Fractal alternative to correlation

Question

I am looking for a fractal-based statistical measure which could be used as alternative to correlation between two variables (I know that hurst exponent can be used for auto-correlation).

Is anyone aware of such measures?

Answer 1

I doubt you're going to find a single answer to this, given the space of fractal dimensions. Most papers (in physics, geology) looking at correlation simply stick to a Pearson correlation with fractal math reserved for identifying dimension/self-similarity, etc.

But you might be interested in the following papers which use a "Correlation Fractal Dimension" as a similarity metric. The second paper mentions a fractal clustering algorithm which employs this metric.

• Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension (Belussi, Faloutsos, 1995)

• Characterizing Datasets Using Fractal Methods (Abrahao, Barbosa, 2003)

Answer 2

I agree with @ars that you are unlikely to get one answer for this (you may also have more success on http://mathoverflow.net, since our community tends to be more applied, while this technique would have very little real-world usage). The Abrahao/Barbosa paper is a good reference. Just to provide some additional sources:

This paper looks at the correlation between fractal dimensions, which seems like a reasonable approach to the problem.

• "The correlation of fractal structures in the photospheric and the coronal magnetic field" (Dimitropoulou, Georgoulis, Isliker, Vlahos, Anastasiadis, Strintzi, Moussas 2009)

This paper uses the multi-fractal spectra to estimate correlation:

• "Continuous wavelet transform based time-scale and multi-fractal analysis of the nonlinear oscillations in a hollow cathode glow discharge plasma" (Nurujjaman, Narayanan, Iyengar 2009)

Regarding the "Correlation Fractal Dimension", this paper provides a fast algorithm:

• "Faster estimation of the correlation fractal dimension using box-counting" (Attikos, Doumpos 2009)