Information 2019, 10, 150 12 of 68
PCA can be used as a pre-processing tool to reduce the dimension of a data set before running a
supervised learning algorithm on it (
x
(i)
s
as inputs). PCA is also a valuable tool as a noise reduction
algorithm and can be helpful in avoiding the over-fitting problem [
70
]. kernel principal component
analysis (KPCA) is another dimensionality reduction method that generalizes linear PCA into the
nonlinear case by using the kernel method [71].
3.1.2. Independent Component Analysis (ICA)
Independent component analysis (ICA) was introduced by H. Jeanny [
72
]. This technique was
then further developed by C. Jutten and J. Herault [
73
]. ICA is a statistical modeling method
where the observed data are expressed as a linear transformation [
74
]. Assume that 4
n
linear
mixtures (x
1
, x
2
, . . . , x
n
) are observed where independent components:
x
j
= a
j1
s
1
+ a
j2
s
2
+ . . . + a
jn
s
n
∀j (13)
The vector-matrix notation is written as:
X = As (14)
Denoting them by a
i
, the model can also be written [75] as follows:
X =
n
∑
i=1
a
i
s
i
(15)
3.2. Linear Discriminant Analysis (LDA)
LDA is a commonly used technique for data classification and dimensionality reduction [
76
]. LDA
is particularly helpful where the within-class frequencies are unequal and their performances have been
evaluated on randomly generated test data. Class-dependent and class-independent transformation
are two approaches to LDA in which the ratio of between class variance to within class variance and
the ratio of the overall variance to within class variance are used respectively [77].
Let
x
i
∈ R
d
which be
d
-dimensional samples and
y
i
∈ {
1, 2, ...,
c}
be associated target or
output [
76
], where
n
is the number of documents and
c
is the number of categories. The number of
samples in each class is calculated as follows:
S
w
=
c
∑
l=1
s
l
(16)
where
S
i
=
∑
x∈w
i
(x −µ
i
)(x −µ
i
)
T
, µ
i
=
1
N
i
∑
x∈w
i
x (17)
The generalization between the class scatter matrix is defined as follows:
S
B
=
c
∑
i=1
N
i
(µ
i
−µ)(µ
i
−µ)
T
(18)
where
µ =
1
N
∑
∀x
x (19)
Respect to c −1 projection vector of w
i
that can be projected into W matrix:
W =
[
w
1
|w
2
|. . . |w
c−1
]
(20)
y
i
= w
T
i
x (21)
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