Hybrid L1 and L2 smoothing presented by Doug Nychka
Abstract: Spline smoothing and more generally Gaussian process smoothing have become a successful methodology for estimating a smooth trend or surface from noisy data. Similarly the LASSO and related L1 penalties have become important tools for variable selection and also admit of a Bayesian version based on the Laplace distribution. This project combines these two approaches as a method to detect discontinuous behavior in an otherwise smooth signal. Every day the Foothills Facility of Denver Water filters more than 250 million gallons of water for the metropolitan area. This process runs continuously and is monitored across an array of filters, each the size of a small swimming pool, at 5 minute intervals. It is important to be able detect anomalous behavior in a filter in a prompt manner or to process past measurements to determine trends. The anomalies take the form of discontinuities or appear as step changes in the smooth filtering cycle. This application is the motivation for a mixed smoothing approach where normal operation is captured by a smoothing spline and the anomalies by basis function coefficients determined by an L1 penalty. As part of this research a frequentist penalty method is compared against its equivalent Bayesian hierarchical model (BHM) based on Gaussian processes and a Laplace prior for the anomaly coefficients. This talk will discuss some of the challenges in implementing both models. Specifically we study how to choose penalty parameters for the frequentist model and how to formulate then BHM in a way that the MCMC sampling algorithm mixes efficiently. Both approaches appear to classify anomalies in the filter cycles well with the spline model being much faster but the BHM providing measures of uncertainty in the detected anomalies. The similarities between these frequentist and Bayesian models relies on the correspondence between splines and Gaussian processes. This was first described by Grace Wahba, a long-time faculty member of the UW statistics department, and George Kimeldorf. Some background for this connection will be given as part of developing the Bayesian model.