Asymmetric Laplace Mixture Modelling of Incomplete Power-Law Distributions: Application to ‘Seismicity Vision’

Abstract

Data used in statistical analyses are often limited to a narrow range over which the quantity of interest is observed to be reliable. The power-law behavior of many natural processes is only observed above a threshold below which information is discarded due to detection limitations. These incomplete data can however also be described by a power law, and the distribution over the full quantity range reformulated as an asymmetric Laplace (AL) distribution. With the detection process heterogeneous in space and time in realistic conditions, the data can be modelled by a mixture of AL components. Using seismicity as example, we describe an asymmetric Laplace mixture model (ALMM), which considers ambiguous overlapping components - as observed in Nature - based on a semi-supervised hard Expectation-Maximization algorithm. We show that the ALMM fits reasonably well incomplete data and that the number of AL components can be related to the seismic network density. We conclude that the full range of data can be used in statistical analyses, including in computer vision. In the case of seismicity, a ten-fold increase in sample size is possible which provides, for example, a better spatial pattern resolution to improve the correlation between fault features and earthquake labels.