CS583-supervised-learning.ppt

Chapter 3:
Supervised Learning

CS583, Bing Liu, UIC 2
Road Map
 Basic concepts
 Decision tree induction
 Evaluation of classifiers
 Rule induction
 Classification using association rules
 Naïve Bayesian classification
 Naïve Bayes for text classification
 Support vector machines
 K-nearest neighbor
 Ensemble methods: Bagging and Boosting
 Summary

An example application
 An emergency room in a hospital measures 17
variables (e.g., blood pressure, age, etc) of newly
admitted patients.
 A decision is needed: whether to put a new patient
in an intensive-care unit.
 Due to the high cost of ICU, those patients who
may survive less than a month are given higher
priority.
 Problem: to predict high-risk patients and
discriminate them from low-risk patients.

Another application
 A credit card company receives thousands of
applications for new cards. Each application
contains information about an applicant,
 age
 Marital status
 annual salary
 outstanding debts
 credit rating
 etc.
 Problem: to decide whether an application should
approved, or to classify applications into two
categories, approved and not approved.

Machine learning and our focus
 Like human learning from past experiences.
 A computer does not have “experiences”.
 A computer system learns from data, which
represent some “past experiences” of an
application domain.
 Our focus: learn a target function that can be used
to predict the values of a discrete class attribute,
e.g., approve or not-approved, and high-risk or low
risk.
 The task is commonly called: Supervised learning,
classification, or inductive learning.

 Data: A set of data records (also called
examples, instances or cases) described by
 k attributes: A1, A2, … Ak.
 a class: Each example is labelled with a pre-
defined class.
 Goal: To learn a classification model from the
data that can be used to predict the classes
of new (future, or test) cases/instances.
The data and the goal

An example: data (loan application)
Approved or not

An example: the learning task
 Learn a classification model from the data
 Use the model to classify future loan applications
into
 Yes (approved) and
 No (not approved)
 What is the class for following case/instance?

Supervised vs. unsupervised Learning
 Supervised learning: classification is seen as
supervised learning from examples.
 Supervision: The data (observations,
measurements, etc.) are labeled with pre-defined
classes. It is like that a “teacher” gives the classes
(supervision).
 Test data are classified into these classes too.
 Unsupervised learning (clustering)
 Class labels of the data are unknown
 Given a set of data, the task is to establish the
existence of classes or clusters in the data

Supervised learning process: two steps
 Learning (training): Learn a model using the
training data
 Testing: Test the model using unseen test data
to assess the model accuracy
,
cases
test
of
number
Total
tions
classifica
correct
of
Number

Accuracy

What do we mean by learning?
 Given
 a data set D,
 a task T, and
 a performance measure M,
a computer system is said to learn from D to
perform the task T if after learning the
system’s performance on T improves as
measured by M.
 In other words, the learned model helps the
system to perform T better as compared to
no learning.

An example
 Data: Loan application data
 Task: Predict whether a loan should be
approved or not.
 Performance measure: accuracy.
No learning: classify all future applications (test
data) to the majority class (i.e., Yes):
Accuracy = 9/15 = 60%.
 We can do better than 60% with learning.

Fundamental assumption of learning
Assumption: The distribution of training
examples is identical to the distribution of test
examples (including future unseen examples).
 In practice, this assumption is often violated
to certain degree.
 Strong violations will clearly result in poor
classification accuracy.
 To achieve good accuracy on the test data,
training examples must be sufficiently
representative of the test data.

Road Map
 Basic concepts
 Rule induction
 Summary

Introduction
 Decision tree learning is one of the most
widely used techniques for classification.
 Its classification accuracy is competitive with
other methods, and
 it is very efficient.
 The classification model is a tree, called
decision tree.
 C4.5 by Ross Quinlan is perhaps the best
known system. It can be downloaded from
the Web.

The loan data (reproduced)
Approved or not

A decision tree from the loan data
 Decision nodes and leaf nodes (classes)

Use the decision tree
No

Is the decision tree unique?
 No. Here is a simpler tree.
 We want smaller tree and accurate tree.
 Easy to understand and perform better.
 Finding the best tree is
NP-hard.
 All current tree building
algorithms are heuristic
algorithms

From a decision tree to a set of rules
 A decision tree can
be converted to a
set of rules
 Each path from the
root to a leaf is a
rule.

Algorithm for decision tree learning
 Basic algorithm (a greedy divide-and-conquer algorithm)
 Assume attributes are categorical now (continuous attributes
can be handled too)
 Tree is constructed in a top-down recursive manner
 At start, all the training examples are at the root
 Examples are partitioned recursively based on selected
attributes
 Attributes are selected on the basis of an impurity function (e.g.,
information gain)
 Conditions for stopping partitioning
 All examples for a given node belong to the same class
 There are no remaining attributes for further partitioning –
majority class is the leaf
 There are no examples left

Decision tree learning algorithm

Choose an attribute to partition data
 The key to building a decision tree - which
attribute to choose in order to branch.
 The objective is to reduce impurity or
uncertainty in data as much as possible.
 A subset of data is pure if all instances belong to
the same class.
 The heuristic in C4.5 is to choose the attribute
with the maximum Information Gain or Gain
Ratio based on information theory.

