1. Statistical Classification Methods
Introduction
k-nearest neighbor
Neural networks
Decision trees
Support Vector Machine

2. What is a Decision Tree? An inductive learning task
Use particular facts to make more generalized conclusions
A predictive model based on a branching series of Boolean tests
These smaller Boolean tests are less complex than a one-stage classifier
Let’s look at a sample decision tree…

3. Predicting Commute Time
Leave At
If we leave at 10 AM and there are no cars stalled on the road, what will our commute time be?
10 AM
9 AM
8 AM
Stall?
Accident?
Long No
Yes No
Yes
Short
Long
Medium
Long

4. Inductive Learning
In this decision tree, we made a series of Boolean decisions and followed the corresponding branch
Did we leave at 10 AM?
Did a car stall on the road?
Is there an accident on the road?
By answering each of these yes/no questions, we then came to a conclusion on how long our commute might take

5. Decision Trees as Rules
We did not have represent this tree graphically
We could have represented as a set of rules. However, this may be much harder to read…

6. Decision Tree as a Rule Set
if hour == 8am
commute time = long
else if hour == 9am
if accident == yes
else
commute time = medium
else if hour == 10am
if stall == yes
commute time = short
Notice that all attributes to not have to be used in each path of the decision.
As we will see, all attributes may not even appear in the tree.

7. How to Create a Decision Tree
We first make a list of attributes that we can measure
These attributes (for now) must be discrete
We then choose a target attribute that we want to predict
Then create an experience table that lists what we have seen in the past

8. Sample Experience Table
Example
Attributes
Target
Hour
Weather
Accident
Stall
Commute D1
8 AM
Sunny No
Long D2
Cloudy
Yes D3
10 AM
Short D4
9 AM
Rainy D5 D6 D7 D8
Medium D9
D10
D11
D12
D13

9. Choosing Attributes
The previous experience decision table showed 4 attributes: hour, weather, accident and stall
But the decision tree only showed 3 attributes: hour, accident and stall
Why is that?

10. Choosing Attributes
Methods for selecting attributes (which will be described later) show that weather is not a discriminating attribute
We use the principle of Occam’s Razor: Given a number of competing hypotheses, the simplest one is preferable

11. Choosing Attributes
The basic structure of creating a decision tree is the same for most decision tree algorithms
The difference lies in how we select the attributes for the tree
We will focus on the ID3 algorithm developed by Ross Quinlan in 1975

12. Decision Tree Algorithms
The basic idea behind any decision tree algorithm is as follows:
Choose the best attribute(s) to split the remaining instances and make that attribute a decision node
Repeat this process for recursively for each child
Stop when:
All the instances have the same target attribute value
There are no more attributes
There are no more instances

13. Identifying the Best Attributes
Refer back to our original decision tree
Leave At
9 AM
10 AM
8 AM
Accident?
Stall?
Long
Yes No
Yes No
Short
Long
Medium
Long
How did we know to split on leave at and then on stall and accident and not weather?

14. ID3 Heuristic
To determine the best attribute, we look at the ID3 heuristic
ID3 splits attributes based on their entropy.
Entropy is the measure of disinformation…

15. Entropy
Entropy is minimized when all values of the target attribute are the same.
If we know that commute time will always be short, then entropy = 0
Entropy is maximized when there is an equal chance of all values for the target attribute (i.e. the result is random)
If commute time = short in 3 instances, medium in 3 instances and long in 3 instances, entropy is maximized

16. Entropy Calculation of entropy
Entropy(S) = ∑(i=1 to l)-|Si|/|S| * log2(|Si|/|S|)
S = set of examples
Si = subset of S with value vi under the target attribute
l = size of the range of the target attribute

17. ID3 ID3 splits on attributes with the lowest entropy
We calculate the entropy for all values of an attribute as the weighted sum of subset entropies as follows:
∑(i = 1 to k) |Si|/|S| Entropy(Si), where k is the range of the attribute we are testing
We can also measure information gain (which is inversely proportional to entropy) as follows:
Entropy(S) - ∑(i = 1 to k) |Si|/|S| Entropy(Si)

18. ID3
Given our commute time sample set, we can calculate the entropy of each attribute at the root node
Attribute
Expected Entropy
Information Gain
Hour
0.6511
Weather
Accident
Stall

19. Decision Trees: Random Forest Classifier
Random forests (RF) are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest.
growing an ensemble of trees and letting them vote for the most popular class.
Decision trees

