232 research outputs found
A Stochastic Description for Extremal Dynamics
We show that extremal dynamics is very well modelled by the "Linear
Fractional Stable Motion" (LFSM), a stochastic process entirely defined by two
exponents that take into account spatio-temporal correlations in the
distribution of active sites. We demonstrate this numerically and analytically
using well-known properties of the LFSM. Further, we use this correspondence to
write an exact expressions for an n-point correlation function as well as an
equation of fractional order for interface growth in extremal dynamics.Comment: 4 pages LaTex, 3 figures .ep
Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution
Textures in images can often be well modeled using self-similar processes
while they may at the same time display anisotropy. The present contribution
thus aims at studying jointly selfsimilarity and anisotropy by focusing on a
specific classical class of Gaussian anisotropic selfsimilar processes. It will
first be shown that accurate joint estimates of the anisotropy and
selfsimilarity parameters are performed by replacing the standard 2D-discrete
wavelet transform by the hyperbolic wavelet transform, which permits the use of
different dilation factors along the horizontal and vertical axis. Defining
anisotropy requires a reference direction that needs not a priori match the
horizontal and vertical axes according to which the images are digitized, this
discrepancy defines a rotation angle. Second, we show that this rotation angle
can be jointly estimated. Third, a non parametric bootstrap based procedure is
described, that provides confidence interval in addition to the estimates
themselves and enables to construct an isotropy test procedure, that can be
applied to a single texture image. Fourth, the robustness and versatility of
the proposed analysis is illustrated by being applied to a large variety of
different isotropic and anisotropic self-similar fields. As an illustration, we
show that a true anisotropy built-in self-similarity can be disentangled from
an isotropic self-similarity to which an anisotropic trend has been
superimposed
Cascades infiniment divisibles voilées : au-delà des lois de puissance
Nous présentons les définitions et synthèses de processus stochastiques respectant des lois d'échelles voilées, qui
s'écartent de façon contrôlée d'un comportement en loi de puissance. Nous définissons des bruit, mouvement et
marche aléatoire issus de cascades infiniment divisibles (IDC) voilées. Nous étudions analytiquement le comportement
des moments des accroissements de ces processus à travers les échelles. Ces résultats théoriques sont illustrés sur
l'exemple d'une cascade log-Normale voilée. Les algorithmes de synthèse et les fonctions Matlab utilisés sont
disponibles sur nos pages web.We address the definitions and synthesis of stochastic processes which possess warped scaling laws that
depart from power law behaviors in a controlled manner. We define warped infinitely divisible cascading
(IDC) noise, motion and random walk. We provide a theoretical derivation of the scaling behavior of the
moments of their increments. We provide numerical simulations of a warped log-Normal cascade to illustrate
these results. Algorithms for synthesis and Matlab functions are available from our web pages
Intermittent turbulence, noisy fluctuations and wavy structures in the Venusian magnetosheath and wake
Recent research has shown that distinct physical regions in the Venusian
induced magnetosphere are recognizable from the variations of strength of the
magnetic field and its wave/fluctuation activity. In this paper the statistical
properties of magnetic fluctuations are investigated in the Venusian
magnetosheath and wake regions. The main goal is to identify the characteristic
scaling features of fluctuations along Venus Express (VEX) trajectory and to
understand the specific circumstances of the occurrence of different types of
scalings. For the latter task we also use the results of measurements from the
previous missions to Venus. Our main result is that the changing character of
physical interactions between the solar wind and the planetary obstacle is
leading to different types of spectral scaling in the near-Venusian space.
Noisy fluctuations are observed in the magnetosheath, wavy structures near the
terminator and in the nightside near-planet wake. Multi-scale turbulence is
observed at the magnetosheath boundary layer and near the quasi-parallel bow
shock. Magnetosheath boundary layer turbulence is associated with an average
magnetic field which is nearly aligned with the Sun-Venus line. Noisy magnetic
fluctuations are well described with the Gaussian statistics. Both
magnetosheath boundary layer and near shock turbulence statistics exhibit
non-Gaussian features and intermittency over small spatio-temporal scales. The
occurrence of turbulence near magnetosheath boundaries can be responsible for
the local heating of plasma observed by previous missions
Numerical Schemes for Rough Parabolic Equations
This paper is devoted to the study of numerical approximation schemes for a
class of parabolic equations on (0, 1) perturbed by a non-linear rough signal.
It is the continuation of [8, 7], where the existence and uniqueness of a
solution has been established. The approach combines rough paths methods with
standard considerations on discretizing stochastic PDEs. The results apply to a
geometric 2-rough path, which covers the case of the multidimensional
fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201
Statistical Tests of Distributional Scaling Properties for Financial Return Series
Existing empirical evidence of distributional scaling in financial returns has helped motivate the use of multifractal processes for modelling return processes. However, this evidence has relied on informal tests that may be unable to reliably distinguish multifractal processes from other related classes. The current paper develops a formal statistical testing procedure for determining which class of fractal process is most consistent with the distributional scaling properties in a given sample of data. Our testing methodology consists of a set of test statistics, together with a model-based bootstrap resampling scheme to obtain sample p-values. We demonstrate in Monte Carlo exercises that the proposed testing methodology performs well in a wide range of testing environments relevant for financial applications. Finally, the methodology is applied to study the scaling properties of a dataset of intraday equity index and exchange rate returns. The empirical results suggest that the scaling properties of these return series may be inconsistent with purely multifractal processes
Wavelets techniques for pointwise anti-Holderian irregularity
In this paper, we introduce a notion of weak pointwise Holder regularity,
starting from the de nition of the pointwise anti-Holder irregularity. Using
this concept, a weak spectrum of singularities can be de ned as for the usual
pointwise Holder regularity. We build a class of wavelet series satisfying the
multifractal formalism and thus show the optimality of the upper bound. We also
show that the weak spectrum of singularities is disconnected from the casual
one (denoted here strong spectrum of singularities) by exhibiting a
multifractal function made of Davenport series whose weak spectrum di ers from
the strong one
Renormalization flow for extreme value statistics of random variables raised to a varying power
Using a renormalization approach, we study the asymptotic limit distribution
of the maximum value in a set of independent and identically distributed random
variables raised to a power q(n) that varies monotonically with the sample size
n. Under these conditions, a non-standard class of max-stable limit
distributions, which mirror the classical ones, emerges. Furthermore a
transition mechanism between the classical and the non-standard limit
distributions is brought to light. If q(n) grows slower than a characteristic
function q*(n), the standard limit distributions are recovered, while if q(n)
behaves asymptotically as k.q*(n), non-standard limit distributions emerge.Comment: 21 pages, 1 figure,final version, to appear in Journal of Physics
Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series
Notwithstanding the significant efforts to develop estimators of long-range
correlations (LRC) and to compare their performance, no clear consensus exists
on what is the best method and under which conditions. In addition, synthetic
tests suggest that the performance of LRC estimators varies when using
different generators of LRC time series. Here, we compare the performances of
four estimators [Fluctuation Analysis (FA), Detrended Fluctuation Analysis
(DFA), Backward Detrending Moving Average (BDMA), and centred Detrending Moving
Average (CDMA)]. We use three different generators [Fractional Gaussian Noises,
and two ways of generating Fractional Brownian Motions]. We find that CDMA has
the best performance and DFA is only slightly worse in some situations, while
FA performs the worst. In addition, CDMA and DFA are less sensitive to the
scaling range than FA. Hence, CDMA and DFA remain "The Methods of Choice" in
determining the Hurst index of time series.Comment: 6 pages (including 3 figures) + 3 supplementary figure
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