Aerospace and Electronic Systems Magazine April 2017 - 29

Nassar, Hussein, and Medhat

Figure 1.

The PLS-DA methodology.

PLS-DA attempts to locate the directions that maximize discrimination power and separations between classes membership [17].
The PLS-DA technique makes it possible to accomplish a rotation of the projection to give latent variables that focus on class
separation/discrimination. The method offers a convenient way of
explicitly taking into account the class membership of observations even at the problem formulation stage. Thus, the objective of
PLS-DA is to find a model that separates classes of observations
on the basis of their X-variables. This model is developed from a
training set of observations of known class membership.
In PLS-DA, the X-matrix consists of multivariate characterization data of the observations. In order to encode a class identity,
one uses as Y-data a matrix of dummy/artificial variables, which
describes the class membership of each observation in the training set. A dummy variable is an artificial variable that assumes
a discrete numerical value in the class description. The dummy
Y-matrix has G columns (for G classes) with ones and zeros, such
that the entry in the gth column is one and the entries in other columns are zero for observations of class g. The dummy variable is
added for each category of observations, then the PLS-DA is used
to relate X and Y [17], [18].
For example, there are three groups of observations to cluster,
denoted F, S, and C. Mathematically, this can be demonstrated by
assigning the first group (e.g. nominal condition data) with a code
value of "1, 0, 0" for each row (observation) of the same group.
Depending on the number of samples or observations from each
group, there will be an equal number of code values with the same
value "1, 0, 0". So, dummy variables are utilized to account for the
class membership of each observation [19], [20].

NONLINEAR SVM METHODOLOGY
SVMs are a group of learning machines for efficiently solving pattern
recognition problems. SVMs try to find the hyperplane, which sepaAPRIL 2017

rates optimally the training patterns according to their classes (i.e.
hyperplane with maximum boundary margin). This is performed by
using what is commonly known in machine learning as the "kernel
trick." Kernel function is chosen to map the data from its original
space to feature space. It can be chosen arbitrarily so as to best suit the
data and at the same time reduce the computational burden involved
with generating the mapped values by direct evaluation. "Support
vectors" correspond to those points that lie along the margin or closest to it. The maximum margin between classes is found by solving
a quadratic optimization problem. SVMs have a good generalization
performance over traditional approaches, since their training is based
on the principle of structural risk minimization (SRM) (i.e., minimizing the upper bound on the expected risk), while the training for
traditional approaches is based on empirical risk minimization (i.e.,
minimizing the number of the training errors). SVMs have a high
computational efficiency in terms of speed and accuracy. They are
also more preferable when dealing with high dimensional data, as
they are more robust than traditional approaches, which may overfit
the data. However, they still have negative aspects in terms of giving
information about the system output and no physical explanation and
interpretation of the process itself. The description of SVMs classification can be explained as follows [21], [22].
Consider the training data {xi, yi}, where i =1,...., N, yi ∈
{+1,−1} corresponding to the class of xi (yi = 1 for class A, yi = −1
for class B). The principle of operation of SVMs classifier will be
modified according to the type of the data (linearly separable data,
nonseparable data (noisy data), and nonlinear data).
The SVMs classifier tries to find the separating hyperplane
between two classes of data with the largest margin (optimal hyperplane) in terms of the normal vector to the hyperplane w, and
the bias b that represents the distance from the origin. So, for mathematical convenience minimizing ||w|| is equivalent to minimizing ½||w||2 and the use of this term makes it possible to perform
quadratic programming (QP) optimization later on. We therefore
need to find
1
such that yi b  wT · xi  1
2



Minimize



(1)

In order to cater for the constraints in this minimization, we
need to allocate them Lagrange multipliers αi, where αi ≥ 0 for ∀i.
Using the dual Lagrangian formulation of the problem enables us
to solve the problem conveniently.

LD


i

i



1
 i j · yi · y j · xi · x j
2 i, j

(2)

subjected to αi ≥ 0 and ∑iαi · yi = 0.
The solution is found by maximizing LD, once α is obtained
from (2) (using a QP solver), the dimensions of the classifier w, b
can be determined. It is worth noting that the dual form requires
only the dot product of each input vector xi to be calculated; this is
important for the Kernel trick described afterwards.
In the case of nonlinear input data, kernel functions are used.
The choice of an appropriate kernel is an important issue because
the ideal kernel would map the data from its original input space
(x) to a higher dimensional feature space φ(x) where linear sepa-

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