20. Random Forest- Example
Training Data
M features
N examples
As the classifier of choice we adopt Random Forest Classifier, due to its robustness to heterogenous and noisy feature.
Random Forest is an ensembe classifier. Briefly, given N data and M features, bootsrap samples are created from the traning data
Decision trees

21. Random Forest Classifier
Create bootstrap samples
from the training data
M features
N examples
....…
Decision trees

22. Random Forest Classifier
Construct a decision tree
M features
N examples
From eachd descision tree .. In spliting the nodes, the Gini Gain is employed
....…
Decision trees

23. Random Forest Classifier
At each node in choosing the split feature
choose only among m<M features
M features
N examples
Different from the regular decision trees in random forest, when the splitting feature is chosen from only a subset of allf eatures.
The robostness of the classifier arises from
bootsraping of the training data and the random selection of features, the the random choose of features.
....…
Decision trees

24. Random Forest Classifier
Create decision tree
from each bootstrap sample
N examples
....…
M features
Decision trees

25. Random Forest Classifier
M features
Take he majority vote
N examples
Finally the given an input data the decision is made as follows given an input data,
....…
....…
Decision trees

26. Random Forest: Algorithm Steps
Training Samples
N samples
M features
Extract m from M
Randomly choose a training set for this tree by estimated the prediction error of the tree
Randomly choose m
Calculate the best split based on m in the training set
Each tree is fully grown and not pruned
A test sample
RF Model
(Many Decision Trees)
Class label
To create training model (RF), many decision trees are needed, for each decision tree, we do following steps:
Decision trees

27. On Class Practice 1 Data Method Software Step Iris.arff RF
Parameter (Select by yourself)
Software
weka
Step
Explorer->Classify->Classifier (Trees – RandomForest )

28. Alternating Decision Tree
Two Components:
Decision nodes - specify a predicate condition.
Prediction nodes - contain a single number.
Note: Only for 2 class

29. ADT: Example Is that a spam email? Prediction > 0 Spam
Prediction < Normal
Decision node
Prediction node
AD Tree

30. ADT: Example Total score = 0.657 > 0 , so the instance is a spam.
An instance to be classified
Feature
Value
char_freq_bang
0.08
word_freq_hp
0.4
capital_run_length_longest 4
char_freq_dollar
word_freq_remove
0.9
word_freq_george
Other features
...
Score for the above instance
Iteration 1 2 3 4 5 6
Instance values
N/A
.08 < .052 = f
.4 < .195 = f
0 < .01 = t
0 < = t
.9 < .225 = f
Prediction
-0.093
0.74
-1.446
-0.38
0.176
1.66
Total score = > 0 , so the instance is a spam.

31. Prediction = SUM(scores)
ADT: Algorithm Steps
M features
Input sample …
Condition 1
Condition 2
Condition 3
Condition M
score_1
score_2
score_1
score_2
score_1
score_2
score_1
score_2
Prediction = SUM(scores)
Condition 1
Condition 2
Classify
AD Tree

32. On Class Practice 2 Data Method Software Step labor.arff ADTree
Parameter (Select by yourself)
Software
weka
Step
Explorer->Classify->Classifier (Trees – ADTree)

33. SVM Overview Intro. to Support Vector Machines (SVM) Properties of SVM
Applications
Gene Expression Data Classification
Text Categorization if time permits
Discussion

34. Linear Classifiers f a yest x f(x,w,b) = sign(w x + b) denotes +1
How would you classify this data?
w x + b<0

35. Linear Classifiers f a yest x f(x,w,b) = sign(w x + b) denotes +1
How would you classify this data?

36. Linear Classifiers f a yest x f(x,w,b) = sign(w x + b) denotes +1
How would you classify this data?

37. Linear Classifiers f a yest x f(x,w,b) = sign(w x + b) denotes +1
Any of these would be fine..
..but which is best?

38. Linear Classifiers f a yest x f(x,w,b) = sign(w x + b) denotes +1
How would you classify this data?
Misclassified
to +1 class

39. Classifier Margin Classifier Margin f f a a yest yest x x
f(x,w,b) = sign(w x + b)
f(x,w,b) = sign(w x + b)
denotes +1
denotes -1
denotes +1
denotes -1
Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.
Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.

