figure 10.

Top: Time series and driven IFS for the logistic map with equal‐size bins. Bottom: Equal‐weight bins.

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Figure 11.

Left: Steps in measuring the length of the Koch curve. Right: Steps in measuring the area of the Koch curve.

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Figure 12.

Left: A small DLA simulation. Right: Natural fractal dendrites, modeled by DLA.

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Figure 13.

Covering the gasket by boxes of side length 1, 1/2, 1/4, and 1/8.

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Figure 14.

Left to right: IFS with memory, random, nonlinear, and self‐affine IFS, on a grid for computing the dimension.

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Figure 15.

Log–log plot for computing the dimension of a nonlinear gasket.

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Figure 16.

The f(α) curves from Eq. (20), with p1 = 0.3 (left) and p1 = 0.45 (right).

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Figure 17.

Examples of parametric plots of f(α) curves.

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Figure 18.

Left: the Koch curve. Center: a randomized Koch curve. Right: a randomized Sierpinski gasket.

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Figure 19.

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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Figure 20.

Lévy flight. Left: the path; right, the first difference.

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Figure 21.

The initiator (left), generator (middle), and first iterate (right) of the cartoon with generator vertices (24).

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Figure 22.

Left: the sixth iterate of the process of Figure 21. Right: a randomized version of the left image.

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Figure 23.

Cartoon and increment graphs for Δt1 = 4/9 (top left), Δt1 = 1/3 (top right), Δt1 = 2/9 (bottom left), and Δt1 = 1/9 (bottom right).

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Figure 24.

A driven IFS example: is the unoccupied address 323 forbidden or just unlikely?

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Figure 25.

The transition graph for this Markov process.

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Figure 26.

The Mandelbrot set (top left), and a collection of magnifications.

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Data Visualization > Computer Graphics
Data Structures > Time Series, Stochastic Processes, and Functional Data
Applications of Computational Statistics > Computational Mathematics
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