Hybrid Smoothing for Anomaly Detection in Time Series

Many industrial and engineering processes monitored as times series have smooth trends that indicate normal behavior and occasionally anomalous patterns that can indicate a problem. This kind of behavior can be modeled by a smooth trend such as a spline or Gaussian process and a disruption based on a sparser representation. Our approach is to expand the process signal into two sets of basis functions: one set uses $L_2$ penalties on the coefficients and the other set uses $L_1$ penalties to control sparsity. From a frequentist perspective, this results in a hybrid smoother that combines cubic smoothing splines and the LASSO, and as a Bayesian hierarchical model (BHM), this is equivalent to priors giving a Gaussian process and a Laplace distribution for anomaly coefficients. For the hybrid smoother we propose two new ways of determining the penalty parameters that use effective degrees of freedom and contrast this with the BHM that uses loosely informative inverse gamma priors. Several reformulations are used to make sampling the BHM posterior more efficient including some novel features in orthogonalizing and regularizing the model basis functions. This methodology is motivated by a substantive application, monitoring the water treatment process for the Denver Metropolitan area. We also test the methods with a Monte Carlo study designed around the kind of anomalies expected in this application. Both the hybrid smoother and the full BHM give comparable results with small false positive and false negative rates. Besides being successful in the water treatment application, this work can be easily extended to other Gaussian process models and other features that represent process disruptions.