Chemometric methods application in pharmaceutical products and processes analysis and control - Computer-Aided Applications in Pharmaceutical Technology - page 64

objective can be achieved by means of a model, where the observed result,
that is the response ( y ), is described as a function of the x -variables ( x 1 ,
x 2 , . . ., x N ). The noise is left in residual e y (Rajalahti and Kvalheim,
2011):

y = f ( x 1 , x 2 , . . ., x N ) + e y [4.5]

Two main groups of multivariate regression methods are those based
on MLR and the so-called FA methods. MLR methods are more easily
understandable and applicable, since the goal is to directly correlate
independent and dependent variables. However, FA methods fi rst require
derivation of the original data into a space with less dimensions (another
coordinate system for data representation is used), which is then followed
by the correlation investigation. The main advantage of FA methods is
that factors (usually known as PCs) capture most of data variation and
are capable of appointing more accurate x-y correlation in comparison
to MLR methods. Typical MLR methods are the classic least squares
method and the inverse least square method. The most prominent FA
methods are PCA (although it is, in effect, a classifi cation technique),
PCR and PLS analysis.

Multiple Linear Regression (MLR)

MLR is one of the oldest regression methods and is used to establish
linear relationships between multiple independent variables and the
dependent variable (sample property) that is infl uenced by them. The
developed model can be represented in the following way:

[4.6]

￿

￿ ￿

where y j is the sample property, b i is the computed coeffi cient for
independent variable x i , and e i,j is the error. Each independent variable is
studied one after another and correlated with the sample property y j .
Regression coeffi cients b i describe the effects of each calculated term.
In the case of N , non-interacting x -variables linearly correlated to y
model can be written as:

y = b 0 + b 1 x 1 + b 2 x 2 + . . . + b N x N + e y [4.7]

Eq. 4.7 can also be written in the matrix form:

y = Xb + e y [4.8]