CLASS IMBALANCE LEARNING METHODS FOR SUPPORT VECTOR MACHINES - Imbalanced Learning: Foundations, Algorithms, and Applications - page 93

B. f(x i ) is based on the distance from the preestimated separating hyperplane: In
this method, f(x i ) is defined based on d sp i , which is the distance to x i from the
pre-estimated separating hyperplane, as introduced in [42]. Here d sp i is estimated
by the distance to x i from the center of the common spherical region, which can
be defined as a hyper-sphere covering the overlapping region of the two classes,
where the separation hyperplane is more likely to pass through. Both linear and
exponential decaying functions are used to define the function f(x i ) , which are
represented by f sph

lin (x i ) and f sph

exp (x i ) as follows:

f sph

lin (x i ) = 1 − (d sph

/( max (d sph

) + δ)) (5.21)

i i

exp (x i ) = 2 /( 1 + exp (d sph

f sph ∗ β)) (5.22)

i

where d sph
i

2 ,and x is the center of the spherical region, which is
estimated by the center of the entire dataset, and δ is a small positive value and
β ∈ [0 , 1].

= x i − x

C. f(x i ) is based on the distance from the actual separating hyperplane: In this
method, f(x i ) is defined based on the distance from the actual separating hyper-
plane to x i , which is found by training a conventional SVM model on the
imbalanced dataset. The data points closer to the actual separating hyperplane are
treated as more informative and are assigned higher membership values, while
the data points far away from the separating hyperplane are treated as less infor-
mative and are assigned lower membership values. The following procedure is
carried out to assign f(x i ) values in this method:

1. Train a normal SVM model with the original imbalanced dataset
2. Find the functional margin d hy i of each example x i (given in Eq. 5.23) (this
is equivalent to the absolute value of the SVM decision value) with respect
to the separating hyperplane found. The functional margin is proportional
to the geometric margin of a training example with respect to the separating
hyperplane.

d hyp = y i (w · (x i ) + b) (5.23)

i

3. Consider both linear and exponential decaying functions to define f(x i ) as
follows:

f hyp

lin (x i ) = 1 − (d hyp

/( max (d hyp
i

) + δ)) (5.24)

i

exp (x i ) = 2 /( 1 + exp (d hyp

f hyp ∗ β)) (5.25)

i

where δ is a small positive value and β ∈ [0 , 1].