Non-parametric Information-Theoretic Measures of One-Dimensional Distribution Functions from Continuous Time Series

We study non-parametric measures for the problem of comparing distributions, which arise in anomaly detection for continuous time series. Non-parametric measures take two distributions as input and produce two numbers as output: the difference between the input distributions and the statistical significance of this difference. Some of these measures, such as Kullback-Leibler measure, are defined for comparing probability distribution functions (PDFs) and some others, such as Kolmogorov-Smirnov measure, are for cumulative distribution functions (CDFs). We first show how to adapt the PDF based measures to compare CDFs, resulting in a total of 23 CDF based measures. We then provide a unified functional form that subsumes all these measures. We present our methodology to determine the significance (of the measures) by simulations only. Finally, we evaluate these measures for the anomaly detection in continuous time series.