State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 327

mapping function M ( x ) could more probably be linearly separable than in the low-
dimensional input space of the vector (see Figure 10.1).

Support Vectors

H opt

x M x

Nonlinear
Mapping

Low-dimensional
Data Space

High-dimensional
Feature Space

Figure 10.1. Nonlinear mapping from input space into feature space

In the high-dimensional feature space, if the data are nonlinearly separable in
the low-dimensional data space, then linear separability of features could be
achieved by constructing a hyperplane as a linear discriminant. In this way, a data
classifier can be built, as illustrated by the following example.
Let a set of labelled training patterns (, )
i

x y be available, with i = 1, 2, … , N ,
where x is an n -dimensional pattern vector and y is the desired output
corresponding to the input pattern vector x , the values of which belong to the
linearly separable classes A with y = - 1 and B with y =+ 1. The postulated
separability condition implies that there exists an n -dimensional weight vector w
and a scalar b such that

i

y = - 1,

T

wx b for 0

(10.1)

i

y =+ 1,

T

wx b t for 0

(10.2)

i

whereby the parametric equation

wx

T

b 0

(10.3)

i

defines the n -dimensional separating hyperplane. The data point nearest to the
hyperplane is called the margin of separation . The objective is to determine a
specific hyperplane that maximizes this margin between the two classes, called the
optimal hyperplane

H , defined by the parametric equation for H in the

opt opt

feature space

T

wx b 0 (10.4)

0 i 0