Conditioning of data outliers to identify and obtain the scaling behaviour of statistically self-similar and multifractal, non-Gaussian processes from finite length time-series
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
We address the generic problem of extracting the scaling exponents of a stationary non-Gaussian process realised by a time-series of finite length, where information about the process is not known a priori. Estimating the scaling exponents relies upon estimating the moments, or more typically structure functions, of the probability density of the differenced time-series. If the probability density is heavy tailed, outliers strongly influence the scaling behaviour of the moments. From an operational point of view, we wish to recover the scaling exponents of the underlying process by excluding a minimal population of the outliers. This method is particularly sensitive in distinguishing and quantifying self-affine, or self-similar, scaling from weak multifractality. We will illustrate this with two synthetically generated reference models: the first of which is manifestly self-similar, an $\alpha$-stable Levy process; and the second, manifestly multifractal, a cascade p-model. We show that for the symmetric $\alpha$-stable Levy process, the Levy exponent is recovered in up to the 6th order moment after only ~0.1-0.5 percent of the data are excluded. The scaling properties of the excluded outliers can also be tested to provide additional information about the system. Unlike the self-similar Levy process, which shows a convergence of all its exponents for each moment order, the multifractal p-model process shows a divergence of its exponents as the outlying data points are excluded. Importantly then, successively removing outlying data points does not convert the multifractal p-model time-series into a self-similar process. Although this is expected for a multifractal, as the Hurst exponent is locally dependent on the position in the time-series, we will show that outliers distort the scaling of these processes too and that conditioning is also needed. We will be discussing a way of distinguishing this multifractal scaling thus presenting a unified treatment of the handling and remedying of extreme data outliers.
As a practical application of the above technique we quantify the scaling of magnetic energy density in the inertial range of solar wind turbulence seen in situ at 1 AU with respect to solar activity. At solar maximum, when the coronal magnetic field is dynamic and topologically complex, we find self-similar scaling in the solar wind, whereas at solar minimum, when the coronal fields are more ordered, we find multifractality. We propose that the self-similar scaling seen at solar maximum is indicative of the non-trivial evolution of the early stages of the development of turbulence being represented near 1 AU in the elliptic at solar maximum, and thus reflects the fractal structure of the processes which drive the interplanetary solar wind at its solar origin. More importantly, this quantifies the solar wind signature that is of direct coronal origin, and distinguishes it from that of local MHD turbulence, with quantitative implications for our understanding of coronal heating of the solar wind.
1 K. Kiyani, S. C. Chapman and B. Hnat , Phys. Rev. E 74(5), 047611 (2006)
2 B. Hnat, S. C. Chapman, G. Rowlands, N. W. Watkins, and W. M. Farrell, Geophys. Res. Lett. 29 (2002)
3 K. Kiyani, S. C. Chapman, B. Hnat, R. M. Nicol, (2007)http://arxiv.org/abs/physics/0702123