Statistical Analysis of Nearest Neighbor Methods for Anomaly Detection
This work addresses anomaly detection, a key problem in data analysis for applications like fraud detection, but it is incremental as it builds on existing nearest-neighbor and distance-to-measure frameworks.
The paper investigates nearest-neighbor methods for anomaly detection, showing through simulations that they compare favorably to state-of-the-art algorithms on synthetic datasets and analyzing their performance on real datasets in relation to dimensionality. It provides theoretical guarantees for misclassification rates using distance-to-measure under Huber's contamination model.
Nearest-neighbor (NN) procedures are well studied and widely used in both supervised and unsupervised learning problems. In this paper we are concerned with investigating the performance of NN-based methods for anomaly detection. We first show through extensive simulations that NN methods compare favorably to some of the other state-of-the-art algorithms for anomaly detection based on a set of benchmark synthetic datasets. We further consider the performance of NN methods on real datasets, and relate it to the dimensionality of the problem. Next, we analyze the theoretical properties of NN-methods for anomaly detection by studying a more general quantity called distance-to-measure (DTM), originally developed in the literature on robust geometric and topological inference. We provide finite-sample uniform guarantees for the empirical DTM and use them to derive misclassification rates for anomalous observations under various settings. In our analysis we rely on Huber's contamination model and formulate mild geometric regularity assumptions on the underlying distribution of the data.