The loan data (reproduced)
Approved or not

Two possible roots, which is better?
 Fig. (B) seems to be better.

Information theory
 Information theory provides a mathematical
basis for measuring the information content.
 To understand the notion of information, think
about it as providing the answer to a question,
for example, whether a coin will come up heads.
 If one already has a good guess about the answer,
then the actual answer is less informative.
 If one already knows that the coin is rigged so that it
will come with heads with probability 0.99, then a
message (advanced information) about the actual
outcome of a flip is worth less than it would be for a
honest coin (50-50).

Information theory (cont …)
 For a fair (honest) coin, you have no
information, and you are willing to pay more
(say in terms of $) for advanced information -
less you know, the more valuable the
information.
 Information theory uses this same intuition,
but instead of measuring the value for
information in dollars, it measures information
contents in bits.
 One bit of information is enough to answer a
yes/no question about which one has no
idea, such as the flip of a fair coin

Information theory: Entropy measure
 The entropy formula,
 Pr(cj) is the probability of class cj in data set D
 We use entropy as a measure of impurity or
disorder of data set D. (Or, a measure of
information in a tree)
,
1
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Pr(
)
Pr(
log
)
Pr(
)
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j
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c
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c
D
entropy

Entropy measure: let us get a feeling
 As the data become purer and purer, the entropy value
becomes smaller and smaller. This is useful to us!

Information gain
 Given a set of examples D, we first compute its
entropy:
 If we make attribute Ai, with v values, the root of the
current tree, this will partition D into v subsets D1, D2
…, Dv . The expected entropy if Ai is used as the
current root:




v
j
j
j
A D
entropy
D
D
D
entropyi
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Information gain (cont …)
 Information gained by selecting attribute Ai to
branch or to partition the data is
 We choose the attribute with the highest gain to
branch/split the current tree.
)
(
)
(
)
,
( D
entropy
D
entropy
A
D
gain i
A
i 


An example
Age Yes No entropy(Di)
young 2 3 0.971
middle 3 2 0.971
old 4 1 0.722
 Own_house is the best
choice for the root.
971
.
0
15
9
log
15
9
15
6
log
15
6
)
( 2
2 




D
entropy
551
.
0
918
.
0
15
9
0
15
6
)
(
15
9
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(
15
6
)
( 2
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entropy
D
entropy
D
entropy house
Own
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 D
entropy
D
entropy
D
entropy
D
entropyAge

We build the final tree
 We can use information gain ratio to evaluate the
impurity as well (see the handout)

Handling continuous attributes
 Handle continuous attribute by splitting into
two intervals (can be more) at each node.
 How to find the best threshold to divide?
 Use information gain or gain ratio again
 Sort all the values of an continuous attribute in
increasing order {v1, v2, …, vr},
 One possible threshold between two adjacent
values vi and vi+1. Try all possible thresholds and
find the one that maximizes the gain (or gain
ratio).

An example in a continuous space

Avoid overfitting in classification
 Overfitting: A tree may overfit the training data
 Good accuracy on training data but poor on test data
 Symptoms: tree too deep and too many branches,
some may reflect anomalies due to noise or outliers
 Two approaches to avoid overfitting
 Pre-pruning: Halt tree construction early
 Difficult to decide because we do not know what may
happen subsequently if we keep growing the tree.
 Post-pruning: Remove branches or sub-trees from a
“fully grown” tree.
 This method is commonly used. C4.5 uses a statistical
method to estimates the errors at each node for pruning.
 A validation set may be used for pruning as well.

An example Likely to overfit the data

Other issues in decision tree learning
 From tree to rules, and rule pruning
 Handling of miss values
 Handing skewed distributions
 Handling attributes and classes with different
costs.
 Attribute construction
 Etc.

Road Map
 Basic concepts
 Rule induction
 Summary

Evaluating classification methods
 Predictive accuracy
 Efficiency
 time to construct the model
 time to use the model
 Robustness: handling noise and missing values
 Scalability: efficiency in disk-resident databases
 Interpretability:
 understandable and insight provided by the model
 Compactness of the model: size of the tree, or the
number of rules.

Evaluation methods
 Holdout set: The available data set D is divided into
two disjoint subsets,
 the training set Dtrain (for learning a model)
 the test set Dtest (for testing the model)
 Important: training set should not be used in testing
and the test set should not be used in learning.
 Unseen test set provides a unbiased estimate of accuracy.
 The test set is also called the holdout set. (the
examples in the original data set D are all labeled
with classes.)
 This method is mainly used when the data set D is
large.

