# Need good material on multifractal analysis

I'm searching for some good reading material on multifractal analysis. Preferably something accessible that doesn't put the stress too much on mathematical proofs but rather on applications. As long as it gives a good review of the status of the field, the interesting results and applications, I would be happy. Also, if anything related to multifractal analysis and statistics or time series comes up, I'll take it as well.

Books, papers, internetpages, videos, etc... accepted!

EDIT: Since the question has been bumped, I decided to put a bounty on it. But I also want to make a bit more precise what I'm looking for.

I have always had the impression when encountering the multifractal techniques that people are able to compute a whole bunch of numbers with some nice and fancy formulas. But I have always missed an "understanding" of what the numbers mean. Why is it useful to do a mutlifractal analysis of fluid flow? Of species abundance distributions? Etc... I feel like the technique is purely descriptive with little theory backing up the connection with some deeper underlying structures. But that may just be due to my limited understanding of the field and that is precisely why I ask for pointers to where I can look for this.

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It's frustrating that this video has no sound (the original video was on google video, which has since been shut down), but when I saw this video (with sound), I found it informative. I admit to not fully groking multifractals, but I've glimpsed at papers found here and here. –  user4143 May 10 '11 at 15:18
You should make that a complete answer instead of just a comment. –  Raskolnikov May 10 '11 at 15:28

I found Multifractals: Theory and Applications by David Harte to be really good.

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Hmmm... The first reviewer doesn't seem very enthusiastic about this book. On the other hand, he gives a link to the page of Rudolf Riedi which seems interesting. I'll take a look at both. +1 and you are first in line for the prize. ;) –  Raskolnikov May 10 '11 at 15:22
When I read it, I already knew why multifractals were interesting and what I needed to analyze & why. What I needed was the 'how', and I got that from this book. Riedi does have some interesting papers though, I remember really liking "The Multiscale Nature of Network Traﬃc: Discovery, Analysis, and Modelling". Also have a look at "Why study multifractal spectra?" by Peter Morters at people.bath.ac.uk/maspm/whystudy.pdf –  PeterR May 10 '11 at 17:42

OK, I'm going to hijack this thread even though there's an answer as I haven't found any quality, localized information about multifractals.

As mentioned in the comments, I first heard about multifractals from a Google Tech Talk by Rogene M. Eichler West, which can be found, without sound, on YouTube, called "Multifractals: Theory, Algorithms, & Applications" . Unfortunately Google Video got discontinued after they bought out YouTube and I can't find the original video that had the sound included.

I still do not understand on a deep level what, how and why multifractals are doing, are better than another method or how they do it, but from what I understand the idea is to generalize the concept of spectrum to include functions that have a scale symmetry, where the scale symmetry can be on many different scales (thus multi-fractal, instead of just being fractal). Just as the Fourier spectra is constructing a profile of the translation invariances of a function, the multifractal spectra gives information about the scale invariances of a function.

The general methodology seems to be, for a given function $f(t)$:

Where $D(\alpha) \stackrel{def}{=} D_F\{x, h(x) = \alpha\}$, and $D_F\{\cdot\}$ is the (Hausdorff?) dimension of a point-set.

I believe the idea is that for chaotic/fractal/discontinuous functions, at any point they can be characterized, locally, by the largest term of their Taylor expansion and the Hölder exponent is a way to characterize this. Once you have the function, $h(t)$, characterizing the Hölder exponent, you use that to construct the singularity spectrum. I believe the singularity spectrum is a synonym for the multi-fractal spectrum.

From what I can tell, the specifics of how to calculate $h(t)$ and $D(\alpha)$ in practice vary from approximating them outright by their definition or by using wavelets to approximate the Hölder exponent and then using a Legendre transform to approximate the multifractal spectrum.

From what I understand, $D(\alpha)$ tends to be (or is always?) concave. I have only the vaguest notion of why this is so. How one relates wavelet transforms to finding the Hölder exponent, how one uses the Legendre transform to find the multi-fractal spectrum, why the multi-fractal spectrum should be concave, what kind of intuitive feeling one should get about a function from viewing the spectrum, amongst many others, I still have no idea about.

The multiplicative cascade seems to be a canonical example of a multifractal process.

Online, "A Brief Overview of Multifractal Time Series" gives a terse run through of multifractals. They claim to be able to tell a healthy heart from one that is suffering from congestive heart failure (see here).

Here are some slides giving a brief overview of multifractals. Near the end of the slides, they give a wavelet transform of the Devil's staircase function and talk a bit about using Wavelet Transform Modulus Maxima Method (WTMM), which appears to be a standard tool when doing this type of analysis (anyone have any good links for this?).

Looking around, I found Wavelets in physics by J. C. van den Berg that had this section web accessible for a definition of the singularity spectrum.

Rudolf H. Riedi seems to have a few papers out there that describe multifractal processes. Here are a few:

While focused on finance, Laurent Calvet and Adlai Fisher have a lot of introduction to terminology in "Multifractility in asset returns: Theory and evidence".

And of course Mandelbrot, along with other authors, has many papers, some of which are:

Fractional Brownian Motion is also mentioned frequently, but I have no real idea of how they relate. Large Deviation Theory also seems to be mentioned, but I don't know how this relates to multifractals either. I believe I've also seen entropy, phase transitions and statistical mechanics mentioned here and there. I would be curious if and what the relation to these subjects and multifractals is.

I feel like I'm stumbling around trying to understand this subject and I have yet to find a cohesive text that brings together enough intuition, math and implementation details so that I feel like I have a firm grasp of what's going on. I would welcome any additional resources or corrections to this answer.

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Hi, great addition to this thread. As far as I can understand it, multifractal analysis is only a technique to analyse "fractal phenomena". But just as you, I've never quite understood what we got from those spectra. However, I've seen them show up quite a lot during my research. The first time was during my PhD, when I was studying a quasifree fermion system. We proved some results about this system and they were tied to the Hamiltonian of the single particle system. When the Hamiltonian had an absolutely continuous spectrum, we could prove some stuff about... –  Raskolnikov Jan 4 '12 at 9:17
...the macroscopic quantities of interest. When the spectrum was singular continuous, we got another result. To check that result for a specific case we used devil staircase type spectra. Further, these type of spectra also show up when studying thinks like the quantum Hall effect. The famuous Hofstadter butterfly is often approached with multifractal techniques. But the closest I've ever been to finding an actual practical application of multifractal analysis, rather than just a descriptive technique was later during a postdoc. –  Raskolnikov Jan 4 '12 at 9:22
I was studying how we could retrieve the abundance distribution of species from Rényi entropies. The thing is, it turned out that Rényi entropies of species abundance of bacteria are more easily measured in the lab using all kind of DNA profiling techniques. But determining the species abundance distribution itself is hard. So, I wondered if I could not just transform back the Rényi entropies (which are just the fractal spectrum) to the original distribution. It turned out not to work really well and I've never managed to make it work properly. Nor could I find anything in the literature. –  Raskolnikov Jan 4 '12 at 9:26
@Raskolnikov, interesting. I wonder if asking on the physics SE site would find some people better able to answer. There's also a quant SE. I know cross posting is forbidden, but maybe one could phrase it in such a way as to be relevant to each of those sites. –  user4143 Jan 4 '12 at 17:17
I actually tried on CrossValidated but I did not get any reply. –  Raskolnikov Jan 4 '12 at 17:36

For now, I'll have to answer my own question. I noticed that on the page of Benoît Mandelbrot, the man himself, there is a nice list of books on fractals. I've seen some titles that look like what I need.

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