Linear Discriminant Function and Linear Classifier
Let 
be a set of training samples from two classes 
and 
with 
samples from 
, and let 
be their corresponding class labels. Here 
means that 
belongs to 
whereas 
means that 
belongs to 
. A linear discriminant function is a linear combination of the components of a feature vector 
which can be written as:
, (1) where the vector 
and the scalar 
are called weight and bias respectively. The hyperplane 
is a decision surface which is used to separate samples with positive class labels from samples with negative ones.
A linear discriminant criterion is an optimization model which is used to seek the weight for a linear discriminant function. The chief goal of classification-oriented LDA is to set up an appropriated linear discriminant criterion and to calculate the optimal projection direction, i.e. the weight. Here “optimal” means that after samples are projected onto the weight, the resultant projections of samples from two distinct classes 
and 
are fully separated.
Once the weight 
has been derived from a certain linear discriminant criterion, the corresponding bias 
can be computed using:
, (2) or
, (3) where 
and 
are respectively the mean training sample and the mean of training samples from the class 
. They are defined as
, (4) and
.
(5)For simplicity, we calculate the bias using the Eq. (2) throughout this chapter.
Let 
denote the mean of the projected training samples from the class 
. Thus, the binary linear classifier based on the weight 
and the bias 
is defined as follow:
, (6) which assigns a class label 
to an unknown sample 
. Here, 
is the sign function. That is, once the weight in a linear discriminant function has been worked out the corresponding binary linear classifier is fixed.