Identifying and analyzing power-law scaling in two-dimensional image datasets

Power-law distributions provide a general description of diverse natural phenomena in which events with a logarithmically increasing size occur with logarithmically decreasing probability. However, experimentally derived correlated two-dimensional information is often difficult to cleanly interpret as discrete events of defined size. Moreover, physical limitation of techniques such as those based on scanning probe microscopy, which can ideally be used to observe power-law behavior, reduce event number and thus render straightforward power-law fits even more challenging. Here we develop and compare different techniques to analyze event distributions from two-dimensional images. We show that tracking interface position allows the associated scaling parameters to be accurately extracted from both experimental and synthetic image-based datasets. We also show how these techniques can differentiate between power-law and non-power-law behavior by comparison of Hill, moments, and kernel estimators of this scaling parameter. We thus present computational tools to analyze power-law fits in two-dimensional datasets and identify the scaling parameters that best describe these distributions.