Bilinear modelling of batch processes. Part I: theoretical discussion

Abstract

When studying the principal component analysis (PCA) or partial least squares (PLS) modelling of batch process data, one realizes that there is a wide range of approaches. In many cases, new modelling approaches are presented just because they work properly for a particular application, for example, on-line monitoring and a given number of processes. A clear understanding of why these approaches perform successfully and which are the advantages and disadvantages in front of the others is seldom supplied. Why does modelling after batch-wise unfolding capture changing dynamics? What are the consequences of variable-wise unfolding? Is there any best unfolding method? When should several models for a single process be used? In this paper, it is shown how these and other related questions can be answered by properly analyzing the dynamic covariance structures of the various approaches.

Multivariate Statistical Approach for Anomaly Detection and Lost Data Recovery in Wireless Sensor Networks

“…Furthermore, the suitability of one or another arrangement depends on the purpose of the modeling process itself [34].…”

Dynamic model-based fault diagnosis for (bio)chemical batch processes

“…Variable-wise unfolded MPCA (MPCAV) only incorporates the instantaneous cross-correlations between the measured variables [8,9]. Leaving serial correlation in the residuals violates the assumptions of the fault detection statistics and may lead to decreased monitoring performance [10].…”

“…For a detailed study on the impact of cross-and autocorrelation on the significance level for hypothesis testing in monitoring statistics, the reader is referred to the work of Xie et al [11]. Batch-wise unfolded MPCA (MPCAB) incorporates the full linear dynamics around the average trajectory, but entails the need for estimating future observations [8,9]. Batch Dynamic PCA (BDPCA) incorporates dynamic behavior by adding time-lagged variables [3].…”

“…If no lagged variables are added, the resulting matrix is unfolded variable-wise. If all possible lag variables are added, a batch-wise unfolded matrix emerges [9,12]. BDPCA is discussed in Section 3.3.…”

Building covariance matrices with the desired structure

“…The multi-stage nature of both batch and continuous processes can also be the result of the different processing units and the distinguishable operations inside a unit. Even in the same unit or stage of a batch process, the correlation structure and process dynamics may change as the batch is being processed [10].…”