40. Maximum Margin f a yest x
Maximizing the margin is good according to intuition and PAC theory
Implies that only support vectors are important; other training examples are ignorable.
Empirically it works very very well.
f(x,w,b) = sign(w x + b)
denotes +1
denotes -1
The maximum margin linear classifier is the linear classifier with the, um, maximum margin.
This is the simplest kind of SVM (Called an LSVM)
Support Vectors are those datapoints that the margin pushes up against
Linear SVM

41. Linear SVM Mathematicallyx+
M=Margin Width
“Predict Class = +1” zone X-
wx+b=1
“Predict Class = -1” zone
wx+b=0
wx+b=-1
What we know:
w . x+ + b = +1
w . x- + b = -1
w . (x+-x-) = 2

42. Linear SVM Mathematically
Goal: 1) Correctly classify all training data
if yi = +1
if yi = -1
for all i
2) Maximize the Margin
same as minimize
We can formulate a Quadratic Optimization Problem and solve for w and b
Minimize
subject to

43. Solving the Optimization Problem
Find w and b such that
Φ(w) =½ wTw is minimized;
and for all {(xi ,yi)}: yi (wTxi + b) ≥ 1
Need to optimize a quadratic function subject to linear constraints.
Quadratic optimization problems are a well-known class of mathematical programming problems, and many (rather intricate) algorithms exist for solving them.
The solution involves constructing a dual problem where a Lagrange multiplier αi is associated with every constraint in the primary problem:
Find α1…αN such that
Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and
(1) Σαiyi = 0
(2) αi ≥ 0 for all αi

44. The Optimization Problem Solution
The solution has the form:
Each non-zero αi indicates that corresponding xi is a support vector.
Then the classifying function will have the form:
Notice that it relies on an inner product between the test point x and the support vectors xi – we will return to this later.
Also keep in mind that solving the optimization problem involved computing the inner products xiTxj between all pairs of training points.
w =Σαiyixi b= yk- wTxk for any xk such that αk 0
f(x) = ΣαiyixiTx + b

45. Dataset with noise OVERFITTING!
Hard Margin: So far we require all data points be classified correctly
- No training error
What if the training set is noisy?
- Solution 1: use very powerful kernels
denotes +1
denotes -1
OVERFITTING!

46. Soft Margin Classification
Slack variables ξi can be added to allow misclassification of difficult or noisy examples.
What should our quadratic optimization criterion be?
Minimize
wx+b=1
wx+b=0
wx+b=-1 e7
e11 e2

47. Hard Margin v.s. Soft Margin
The old formulation:
The new formulation incorporating slack variables:
Parameter C can be viewed as a way to control overfitting.
Find w and b such that
Φ(w) =½ wTw is minimized and for all {(xi ,yi)}
yi (wTxi + b) ≥ 1
Find w and b such that
Φ(w) =½ wTw + CΣξi is minimized and for all {(xi ,yi)}
yi (wTxi + b) ≥ 1- ξi and ξi ≥ 0 for all i

48. Linear SVMs: Overview The classifier is a separating hyperplane.
Most “important” training points are support vectors; they define the hyperplane.
Quadratic optimization algorithms can identify which training points xi are support vectors with non-zero Lagrangian multipliers αi.
Both in the dual formulation of the problem and in the solution training points appear only inside dot products:
Find α1…αN such that
Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and
(1) Σαiyi = 0
(2) 0 ≤ αi ≤ C for all αi
f(x) = ΣαiyixiTx + b

49. Non-linear SVMs
Datasets that are linearly separable with some noise work out great:
But what are we going to do if the dataset is just too hard?
How about… mapping data to a higher-dimensional space: x x x2 x

50. Non-linear SVMs: Feature spaces
General idea: the original input space can always be mapped to some higher-dimensional feature space where the training set is separable:
Φ: x → φ(x)

51. The “Kernel Trick”
The linear classifier relies on dot product between vectors K(xi,xj)=xiTxj
If every data point is mapped into high-dimensional space via some transformation Φ: x → φ(x), the dot product becomes:
K(xi,xj)= φ(xi) Tφ(xj)
A kernel function is some function that corresponds to an inner product in some expanded feature space.
Example:
2-dimensional vectors x=[x1 x2]; let K(xi,xj)=(1 + xiTxj)2,
Need to show that K(xi,xj)= φ(xi) Tφ(xj):
K(xi,xj)=(1 + xiTxj)2,
= 1+ xi12xj xi1xj1 xi2xj2+ xi22xj22 + 2xi1xj1 + 2xi2xj2
= [1 xi12 √2 xi1xi2 xi22 √2xi1 √2xi2]T [1 xj12 √2 xj1xj2 xj22 √2xj1 √2xj2]
= φ(xi) Tφ(xj), where φ(x) = [1 x12 √2 x1x2 x22 √2x1 √2x2]

