Automatic algorithm to decompose discrete paths of fractional Brownian motion into self-similar intrinsic components
T. Popoviciu Institute of Numerical Analysis, Romanian
Academy, P.O. Box
2 Mathematics Department, Friedrich Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany
Received: 29 June 2015
Received in final form: 20 August 2015
Published online: 7 October 2015
Fractional Brownian motion (fBm) is a nonstationary self-similar continuous stochastic process used to model many natural phenomena. A realization of the fBm can be numerically approximated by discrete paths which do not entirely preserve the self-similarity. We investigate the self-similarity at different time scales by decomposing the discrete paths of fBm into intrinsic components. The decomposition is realized by an automatic numerical algorithm based on successive smoothings stopped when the maximum monotonic variation of the averaged time series is reached. The spectral properties of the intrinsic components are analyzed through the monotony spectrum defined as the graph of the amplitudes of the monotonic segments with respect to their lengths (characteristic times). We show that, at intermediate time scales, the mean amplitude of the intrinsic components of discrete fBms scales with the mean characteristic time as a power law identical to that of the corresponding continuous fBm. As an application we consider hydrological time series of the transverse component of the transport process generated as a superposition of diffusive movements on advective transport in random velocity fields. We found that the transverse component has a rich structure of scales, which is not revealed by the analysis of the global variance, and that its intrinsic components may be self-similar only in particular cases.
Key words: Computational Methods
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