An Introduction to Support Vector
Machine Classification
Bioinformatics Lecture 7/2/2003
by
Pierre Dönnes
Outline
•
What do we mean with classification, why is it
useful
•
Machine learning

basic concept
•
Support Vector Machines (SVM)
–
Linear SVM
–
basic terminology and some
formulas
–
Non

linear SVM
–
the Kernel trick
•
An example: Predicting protein subcellular
location with SVM
•
Performance measurments
Classification
•
Everyday, all the time we classify
things.
•
Eg crossing the street:
–
Is there a car coming?
–
At what speed?
–
How far is it to the other side?
–
Classification: Safe to walk or not!!!
•
Decision tree learning
IF (
Outlook = Sunny
) ^ (
Humidity = High
)
THEN
PlayTennis =NO
IF (
Outlook = Sunny)^ (Humidity = Normal)
THEN
PlayTennis = YES
Training examples:
Day Outlook Temp. Humidity Wind PlayTennis
D1 Sunny Hot High Weak No
D2 Overcast Hot High Strong Yes ……
Outlook
Sunny
Overcast
Rain
Humidity
Yes
Wind
High
Normal
Yes
No
Strong
Weak
Yes
No
Classification tasks in
Bioinformatics
•
Learning Task
–
Given
: Expression profiles of leukemia patients and
healthy persons.
–
Compute
: A model distinguishing if a person has
leukemia from expression data.
•
Classification Task
–
Given
: Expression profile of a new patient + a
learned model
–
Determine
: If a patient has leukemia or not.
Problems in classifying
biological data
•
Often high dimension of data.
•
Hard to put up simple rules.
•
Amount of data.
•
Need automated ways to deal with the
data.
•
Use computers
–
data processing,
statistical analysis, try to learn patterns
from the data (Machine Learning)
Examples are:

Support Vector Machines

Artificial Neural Networks

Boosting

Hidden Markov Models
Black box view of
Machine Learning
Magic black box
(learning machine)
Training data
Model
Training data:

Expression patterns of some cancer +
expression data from healty person
Model:

The model can distinguish between healty
and sick persons. Can be used for prediction.
Test

data
Prediction = cancer or not
Model
Tennis example 2
Humidity
Temperature
= play tennis
= do not play tennis
Linear Support Vector Machines
x1
x2
=+1
=

1
Data: <
x
i
,y
i
>, i=1,..,l
x
i
R
d
y
i
{

1,+1}
=

1
=+1
Data: <
x
i
,y
i
>, i=1,..,l
x
i
R
d
y
i
{

1,+1}
All hyperplanes in R
d
are parameterize by a vector (
w
) and a constant b.
Can be expressed as
w
•x
+b=0 (remember the equation for a hyperplane
from algebra!)
Our aim is to find such a hyperplane
f(x)=sign(
w
•x
+b),
that
correctly classify our data.
f(x)
Linear SVM 2
d
+
d

Definitions
Define the hyperplane H such that:
x
i
•
w
+b
+1 when
y
i
=+1
x
i
•
w
+b

1 when
y
i
=

1
d+ = the shortest distance to the closest poitive point
d

= the shortest distance to the closest negative point
The
margin
of a separating hyperplane is d
+
+ d

.
H
H1 and H2 are the planes:
H1:
x
i
•
w
+b
= +1
H2:
x
i
•
w
+b
=

1
The points on the planes
H1 and H2 are the
Support Vectors
H1
H2
Maximizing the margin
d+
d

We want a classifier with as big margin as possible.
Recall the distance from a point(x
0
,y
0
) to a line:
Ax+By+c = 0 isA x
0
+B y
0
+c/sqrt(A
2
+B
2
)
The distance between H and H1 is:

w
•x
+b/w=1/w
The distance between H1 and H2 is: 2/w
In order to maximize the margin, we need to minimize w. With the
condition that there are no datapoints between H1 and H2:
x
i
•
w
+b
+1 when
y
i
=+1
x
i
•
w
+b

1 when
y
i
=

1
Can be combined into yi(
x
i
•
w)
ㄠ
H1
H2
H
The Lagrangian trick
Reformulate the optimization problem:
A ”trick” often used in optimization is to do an Lagrangian
formulation of the problem.The constraints will be replace
by constraints on the Lagrangian multipliers and the training
data will only occur as dot products.
Gives us the task:
Max Ld =
i
–
½
i
j
x
i
•x
j
,
Subject to:
w
=
i
y
i
x
i
i
y
i
= 0
What we need to see: x
i
and x
j
(input vectors) appear only in the form
of dot product
–
we will soon see why that is important.
Problems with linear SVM
=