Evaluation methods (cont…)
 n-fold cross-validation: The available data is
partitioned into n equal-size disjoint subsets.
 Use each subset as the test set and combine the rest
n-1 subsets as the training set to learn a classifier.
 The procedure is run n times, which give n accuracies.
 The final estimated accuracy of learning is the
average of the n accuracies.
 10-fold and 5-fold cross-validations are commonly
used.
 This method is used when the available data is not
large.

 Leave-one-out cross-validation: This
method is used when the data set is very
small.
 It is a special case of cross-validation
 Each fold of the cross validation has only a
single test example and all the rest of the
data is used in training.
 If the original data has m examples, this is m-
fold cross-validation

 Validation set: the available data is divided into
three subsets,
 a training set,
 a validation set and
 a test set.
 A validation set is used frequently for estimating
parameters in learning algorithms.
 In such cases, the values that give the best
accuracy on the validation set are used as the final
parameter values.
 Cross-validation can be used for parameter
estimating as well.

Classification measures
 Accuracy is only one measure (error = 1-accuracy).
 Accuracy is not suitable in some applications.
 In text mining, we may only be interested in the
documents of a particular topic, which are only a
small portion of a big document collection.
 In classification involving skewed or highly
imbalanced data, e.g., network intrusion and
financial fraud detections, we are interested only in
the minority class.
 High accuracy does not mean any intrusion is detected.
 E.g., 1% intrusion. Achieve 99% accuracy by doing nothing.
 The class of interest is commonly called the
positive class, and the rest negative classes.

Precision and recall measures
 Used in information retrieval and text classification.
 We use a confusion matrix to introduce them.

Precision and recall measures (cont…)
 Precision p is the number of correctly classified
positive examples divided by the total number of
examples that are classified as positive.
 Recall r is the number of correctly classified positive
examples divided by the total number of actual
positive examples in the test set.
.
.
FN
TP
TP
r
FP
TP
TP
p





An example
 This confusion matrix gives
 precision p = 100% and
 recall r = 1%
because we only classified one positive example correctly
and no negative examples wrongly.
 Note: precision and recall only measure
classification on the positive class.

F1-value (also called F1-score)
 It is hard to compare two classifiers using two measures. F1
score combines precision and recall into one measure
 The harmonic mean of two numbers tends to be closer to the
smaller of the two.
 For F1-value to be large, both p and r much be large.

Receive operating characteristics curve
 It is commonly called the ROC curve.
 It is a plot of the true positive rate (TPR)
against the false positive rate (FPR).
 True positive rate:
 False positive rate:

Sensitivity and Specificity
 In statistics, there are two other evaluation
measures:
 Sensitivity: Same as TPR
 Specificity: Also called True Negative Rate (TNR)
 Then we have

Example ROC curves

Area under the curve (AUC)
 Which classifier is better, C1 or C2?
 It depends on which region you talk about.
 Can we have one measure?
 Yes, we compute the area under the curve (AUC)
 If AUC for Ci is greater than that of Cj, it is
said that Ci is better than Cj.
 If a classifier is perfect, its AUC value is 1
 If a classifier makes all random guesses, its AUC
value is 0.5.

Drawing an ROC curve

Another evaluation method:
Scoring and ranking
 Scoring is related to classification.
 We are interested in a single class (positive
class), e.g., buyers class in a marketing
database.
 Instead of assigning each test instance a
definite class, scoring assigns a probability
estimate (PE) to indicate the likelihood that the
example belongs to the positive class.

Ranking and lift analysis
 After each example is given a PE score, we can
rank all examples according to their PEs.
 We then divide the data into n (say 10) bins. A
lift curve can be drawn according how many
positive examples are in each bin. This is called
lift analysis.
 Classification systems can be used for scoring.
Need to produce a probability estimate.
 E.g., in decision trees, we can use the confidence value at
each leaf node as the score.

An example
 We want to send promotion materials to
potential customers to sell a watch.
 Each package cost $0.50 to send (material
and postage).
 If a watch is sold, we make $5 profit.
 Suppose we have a large amount of past
data for building a predictive/classification
model. We also have a large list of potential
customers.
 How many packages should we send and
who should we send to?

