Multivariate statistical process control methods for batch production: a review focused on applications

Figures & data

Table 1. Distribution of reviewed articles based on databases

Databases	Articles
Science Direct	11
Wiley Online Library	6
ACS Publications	6
Taylor & Francis Online	5
IEEE Xplore	5
Springer	1
Others	3

Figure 1. Three-way array dataset

Figure 2. Errors from estimation in dimensionality reduction

Figure 3. Slicing perspectives in three-way arrays

Table 2. Different unfolding perspectives for a three-way array

Slicing dimension	Rightwards One next to the other		Downwards One below to the other
	Matrix	Order	Matrix	Order
$I$	${\mathbf{X}}^{\left(I_R\right)}$	J$\times IK$	${\mathbf{X}}^{\left(I_D\right)}$	$IJ\times K$
$J$	${\mathbf{X}}^{\left(J_R\right)}$	$I\times JK$	${\mathbf{X}}^{\left(J_D\right)}$	$IJ\times K$
$K$	${\mathbf{X}}^{\left(K_R\right)}$	$I\times JK$	${\mathbf{X}}^{\left(K_D\right)}$	$IK\times J$
Slices transposed
$I$	${\mathbf{X}}^{{\left(I_R\right)}^{\mathrm{T}}}$	$K\times IJ$	${\mathbf{X}}^{{\left(I_D\right)}^{\mathrm{T}}}$	$IK\times J$
$J$	${\mathbf{X}}^{{\left(J_R\right)}^{\mathrm{T}}}$	$K\times IJ$	${\mathbf{X}}^{{\left(J_D\right)}^{\mathrm{T}}}$	$JK\times I$
$K$	${\mathbf{X}}^{{\left(K_R\right)}^{\mathrm{T}}}$	$J\times IK$	${\mathbf{X}}^{{\left(K_D\right)}^{\mathrm{T}}}$	$JK\times I$

Figure 4. Schematic overview of MSPC methods and articles’ distribution by dimensionality criteria