1
=+1
What if the decison function is not a linear?
Non

linear SVM 1
The Kernel trick
=

1
=+1
Imagine a function
that maps the data into another space:
=Rd
=

1
=+1
Remember the function we want to optimize:
Ld =
i
–
½
i
j
x
i
•x
j
,
x
i
and x
j
as a dot product. We will have
(x
i
)
•
(
x
j
) in the non

linear case.
If there is a ”kernel function” K such as K(xi,xj) =
(xi)
•
(xj), we
do not need to know
explicitly. One example:
Rd
Non

linear svm2
The function we end up optimizing is:
Max Ld =
i
–
½
i
j
K(xi
•x
j
)
,
Subject to:
w
=
i
y
i
x
i
i
y
i
= 0
Another kernel example: The polynomial kernel
K(xi,xj) = (
xi
•xj + 1)
p
, where p is a tunable parameter.
Evaluating K only require one addition and one exponentiation
more than the original dot product.
Solving the optimization
problem
•
In many cases any general purpose
optimization package that solves linearly
constrained equations will do.
–
Newtons’ method
–
Conjugate gradient descent
•
Other methods involves nonlinear
programming techniques.
Overtraining/overfitting
=

1
=+1
An example: A botanist really knowing trees.Everytime he sees a new tree,
he claims it is not a tree.
A well known problem with machine learning methods is overtraining.
This means that we have learned the training data very well, but
we can not classify unseen examples correctly.
Overtraining/overfitting 2
It can be shown that: The portion, n, of unseen data that will be
missclassified is bound by:
n
No of support vectors / number of training examples
A measure of the risk of overtraining with SVM (there are also other
measures).
Ockham
´
s razor principle: Simpler system are better than more complex ones.
In SVM case: fewer support vectors mean a simpler representation of the
hyperplane.
Example: Understanding a certain cancer if it can be described by one gene
is easier than if we have to describe it with 5000.
A practical example, protein
localization
•
Proteins are synthesized in the cytosol.
•
Transported into different subcellular
locations where they carry out their
functions.
•
Aim: To predict in what location a
certain protein will end up!!!
Subcellular Locations
Method
•
Hypothesis: The amino acid composition of proteins
from different compartments should differ.
•
Extract proteins with know subcellular location from
SWISSPROT.
•
Calculate the amino acid composition of the proteins.
•
Try to differentiate between: cytosol, extracellular,
mitochondria and nuclear by using SVM
Input encoding
Prediction of nuclear proteins:
Label the known nuclear proteins as +1 and all others
as
–
1.
The input vector xi represents the amino acid
composition.
Eg xi =(4.2,6.7,12,….,0.5)
A , C , D,….., Y)
Nuclear
All others
SVM
Model
Cross

validation
Cross validation:
Split the data into n sets, train on n

1 set, test on the set left
out of training.
Nuclear
All others
1
2
3
1
2
3
Test set
1
1
Training set
2
3
2
3
Performance measurments
Model
=+1
=

1
Predictions
+1

1
Test data
TP
FP
TN
FN
SP = TP /(TP+FP), the fraction of predicted +1 that actually are +1.
SE = TP /(TP+FN), the fraction of the +1 that actually are predicted as +1.
In this case: SP=5/(5+1) =0.83
SE = 5/(5+2) = 0.71
Results
•
We definetely get some predictive
power out of our models.
•
Seems to be a difference in composition
of proteins from different subcellular
locations.
•
Another questions: What about nuclear
proteins. Is there a difference between
DNA

binding proteins and others???
Conclusions
•
We have (hopefully) learned some basic
concepts and terminology of SVM.
•
We know about the risk of overtraining
and how to put a measure on the risk
of bad generalization.
•
SVMs can be useful for example in
predicting subcellular location of
proteins.
You can’t input anything into a
learning machine!!!
Image classification of tanks. Autofire when an enemy tank is spotted.
Input data: Photos of own and enemy tanks.
Worked really good with the training set used.
In reality it failed completely.
Reason: All enemy tank photos taken in the morning. All own tanks in dawn.
The classifier could recognize dusk from dawn!!!!
References
http://www.kernel

machines.org/
AN INTRODUCTION TO SUPPORT VECTOR MACHINES
(and other kernel

based learning methods)
N. Cristianini and J. Shawe

Taylor
Cambridge University Press
2000 ISBN: 0 521 78019 5
http://www.support

vector.net/
Papers by Vapnik
C.J.C. Burges: A tutorial on Support Vector Machines. Data Mining and
Knowledge Discovery 2:121

167, 1998.
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