An example
 Assume that the test set has 10000
instances. Out of this, 500 are positive cases.
 After the classifier is built, we score each test
instance. We then rank the test set, and
divide the ranked test set into 10 bins.
 Each bin has 1000 test instances.
 Bin 1 has 210 actual positive instances
 …

Lift curve
210 120 60 40 22 18 12 7 6 5
42% 24% 12% 8% 4.40% 3.60% 2.40% 1.40% 1.20% 1%
42% 66% 78% 86% 90.40% 94% 96.40% 97.80% 99% 100%
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100
Percent of testing cases
Percent
of
total
positive
cases
lift
random
Bin 1 2 3 4 5 6 7 8 9 10

Road Map
 Basic concepts
 Rule induction
 Summary

Introduction
 We showed that a decision tree can be
converted to a set of rules.
 Can we find if-then rules directly from data
for classification?
 Yes.
 Rule induction systems find a sequence of
rules (also called a decision list) for
classification.
 The commonly used strategy is sequential
covering.

Sequential covering
 Learn one rule at a time, sequentially.
 After a rule is learned, the training examples
covered by the rule are removed.
 Only the remaining data are used to find
subsequent rules.
 The process repeats until some stopping
criteria are met.
Note: a rule covers an example if the example
satisfies the conditions of the rule.
 We introduce two specific algorithms.

Algorithm 1: ordered rules
 The final classifier:
<r1, r2, …, rk, default-class>

Algorithm 2: ordered classes
 Rules of the same class are together.

Algorithm 1 vs. Algorithm 2
 Differences:
 Algorithm 2: Rules of the same class are found
together. The classes are ordered. Normally,
minority class rules are found first.
 Algorithm 1: In each iteration, a rule of any class
may be found. Rules are ordered according to the
sequence they are found.
 Use of rules: the same.
 For a test instance, we try each rule sequentially.
The first rule that covers the instance classifies it.
 If no rule covers it, default class is used, which is
the majority class in the data.

Learn-one-rule-1 function
 Let us consider only categorical attributes
 Let attributeValuePairs contains all possible
attribute-value pairs (Ai = ai) in the data.
 Iteration 1: Each attribute-value is evaluated
as the condition of a rule. I.e., we compare all
such rules Ai = ai  cj and keep the best one,
 Evaluation: e.g., entropy
 Also store the k best rules for beam search (to
search more space). Called new candidates.

Learn-one-rule-1 function (cont …)
 In iteration m, each (m-1)-condition rule in the
new candidates set is expanded by attaching
each attribute-value pair in attributeValuePairs
as an additional condition to form candidate
rules.
 These new candidate rules are then evaluated
in the same way as 1-condition rules.
 Update the best rule
 Update the k-best rules
 The process repeats unless stopping criteria
are met.

Learn-one-rule-1 algorithm

Learn-one-rule-2 function
 Split the data:
 Pos -> GrowPos and PrunePos
 Neg -> GrowNeg and PruneNeg
 Grow sets are used to find a rule (BestRule),
and the Prune sets are used to prune the rule.
 GrowRule works similarly as in learn-one-rule-
1, but the class is fixed in this case. Recall the
second algorithm finds all rules of a class first
(Pos) and then moves to the next class.

Learn-one-rule-2 algorithm

Rule evaluation in learn-one-rule-2
 Let the current partially developed rule be:
R: av1, .., avk  class
 where each avj is a condition (an attribute-value pair).
 By adding a new condition avk+1, we obtain the rule
R+: av1, .., avk, avk+1 class.
 The evaluation function for R+ is the following
information gain criterion (which is different from
the gain function used in decision tree learning).
 Rule with the best gain is kept for further extension.

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
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1
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2
2 log
log
)
,
(
n
p
p
n
p
p
p
R
R
gain

Rule pruning in learn-one-rule-2
 Consider deleting every subset of conditions from
the BestRule, and choose the deletion that
maximizes the function:
where p (n) is the number of examples in PrunePos
(PruneNeg) covered by the current rule (after a
deletion).
n
p
n
p
PruneNeg
PrunePos
BestRule
v



)
,
,
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Discussions
 Accuracy: similar to decision tree
 Efficiency: Run much slower than decision tree
induction because
 To generate each rule, all possible rules are tried on the
data (not really all, but still a lot).
 When the data is large and/or the number of attribute-value
pairs are large. It may run very slowly.
 Rule interpretability: Can be a problem because
each rule is found after data covered by previous
rules are removed. Thus, each rule may not be
treated as independent of other rules.

Road Map
 Basic concepts
 Rule induction
 Summary

Three approaches
 Three main approaches of using association
rules for classification.
 Using class association rules to build classifiers
 Using class association rules as
attributes/features
 Using normal association rules for classification

Using Class Association Rules
 Classification: mine a small set of rules
existing in the data to form a classifier or
predictor.
 It has a target attribute: Class attribute
 Association rules: have no fixed target, but
we can fix a target.
 Class association rules (CAR): has a target
class attribute. E.g.,
Own_house = true  Class =Yes [sup=6/15, conf=6/6]
 CARs can obviously be used for classification.

