  Many multivariate statistical analysis methods and their corresponding
probabilistic counterparts have been adopted to develop process monitoring
models in recent decades. However, the insightful connections between them have
rarely been studied. In this study, a generalized probabilistic monitoring
model (GPMM) is developed with both random and sequential data. Since GPMM can
be reduced to various probabilistic linear models under specific restrictions,
it is adopted to analyze the connections between different monitoring methods.
Using expectation maximization (EM) algorithm, the parameters of GPMM are
estimated for both random and sequential cases. Based on the obtained model
parameters, statistics are designed for monitoring different aspects of the
process system. Besides, the distributions of these statistics are rigorously
derived and proved, so that the control limits can be calculated accordingly.
After that, contribution analysis methods are presented for identifying faulty
variables once the process anomalies are detected. Finally, the equivalence
between monitoring models based on classical multivariate methods and their
corresponding probabilistic graphic models is further investigated. The
conclusions of this study are verified using a numerical example and the
Tennessee Eastman (TE) process. Experimental results illustrate that the
proposed monitoring statistics are subject to their corresponding
distributions, and they are equivalent to statistics in classical deterministic
models under specific restrictions.
