155 research outputs found
Stochastic 2-microlocal analysis
A lot is known about the H\"older regularity of stochastic processes, in
particular in the case of Gaussian processes. Recently, a finer analysis of the
local regularity of functions, termed 2-microlocal analysis, has been
introduced in a deterministic frame: through the computation of the so-called
2-microlocal frontier, it allows in particular to predict the evolution of
regularity under the action of (pseudo-) differential operators. In this work,
we develop a 2-microlocal analysis for the study of certain stochastic
processes. We show that moments of the increments allow, under fairly general
conditions, to obtain almost sure lower bounds for the 2-microlocal frontier.
In the case of Gaussian processes, more precise results may be obtained: the
incremental covariance yields the almost sure value of the 2-microlocal
frontier. As an application, we obtain new and refined regularity properties of
fractional Brownian motion, multifractional Brownian motion, stochastic
generalized Weierstrass functions, Wiener and stable integrals.Comment: 35 page
Estimation locale : compromis régression-variance
International audienceWe present a method allowing under some hypothesis to decrease the variance of an optimal estimator without causing damage to its risk, using many estimators jointly, thanks to a very basic technique leading to a hybrid estimator that takes advantage of the best of them. This method can be applied in various fields, typically on estimators that were obtained using linear regressions
Localisable moving average stable and multistable processes
We study a particular class of moving average processes which possess a
property called localisability. This means that, at any given point, they admit
a ``tangent process'', in a suitable sense. We give general conditions on the
kernel g defining the moving average which ensures that the process is
localisable and we characterize the nature of the associated tangent processes.
Examples include the reverse Ornstein-Uhlenbeck process and the multistable
reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is,
at each time t, a L\'evy stable motion with stability index possibly varying
with t. We also consider the problem of path synthesis, for which we give both
theoretical results and numerical simulations
On various multifractal spectra
International audienceWe introduce two classes of multifractal spectra, called respectively dimension and continuous spectra. Dimension spectra offer an interesting alternative to the classical Hausdorff spectrum: They are much easier to estimate yet still give relevant information about the geometry of the H¨older function. Continuous spectra are a generalization of the large deviation spectrum that allow to obtain partition free results. Both classes of spectra allow to perform efficient multifractal analysis in an experimental framewor
The local Hölder function of a continuous function
AbstractThis work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0
Wavelet packet based digital watermarking
International audienceWe present a method for digital image watermarking based on the modification of certain subsets of the wavelet packet decomposition. These subsets are determined both from a secret key and an image dependent procedure that chooses a best basis from an energy criterion. The mark is set by imposing a parity constraint at each level of the decomposition. We elaborate on the choice of some of the parameters of the model, showing how they can be tuned so as to obtain good resistance to attacks. Examples are displayed to assess the validity of our approach
Weakly Self Affine Functions and Applications in Signal Processing
International audienceWe study a class of functions, called weakly self afire functions, which are a generalization of Fractal Interpolation Functions where the concentrate ratios are allowed to envolve in scale. We show how to compute the milifractal spectrum of such functions, and mention an application to the multifractal segmentation of signals
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