Decision tree vs. CARs
 The decision tree below generates the following 3 rules.
Own_house = true  Class =Yes [sup=6/15, conf=6/6]
Own_house = false, Has_job = true  Class=Yes [sup=5/15, conf=5/5]
Own_house = false, Has_job = false  Class=No [sup=4/15, conf=4/4]
 But there are many other
rules that are not found by
the decision tree

There are many more rules
 CAR mining finds all of
them.
 In many cases, rules
not in the decision tree
(or a rule list) may
perform classification
better.
 Such rules may also be
actionable in practice

Decision tree vs. CARs (cont …)
 Association mining require discrete attributes.
Decision tree learning uses both discrete and
continuous attributes.
 CAR mining requires continuous attributes
discretized. There are several such algorithms.
 Decision tree is not constrained by minsup or
minconf, and thus is able to find rules with
very low support. Of course, such rules may
be pruned due to the possible overfitting.

Considerations in CAR mining
 Multiple minimum class supports
 Deal with imbalanced class distribution, e.g.,
some class is rare, 98% negative and 2%
positive.
 We can set the minsup(positive) = 0.2% and
minsup(negative) = 2%.
 If we are not interested in classification of
negative class, we may not want to generate
rules for negative class. We can set
minsup(negative)=100% or more.
 Rule pruning may be performed.

Building classifiers
 There are many ways to build classifiers using
CARs. Several existing systems available.
 Strongest rules: After CARs are mined, do
nothing.
 For each test case, we simply choose the most
confident rule that covers the test case to classify it.
Microsoft SQL Server has a similar method.
 Or, using a combination of rules.
 Selecting a subset of Rules
 used in the CBA system.
 similar to sequential covering.

CBA: Rules are sorted first
Definition: Given two rules, ri and rj, ri  rj (also
called ri precedes rj or ri has a higher
precedence than rj) if
 the confidence of ri is greater than that of rj, or
 their confidences are the same, but the support of
ri is greater than that of rj, or
 both the confidences and supports of ri and rj are
the same, but ri is generated earlier than rj.
A CBA classifier L is of the form:
L = <r1, r2, …, rk, default-class>

Classifier building using CARs
 This algorithm is very inefficient
 CBA has a very efficient algorithm (quite sophisticated)
that scans the data at most two times.

Using rules as features
 Most classification methods do not fully explore
multi-attribute correlations, e.g., naïve Bayesian,
decision trees, rules induction, etc.
 This method creates extra attributes to augment the
original data by
 Using the conditional parts of rules
 Each rule forms an new attribute
 If a data record satisfies the condition of a rule, the attribute
value is 1, and 0 otherwise
 One can also use only rules as attributes
 Throw away the original data

Using normal association rules for
classification
 A widely used approach
 Main approach: strongest rules
 Main application
 Recommendation systems in e-commerce Web site (e.g.,
amazon.com).
 Each rule consequent is the recommended item.
 Major advantage: any item can be predicted.
 Main issue:
 Coverage: rare item rules are not found using classic algo.
 Multiple min supports and support difference constraint
help a great deal.

Road Map
 Basic concepts
 Rule induction
 Summary

Bayesian classification
 Probabilistic view: Supervised learning can naturally
be studied from a probabilistic point of view.
 Let A1 through Ak be attributes with discrete values.
The class is C.
 Given a test example d with observed attribute
values a1 through ak.
 Classification is basically to compute the following
posteriori probability. The prediction is the class cj
such that
is maximal

Apply Bayes’ Rule
 Pr(C=cj) is the class prior probability: easy to
estimate from the training data.





















|
|
1
|
|
|
|
1
1
|
|
|
|
1
1
|
|
|
|
1
1
|
|
|
|
1
1
|
|
|
|
1
1
)
Pr(
)
|
,...,
Pr(
)
Pr(
)
|
,...,
Pr(
)
,...,
Pr(
)
Pr(
)
|
,...,
Pr(
)
,...,
|
Pr(
C
r
r
r
A
A
j
j
A
A
A
A
j
j
A
A
A
A
j
c
C
c
C
a
A
a
A
c
C
c
C
a
A
a
A
a
A
a
A
c
C
c
C
a
A
a
A
a
A
a
A
c
C

Computing probabilities
 The denominator P(A1=a1,...,Ak=ak) is
irrelevant for decision making since it is the
same for every class.
 We only need P(A1=a1,...,Ak=ak | C=ci), which
can be written as
Pr(A1=a1|A2=a2,...,Ak=ak, C=cj)* Pr(A2=a2,...,Ak=ak |C=cj)
 Recursively, the second factor above can be
written in the same way, and so on.
 Now an assumption is needed.

