Outlier Detection from a Mixture Distribution When Training Data Are Unlabeled

We consider the difficult task of using seismic signals (or any other discriminants) for detecting nuclear explosions from the large number of background signals such as earthquakes and mining blasts. Given a ground-truth database (i.e., labeled data), Fisk et aL (1996) consider the problem of detecting outliers (nuclear explosions) from a single background-signal population, and their approach has been applied successfully in several regions around the world. Wang et al. (1997) attack the problem in terms of modeling the background as a mixture distribution and looking for outliers (nuclear events) from that mixture. However, those authors only considered the case in which at least some fraction of the training sample was labeled, that is, at least some ground-truth information was available, and the number of distinct classes of events was known. In the current article, we extend these results to the case in which no events in the training sample are labeled and also to the case in which the number of event types represented in the training sample is unknown. One can view the mixture approach as a robust method for fitting a density to training data that may not be normally distributed whether or not the data consist of identifiable components that have a physical interpretation. The technique is demonstrated using simulated data as well as two sets of seismic data.