A Wavelet Based

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A Wavelet Based Multifractal Formalism
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Auto correlation of fractal functions
Quantities varying with time often turn out to
have fractal graphs. One way in which their
fractal nature is often manifested is by a power
law behavior of the correlation between
measurements separated by time h.
Let f(-?, ?) ? R be continuous bounded function
we define average denoted by and defined as
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A measure of the correlation between f at times
separated by h is provided by the autocorrelation
function .
From ( ) implies that C(h) is positive if
tend to have the same sign, and is
negative if they tend to have opposite signs. If
there is no correlation, C(h)0
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The auto-correlation function is closely
connected with the power spectrum of f, defined by
Modified for wavelet
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It can be verified that
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Autocorrelation provides useful methods for
estimating the dimension of the graph of a
function or signal f. Power spectrum can be used
to estimate Auto correlation.
Interesting case is that when the power spectrum
obeys the law for
large in which case
for small h, for some constant b.
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A Wavelet based multifractal formalism for
fractal functions
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Random Variable Normal Distribution of mean zero
A method of constructing functions with
characteristics similar to index Brownian
function is to randomize the Weistrass function
Variance
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In the following manner
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Wavelet Transform Modulus Maximum Method
This method allows us to determine the
singularity spectrum of the considered signal and
there by to achieve a complete multi-fractal
analysis. Finding the distribution of
singularities in a signal f is of vital
importance.
Devils Staircase
It is the integral of the cantor measure It is
a continuous function that increases from 0 to 1
on 0,1. The recursive construction of the
cantor measure implies that f is self similar.
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if if if
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Two subdivisions of the uniform measure on 0,1
with left and right weights p1 and p2. The cantor
measure is the limit of an infinite number
of these subdivisions.
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Figure 1
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Definition 1 ( Self similar Set)
A set is said to be a self similar if it
the union of disjoint subsets S1., SR that can
be obtained from S with a scaling, translation
and rotation.
It may be observed that the self-similarity often
implies an infinite multiplication of details,
which creates irregular structures.
The triadic cantor set and the Van Koch curve are
well known examples.
Definition 2 ( Spectrum)
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This concept was introduced by Frisch and Parisi
in 1985 Turbulence and Predictability in
geophysical fluid dynamics and climatic dynamics,
fully developed turbulence and intermittency,
page 84 , North Holland Amsterdam (editors) M.
Ghil, R. Benzi, G. Parisi. It was extended to
multifractals by Arneodo, Bacry and Muzy The
multifractal formalism revisited with Wavelets.
Internatinal Journal of Bifurcation and Chaos,
4(1994) p.245.
From the definition of capacity dimension, a
special form of Hausdorff dimension, pp.
201-205, Mallat, A wavelet tour of signal
processing, Academic press 1999 it follows that
if we make a disjoint cover of the support of f
with intervals of size s then the number of
intervals that intersect is
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Figure 2
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Partition Function
One cannot compute the Lipschitz regularity of a
multifractal because its singularities are not
isolated and finite numerical resolution is not
sufficient to discriminate them. To overcome this
difficulty Areodo, Bacry and Muzy 1994
introduces the concept of wavelet transform
modulus maximum using a global partition function.
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Let be a wavelet with n vanishing movements.
Mallat theorem 6.5, Book 1999 our discussion
earlier proves that if f has point wise Lipschitz
regularity at v then the wavelet
transform has a sequences of
modulus maxima that converges towards v at fine
scales. The set of maxima at the scale a can thus
be interpreted as a cover of the support of f
wavelets of scale a. At there maxima locations
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This typically means that
The following theorem relates to the
Legendre transform of for self-similar
signals Bacry, Muzy Arneodo, Singularity
spectrum of fractal signals exact results, J. of
stat physics. 70(314) p.635, 1993, For Particular
class of fractals Jaffard, Multifractals
formalism for functions parts I and II, SIAM J.
of Math Anal.- 28(4)944-998, 1997
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Theorem A Areodo, Bacry, Jaffard, Muzy
Remark
This theorem proves that the scaling exponent
is the Legendre transform of . It is
necessary to use a wavelet with enough vanishing
moments to measure all lipschitz exponent up to
. In numerical calculations is computed
by evaluating the sum Z(q,a). To recover the
spectrum of singularity we are required
to invert the Legendre transform (). We have
the following result in this direction
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Theorem B Mallat, 1999
The scaling exponent is a convex and
increasing function of q The legendre transform
() is inevitable if and only if is
convex, in which case
The spectrum of self-similar signals is
convex Jaffard 1997
Remark
The spectrum of self similarity signals (self
similar multifractals) is convex and can
therefore be calculated from with the
inverse Legendre formula (). This formual is
valid for a much wider class of multifractals,
for example fractional Brownian motions.
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For detail properties of singularities spectrum
we refer to Meyer Wavelets, Vibrations and
Scalings, CRM, AMS 1997. Figure 3 illustrates
properties of convex spectrum.
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Procedure for Numerical Calculation

