Dear all (Mathematics and Ecologists mainly):
The Renyi spectrum of fractals contain all the main fractal dimensions in a multivariate structure. This structure is very useful for comparative analysis. I want to know what values of Q in this Renyi spectrum correspond exactly to the fractal dimensions of gyration and variance. I do not find this information in any published paper at the moment.
Many thanks.
This is an interesting question. Here is an article by Alfred Renyi that may be of interest to followers of this thread:
A. Renyi, A Socratic dialogue on mathematics, 2006:
http://www.bgc-jena.mpg.de/~bsmolny/pmwiki/uploads/DasJahr2007/test1AlephBibNo1236.pdf
More to the point for the question, see
I. Varga, J. Pipek, M. Janssen, K. Pracz, Multifractality beyond the Parabolic Approximation: Deviations from the Log-normal
Distribution at Criticality in Quantum Hall Systems, 1996
[available at the RG page for Imre Varga]
Daniel, the work by Renyi on spectra of fractals is well-worth careful study. I was surprised to find Renyi's 2006 Socratic dialogue, which reveals his ways of thinking. That Socratic dialogue goes along nicely with the more abstract treatment of spectra of fractals.
Thanx, James, for the reference to this extraordinary anthology of Hersh's.
OK but we have:
D0: capacity dimension
D1: information dimension
D2: correlation dimension
D2 < D1 < D0. D0 is similar to general Db (box-cox dimension). D1 measure complexity along scale in probabilistic terms and D2 measure the joined variation in X and Y or the level of aggregation. The question is: What is the significance of D3, D4, D5, etc... and particularly that Q-moment correspond to Gyration dimension Dg?. Regards
Hi, I think this is old, but, I dont know the exactly the definition of the gyration dimension, but the quantity that I know is associated with the radius of giration is the correlation length, if you check the definition https://en.wikipedia.org/wiki/Radius_of_gyration you will see it's almost the same, so the dimension that you are looking for is the correlation dimension D2.
The meaning of the higher moments D4 5 ... is obscure, but I wrote:
"A caveat of generalized dimensions is that they do not
have a clear interpretation in geometrical terms, except
for three special cases: D 0 is the standard fractal dimen-
sion, D 1 is the information dimension that describes how
the information content changes with scale, and is related to
the Shannon index of diversity. Finally D 2 is the correlation
dimension that measures how density accumulates as the
area changes, D 2 is related to the Simpson diversity index.
In fact D q might be considered as a measure of the the speed
of increasing of Hill’s generalized diversity index (Hill 1973)
as the area changes (Ricotta 2000). "
From:
Saravia, L. A., Giorgi, A., & Momo, F. (2012). Multifractal growth in periphyton communities. Oikos, 121(11), 1810–1820. http://doi.org/10.1111/j.1600-0706.2011.20423.x
https://www.researchgate.net/publication/256296052_Multifractal_growth_in_periphyton_communities
Cheers!
Article Multifractal growth in periphyton communities