View source for Wavelet-based multifractal analysis

== The wavelet transform modulus maxima method for multifractal analysis ==

=== Singularity spectrum ===

As originally defined by Parisi & Frisch (1985), the multifractal formalism of multi-affine functions
amounts to compute the so-called ''singularity spectrum'' <math>D(h)</math> defined as the Hausdorff
dimension of the set where the H&ouml;lder exponent is equal to
<math>h</math> (Arneodo et al., 1995b; Bacry et al., 1993; Muzy
et al., 1994):

: <math>\label{eq6}
D(h)=dim_H\{ x\;,\; h(x)=h \} \; ,
</math>

where <math>h</math> can take, ''a priori'', positive as well as
negative real values (''e.g.'', the Dirac distribution
<math>\delta(x)</math> corresponds to the H&ouml;lder exponent <math>h(0)=-1</math>) (Jaffard, 1997a).

=== The WTMM method ===

[[Image:fig_wbmf_tq_1d.gif|thumb|400px|right|fig:tq_1d|Determination of the <math>\tau(q)</math> and <math>D(h)</math> multifractal spectra of the “G” DNA walks of a set of
    2184 human introns (<math>L\geq800</math>bp) (<math>{\color{Red}\circ}</math>) and of the turbulent velocity signals (<math>\color{Blue}\bullet</math>)
    using the WTMM method.
    <math>\log_2Z(q,a)</math> vs <math>\log_2a</math> for, from bottom to top, <math>q=</math>-2, -1, 0, 1, 2, 3 and  4 in (a)
    and for <math>q=</math>-3, 0, 3 and 6 in (b).
    (c) <math>\tau(q)</math> vs <math>q\ ;</math> the solid lines correspond respectively to a monofractal spectrum
    <math>\tau(q)=qH-1</math> with <math>H=0.60\pm0.02</math> (red) and to a quadratic multifractal spectrum
    <math>\tau(q)=c_1q-\frac{c_2}{2}q^2-1</math> with <math>c_1=0.36\pm0.02</math> and a non zero value for the
    intermittent coefficient <math>c_2=0.028\pm0.003</math> (blue).
    (d) <math>D(h)</math> vs <math>h\ ;</math> while the singularity spectrum of the monofractal DNA walk landscape
    reduces to a point (<math>D(h=0.6)=1</math> and 0 elsewhere, <math>\color{Red}\circ</math>),
    it has a parabolic shape and extends from <math>h_{min}=0.12</math> to <math>h_{max}=0.60</math> as the signature
    of the “intermittent” multifractal nature of Eulerian turbulence (<math>\color{Blue}\bullet</math>).
    The analyzing wavelet is <math>g^{(2)}</math> (Eq. \eqref{dergauss}).]]

A natural way of performing a multifractal
analysis of fractal functions consists in generalizing the “classical”
multifractal formalism (Collet et al., 1987; Grassberger et al., 1988; Halsey et al., 1986;
Paladin & Vulpiani, 1987; Rand, 1989)
using wavelets instead of boxes.
By taking advantage of the freedom in the choice of the
“generalized oscillating boxes” that are
the wavelets, one can hope to get rid of possible smooth behavior
that could mask singularities or perturb the estimation
of their strength <math>h\ .</math> But the major difficulty with
respect to box-counting techniques (Argoul et al.,
1990; Farmer et al., 1983; Grassberger & Procaccia, 1983; Grassberger et al., 1988; Meneveau
& Sreenivasan, 1991) for singular measures,
consists in defining a covering of the support of
the singular part of the function with our set of
wavelets of different sizes.
As emphasized in (Arneodo et al., 1995b; Bacry et al., 1993; Muzy et al., 1991, 1993, 1994),
the branching structure of the WT skeletons of fractal functions in
the <math>(x,a)</math> half-plane enlightens the hierarchical organization
of their singularities ( <figref>Fig_to_squel.gif</figref>(e,f)).
The WT skeleton can thus be used as a guide to position, at a considered
scale <math>a\ ,</math> the oscillating boxes in order to obtain a partition of the
singularities of <math>f\ .</math>
The wavelet transform modulus maxima (WTMM) method amounts to compute
the following partition function in terms of WTMM
coefficients (Arneodo et al., 1995b; Bacry et al.,
1993; Muzy et al., 1991, 1993, 1994):

: <math>\label{eq10}
Z(q,a)=\sum_{l\in \mathcal{L}(a)} {\left( \sup_{\stackrel{\scriptstyle{(x,a')\in l}}{\scriptstyle{a'\leq a}}}
      \vert T_\psi[f](x,a')\vert \right)}^q \; ,
</math>

where <math>q\in\mathbb{R}</math> and the <math>\sup</math> can be regarded as a way to define a scale
adaptive “Hausdorff-like” partition.
Now from the deep analogy that links the multifractal formalism to
thermodynamics (Arneodo et al., 1995b; Bohr & Tel, 1988), one
can define the exponent
<math>\tau(q)</math> from the power-law behavior of the partition function:

: <math>\label{eq11}
Z(q,a) \; \sim \; a^{\tau(q)} \; ,\;\; a \rightarrow 0^+\; ,
</math>

where <math>q</math> and <math>\tau(q)</math> play respectively the role of the inverse
temperature and the free energy.
The main result of this wavelet-based multifractal
formalism is that in place of the energy and the entropy
(''i.e.'' the variables conjugated to <math>q</math> and <math>\tau</math>),
one has <math>h\ ,</math> the H&ouml;lder exponent, and <math>D(h)\ ,</math> the
singularity spectrum. This means that the
singularity spectrum of <math>f</math> can be determined from the Legendre transform
of the partition function scaling exponent <math>\tau(q)</math>(Bacry et al., 1993; Jaffard,
1997a,b):

: <math>\label{eq12}
D(h)=\min_q(qh-\tau(q)) \; .
</math>


=== Monofractal versus multifractal functions ===

From the properties of the Legendre transform, it is easy to see that
''homogeneous'' monofractal functions that involve singularities of unique H&ouml;lder exponent
<math>h=\partial \tau/\partial q\ ,</math> are characterized by a <math>\tau(q)</math> spectrum which is a ''linear''
function of <math>q</math> (<figref>Fig_wbmf_tq_1d.gif</figref>(c)).
On the contrary, a ''nonlinear'' <math>\tau(q)</math> curve is the signature of nonhomogeneous functions that
exhibit ''multifractal'' properties, in the sense that the H&ouml;lder exponent <math>h(x)</math> is a
fluctuating quantity that depends upon the spatial position <math>x</math> (<figref>Fig_wbmf_tq_1d.gif</figref>(c)).
As illustrated in <figref>Fig_wbmf_tq_1d.gif</figref>(d), the <math>D(h)</math> singularity spectrum of a multifractal
function displays a single humped shape that characterizes intermittent fluctuations corresponding
to H&ouml;lder exponent values spanning a whole interval <math>[h_{min}, h_{max}]\ ,</math> where <math>h_{min}</math> and
<math>h_{max}</math> are the H&ouml;lder exponents of the strongest and weakest singularities respectively.