• Maxima compute and the modulus
  maxima at each scale a. Chain the wavelet maxima
  across scales

• Partition function compute

• Scaling compute with a linear regression
  of as a function of

• Spectrum Compute

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Figure 3
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Figure 4
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Smooth Perturbations
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Theorem C (ARNEODO, BACRY, MUZY)
Let be a wavelet with exactly n vanishing
moments. Suppose that f is a self-similar
function.

• If g is a polynomial of degree pltn then
  for all

• if g(n) is almost every where non-zero then

if if
Where qc is defined by
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Fractal Noise
Fractional Brownian motions are statistically
self-similar Gaussian processes that give
interesting models for a wide class of natural
phenomena. Despite their non-stationarity, one
can define a power spectrum that has a power
decay. Realizations of fractional Brownian
motions are almost everywhere singular, with the
same Lipschitz regularity at all points.
We often encounter fractal noise processes that
are not Gaussian although their power spectrum
has a power decay. Realizations of these
processes may include singularities of various
types.
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Definition (Fractional Brownian Motion)
A fractional Brownian motion of Hurst exponent
0ltHlt1 is a zero-mean Gaussian process BH such
that
and
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Figure 5
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Developing () for also gives
The covariance does not depend only on
which proves that a fractional Brownian motion
is non-stationary.
The statistical self-similarity appears when
scaling this process. One can derive from ()
that for any sgt0
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Since and are two
Gaussian processes with same mean and same
covariance, they have the same probability
distribution
where denotes an equality of
finite-dimensional distribution
Power Spectrum
Although BH is not stationary, one can define a
generalized power spectrum. This power spectrum
is introduced by proving that the increments of a
fractional Brownian motion are stationary , and
by computing their power spectrum.
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Theorem D
Let
The increment
is a stationary process whose power spectrum is
Wavelet Transform
The wavelet transform of a fractional Brownian
motion is
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The self similarity of the power spectrum and the
fact that BH is Gaussian are sufficient to prove
that WBH(u,s) is self similar across scales
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Where the equivalence means that they have same
finite distributions. Interesting
characterizations of fractional Brownian motion
properties are also obtained by decomposing these
processes in wavelets bases.
Example 1
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Fractal Noises
Some physical phenomena produce more general
fractal noises X(t), which are not Gaussian
processes, but which have stationary increments.
As for fractional Brownian motions, one can
define a generalized power spectrum that has a
power decay
Hydrodynamic Turbulence
Fully developed turbulence appears in
incompressible flows at high Reynolds number.
Under standing the properties of hydrology
turbulence is a major problem of modern physics,
which remains mostly open despite an intense
effort since the first theory of Kolmogorov in
1941.
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The number of degrees of liberty of
three-dimensional turbulence is considerable,
which produces extremely complex spatio-temporal
behavior. No formalism is yet able to build a
statistical physics framework based on the Navier
stokes equations, that would enable us to
understand the global behavior of turbulent
flows, at it is done in thermodynamics.
In 1941, Kolomogorov formulated a statistical
theory of turbulence. The velocity filed is
modeled as a process V(x) whose increment have a
variance .
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The constant is supposed to be independent of
the location. This indicates that the velocity
field is statistically homogeneous with Lipschitz
regularity . The theory
predicts that a one dimensional trace of a three
dimensional velocity field is a fractal noise
process with stationary increments, and whose
spectrum delays with a power exponent
The success of this theory comes from numerous
experimental verifications of this power spectrum
decay. However, the theory does not take into
account the existence of coherent structures such
as vortices. These phenomena contradicts the
hypothesis of homogeneity, which is at the root
of kolmogorovs 1941 theory.
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