Multivariate Image Analysis in Mineral Processing - Advanced Control and Supervision of Mineral Processing Plants - page 88

flotation examples in which froth grade is predicted and froth health is monitored,
and one application to ROM in which it is attempted to estimate the proportions of
the various mineral types on a conveyor belt. Some conclusions are drawn in Section
3.6.

3.2 Background on Latent Variable Methods

3.2.1 Principal Component Analysis

Principal component analysis is a classical multivariate data analysis approach and
a tutorial with some chemical examples can be found in [15]. Figure 3.2 (a) shows
a data table, X , that consists of I measurements taken on J different variables. PCA
takes advantage of the correlation structure among the J variables to summarize X
into a few principal components or latent variables. The number of principal com-
ponents, A , is generally smaller than the number variables J and is often viewed as
an estimate of the effective rank of X .

J t

X

(a)

I

p T

J t u

M

X

Y

(b)

I

w T q T

p T

Figure 3.2 Data vectors and matrices involved in projection methods. (a) PCA. (b) PLS

From a rigorous mathematical point of view, the PCA model building procedure
starts with finding the direction in X , or alternatively, finding a linear combination
of x -variables p 1 that explains the most variance in X :