Sparse PARAFAC2 decomposition: Application to fault detection and diagnosis in batch processes

Abstract The PARAFAC2 decomposition is often used to modeling a set of matrices that have the same number of columns but different numbers of rows. However, the PARAFAC2 model lacks of interpretability because most of elements in factor vectors are nonzero. To overcome this deficiency, a sparse PARAFAC2 (SPARAFAC2) decomposition is developed. SPARAFAC2 yields sparse factor vectors (SFVs) with only a few nonzero elements. Because of the sparsity in factor vectors, the SPARAFAC2 model has much better interpretability than the ordinary PARAFAC2 model. SPARAFAC2 is attractive for the applications in batch processes, because it not only can directly handle the three-way structure of batch data and naturally solve the unequal batch length problem, but also can reveal meaningful connections between process variables. Therefore, based on the SPARAFAC2 decomposition, fault detection and diagnosis methods are proposed for batch processes. To improve the fault detection capability, a cumulative percent contribution criterion is used to adaptively select SFVs for each sample from the fault detection point of view. Two fault detection indices are then defined using the selected SFVs. A contribution-based fault diagnosis method is also proposed. This method identifies faulty variables by evaluating contributions of SFVs and active variables (with nonzero elements) in each SFV to the detection of faults. The effectiveness of the proposed methods is demonstrated with a case study in an industrial-scale fermentation process.