A dynamic modeling approach for anomaly detection using stochastic differential equations

In this paper the stochastic differential equation (SDE) is utilized as a quantitative description of a natural phenomenon to distinguish normal and anomalous samples. In this framework, discrete samples are modeled as a continuous time-dependent diffusion process with time varying drift and diffusion coefficients. We employ a local non-parametric technique using kernel regression and polynomial fitting to learn coefficients of stochastic models. Next, a numerical discrete construction of likelihood over a sliding window is established using Girsanov's theorem to calculate an anomalous score for test observations. One of the benefits of the method is to estimate the ratio of probability density functions (PDFs) through the Girsanov's theorem instead of evaluating PDFs themselves. Another feature of employing SDE model is its generality, in the sense that it includes most of the well-known stochastic models. Performance of the new approach in comparison to other methods is demonstrated through simulated and real data. For real-world cases, we test our method on detecting anomalies in twitter user engagement data and discriminating speech samples from non-speech ones. In both simulated and real data, proposed algorithm outperforms other methods.