Analysis of Random Cascades using Wavelet
Analysis: From Theoretical Concepts to Experimental Applications
Centre de Recherche Paul Pascal
Avenue Schweitzer, 33600 Pessac, France
- Multiplicative cascade models have enjoyed an increasing
interest in recent years as the paradigm of multifractal objects1-4.
The notion of cascade actually refers to a self-similar process
whose properties are defined multiplicatively from coarse to
fine scales. In that respect, it occupies a central place in
the statistical theory of turbulence3,5. Since Richardson's famous
poem, the turbulent cascade picture has been often invoked to
account for the intermittency phenomenon observed in fully developed
turbulent flows5,6: energy is transfered from large eddies down
to small scales (where it is dissipated) through a cascade process
in which the transfer rate at a given scale is not spatially
homogeneous, as supposed in the theory developed by Kolmogorov7
in 1941, but undergoes local intermittent fluctuations5. Over
the past thirty years, refined models including the log-normal
model of Kolmogorov8 and Obukhov9, multiplicative hierarchical
cascade models like the random beta-model, the alpha-model, the
p-model (for review, see Ref. ), the log-stable models10 and
more recently the log-infinitely divisible cascade models11-14
with the rather popular log-Poisson model advocated by She and
Leveque15, have grown in the literature as reasonable models
to mimic the energy cascading process in turbulent flows. On
a very general ground, a self-similar cascade is defined by the
way the scales are refined and by the statistics of the multiplicative
factors at each step of the process3,4,10. One can thus distinguish
discrete cascades that involve discrete scale ratios leading
to log-periodic corrections to scaling (discrete scale invariance16),
from continous cascades without preferable scale factors (continuous
scale invariance). As far as the fragmentation process is concerned,
one can specify whether some conservation laws are operating
or not4; in particular, one can discriminate between conservative
(the measure is conserved at each cascade step) and non conservative
(only some fraction of the measure
- is transfered at each step) cascades. More fundamentally,
there are two main classes of self-similar cascade processes:
deterministic cascades that generally correspond to solvable
models and random cascades that are likely to provide more realistic
models but for which some theoretical care is required as far
as their multifractal limit and some basic multifractal properties
(including multifractal phase transitions) are concerned4. As
a notable member of the later class, the independent random cascades
introduced by Mandelbrot6 as a general model of random curdling
in fully developed turbulence, have a special status since they
are the main cascade model for which deep mathematical results
have been obtained17,18. More recently, the concept of self-similar
cascades leading to multifractal measures has been generalized
to the construction of scale-invariant signals using orthonormal
wavelet basis19-22. Instead of reditributing the measure over
sub-intervals with multiplicative weights, one allocates the
wavelet coefficients in a multiplicative way on a dyadic grid.
This method allows us to generate multifractal functions from
a given deterministic or probabilistic multiplicative process.
- The main goal of this paper is to emphasize the continuous
wavelet transform as a very efficient tool to get deep insight
into the hierarchical structural complexity of multifractal objects23,24.
Our point is to show that the main information about the underlying
cascading process can actually be extracted from the wavelet
transform skeleton defined by the wavelet transform modulus maxima25
(WTMM). We will discuss two situations which are typical, namely
the case of deterministic discrete cascades and the case of random
continuous cascades. For both cases, we will illustrate our purpose
with some applications to experimental data ranging from fractal
growth phenomena to fully developed turbulence and to financial
- Deterministic discrete cascades: In many cases, the self-similarity
properties of fractal objects can be expressed in terms of a
dynamical system which leaves the object invariant. The inverse
problem consists in recovering this dynamical system (or its
main characteristics) from the data representing the fractal
object. This problem has been previously approached within the
theory of Iterated Function Systems26 (IFS). But the methods
developed in this context are based on the search for a "best-fit"
within a prescribed class of IFS attractors (mainly linear homogeneous
attractors). In that sense, they approximate the self-similarity
properties more than they reveal them. We show that, in many
situations, the WTMM skeleton can be used to extract some one-dimensional
(1D) map which accounts for its construction process27,28. We
illustrate our theoretical considerations on pedagogical examples
including Bernoulli invariant measures of linear and nonlinear
expanding Markov maps as well as the invariant measure of period-doubling
dynamical systems at the onset of chaos. We apply this wavelet
based technique to analyze the fractal properties of DLA azimuthal
Cantor sets defined by intersecting the inner frozen region of
large mass off\--lattice Diffusion\--Limited\--Aggregates (DLA)
with a circle28-30. This study clearly reveals the existence
of an underlying multiplicative process that is likely to account
for the Fibonacci structural ordering recently discovered in
the apparently disordered arborescent DLA morphology31. These
results demonstrate the statistical relevance of the golden mean
arithmetic to the fractal hierarchy of DLA azimuthal Cantor sets.
- Random continuous cascades: We start introducing various
statistical quantities such as i) the statistical singularity
spectrum which quantifies the statistical contribution of each
singularity in the signals22; ii) the self-similarity kernel14,20,21,32,33
which, from a statistical point of view, characterizes the nature
of the cascade process from a given scale to a finer scale and
iii) the space-scale correlation functions34 which can be proved
to follow a power-law behavior when varying the spatial distance
of the two wavelet coefficients. We show mathematically and check
numerically on various computer synthetized signals22,35 that
these statistical quantities can be extracted directly from the
considered fractal function using its WTMM skeleton with an arbitrary
analyzing wavelet. This mathematical study actually provides
algorithms that are readily applicable to experimental situations.
We report on recent applications of our methodology in the context
of fully-developed turbulence20,21,34,35. Our results show that
the commonly accepted multifractal description of the intermittency
phenomenon is not valid for finite Reynolds number flows. In
particular, the search for a cascade process reveals a lack of
scale-invariance for both the velocity and the dissipation fields.
The statistical study of the WTMM yields a very convincing log-normal
law on a well defined range of scales that can be further used
as an objective definition of the inertial range. But in contrast
to the Kolmogorov-Obukhov log-normal cascade model8,9, we find
that the number of cascade steps does not evolve logarithmically
as a function of the scale which is the signature of the breaking
of scale-invariance20,21. We further comment on the possible
asymptotic validity of the log-normal multifractal description.
- These very first applications are very promising as fas
as further experimental investigations of multilplicative cascade
processes are concerned. It is likely that similar wavelet-based
statistical analysis will lead to significant progress in other
fields than fractal growth phenomena and fully developed turbulence.
To conclude, we report on preliminary results of a similar investigation
of financial time series36. Underlying the fluctuations of volatility
(standard deviation) of price variations, we have discovered
the existence of a causal information cascade from large to small
time scales that can be visualized with the wavelet representation.
Let us emphasize that the fact the variations of prices over
one month scale influence in the future the daily prices variations,
is likely to be extraordinarily rich in consequences and this,
not only for the fundamental understanding of the nature of financial
markets, but also (and may be more important) for practical applications.
Indeed, the nature of the correlations across scales that are
implied by this causal cascade have profound implications on
the market risk, a problem of upmost concern for all financial
institutions as well as individuals.
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