Distribution-Free Detection of Structured Anomalies: Permutation and Rank-Based Scans

The scan statistic is by far the most popular method for anomaly detection, being popular in syndromic surveillance, signal and image processing, and target detection based on sensor networks, among other applications. The use of the scan statistics in such settings yields an hypotheses testing procedure, where the null hypothesis corresponds to the absence of anomalous behavior. If the null distribution is known, then calibration of a scan-based test is relatively easy, as it can be done by Monte-Carlo simulation. When the null distribution is unknown, it is not clear what the best way to proceed is. We propose two procedures. One is a calibration by permutation and the other is a rank-based scan test, which is distribution-free and less sensitive to outliers. Furthermore, the rank-scan test requires only a one-time calibration for a given data size making it computationally more appealing. In both cases, we quantify the performance loss with respect to an oracle scan test that knows the null distribution, and show one incurs only a very small loss in the context of a natural exponential family. These results include the classical normal location model, as well as Poisson model popular in syndromic surveillance. We perform numerical experiments on simulated data further supporting our theory, and also experiments with a real dataset from genomics.