Anomaly detection is a classical problem in Statistical Learning with wide-reaching applications in security, networks, genomics and others. In this work, we formulate the anomaly classification problem as an extension to the detection problem: how to distinguish between samples from multiple heterogenous classes that are anomalies relative to a well-defined, homogenous, normal class. Our formulation of this learning setting arises from studies in cancer genomics, where this problem follows from prognosis and diagnosis applications.

Standard binary and multi-class classification schemes are not well suited to the anomaly classification task since they attempt to directly model these highly unstable, heterogeneous classes. In this work, we show that robust classifiers can be obtained by modeling the degree of deviation from the normal class as a stable characteristic of each anomaly class. To do so, we formalize the anomaly classification problem, characterize it statistically and computationally via kernel methods and propose a class of robust learning methods, anti-profiles, specifically designed for this task.

We focus on an open area of research in cancer genomics which motivates this project: the classification of tumors for prognosis and diagnosis. We provide experimental results obtained by applying the anti-profile method to gene expression data. In addition we extend the anti-profile approach to use kernel functions, and develop a support-vector machine (SVM) based method for classification of anomalies based on their deviation from a stable normal class. We provide experimental results obtained by applying this method to genetic data to classify different stages of tumor progression, and show that this method provides much more stable classifiers than the application of regular classifiers. In addition we show that this approach can be applied to anomaly classification problems in other application domains.

We conclude by developing an SVM for censored survival information and demonstrate that the anti-profile method can produce stable classifiers for modeling the clinical outcome of clinical studies of cancer.