Matthew Davidow

Matthew Davidow, David Matteson

Practical anomaly detection requires applying numerous approaches due to the inherent difficulty of unsupervised learning. Direct comparison between complex or opaque anomaly detection algorithms is intractable; we instead propose a framework for associating the scores of multiple methods. Our aim is to answer the question: how should one measure the similarity between anomaly scores generated by different methods? The scoring crux is the extremes, which identify the most anomalous observations. A pair of algorithms are defined here to be similar if they assign their highest scores to roughly the same small fraction of observations. To formalize this, we propose a measure based on extremal similarity in scoring distributions through a novel upper quadrant modeling approach, and contrast it with tail and other dependence measures. We illustrate our method with simulated and real experiments, applying spectral methods to cluster multiple anomaly detection methods and to contrast our similarity measure with others. We demonstrate that our method is able to detect the clusters of anomaly detection algorithms to achieve an accurate and robust ensemble algorithm.

Matthew Davidow, David S. Matteson

Anomaly detection aims to identify observations that deviate from the typical pattern of data. Anomalous observations may correspond to financial fraud, health risks, or incorrectly measured data in practice. We show detecting anomalies in high-dimensional mixed data is enhanced through first embedding the data then assessing an anomaly scoring scheme. We focus on unsupervised detection and the continuous and categorical (mixed) variable case. We propose a kurtosis-weighted Factor Analysis of Mixed Data for anomaly detection, FAMDAD, to obtain a continuous embedding for anomaly scoring. We illustrate that anomalies are highly separable in the first and last few ordered dimensions of this space, and test various anomaly scoring experiments within this subspace. Results are illustrated for both simulated and real datasets, and the proposed approach (FAMDAD) is highly accurate for high-dimensional mixed data throughout these diverse scenarios.

Phillip A. Jang, Andrew E. Loeb, Matthew B. Davidow et al.

Gaussian processes are rich distributions over functions, with generalization properties determined by a kernel function. When used for long-range extrapolation, predictions are particularly sensitive to the choice of kernel parameters. It is therefore critical to account for kernel uncertainty in our predictive distributions. We propose a distribution over kernels formed by modelling a spectral mixture density with a Lévy process. The resulting distribution has support for all stationary covariances--including the popular RBF, periodic, and Matérn kernels--combined with inductive biases which enable automatic and data efficient learning, long-range extrapolation, and state of the art predictive performance. The proposed model also presents an approach to spectral regularization, as the Lévy process introduces a sparsity-inducing prior over mixture components, allowing automatic selection over model order and pruning of extraneous components. We exploit the algebraic structure of the proposed process for $\mathcal{O}(n)$ training and $\mathcal{O}(1)$ predictions. We perform extrapolations having reasonable uncertainty estimates on several benchmarks, show that the proposed model can recover flexible ground truth covariances and that it is robust to errors in initialization.