Figure 5. K-unfolding and MPCA method scheme

Figure 6. MPLS method scheme

Figure 7. STATIS method scheme

Figure 8. PARAFAC decomposition

Figure 9. Tucker3 decomposition

Figure 10. MCCA method scheme

Figure 11. MICA decomposition

Figure 12. MSFA method scheme

Figure 13. Parallel coordinates confidence region

Table 3. MSPC applications for batch process control

Nº	TITLE	AUTHOR	YEAR	METHOD
1	Monitoring batch processes using multiway principal component analysis	Nomikos, MacGregor	1994	MPCA
2	Multi-way PCA applied to an industrial batch process	Kosanovich, Piovoso, Dahl, MacGregor, Nomikos	1994	MPCA
3	Multivariate SPC Charts for Monitoring Batch Processes	Nomikos, MacGregor	1995	MPCA
4	Analysis, monitoring and fault diagnosis of batch processes using multiblock and multiway PLS	Kourti, Nomikos, MacGregor	1995	MPLS (+Multiblock)
5	Multi-way partial least squares in monitoring batch processes	Nomikos, MacGregor	1995	MPLS
6	Application of multi-way principal components analysis to nuclear waste storage tank monitoring	Gallagher, Wise, Stewart	1996	MPCA
7	Batch process monitoring for consistent production	Martin, Morris, Papazoglou, Kiparissides	1996	MPCA
8	Improved Process Understanding Using Multiway Principal Component Analysis	Kosanovich, Dahl, Piovoso	1996	MPCA
9	Batch tracking via nonlinear principal component analysis	Dong, McAvoy	1996	MPCA (+NLPCA +Multistage)
10	Multivariate Statistical Analysis of an Emulsion Batch Process	Neogi, Schlags	1998	MPCA, MPLS
11	Monitoring a PVC Batch Process with Multivariate Statistical Process Control Charts	Tates, Louwerse, Smilde, Koot, Berndt	1999	MPCA
12	Manufacturing performance enhancement through multivariate statistical process control	Martin, Morris, Kiparissides	1999	MPCA, MPLS (IMPLS)
13	Post analysis on different operating time processes using orthonormal function approximation and multiway principal component analysis	Chen, Liu	2000	MPCA
14	Multivariate statistical process control of batch processes based on three-way models	Louwerse, Smilde	2000	MPCA, PARAFAC, TUCKER3
15	Multilevel PCA and Inductive learning for knowledge extraction from operational data of batch processes	Yuan, Wang	2001	MPCA
16	Application of PARAFAC2 to fault detection and diagnosis in semiconductor etch	Wise	2001	PARAFAC, MPCA, MPLS
17	Online Batch/Fed-Batch Process Performance Monitoring, Quality Prediction, and Variable-Contribution Analysis for Diagnosis	Ündey, Ertunç, Çinar	2003	MPLS (+ILC, MVTs)
18	On-line monitoring of batch processes using a PARAFAC representation	Meng	2003	PARAFAC + MPCA
19	Model Predictive Monitoring for Batch Processes	García-Muñoz, Kourti, MacGregor	2004	MPCA
20	Fault detection of batch processes using multiway kernel principal component analysis	Jong-Min Lee, Yoo, In-Beum Lee	2004	MPCA (+Multiway Kernel PCA)
21	Statistical Process Control Charts for Batch Operations Based on Independent Component Analysis	Albazzaz, Wang	2004	MICA
22	On-line monitoring of batch processes using multiway independent component analysis	Yoo, Jong-Min Lee, Vanrolleghem, In-Beum Lee	2004	MICA
23	Monitoring batch processes with the STATIS approach	Gourvénec, Stanimirova, Saby, Airiau, Massart	2005	STATIS
24	Online monitoring and diagnosis of batch processes: empirical model-based framework and a case study	Cho, Kim, Jeong	2006	MPCA
25	Study of the application of multiway multivariate techniques to model data from an industrial fermentation process	Ferreira, Lopes, Menezes	2007	MPCA, MPLS
26	Online monitoring of multi-phase batch processes using phase-based multivariate statistical process control	Doan, Srinivasan	2008	MPCA (+Dynamic PCA)
27	Two-Step MPLS-Based Iterative Learning Control for Batch Processes	Chen, Lin	2008	MPLS
28	Batch Process Monitoring by Three-way Data Analysis Approach	Niang, Fogliatto, Saporta	2009	STATIS
29	Multivariate Statistical Control of Batch Processes with Variable Duration	Fogliatto, Niang	2009	STATIS
30	Multiway Independent Component Analysis Mixture Model and Mutual Information Based Fault Detection and Diagnosis Approach of Multiphase Batch Processes	Yu, Chen, Rashid	2013	MICA
31	Non parametric on-line control of batch processes based on STATIS and clustering	Niang, Fogliatto, Saporta	2013	STATIS
32	Multivariate quality control of batch processes using STATIS	Filho, Luna	2015	STATIS
33	Batch Process Monitoring Based on Multiway Global Preserving Kernel Slow Feature Analysis	Zhang, Tian, Deng	2017	MSFA (+Global preserving Kernel SFA, BDKPCA, MKICA, MKPCA)
34	Joint-Individual Monitoring of Parallel-Running Batch Processes Based on MCCA	Wang, Jiang, Li, Lizhi	2017	MCCA
35	A Method for Predicting Future Observations in the Monitoring of a Batch Process	Cho, Kim	2003	MPCA
36	Multivariate Statistical Monitoring of Key Operation Units of Batch Processes Based on Time-Slice CCA	Jiang, Gao, Yi, Yan	2018	MCCA
37	Batch process control and monitoring: a Dual STATIS and Parallel Coordinates (DS-PC) approach	Ramos-Barberán, Hinojosa-Ramos, Ascencio-Moreno, Vera, Ruiz-Barzola, Galindo-Villardón	2018	STATIS, PARCOORDS (+Dual STATIS)