Conditional independence assumption
 All attributes are conditionally independent
given the class C = cj.
 Formally, we assume,
Pr(A1=a1 | A2=a2, ..., A|A|=a|A|, C=cj) = Pr(A1=a1 | C=cj)
and so on for A2 through A|A|. I.e.,








|
|
1
|
|
|
|
1
1 )
|
Pr(
)
|
,...,
Pr(
A
i
j
i
i
i
A
A c
C
a
A
c
C
a
A
a
A

Final naïve Bayesian classifier
 We are done!
 How do we estimate P(Ai = ai| C=cj)? Easy!.
 

 











|
|
1
|
|
1
|
|
1
|
|
|
|
1
1
)
|
Pr(
)
Pr(
)
|
Pr(
)
Pr(
)
,...,
|
Pr(
C
r
A
i
r
i
i
r
A
i
j
i
i
j
A
A
j
c
C
a
A
c
C
c
C
a
A
c
C
a
A
a
A
c
C

Classify a test instance
 If we only need a decision on the most
probable class for the test instance, we only
need the numerator as its denominator is the
same for every class.
 Thus, given a test example, we compute the
following to decide the most probable class
for the test instance





|
|
1
)
|
Pr(
)
Pr(
max
arg
A
i
j
i
i
j
c
c
C
a
A
c
c
j

An example
 Compute all probabilities
required for classification

An Example (cont …)
 For C = t, we have
 For class C = f, we have
 C = t is more probable. t is the final class.
25
2
5
2
5
2
2
1
)
|
Pr(
)
Pr(
2
1






 

j
j
j t
C
a
A
t
C
25
1
5
2
5
1
2
1
)
|
Pr(
)
Pr(
2
1






 

j
j
j f
C
a
A
f
C

Additional issues
 Numeric attributes: Naïve Bayesian learning
assumes that all attributes are categorical.
Numeric attributes need to be discretized.
 Zero counts: An particular attribute value
never occurs together with a class in the
training set. We need smoothing.
 Missing values: Ignored
i
j
ij
j
i
i
n
n
n
c
C
a
A






 )
|
Pr(

On naïve Bayesian classifier
 Advantages:
 Easy to implement
 Very efficient
 Good results obtained in many applications
 Disadvantages
 Assumption: class conditional independence,
therefore loss of accuracy when the assumption
is seriously violated (those highly correlated
data sets)

Road Map
 Basic concepts
 Rule induction
 Summary

Text classification/categorization
 Due to the rapid growth of online documents in
organizations and on the Web, automated document
classification has become an important problem.
 Techniques discussed previously can be applied to
text classification, but they are not as effective as
the next three methods.
 We first study a naïve Bayesian method specifically
formulated for texts, which makes use of some text
specific features.
 However, the ideas are similar to the preceding
method.

Probabilistic framework
 Generative model: Each document is
generated by a parametric distribution
governed by a set of hidden parameters.
 The generative model makes two
assumptions
 The data (or the text documents) are generated by
a mixture model,
 There is one-to-one correspondence between
mixture components and document classes.

Mixture model
 A mixture model models the data with a
number of statistical distributions.
 Intuitively, each distribution corresponds to a data
cluster and the parameters of the distribution
provide a description of the corresponding cluster.
 Each distribution in a mixture model is also
called a mixture component.
 The distribution/component can be of any
kind

An example
 The figure shows a plot of the probability
density function of a 1-dimensional data set
(with two classes) generated by
 a mixture of two Gaussian distributions,
 one per class, whose parameters (denoted by i) are
the mean (i) and the standard deviation (i), i.e., i
= (i, i).
class 1 class 2

Mixture model (cont …)
 Let the number of mixture components (or
distributions) in a mixture model be K.
 Let the jth distribution have the parameters j.
 Let  be the set of parameters of all
components,  = {1, 2, …, K, 1, 2, …, K},
where j is the mixture weight (or mixture
probability) of the mixture component j and j
is the parameters of component j.
 How does the model generate documents?

Document generation
 Due to one-to-one correspondence, each class
corresponds to a mixture component. The mixture
weights are class prior probabilities, i.e., j = Pr(cj|).
 The mixture model generates each document di by:
 first selecting a mixture component (or class) according to
class prior probabilities (i.e., mixture weights), j = Pr(cj|).
 then having this selected mixture component (cj) generate
a document di according to its parameters, with distribution
Pr(di|cj; ) or more precisely Pr(di|cj; j).
)
;
|
Pr(
)
Θ
|
Pr(
)
|
Pr(
|
|
1


 

C
j
j
i
j
i c
d
c
d (23)

Model text documents
 The naïve Bayesian classification treats each
document as a “bag of words”. The
generative model makes the following further
assumptions:
 Words of a document are generated
independently of context given the class label.
The familiar naïve Bayes assumption used before.
 The probability of a word is independent of its
position in the document. The document length is
chosen independent of its class.

Multinomial distribution
 With the assumptions, each document can be
regarded as generated by a multinomial
distribution.
 In other words, each document is drawn from
a multinomial distribution of words with as
many independent trials as the length of the
document.
 The words are from a given vocabulary V =
{w1, w2, …, w|V|}.

