CLASS IMBALANCE LEARNING METHODS FOR SUPPORT VECTOR MACHINES - Imbalanced Learning: Foundations, Algorithms, and Applications - page 91

s . t . i (w · (x i ) + b) ≥ 1 − ξ i (5.15)

ξ i ≥ 0 ,i = 1 ,...,l

In this reformulation of the objective function, the membership m i of a
data point x i is incorporated into the penalty term, such that a smaller m i
could reduce the effect of the associated slack variable ξ i in the objective
function (if the corresponding data point x i is treated as less important). In
another view, if we consider C as the cost assigned for a misclassification,
now each data point is assigned with a different misclassification cost, m i C ,
which is based on the importance of the data point in its own class, such that
more important data points are assigned higher costs, while less important
ones are assigned lower costs. Therefore, the FSVM algorithm can find a
more robust separating hyperplane through maximizing the margin by allowing
some misclassification for less important examples, such as the outliers and
noise.
In order to solve the FSVM optimization problem, Equation 5.15 can be
transformed into the following dual Lagrangian form:

⎨
⎩

⎬
⎭

l l l

1
2

max
α i

α i − α i α j y i y j K(x i ,x j ) (5.16)

i = 1 i = 1 j = 1

l

s.t. y i α i = 0 , 0 ≤ α i ≤ m i C, i = 1 ,...,l

i = 1

The only difference between the original SVM dual optimization problem
given in Equation 5.7 and the FSVM dual optimization problem given in Equation
5.16 is the upper bound of the values that α i could take. By solving this dual
problem in Equation 5.16 for optimal α i , w and b can be recovered in the same
way as in the normal SVM learning algorithm. The same SVM decision function
in Equation 5.9 applies for the FSVMs method as well.

5.5.6.2 FSVM-CIL Method However, the standard FSVM method is still sen-
sitive to the class imbalance problem, as the assigned misclassification costs
do not consider the imbalance of the dataset. Batuwita and Palade [40] have
improved the standard FSVM method by combining it with the DEC method,
which is called the FSVM-CIL . In the FSVM-CIL method, the membership val-
ues for data points are assigned in such a way to satisfy the following two
goals:

1. To suppress the effect of between-class imbalance.
2. To reflect the within-class importance of different training examples in
order to suppress the effect of outliers and noise.