Rotation algorithm: Generation of Gaussian self-similar stochastic processes

Abstract

In this paper, we introduce a simple and practical method to generate Gaussian self-similar stochastic processes [fractional Gaussian noises (fGns) and fractional Brownian motions (fBms)] by interpolating between two known series of them. We apply the rotation algorithm to different cases including different pairs of fBms (fGns) and also different pairs each composed of an fBm and an fGn. Our results show that the sensitivity of our method for two fBms (fGns) is higher when approaching the series with a larger (smaller) Hurst exponent and, for the case with one fBm and one fGn by approaching the Hurst exponent of the selected fBm, the Hurst exponent of the produced series changes more. Surprisingly, by using this method, we can generate (positively and/or negatively) correlated series from two uncorrelated ones (one Brownian motion and one white Gaussian noise) or it is possible to generate uncorrelated signals (one or two depending on the choice of two input signals) from two correlated ones. For two fGns, using the rotation algorithm, the evolution starts from larger scales in the system, while for two fBms, the evolution starts from smaller scales in the system.