Use probability function of multinomial
distribution
where Nti is the number of times that word wt
occurs in document di and





|
|
1 !
)
;
|
Pr(
|!
|
|)
Pr(|
)
;
|
Pr(
V
t ti
ti
N
j
t
i
i
j
i
N
c
w
d
d
c
d
|
|
|
|
1
i
V
t
it d
N 


.
1
)
;
|
Pr(
|
|
1




V
t
j
t c
w
(24)
(25)

Parameter estimation
 The parameters are estimated based on empirical
counts.
 In order to handle 0 counts for infrequent occurring
words that do not appear in the training set, but may
appear in the test set, we need to smooth the
probability. Lidstone smoothing, 0    1
.
)
|
Pr(
)
|
Pr(
)
ˆ
;
|
Pr( |
|
1
|
|
1
|
|
1
 

 


 V
s
D
i i
j
si
D
i i
j
ti
j
t
d
c
N
d
c
N
c
w
.
)
|
Pr(
|
|
)
|
Pr(
)
ˆ
;
|
Pr( |
|
1
|
|
1
|
|
1
 

 




 V
s
D
i i
j
si
D
i i
j
ti
j
t
d
c
N
V
d
c
N
c
w


(26)
(27)

Parameter estimation (cont …)
 Class prior probabilities, which are mixture
weights j, can be easily estimated using
training data
|
|
)
|
Pr(
)
ˆ
|
Pr(
|
|
1
D
d
c
c
D
i
i
j
j


 (28)

Classification
 Given a test document di, from Eq. (23) (27) and (28)
 

 











|
|
1
|
|
1 ,
|
|
1 ,
)
ˆ
;
|
Pr(
)
ˆ
|
Pr(
)
ˆ
;
|
Pr(
)
ˆ
|
Pr(
)
ˆ
|
Pr(
)
ˆ
;
|
Pr(
)
ˆ
|
Pr(
)
ˆ
;
|
Pr(
C
r
d
k r
k
d
d
k k
d
i
i
r
i
j
i
j
i
j
i
j
i
j
c
w
c
c
w
c
d
c
d
c
d
c

Discussions
 Most assumptions made by naïve Bayesian
learning are violated to some degree in
practice.
 Despite such violations, researchers have
shown that naïve Bayesian learning produces
very accurate models.
 The main problem is the mixture model
assumption. When this assumption is seriously
violated, the classification performance can be
poor.
 Naïve Bayesian learning is extremely efficient.

Road Map
 Basic concepts
 Rule induction
 Summary

Introduction
 Support vector machines were invented by V.
Vapnik and his co-workers in 1970s in Russia and
became known to the West in 1992.
 SVMs are linear classifiers that find a hyperplane to
separate two class of data, positive and negative.
 Kernel functions are used for nonlinear separation.
 SVM not only has a rigorous theoretical foundation,
but also performs classification more accurately than
most other methods in applications, especially for
high dimensional data.
 It is perhaps the best classifier for text classification.

Basic concepts
 Let the set of training examples D be
{(x1, y1), (x2, y2), …, (xr, yr)},
where xi = (x1, x2, …, xn) is an input vector in a
real-valued space X  Rn and yi is its class label
(output value), yi  {1, -1}.
1: positive class and -1: negative class.
 SVM finds a linear function of the form (w: weight
vector)
f(x) = w  x + b















0
1
0
1
b
if
b
if
y
i
i
i
x
w
x
w

The hyperplane
 The hyperplane that separates positive and negative
training data is
w  x + b = 0
 It is also called the decision boundary (surface).
 So many possible hyperplanes, which one to choose?

Maximal margin hyperplane
 SVM looks for the separating hyperplane with the largest
margin.
 Machine learning theory says this hyperplane minimizes the
error bound

Linear SVM: separable case
 Assume the data are linearly separable.
 Consider a positive data point (x+, 1) and a negative
(x-, -1) that are closest to the hyperplane
<w  x> + b = 0.
 We define two parallel hyperplanes, H+ and H-, that
pass through x+ and x- respectively. H+ and H- are
also parallel to <w  x> + b = 0.

