  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 L2 penalties on the coefficients, and the
other set uses L1 penalties to control sparsity. From a frequentist
perspective, this results in a hybrid smoother that combines cubic smoothing
splines and the LASSO. 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, offline monitoring of a water treatment process for municipal
water filtration. We also test the robustness of these methods with a Monte
Carlo study designed to inspect a range trended time series under an array of
conditions and compare this new approach to multiple existing modern methods.
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 in offline
data.
