Shape Constrained Splines with Discontinuities for Anomaly Detection in a Batch Process

A previously developed technique for qualitative trend analysis (QTA) based on shape constrained splines (SCS) has been favourably compared to pre-existing techniques for the purpose of batch process diagnosis. Thanks to the branch-and-bound algorithm, this approach leads to a deterministic and global solution for QTA. One limitation of this method is that local discontinuities in otherwise continuous derivatives of the fitted spline function are not permitted. Recent work however allows to relax the optimization problem further so that provable bounds can be computed for this more complicated case. In this contribution, the resulting shape constrained splines with discontinuities (SCSD) method is applied for anomaly detection in batch process data. Importantly, the QTA approach to anomaly detection proves worthwhile because (i) tuning of the SCSD method is limited to setting an upper control limit, (ii) the resulting sum of squared errors statistic shows almost no drift for SCSD in contrast to the similar Q-statistic computed by principal component analysis (PCA), and (iii) true positive rates by means of SCSD are over 80% while the PCA method delivers at most 70% for false positive rates up to 5% based on a data set consisting of 410 batches.