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Fractional Brownian processes have been proposed as a mathematical model for
a wide diversity
of stochastic phenomena occurring in real extended physical systems
exhibiting different
degrees of correlation. The variance at large t scales as a power law,
s2 ~ t2H, where H is the Hurst exponent, ranging from 0 to 1. The value H=0.5
corresponds to the ordinary uncorrelated Brownian motion, while H<0.5 and H>0.5 correspond respectively
to anticorrelated and correlated signals. The analysis of the Hurst exponent is
nowadays considered a
practical instrument in fields as biophysics (DNA sequence, gait
fluctuations), econophysics,
cloud breaking and many others. For example, one can discriminate heartbeat
intervals of healthy
and sick hearts on the basis of the value of H. Stock price volatility
shows a degree of persistence larger
than that of the return series (H = 0.5), a fact which is exploited when
practical investment tools have to be developed.
The validation of climate models is based on the analysis of long-term
correlation of atmospheric series. In order for
the above mentioned classifications to be reliable, several techniques have
been thus proposed with the main purpose to
extract as accurate as possible values of the Hurst exponent from the data
set. Among the number of different techniques that have
been proposed to estimate the correlation exponent of fractal stochastic
signals we only mention the spectral analysis, the
correlograms and semivariograms, the rescaled range analysis, the Fano
factor, the Allan variance, the Detrended Fluctuation
Analysis and very recently the Detrended Moving Average analysis. These
techniques calculate appropriate statistical
functions over the signal in the time or in the frequency domain. We are investigating
possible estensions and applications of these techniques.
A. Carbone, B.M. Chiaia, B. Frigo, C. Turk,
Multiscale Modelling of Snow Microstructure
Int. J. of Multiscale Comput. Engin. (2012)
S. Arianos, A. Carbone, C. Turk
Self-similarity of higher order moving averages [PDF]
Phys. Rev. E 84, 046113 (2011)
A. Carbone, B.M. Chiaia, B. Frigo, C. Turk
Snow metamorphism: a fractal approach
Int. J. for Multisc. Comp. Eng. (2010)
C. Turk, A. Carbone, B.M. Chiaia
Fractal Heterogeneous Media [PDF]
Phys. Rev. E 81, 026706-1-7 (2010)
A. Carbone, B.M. Chiaia, B. Frigo, C. Turk
Fractal Model for Snow
Mater. Sc. For. 638-642, pp. 2555-2560 (2010)
S. Arianos and A.Carbone
Cross-correlation of long-range correlated series
[PDF]
J. Stat. Mech: Theory and Experiment P03037, (2009)
S. Arianos, E. Bompard, A. Carbone and F. Xue
Power Grids Vulnerability: A Complex Network Approach
[PDF]
Chaos 19, 013119 (2009)
A.Carbone
Algorithm to estimate the Hurst exponent of high-dimensional fractals
Phys. Rev. E 76, (2007) [PDF]
S. Arianos and A.Carbone
Detrending Moving Average (DMA) Algorithm: a closed form approximation of the scaling law [PDF]
Physica A 382, 9 (2007)
A.Carbone and H.E. Stanley
Scaling properties and entropy of long range correlated series [PDF]
Physica A 384, 21 (2007)
L. Xu, P. Ch. Ivanov, C. Zhi, K. Hu, A. Carbone, H. E. Stanley
Quantifying signals with power-law correlations
Phys. Rev. E, 71, 051101, (2005) [PDF]
A.Carbone, G. Castelli and H. E. Stanley
Analysis of the clusters formed by the moving average of a long-range correlated stochastic series
Phys. Rev. E, 69, 026105, (2004) [PDF]
A.Carbone, G. Castelli and H. E. Stanley
Time-Dependent Hurst Exponent in Financial Time Series
Physica A, 344, 267, (2004) [PDF]
A.Carbone and H. E. Stanley
Directed Self-Organized Critical Patterns Emerging from Fractional Brownian Paths
Physica A,
340, 544, (2004)
[PDF]
A.Carbone and G. Castelli
Scaling of long-range correlated noisy signals: application to financial markets
Noise in Complex Systems and Stochastic Dynamics, L. Schimansky-Geier, D. Abbott, A. Neiman, C. Van den Broeck, Eds.,
Proc. of SPIE, 407, 5114 (2003)
[PDF]
E.Alessio, A.Carbone , G.Castelli, V.Frappietro
Second-order moving average and scaling of stochastic time series,
Eur. J. Phys. B, 27, 197, (2002) [PDF]
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