Chapter 9
:
Support Vector Machine
Dr. Essam Al D
a
o
ud
1
Chapter
9
Support Vector Machine
The SVM classification function F(x) takes the form
w is the weight vector and b is the bias, which will be computed by SVM in the
training process.
The l
inear classifiers (hyperplane) in two

dimensional spaces
can be
rep
resented in the following figure:
Hard

margin SVM Classification
T
he hyperplane maximizing the margin in a two

dimensional space
Thus, SVM problem can be written as following:
Chapter 9
:
Support Vector Machine
Dr. Essam Al D
a
o
ud
2
the primal problem deals with a convex cost function and linear
con
straints. Given
such a constrained optimization problem, it is possible to construct
another problem
called dual problem.
Having determined the optim
um Lagrange multipliers
, we may
co
mpute the optimum
weight vectors:
Soft

margin SVM Classifica
tion
S
oft margin SVM allows mislabeled data points while still
maximizing the m
argin.
The method introduces slack variables
,
which measure the degree of
misclassification. The following is the optimization problem for soft
margin SVM.
Kernel Trick f
or Nonlinear Classification
If the training data is not linearly separable, there is no straight hyperplane that can
separate the classes. In order to learn a nonlinear function in that case, linear SVMs
must be extended to nonlinear SVMs for the classifi
cation of nonlinearly separable
.
The dual problem is now defined using the kernel function as follows
Chapter 9
:
Support Vector Machine
Dr. Essam Al D
a
o
ud
3
Kernel functions must be continuous, symmetric, and most preferably should have a
positive (semi

) definite
Gram matrix ,meaning their kernel mat
rices have no non

negative Eigen values. The use of a positive definite kernel insures that the
optimization problem will be convex and solution will be unique.
The followings are popularly used kernel functions.
1. Linear Kernel
2. Polynomial Kernel
3. Gaussian Kern
el
4. Exponential Kernel
Chapter 9
:
Support Vector Machine
Dr. Essam Al D
a
o
ud
4
SVM steps
1

Construct the Gram matrix by using a kernel function
2

Solve the Maximization problem ( famous algorithm is
Sequentia
l Minimal
Optimization
SMO)
3

Find the support vectors
4

find b as following
Where
(x
s
,y
s
)
is any one of the support vectors
5

Classify the test vector z according to the
Example
Classify (

1,0) and (1,0) by SVM
and polynomial kernel
solution
Chapter 9
:
Support Vector Machine
Dr. Essam Al D
a
o
ud
5
The maximization problem of polynomial kernel is
Let the solution of above problem
be:
Then the support vectors are 1 and 2
, and b
Chapter 9
:
Support Vector Machine
Dr. Essam Al D
a
o
ud
6
Matlab
load
fisheriris
data = [meas(:
,1), meas(:,2)];
groups = ismember(species,
'setosa'
);
[train, test] = crossvalind(
'holdOut'
,groups);
cp = classperf(groups);
svmStruct =
svmtrain(data(train,:),groups(train),
'KERNEL_FUNCTION'
,
'linear'
,
'showplot'
,true);
classes = svmclassify(svmStr
uct,data(test,:),
'showplot'
,true);
classperf(cp,classes,test);
cp.CorrectRate
figure
svmStruct = svmtrain(data(train,:),groups(train),
'showplot'
,true,
'boxconstraint'
,1e6);
classes = svmclassify(svmStruct,data(test,:),
'showplot'
,true);
classperf(cp,
classes,test);
cp.CorrectRate
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