Multi-way principal components analysis

Henrion R, Henrion G, Onuoha GC, Multi-way principal components analysis of a complex data array... 

Multi-way Principal Component Analysis (MPCA) is strongly related to the standard data analysis method Principal Component Analysis (PCA). This bilinear modelling technique, based in the eigenvector decomposition of the covariance matrix, does not consider the way in which data has been acquired. This means that external information, such the ordering in time of the data acquisition, is not taken into account for the modelling process. Although this is unnecessary in a wide amount of cases, there are some for which it becomes an evident loss of information. Multi-way are part of these. 

Very often in DCS-operated batch polymer reactors the primary process variables such as pressure, temperature, level, and flow (Section 12.2.1-12.2.4) are recorded during the batch as well as the quality variables at the end of the batch. However, it may be very difficult to obtain a kinetic model of the polymerization process due to the complexity of the reaction mechanism, which is frequently encountered in the batch manufacture of specialty polymers. In this case it is possible to use advanced statistical techniques such as multi-way principal component analysis (PCA) and multi-way partial least squares (PLS), along with an historical database of past successful batches to construct an empirical model of the batch [8, 58, 59]. This empirical model is used to monitor the evolution of future batch runs. Subsequent unusual events in the future can be detected during the course of the batch by referencing the measured process behavior against this incorrective action during the batch in order to bring it on aim. 

Wold, S., Geladi, P., and Ohman, J., Multi-way principal components and PLS analysis, J. Chemometrics 1, 41-56 (1987b). 

S Wold, P Geladi, K Esbensen, and J Ohman. Multi-way principal component and PLS analysis. J. Chemometrics, 1 41-56, 1987. 

One way to think about the factors obtained from the principal component analysis which are independent is to interpret them as defining a multi-dimensional space. For further analyses and in order to locate individuals within the 14-dimensional space, factor scores were calculated. First, the loading of each variable on a factor was multiplied by the individual s original value for that variable. In the next step of the procedure, the same calculation was repeated for all variables in the factor for that individual. These scores were then summed. The process was repeated for all factors for that same individual and then repeated for all other individuals. Finally, all scores were standardised to a mean of 0 with a standard deviation of 1. These procedures facilitate further statistical treatment of the motivational patterns and other variables of interest such as travel experience. 

Ledyard Tucker was one of the pioneers in multi-way analysis. He proposed [Tucker 1964, Tucker 1966] a series of models nowadays called A-mode principal component analysis or Tucker models. An extensive treatment of Tucker models is given by Kroonenberg and de Leeuw [1980] and Kroonenberg [1983], In the following, three different Tucker models will be treated. 

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