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I found this statement in one paper about statistical process control and monitoring:

The P loading matrices contain all the structural information about how the variable measurements should deviate from their mean values at each observation

Why is this true?

Edit: Some reference for better understanding of the context of the question:

  1. Monitoring Batch Processes Using Multiway Principal Component Analysis - Nomikos & McGregor (the paper from which the cite came. pp 8)

  2. Statistical process control of multivariate processes - McGregor & Kourti

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    $\begingroup$ Any reference, please? A quick search on the internet highlights some hits related to batch process monitoring, but it would be better to clarify your question $\endgroup$ – chl Nov 2 '20 at 10:45
  • $\begingroup$ Added in the edits ;) $\endgroup$ – Ralphns Nov 2 '20 at 11:06
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    $\begingroup$ Clearly, the paper's authors are conducting PCA on the centered variables. Thus, the only part of the question that needs an answer is whether the PCA contains "all the structural information." That will be true if and only if the multivariate distribution of the centered data is assumed to be in a family that is determined by the covariance matrix (which is the case for a Multivariate Normal distribution, for instance). Thus, you should look into your paper for information about what distributional assumptions the authors are making. $\endgroup$ – whuber♦ Nov 2 '20 at 14:05

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