Sampling power-law distributions

Abstract

Power-law distributions describe many phenomena related to rock fracture. Data collected to measure the parameters of such distributions only represent samples from some underlying population. Without proper consideration of the scale and size limitations of such data, estimates of the population parameters, particularly the exponent D, are likely to be biased. A Monte Carlo simulation of the sampling and analysis process has been made, to test the accuracy of the most common methods of analysis and to quantify the confidence interval for D. The cumulative graph is almost always biased by the scale limitations of the data and can appear non-linear, even when the sample is ideally power law. An iterative correction procedure is outlined which is generally successful in giving unbiased estimates of D. A standard discrete frequency graph has been found to be highly inaccurate, and its use is not recommended. The methods normally used for earthquake magnitudes, such as a discrete frequency graph of logs of values and various maximum likelihood formulations can be used for other types of data, and with care accurate results are possible. Empirical equations are given for the confidence limits on estimates of D, as a function of sample size, the scale range of the data and the method of analysis used. The predictions of the simulations are found to match the results from real sample D-value distributions. The application of the analysis techniques is illustrated with data examples from earthquake and fault population studies