
196 Control for Aluminum Production
multivariable process, whereas t
r
is the R principal components that
form thePCA model, and E is assumed to be random errors.
15
The prin-
cipal components (PCs) are dened by the R loading vectors (p
r
) that are
the eigenvectors of X (K × J), and these vectors are used to transform the
data, X (K × J), into its R principal components. The number of princi-
pal components for the PCA model is lower than the number of process
variables, i.e., R is less than J. The R loading vectors provide a direction
of maximum variability in the process so that one can observe the pro-
cess using the model built from a few pri ...