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1. Overview

This menu allows you estimate to various multifractal spectra for 1D signals. Multifractal spectra for images may also be computed in Fraclab: Choose the Segmentation menu, and then Image multifractal segmentation. We refer to the help corresponding to this window for further details.

There exists three main multifractal spectra: The Hausdorff, large deviation, and Legendre spectra. Basically, any of the three spectra provides an information as to which singularities occur in your signal, and which are dominant: a spectrum is a 1D curve, where abscissa represents the Hölder exponents actually present in your signal, and ordinates are related to "the amount" of points where you will encounter a given singularity. For instance, if the spectrum f has just one maximum at exponent a, with f(a) = 1, then if you pick a point at random in the signal, it will almost surely have exponent a. If b is such that f(b) = 0, then there is a very sparse set of points which have exponent b. An exponent c which does not occur has f(c) = -infinity.

This is the rough picture. There are essential differences between the three spectra, that we cannot detail in this help. We just briefly mention their main features.

The Hausdorff spectrum gives geometrical information pertaining to the dimension of sets of points in a signal having a given Hölder exponent. This is the most precise spectrum from a mathematical point of view, but is also unfortunately the most difficult to estimate. We hope to incorporate an estimator of this spectrum for 1D signals in forthcoming versions of Fraclab. Note that in the Image multifractal segmentation sub-menu of the Segmentation window, such a estimator is already proposed.

The second spectrum is the large deviation spectrum, denoted fg. This spectrum yields a statistical information, related to the probability of finding a point with a given Hölder exponent in the signal. More precisely, fg measures how this probability behaves under changes of resolution.

The third spectrum is the Legendre spectrum, denoted fl. It is just a concave approximation to the large deviation spectrum. Its main interest is that it usually yields much more robust estimates, though at the expense of a loss of information.

The 1D signals Multifractal Spectra allows you to compute fl and fg for 1D signals which are either measures (i.e. arrays of non negative data adding up to 1) or functions.


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