Support vector machines (SVMs) are a supervised machine learning algorithm used for classification and regression analysis. SVMs find the optimal boundary, known as a hyperplane, that separates classes of data. This hyperplane maximizes the margin between the two classes. Extensions to the basic SVM model include soft margin classification to allow some misclassified points, methods for multi-class classification like one-vs-one and one-vs-all, and the use of kernel functions to handle non-linear decision boundaries. Real-world applications of SVMs include face detection, text categorization, image classification, and bioinformatics.

1. SUPPORT VECTOR MACHINE
1. INTRODUCTION
2. LINEARLY SEPERABLE CASE
3. EXTENSION TO THE SVM MODEL
4. APPLICATION OF SVM IN REAL WORLD

2. SUPPORT VECTOR MACHINE
• It is a supervised machine learning algorithm
that helps in both classification and regression
problem statements.It finds an optimal boundary
known as hyperplane between different classes.
• It is a vector-space-based machine-learning
method where the goal is to find a decision
boundary between two classes that is maximally
far from any point in the training data (possibly
discounting some points as outliers or noise).

3. LINEARLY SEPARABLE CASE
• If the training data is linearly separable, we can
select two parallel hyperplanes that separate
the two classes of data, so that the distance
between them is as large as possible.
• The region bounded by these two hyperplanes
is called the "margin", and the maximum-
margin hyperplane is the hyperplane that lies
halfway between them.

5. • With a normalized or standardized dataset, these
hyperplanes can be described by the equations:-
Case 1:- w ^T x -b=1 (anything on or above this
boundary is of one class, with label 1)
And
 Case 2:- w^T x -b=-1 (anything on or below this
boundary is of the other class, with label −1).
• The distance between these two hyperplanes
is 2|w|,so to maximize the distance between
the planes we want to minimize|w|.

6. • The distance is computed using the distance from a
point to a plane equation. We also have to prevent
data points from falling into the margin, we add the
following constraint: for each i either
w^Tx_i-b>= 1}, if y_i =1,
or
w^Tx_i-b<=-1}, if y_i=-1.
• These constraints state that each data point must lie
on the correct side of the margin. This can be
rewritten as:-
Y_i(w^Tx_i-b)>=1, for all 1<=i<=n.

7. • We can put this together to get the optimization
problem:
"Minimize |w| subject to y_i (w ^Tx _i-b) >=1 for
i=1,……..,n.”
• The w and b that solve this problem determine
our classifier,x→sgn (w ^Tx -b) where sgn(.) is
the sign function.
• An important consequence of this geometric
description is that the max-margin hyperplane is
completely determined by those ͢ x_i that lie
nearest to it. These x _i are called support
vectors.

9. EXTENSION TO THE SVM MODEL
 SOFT MARGIN CLASSIFICATION
MULTICLASS SUPPORT VECTOR MACHINE
NONLINEAR SUPPORT VECTOR MACHINE

10. SOFT MARGIN CLASSIFICATION
An additional set of coefficients are
introduced that give the margin wiggle room
in each dimension. These coefficients are
called slack variables.
 This increases the complexity of the model as
there are more parameters for the model to fit
to the data to provide this complexity.
A tuning parameter is introduced called simply
C that defines the magnitude of the wiggle
allowed across all dimensions.

11. Real data is messy and connot be separated
perfectly with a hyperplane.
 The constrains of maximizing the margin of the
line that separates the classes must be relaxed.
This is called as the soft margin classifier.
 This change allows some points in the training
data to violate the separating line.
The C parameters defines the amount of violation
of the margin allowed. A C=0 is no violation and
we are back to the inflexible Maximal-Margin
Classifier .
The larger the value of C the more violations of
the hyperplane are permitted.

12. During the learning of the hyperplane from data,
all training instances that lie within the distance
of the margin will affect the placement of the
hyperplane and are referred to as support
vectors.
C affects the number of instances that are
allowed to fall within the margin. C influences
the number of support vectors used by the
model.
• The smaller the value of C, the more sensitive the algorithm
is to the training data (higher variance and lower bias).
• The larger the value of C, the less sensitive the algorithm is
to the training data (lower variance and higher bias

13. MULTICLASS SUPPORT VECTOR
MACHINE
• SVM doesn’t support multiclass classification
natively. It supports binary classification and
separating data points into two classes.
• For multiclass classification, the same
principle is utilized after breaking down the
multi classification problem into multiple
binary classification problems.

14. • The popular methods which are used to
perform multi-classification on the problem
statements using SVM are as follows:
• One vs One (OVO) approach
• One vs All (OVA) approach
• Directed Acyclic Graph (DAG) approach

15. One vs One (OVO)
• This technique breaks down our multiclass
classification problem into subproblems which are
binary classification problems. So, after this strategy,
we get binary classifiers per each pair of classes. For
final prediction for any input use the concept
of majority voting along with the distance from the
margin as its confidence criterion.
• The major problem with this approach is that we
have to train too many SVMs.
• In the One-to-One approach, we try to find the
hyperplane that separates between every two
classes, neglecting the points of the third class

16. • For example, here Red-Blue line tries to
maximize the separation only between blue
and red points while It has nothing to do with
the green points.

17. One vs All (OVA)
• In this technique to predict the output for new
input, just predict with each of the build SVMs
and then find which one puts the prediction the
farthest into the positive region (behaves as a
confidence criterion for a particular SVM).
• In the One vs All approach, we try to find a
hyperplane to separate the classes. This means
the separation takes all points into account and
then divides them into two groups in which there
is a group for the one class points and the other
group for all other points.

18. • For example, here the Greenline tries to
maximize the gap between green points and
all other points at once.

19. • A single SVM does binary classification and
can differentiate between two classes. So
according to the two above approaches, to
classify the data points from L classes data set.
• In the One vs All approach, the classifier can
use L SVMs.
• In the One vs One approach, the classifier can
use L(L-1)/2 SVMs.

20. Directed Acyclic Graph (DAG)
• This approach is more hierarchical in nature
and it tries to addresses the problems of the
One vs One and One vs All approach.
• This is a graphical approach in which we
group the classes based on some logical
grouping.

21. NONLINEAR SUPPORT VECTOR
MACHINE
• When we cannot separate data with a straight
line we use Non – Linear SVM.
• for which, we have Kernel functions. They
transform non-linear spaces into linear spaces.
• It transforms data into another dimension so that
the data can be classified.
• It transforms two variables x and y into three
variables along with z. Therefore, the data have
plotted from 2-D space to 3-D space. So we can
easily classify the data by drawing the best
hyperplane between them.

23. KERNELS FUNCTIONS
• It is a Mathematical functions for transforming
data.
• It uses some linear algebra.
Example:
K(x, y) = <f(x), f(y)>
• Different SVM algorithms use different types
of kernel functions.

24. • Various kernels available are:-
1. Linear kernel
2. Non linear kernel
3. Radial basis function (RBF)
4. Sigmoid
5. Polynomial
6. Exponential

25. APPLICATION OF SVM IN REAL WORLD
Some common applications of SVM are-
• Face detection
• Text and hypertext categorization
• Classification of images
• Bioinformatics
• Protein fold and remote homology detection
• Handwriting recognition
• Generalized predictive control(GPC)