4,480 research outputs found
Intermittent process analysis with scattering moments
Scattering moments provide nonparametric models of random processes with
stationary increments. They are expected values of random variables computed
with a nonexpansive operator, obtained by iteratively applying wavelet
transforms and modulus nonlinearities, which preserves the variance. First- and
second-order scattering moments are shown to characterize intermittency and
self-similarity properties of multiscale processes. Scattering moments of
Poisson processes, fractional Brownian motions, L\'{e}vy processes and
multifractal random walks are shown to have characteristic decay. The
Generalized Method of Simulated Moments is applied to scattering moments to
estimate data generating models. Numerical applications are shown on financial
time-series and on energy dissipation of turbulent flows.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1276 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study
We analyze, within the wavelet theory framework, the wandering over a screen
of the centroid of a laser beam after it has propagated through a time-changing
laboratory-generated turbulence. Following a previous work (Fractals 12 (2004)
223) two quantifiers are used, the Hurst parameter, , and the Normalized
Total Wavelet Entropy, . The temporal evolution of both
quantifiers, obtained from the laser spot data stream is studied and compared.
This allows us to extract information of the stochastic process associated to
the turbulence dynamics.Comment: 11 pages, 3 figures, accepted to be published in Physica
Modified detrended fluctuation analysis based on empirical mode decomposition
Detrended fluctuation analysis (DFA) is a simple but very efficient method
for investigating the power-law long-term correlations of non-stationary time
series, in which a detrending step is necessary to obtain the local
fluctuations at different timescales. We propose to determine the local trends
through empirical mode decomposition (EMD) and perform the detrending operation
by removing the EMD-based local trends, which gives an EMD-based DFA method.
Similarly, we also propose a modified multifractal DFA algorithm, called an
EMD-based MFDFA. The performance of the EMD-based DFA and MFDFA methods is
assessed with extensive numerical experiments based on fractional Brownian
motion and multiplicative cascading process. We find that the EMD-based DFA
method performs better than the classic DFA method in the determination of the
Hurst index when the time series is strongly anticorrelated and the EMD-based
MFDFA method outperforms the traditional MFDFA method when the moment order
of the detrended fluctuations is positive. We apply the EMD-based MFDFA to the
one-minute data of Shanghai Stock Exchange Composite index, and the presence of
multifractality is confirmed.Comment: 6 RevTex pages including 5 eps figure
Phase Diagram for Turbulent Transport: Sampling Drift, Eddy Diffusivity and Variational Principles
We study the long-time, large scale transport in a three-parameter family of
isotropic, incompressible velocity fields with power-law spectra. Scaling law
for transport is characterized by the scaling exponent and the Hurst
exponent , as functions of the parameters. The parameter space is divided
into regimes of scaling laws of different {\em functional forms} of the scaling
exponent and the Hurst exponent. We present the full three-dimensional phase
diagram.
The limiting process is one of three kinds: Brownian motion (),
persistent fractional Brownian motions () and regular (or smooth)
motion (H=1).
We discover that a critical wave number divides the infrared cutoffs into
three categories, critical, subcritical and supercritical; they give rise to
different scaling laws and phase diagrams. We introduce the notions of sampling
drift and eddy diffusivity, and formulate variational principles to estimate
the eddy diffusivity. We show that fractional Brownian motions result from a
dominant sampling drift
Fractional Brownian fields, duality, and martingales
In this paper the whole family of fractional Brownian motions is constructed
as a single Gaussian field indexed by time and the Hurst index simultaneously.
The field has a simple covariance structure and it is related to two
generalizations of fractional Brownian motion known as multifractional Brownian
motions. A mistake common to the existing literature regarding multifractional
Brownian motions is pointed out and corrected. The Gaussian field, due to
inherited ``duality'', reveals a new way of constructing martingales associated
with the odd and even part of a fractional Brownian motion and therefore of the
fractional Brownian motion. The existence of those martingales and their
stochastic representations is the first step to the study of natural wavelet
expansions associated to those processes in the spirit of our earlier work on a
construction of natural wavelets associated to Gaussian-Markov processes.Comment: Published at http://dx.doi.org/10.1214/074921706000000770 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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