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?
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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)
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.
This paper uses the multi-fractal spectra to estimate correlation:
Regarding the "Correlation Fractal Dimension", this paper provides a fast algorithm: