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| Titel |
Improved Estimation of Moment Scaling in Multifractal Processes |
| VerfasserIn |
D. Veneziano, P. Furcolo |
| Konferenz |
EGU General Assembly 2009
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 11 (2009) |
| Datensatznummer |
250024692
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| Zusammenfassung |
| Consider a stationary multifractal measure in the unit interval and denote by Én, n -¥ 0, the
average measure density in a generic sub-interval of length 2-n. Multifractality implies that
the non-diverging moments of Én satisfy E[Énq] - 2nK(q) where K(q) is a convex
moment-scaling function. The problem we address is how to most accurately estimate K(q)
from observations at some resolution 2ndat. A popular estimator of K(q) is the slope of the
least-square regression of log 2-¨Énq-© against n for n = n1,...,ndat, where -¨-© denotes sample
averaging and n1 -¥ 0 is a parameter. Not much is known about the performance of this
estimator, except that above some moment order q-² the estimator tends to be linear in q. Our
objective is to determine the bias and variance of this and other estimators and suggest
improvements.
For positive moments and in general for noiseless data, we consider extended
estimators of the following type. Denote by n0 -¥ 0 a resolution level below ndat.
We partition the unit interval into 2n0 sub-intervals of length 2-n0 and view each
sub-record as a shorter realization of the process. We apply the conventional regression
estimator of K(q) to each sub-record (under the constraint n1 -¥ n0) and obtain
the final estimator as the average of the sub-record estimators. The conventional
estimator corresponds to n0 = 0. Since the sub-record estimators are statistically
independent, averaging them is an effective means to reduce the error variance.
For given ndat, the modified estimator depends on (n0, În1), where n0 -¥ 0 and
În1 = n1 - n0 ranges from 0 to ndat - n0 - 1. From extensive Monte Carlo
simulation, we have found that increasing n0 increases the bias but reduces the error
variance while increasing În1 has beneficial effects on the bias. Since n0 + În1
cannot exceed ndat - 1, there are tradeoffs between the two parameters and the
choice that minimizes the RMS error is nontrivial. Conventional estimators (with
n0 = 0) tend to have lower bias but significantly higher variance than the optimal
estimators.
Especially for the negative moments, a practical concern is the robustness of the estimator of
K(q) against noise and other data inaccuracies. To reduce the effect of these imperfections,
one typically uses variants of the so-called wavelet-transform-modulus-maxima (WTMM)
method. We show that these methods are generally biased and propose simple unbiased
alternatives.
Finally we consider the case when K(q;θ) has known parametric form with unknown
parameters θ. In this case it is typical to first estimate K(q) nonparametrically as indicated
above and then find θ to best fit the nonparametric estimates. It is often feasible to use
simulation to assess the bias of such estimators, either systematically as a function of the true
value of θ or, after a first estimate of θ has been obtained, in the vicinity of that estimate. This
allows one to correct for bias and select the parameters n0 and În1 to minimize the
error variance. The minimum is typically attained for (n0 = ndat - 1; În1 = 0).
Optimal bias-corrected estimators of θ are much more accurate than conventional
estimators.
We illustrate all suggested estimators for the case of lognormal multifractal measures. |
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