Estimation of the Scaling Exponent due to Fractal Behaviour of a Time Series

Analysis of time/space series signals from various branches of science viz. physics, earth and planetary sciences, economics etc. shows that they follow power law behaviour, also known as Zipf’s law or the Pareto distribution. Understanding of power law behaviour is very important for the processing and modelling of the time series data. In order to use power law behaviour of the time series in forward and inverse modelling we need more accurate measure of the power law behaviour, often known as scaling exponent. To estimate the scaling exponent we compute likelihood of the assumed probability distribution function. Further, to find the scaling exponent which best fits the data, we compute the probability of best scaling exponent for given time series using Bayes’ law, which turns out to be α = 1 + n (∑ln (xi/xmin))-1. This method is used in time/space domain in contrast to other conventional method such as FFT, maximum entropy method etc. which are used in frequency domain. This method gives more accurate estimation of the exponents as it is free from the errors due to fitting of line to estimate the exponent (slope) which is carried out in frequency domain. The usefulness of the methods is illustrated through some geophysical examples.