52. Every semi-positive definite symmetric function is a kernel
What Functions are Kernels?
For some functions K(xi,xj) checking that
K(xi,xj)= φ(xi) Tφ(xj) can be cumbersome.
Mercer’s theorem:
Every semi-positive definite symmetric function is a kernel
Semi-positive definite symmetric functions correspond to a semi-positive definite symmetric Gram matrix:
K(x1,x1)
K(x1,x2)
K(x1,x3) …
K(x1,xN)
K(x2,x1)
K(x2,x2)
K(x2,x3)
K(x2,xN)
K(xN,x1)
K(xN,x2)
K(xN,x3)
K(xN,xN) K=

53. Examples of Kernel Functions
Linear: K(xi,xj)= xi Txj
Polynomial of power p: K(xi,xj)= (1+ xi Txj)p
Gaussian (radial-basis function network):
Sigmoid: K(xi,xj)= tanh(β0xi Txj + β1)

54. Non-linear SVMs Mathematically
Dual problem formulation:
The solution is:
Optimization techniques for finding αi’s remain the same!
Find α1…αN such that
Q(α) =Σαi - ½ΣΣαiαjyiyjK(xi, xj) is maximized and
(1) Σαiyi = 0
(2) αi ≥ 0 for all αi
f(x) = ΣαiyiK(xi, xj)+ b

55. Nonlinear SVM - Overview
SVM locates a separating hyperplane in the feature space and classify points in that space
It does not need to represent the space explicitly, simply by defining a kernel function
The kernel function plays the role of the dot product in the feature space.

56. Properties of SVM Flexibility in choosing a similarity function
Sparseness of solution when dealing with large data sets
- only support vectors are used to specify the separating hyperplane
Ability to handle large feature spaces
- complexity does not depend on the dimensionality of the feature space
Overfitting can be controlled by soft margin approach
Nice math property: a simple convex optimization problem which is guaranteed to converge to a single global solution
Feature Selection

57. SVM Applications
SVM has been used successfully in many real-world problems
- text (and hypertext) categorization
- image classification
- bioinformatics (Protein classification,
Cancer classification)
- hand-written character recognition

58. Application 1: Cancer Classification
High Dimensional
- p>1000; n<100
Imbalanced
- less positive samples
Many irrelevant features
Noisy
Genes
Patients
g-1
g-2 ……
g-p
P-1
p-2
…….
p-n
FEATURE SELECTION
In the linear case,
wi2 gives the ranking of dim i
SVM is sensitive to noisy (mis-labeled) data 

59. Weakness of SVM
It is sensitive to noise
- A relatively small number of mislabeled examples can dramatically decrease the performance
It only considers two classes
- how to do multi-class classification with SVM?
- Answer:
1) with output arity m, learn m SVM’s
SVM 1 learns “Output==1” vs “Output != 1”
SVM 2 learns “Output==2” vs “Output != 2” :
SVM m learns “Output==m” vs “Output != m”
2)To predict the output for a new input, just predict with each SVM and find out which one puts the prediction the furthest into the positive region.

60. Application 2: Text Categorization
Task: The classification of natural text (or hypertext) documents into a fixed number of predefined categories based on their content.
- filtering, web searching, sorting documents by topic, etc..
A document can be assigned to more than one category, so this can be viewed as a series of binary classification problems, one for each category

61. Representation of Text
IR’s vector space model (aka bag-of-words representation)
A doc is represented by a vector indexed by a pre-fixed set or dictionary of terms
Values of an entry can be binary or weights
Normalization, stop words, word stems
Doc x => φ(x)

62. Text Categorization using SVM
The distance between two documents is φ(x)·φ(z)
K(x,z) = 〈φ(x)·φ(z) is a valid kernel, SVM can be used with K(x,z) for discrimination.
Why SVM?
-High dimensional input space
-Few irrelevant features (dense concept)
-Sparse document vectors (sparse instances)
-Text categorization problems are linearly separable

63. Some Issues Choice of kernel Choice of kernel parameters
- Gaussian or polynomial kernel is default
- if ineffective, more elaborate kernels are needed
- domain experts can give assistance in formulating appropriate similarity measures
Choice of kernel parameters
- e.g. σ in Gaussian kernel
- σ is the distance between closest points with different classifications
- In the absence of reliable criteria, applications rely on the use of a validation set or cross-validation to set such parameters.
Optimization criterion – Hard margin v.s. Soft margin
- a lengthy series of experiments in which various parameters are tested

64. Additional Resources
An excellent tutorial on VC-dimension and Support Vector Machines:
C.J.C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2): , 1998.
The VC/SRM/SVM Bible:
Statistical Learning Theory by Vladimir Vapnik, Wiley-Interscience; 1998