148
5: Mahesh Pal, Pakorn Watanachaturaporn
Table 5.1. Examples of some kernel functions
Kernel Function
Definition
Parameters
Linear
X· Xj
Polynomial with
(X'Xi+1)d
d is a positive integer
degree d
Radial Basis
exp ( _lIx~~1I2)
a is a user defined value
Function
Sigmoid
tanh (K (x. Xi) + e)
K and e are user defined values
In fact, every condition of the linearly separable case can be extended to the
nonlinear case with a suitable kernel function. The kernel function for a linearly
separable case will simply be a dot product of two data vectors, K (Xi, Xj) = Xi ·Xj.
Other examples of well-known kernel functions are provided in Table 5.1.
The selection of a suitable kernel function is essential for a particular problem. For example, the performance of the simple dot product linear kernel
function may deteriorate when decision boundaries between the classes are
non-linear. The performance of sigmoid, polynomial and radial basis kernel
functions may depend on the selection of appropriate values of the user-defined
parameters, which may vary from one dataset to another. An experimental investigation on the choice of kernel functions for the classification of multi and
hyperspectral datasets has been provided in Chap. 10.
5.4
SVMs for Multiclass Classification
Originally, SVMs were developed to perform binary classification, where the
class labels can either be + 1 or -1. However, applications of binary classification are very limited. The classification of a dataset into more than two classes,
called multiclass classification, is of more practical relevance and has numerous
applications. For example, in a digit recognition application, the system has to
classify ten classes of interest from 0 to 9. In remote sensing, land cover classification is generally a multiclass problem. A number of methods to generate
multiclass SVMs from binary SVMs have been proposed by researchers. It is
still a continuing research topic. In this section, we provide a brief description
of some multiclass methods that may be employed to perform classification
using SVMs.
5: Mahesh Pal, Pakorn Watanachaturaporn
Table 5.1. Examples of some kernel functions
Kernel Function
Definition
Parameters
Linear
X· Xj
Polynomial with
(X'Xi+1)d
d is a positive integer
degree d
Radial Basis
exp ( _lIx~~1I2)
a is a user defined value
Function
Sigmoid
tanh (K (x. Xi) + e)
K and e are user defined values
In fact, every condition of the linearly separable case can be extended to the
nonlinear case with a suitable kernel function. The kernel function for a linearly
separable case will simply be a dot product of two data vectors, K (Xi, Xj) = Xi ·Xj.
Other examples of well-known kernel functions are provided in Table 5.1.
The selection of a suitable kernel function is essential for a particular problem. For example, the performance of the simple dot product linear kernel
function may deteriorate when decision boundaries between the classes are
non-linear. The performance of sigmoid, polynomial and radial basis kernel
functions may depend on the selection of appropriate values of the user-defined
parameters, which may vary from one dataset to another. An experimental investigation on the choice of kernel functions for the classification of multi and
hyperspectral datasets has been provided in Chap. 10.
5.4
SVMs for Multiclass Classification
Originally, SVMs were developed to perform binary classification, where the
class labels can either be + 1 or -1. However, applications of binary classification are very limited. The classification of a dataset into more than two classes,
called multiclass classification, is of more practical relevance and has numerous
applications. For example, in a digit recognition application, the system has to
classify ten classes of interest from 0 to 9. In remote sensing, land cover classification is generally a multiclass problem. A number of methods to generate
multiclass SVMs from binary SVMs have been proposed by researchers. It is
still a continuing research topic. In this section, we provide a brief description
of some multiclass methods that may be employed to perform classification
using SVMs.