Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of ”mixed” partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some “mixed asymptotic” exponent χ respectively above and beyond two critical value. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a “box-counting” multifractal formalism that can be seen as a rigorous formulation of Mandelbrot’s negative dimension theory. Numerical illustrations of our purpose on specific examples are also
provided.
Log-normal continuous random cascades form a class of multifractal processes that has already been successfully used in various fields. Several statistical issues related to this model are studied. We first make a quick but extensive review of their main properties and show that most of these properties can be analytically studied. We then develop an approximation theory of these processes in the limit of small intermittency, i.e., when the degree of multifractality is small. This allows us to prove that the probability distributions associated with these processes possess some very simple aggregation properties accross time-scales. Such a control of the process properties at different time-scales, allows us to address the problem of parameter estimation. We show that one has to distinguish two different asymptotic regimes: the first one, referred to as the ”low frequency regime”, corresponds to taking a sample whose overall size increases whereas the second one, referred to as the ”high frequency regime”, corresponds to sampling the process at an increasing sampling rate. We show that, the first regime leads to convergent estimators whereas, in the high frequency regime, the situation is much more intricate: only the intermittency coefficient can be estimated using a consistent estimator. However, we show that, in practical situations, one candetect the nature of the asymptotic regime (low frequency versus high frequency) and consequently decide whether the estimations of the other parameters are reliable or not. We finally illustrate how both our results on parameter estimation and on aggregation properties, allow one to successfully use these models for modelization and prediction of financial time series.
In this note, we present results on the behavior for the partition function of multiplicative cascades in the case where the total time of observation is large, compared to the scale of decay for the correlation of the cascade process.
In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe “negative dimensions” in random multifractals. For that purpose, we define a new way to study scaling where the observation scale τ and the total sample length L are respectively going to zero and to infinity. This “mixed” asymptotic regime is parametrized by an exponent χ that corresponds to Mandelbrot “supersampling exponent”. In order to study the scaling exponents in the mixed regime, we use a formalism introduced in the context of the physics of disordered systems relying upon traveling wave solutions of some non-linear iteration equation. Within our approach, we show that for random multiplicative cascade models, the parameter χ can be interpreted as a negative dimension and, as anticipated by Mandelbrot, allows one to uncover the “hidden” negative part of the singularity spectrum, corresponding to “latent” singularities. We illustrate our purpose on synthetic cascade models. When applied to turbulence data, this formalism allows us to distinguish two popular phenomenological models of dissipation intermittency: We show that the mixed scaling exponents agree with a log-normal model and not with log-Poisson statistics.
In this paper, we make a short overview of continuous time
multifractal processes recently introduced to model asset return
fluctuations. We show that these models account in a very parcimonious
manner for most of ``stylized facts'' of financial time series.
We review in more details the simplest of such models namely
the log-normal Multifractal Random Walk. It can simply be considered
as a stochastic volatility model where the (log-) volatility memory has
a peculiar ''logarithmic'' shape. This model possesses some
appealing stability properties as respect to time aggregation.
We describe how one can
estimate it using a GMM method and we present some applications to
volatility and VaR forecasting.
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Removing noise from audio signals requires a non-diagonal processing of time-frequency coefficients to avoid producing ``musical noise''. State of the art algorithms perform a parameterized filtering of spectrogram coefficients with empirically fixed parameters. A block thresholding estimation procedure is introduced, which adjusts all parameters adaptively to signal property by minimizing a Stein estimation of the risk. Numerical experiments demonstrate the performance and robustness of this procedure through objective and subjective evaluations.
We investigate a new audio denoising algorithm. Complex wavelets protect phase of signals and are thus preferred in audio signal processing to real wavelets. The block attenuation eliminates the residual noise artifacts in reconstructed signals and provides a good approximation of the attenuation with oracle. A connection between the block attenuation and the decision-directed \textit{a priori} SNR estimator of Ephraim and Malah is studied. Finally we introduce an adaptive block technique based on the dyadic CART algorithm. The experiments show that not only the proposed method does eliminate the residual noise artifacts, but it also preserves transients of signals better than short-time Fourier based methods do.
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We discuss a possible scenario explaining in what respect the
observed fat tails of asset returns or volatility fluctuations
can be related to volatility long-range correlations. Our approach is based
on recently introduced multifractal models for asset returns that account for
the volatility correlations
through a multiplicative random cascade. Within the framework of these models,
it can be shown that the sample size required for a correct estimation of
the behavior of extreme return fluctuations is generally huge and outside
the range of accessible size of data.
Consequently, in many cases, the extreme tail probability appears as a power-law,
with a rather small (underestimated) tail exponent.
We point out that increasing the amount of data by using smaller and smaller (intraday) scales,
does not contribute to reduce the bias and, as observed empirically, the tail exponent turns
out to be rather stable across scales.
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In this paper we discuss the problem of the estimation of
extreme event occurrence probability for data drawn from some
multifractal process. We also study
the heavy (power-law) tail behavior of probability density function
associated with such data.
We show that because of strong correlations, standard extreme value approach
is not valid and classical tail exponent estimators should be interpreted
cautiously. Extreme statistics associated with multifractal
random processes turn out to be characterized by non self-averaging properties.
Our considerations rely upon some analogy between random
multiplicative cascades and the physics of disordered systems
and also on recent mathematical results about the so-called
multifractal formalism.
Applied to financial time series, our
findings allow us to propose an unified framemork
that accounts for the observed
multiscaling properties of return fluctuations, the volatility
clustering phenomenon and the observed ``inverse cubic law'' of the
return pdf tails.
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We define a large class of multifractal random measures and processes with
arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes
generalize within a unified framework both the recently defined log-normal Multifractal
Random Walk processes (MRW) (Bacry etal.) and the log-Poisson ``product of cynlindrical
pulses" (Barral and Mandelbrot). Their construction involves some ``continuous
stochastic multiplication'' (Schmitt and Marsan) from coarse to fine scales.
They are obtained as limit processes when the finest scale goes to zero. We
prove the existence of these limits and we study their main statistical properties
including non degeneracy, convergence of the moments and multifractal scaling.
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We introduce a dictionary of elementary waveforms, called harmonic atoms
that extends the Gabor dictionary and fits well the natural harmonic structures
of audio signals. By modifying the ``standard'' matching pursuit, we define
a new pursuit along with a fast algorithm, namely the Fast Harmonic Matching
Pursuit, to approximate N-dimensional audio signals with a linear combination
of M harmonic atoms. Our algorithm has a computational complexity of
O(MKN), where K is the number of partials in a given harmonic
atom. The decomposition method is demonstrated on musical recordings, and we
describe a simple note detection algorithm that shows how one could use a harmonic
matching pursuit to detect notes even in difficult situations, e.g., very different
note durations, lots of reverberation, and overlapping notes.
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We define a large class of continuous time multifractal random measures and
processes with arbitrary log-infinitely divisible exact or asymptotic scaling
law. These processes generalize within a unified framework both the recently
defined log-normal Multifractal Random Walk (MRW) (Bacry et.al) and the log-Poisson
"product of cynlindrical pulses" (Barral and Mandelbrot). Our construction is
based on some "continuous stochastic multiplication" from coarse to fine scales
that can be seen as a continuous interpolation of discrete multiplicative cascades.
We prove the stochastic convergence of the defined processes and study their
main statistical properties. The question of genericity (universality) of limit
multifractal processes is addressed within this new framework. We finally provide
some methods for numerical simulations and discuss some specific examples.
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