Need good material on multifractal analysis

Question

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.

Answer 1

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

Answer 2

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.

Answer 3

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 $f(t)$:

• Find the Hölder exponent, $h(t)$, as a function of time, $t$
• Find the singularity spectrum, $D(\alpha)$

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:

• "Multifractal Processes"
• "Introduction to Multifractals"
• Along with Jacques L. Véhel "TCP traffic is multifractal: a numerical study."

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:

• "Large Deviations and the Distribution of Price Changes"
• "A Multifractal Model of Asset Returns"

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.