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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.

Principal component analysis (PCA) and partial least squares (PLS) are bilinear modelling tools which have been successfully applied to three-way batch process data for monitoring and quality prediction. Most modelling approaches in the literature are based on a fixed model structure. The approach proposed in this paper, named the Multi-phase (MP) analysis framework, provides the flexibility to adjust the model structure to the dynamic nature of the process under study. The existence of several phases, with dynamics of different order and changes in the correlation structure among variables, is effectively identified. This adjustment of the model structure to the features of the process yields performance improvements in several applications, such as the on-line monitoring and final quality prediction, as shown when comparing the MP models with various well-established modelling approaches. Also, the MP approach provides a set of valuable tools for process understanding and data handling. Data from two processes, a fermentation process and a waste-water treatment process, are used to illustrate the capabilities of the proposed modelling framework. † Henceforth, the terms 'unfolding methods' and 'number of LMVs' will be used indistinctly throughout the paper, since each different number of LMVs implies a different unfolding procedure following Figure 1(c). 22: 632-643 ¶ Sub-models S 1 and S 2 may represent a single phase of a process or else comprise several sub-models.

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