Improved moment scaling estimation for multifractal signals

Improved moment scaling estimation for multifractal signals

A fundamental problem in the analysis of multifractal processes is to estimate the scaling exponent <i>K(q)</i> of moments of different order <i>q</i> from data. Conventional estimators use the empirical moments <i><html&am...

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Main Authors: D. Veneziano, P. Furcolo
Format: Article
Language:English
Published: Copernicus Publications 2009-11-01
Series:Nonlinear Processes in Geophysics
Online Access:http://www.nonlin-processes-geophys.net/16/641/2009/npg-16-641-2009.pdf
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spelling doaj-31532cd03aca406f8947cc0d446895272020-11-24T21:27:45ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462009-11-01166641653Improved moment scaling estimation for multifractal signalsD. VenezianoP. FurcoloA fundamental problem in the analysis of multifractal processes is to estimate the scaling exponent <i>K(q)</i> of moments of different order <i>q</i> from data. Conventional estimators use the empirical moments <i><html><body> μ<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html><sub>r</sub><sup>q</sup>=⟨ | ε<sub>r</sub>(τ)|<sup>q</sup>⟩</i> of wavelet coefficients ε<sub>r</sub>(τ), where τ is location and <i>r</i> is resolution. For stationary measures one usually considers "wavelets of order 0" (averages), whereas for functions with multifractal increments one must use wavelets of order at least 1. One obtains <i><html><body> K<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html>(q)</i> as the slope of log(<i><html><body> μ<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html><sub>r</sub><sup>q</sup></i>) against log(<i>r</i>) over a range of <i>r</i>. Negative moments are sensitive to measurement noise and quantization. For them, one typically uses only the local maxima of <i>| ε<sub>r</sub>(τ)|</i> (modulus maxima methods). For the positive moments, we modify the standard estimator <i><html><body> K<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html>(q)</i> to significantly reduce its variance at the expense of a modest increase in the bias. This is done by separately estimating <i>K(q)</i> from sub-records and averaging the results. For the negative moments, we show that the standard modulus maxima estimator is biased and, in the case of additive noise or quantization, is not applicable with wavelets of order 1 or higher. For these cases we propose alternative estimators. We also consider the fitting of parametric models of <i>K(q)</i> and show how, by splitting the record into sub-records as indicated above, the accuracy of standard methods can be significantly improved. http://www.nonlin-processes-geophys.net/16/641/2009/npg-16-641-2009.pdf
collection DOAJ
language English
format Article
sources DOAJ
author D. Veneziano
P. Furcolo
spellingShingle D. Veneziano
P. Furcolo
Improved moment scaling estimation for multifractal signals
Nonlinear Processes in Geophysics
author_facet D. Veneziano
P. Furcolo
author_sort D. Veneziano
title Improved moment scaling estimation for multifractal signals
title_short Improved moment scaling estimation for multifractal signals
title_full Improved moment scaling estimation for multifractal signals
title_fullStr Improved moment scaling estimation for multifractal signals
title_full_unstemmed Improved moment scaling estimation for multifractal signals
title_sort improved moment scaling estimation for multifractal signals
publisher Copernicus Publications
series Nonlinear Processes in Geophysics
issn 1023-5809
1607-7946
publishDate 2009-11-01
description A fundamental problem in the analysis of multifractal processes is to estimate the scaling exponent <i>K(q)</i> of moments of different order <i>q</i> from data. Conventional estimators use the empirical moments <i><html><body> μ<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html><sub>r</sub><sup>q</sup>=⟨ | ε<sub>r</sub>(τ)|<sup>q</sup>⟩</i> of wavelet coefficients ε<sub>r</sub>(τ), where τ is location and <i>r</i> is resolution. For stationary measures one usually considers "wavelets of order 0" (averages), whereas for functions with multifractal increments one must use wavelets of order at least 1. One obtains <i><html><body> K<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html>(q)</i> as the slope of log(<i><html><body> μ<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html><sub>r</sub><sup>q</sup></i>) against log(<i>r</i>) over a range of <i>r</i>. Negative moments are sensitive to measurement noise and quantization. For them, one typically uses only the local maxima of <i>| ε<sub>r</sub>(τ)|</i> (modulus maxima methods). For the positive moments, we modify the standard estimator <i><html><body> K<span style="margin-left: -.6em; vertical-align: super;">^</span></body></html>(q)</i> to significantly reduce its variance at the expense of a modest increase in the bias. This is done by separately estimating <i>K(q)</i> from sub-records and averaging the results. For the negative moments, we show that the standard modulus maxima estimator is biased and, in the case of additive noise or quantization, is not applicable with wavelets of order 1 or higher. For these cases we propose alternative estimators. We also consider the fitting of parametric models of <i>K(q)</i> and show how, by splitting the record into sub-records as indicated above, the accuracy of standard methods can be significantly improved.
url http://www.nonlin-processes-geophys.net/16/641/2009/npg-16-641-2009.pdf
work_keys_str_mv AT dveneziano improvedmomentscalingestimationformultifractalsignals
AT pfurcolo improvedmomentscalingestimationformultifractalsignals
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