fracta geometry

8/9/2019 Fracta Geometry

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Fractal Geometry

Will Martino 2012


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Before we start, some cartoons

Of the 8 graphs on the

right:

Two are financial data

(successive differences)

One is a Levy Stable

One is Brownian motion

One is fractional Brownian Three are fractal 'forgeries'


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Before we start, some cartoons

Of the 8 graphs on the

right:

Two are financial data

(successive differences)

One is a Levy Stable

One is Brownian motion

One is fractional Brownian Three are fractal 'forgeries'


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Financial Forgeries


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Financial Forgeries

Add Randomness (in application of generator segments)

By the way, you've just simulated Brownian motion


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Coarse Holder Exponent

(aka The Roughness Exponent)

When we impose the condition

that there is only one H for a

given cartoon, the cartoon

generated is considered a

unifractal


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Making Financial Forgeries

To make convincing financialforgeries throw out the

unifractal condition (one H)

Price vs Trading Time (TT) is unifractal

Clock vs Trading time is Multifractal

Price vs Clock is transformed via Trading

Time

TT compresses or extends volatility

in clock time


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Other methods of generation

Iterated Function Systems (IFS)

IFS with memory

Driven IFS

This can be driven via experimental data

Driven IFS generated from a Chromosome #7

Interesting visualizations interpretations

still under active research

We will go further into this topic later


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Quantifying Fractals

Given what we've see, what is the

appropriate way to measure sets that

are self similar?

The Koch curve's length depends on

how long of a yardstick you use


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Quantifying Fractals

Well, if length doesn't work what

about area?

Predictably, this also doesn't work


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I think this is what the Buddhists would call a question wrongly put.

John Greene


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Flavors of Dimensions

Fractals simply are not normalgeometric shapes. Given they're

infinite recursive self similarity, we need a new approach

Box-Counting Dimension a set

covering approach


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Flavors of Dimensions

The Similarity Dimension

Intuition: focus on counting the

number of copies (N) scaled by a

contraction factor (r) a self-similar

shape needs to construct itself

The Moran Equation

Allows us to apply the dim-similarity

to sets of non-uniform contractionfactor construction


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Hausdorff Measure


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Hausdorff Dimension

The t for which the measure makes its transition between

zero and infinity is defined as the Hausdorff dimension

In too low a dimension, the Hausdorff measure is zero and in

too high of one it is infinite


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Quick Overview of (Driven) IFS

IFS is the repeated application of transforms according to aspecified regime (either ordered or probabilistic)

In this example, the transforms we will use are:


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Intuition Recap

Moran equation relates the self-similarity of a set to its

dimension

Coarse Holder exponent measures the roughness of a set

The Hausdorff is the 'gold standard' of generalized

dimensional measures but it can be very hard to directly

compute

Now that we have the tools needed, we can put everything

together to develop the main tool used in my research


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Moran, Holder and Hausdorff


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Moran, Holder and Hausdorff

Where a(q) is the local Holder exponent and f(a) is the

Hausdorff dimension of the set for a given Holder exponent


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f(a) Curves

Via IFS with Memory, the Moran

equation further generalizes to:

Where rho is the spectral radius ofthe transition probability matrix of A

This enables us to plot the f(a) by

binning the data and extracting the

transition matrix


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Binning


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Speculation

What else? We end with a speculation. Recall Noethers theorem that if theLagrangian of a physical system exhibits a symmetry, then a conservation law is

associated with this symmetry. Familiar examples include:

space translation symmetry conservation of linear momentum

space rotation symmetry conservation of anguar momentum

time translation symmetry conservation of energy

gauge symmetry in electromagnetism conservation of charge

If fractality, or some class of fractality, is mediated by a Lagrangian, then there is a

conservation law, as yet unknown, corresponding to scaling symmetry.

- Prof. Michael Frame


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Citations

Much of this derives itself from my tutelage under Prof.

Michael Frame, his online textbook on the subject and the

following publications:

fracta geometry

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