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Wednesday, September 11, 2013

Fractals, and probability

Fractals can be experienced from geometric visualizations, inside natural mathematical functions.  They support a range of applications from design sciences, to medicine, to finance.  In this note we show ways fractals and probability models can help inform one another.  In addition to a reputable Mandelbrot set, we have assembled three fractal "likelihood functions", formulated with the probability concepts of correlation and variance in mind.  We ultimately test whether any of the fractal's geometric space would be penetrable to escaping the central orbit.  And we again apply a compound probability function that quickly arrives at its estimate, in under only a half dozen iterations.  This is a far lower magnitude than most mathematical algorithms trying to narrowly predict the changes in the series patterns.  See Credibility of Fibonacci (a constant itself in addition to some risk measures that are rooted in fractal math) blog note for a complementary discussion on confidence intervals.

We illuminate the video with 10,000 simulations, per function, all sampled from a bi-normal scattering within a radius of 2 from the center.  We also zoom in on each function's center.  Even at a small number of iterations, one can see the contained range, from the middle two fraction equations.  We use the color blue for low a probability (e.g., <25% chance) of escape.  And on the other side, we use the color red for a high probability (e.g., >75% chance) of escape.

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    1. Thanks much Psychologist in Azusa. Please make sure to follow the blog (one can do that a few ways, including using the RSS form on this site). Also let us know of the sorts of topics on which you would enjoy seeing statistical analysis. Recently my interests are in doing research on mortality, and also in the microsciences.

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