Compute the margin
 Now let us compute the distance between the two
margin hyperplanes H+ and H-. Their distance is the
margin (d+ + d in the figure).
 Recall from vector space in algebra that the
(perpendicular) distance from a point xi to the
hyperplane w  x + b = 0 is:
where ||w|| is the norm of w,
||
||
|
|
w
x
w b
i 



2
2
2
2
1 ...
||
|| n
w
w
w 






 w
w
w
(36)
(37)

Compute the margin (cont …)
 Let us compute d+.
 Instead of computing the distance from x+ to the
separating hyperplane w  x + b = 0, we pick up
any point xs on w  x + b = 0 and compute the
distance from xs to w  x+ + b = 1 by applying the
distance Eq. (36) and noticing w  xs + b = 0,
||
||
1
||
||
|
1
|
w
w
x
w s








b
d
||
||
2
w


 
 d
d
margin
(38)
(39)

A optimization problem!
Definition (Linear SVM: separable case): Given a set of
linearly separable training examples,
D = {(x1, y1), (x2, y2), …, (xr, yr)}
Learning is to solve the following constrained
minimization problem,
summarizes
w  xi + b  1 for yi = 1
w  xi + b  -1 for yi = -1.
r
i
b
y i
i ...,
2,
1,
,
1
)
(
:
Subject to
2
:
Minimize









x
w
w
w
r
i
b
y i
i ...,
2,
1,
,
1
( 




 x
w
(40)

Solve the constrained minimization
 Standard Lagrangian method
where i  0 are the Lagrange multipliers.
 Optimization theory says that an optimal
solution to (41) must satisfy certain conditions,
called Kuhn-Tucker conditions, which are
necessary (but not sufficient)
 Kuhn-Tucker conditions play a central role in
constrained optimization.
]
1
)
(
[
2
1
1









 

b
y
L i
r
i
i
i
P x
w
w
w  (41)

Kuhn-Tucker conditions
 Eq. (50) is the original set of constraints.
 The complementarity condition (52) shows that only those
data points on the margin hyperplanes (i.e., H+ and H-) can
have i > 0 since for them yi(w  xi + b) – 1 = 0.
 These points are called the support vectors, All the other
parameters i = 0.

Solve the problem
 In general, Kuhn-Tucker conditions are necessary
for an optimal solution, but not sufficient.
 However, for our minimization problem with a
convex objective function and linear constraints, the
Kuhn-Tucker conditions are both necessary and
sufficient for an optimal solution.
 Solving the optimization problem is still a difficult
task due to the inequality constraints.
 However, the Lagrangian treatment of the convex
optimization problem leads to an alternative dual
formulation of the problem, which is easier to solve
than the original problem (called the primal).

Dual formulation
 From primal to a dual: Setting to zero the
partial derivatives of the Lagrangian (41) with
respect to the primal variables (i.e., w and
b), and substituting the resulting relations
back into the Lagrangian.
 I.e., substitute (48) and (49), into the original
Lagrangian (41) to eliminate the primal variables
(55)
,
2
1
1
,
1




 
 

j
i
r
j
i
j
i
j
i
r
i
i
D y
y
L x
x




Dual optimization prolem
 This dual formulation is called the Wolfe dual.
 For the convex objective function and linear constraints of
the primal, it has the property that the maximum of LD
occurs at the same values of w, b and i, as the minimum
of LP (the primal).
 Solving (56) requires numerical techniques and clever
strategies, which are beyond our scope.

The final decision boundary
 After solving (56), we obtain the values for i, which
are used to compute the weight vector w and the
bias b using Equations (48) and (52) respectively.
 The decision boundary
 Testing: Use (57). Given a test instance z,
 If (58) returns 1, then the test instance z is classified
as positive; otherwise, it is classified as negative.
0









 

b
y
b
sv
i
i
i
i x
x
x
w  (57)
















 
sv
i
i
i
i b
y
sign
b
sign z
x
z
w 
)
( (58)

Linear SVM: Non-separable case
 Linear separable case is the ideal situation.
 Real-life data may have noise or errors.
 Class label incorrect or randomness in the application
domain.
 Recall in the separable case, the problem was
 With noisy data, the constraints may not be
satisfied. Then, no solution!
r
i
b
y i
i ...,
2,
1,
,
1
)
(
:
Subject to
2
:
Minimize









x
w
w
w

Relax the constraints
 To allow errors in data, we relax the margin
constraints by introducing slack variables, i
( 0) as follows:
w  xi + b  1  i for yi = 1
w  xi + b  1 + i for yi = -1.
 The new constraints:
Subject to: yi(w  xi + b)  1  i, i =1, …, r,
i  0, i =1, 2, …, r.

Geometric interpretation
 Two error data points xa and xb (circled) in wrong
regions

Penalize errors in objective function
 We need to penalize the errors in the
objective function.
 A natural way of doing it is to assign an extra
cost for errors to change the objective
function to
 k = 1 is commonly used, which has the
advantage that neither i nor its Lagrangian
multipliers appear in the dual formulation.





 r
i
k
i
C
1
)
(
2
:
Minimize 
w
w (60)

New optimization problem
 This formulation is called the soft-margin
SVM. The primal Lagrangian is
where i, i  0 are the Lagrange multipliers
r
i
r
i
b
y
C
i
i
i
i
r
i
i
...,
2,
1,
,
0
...,
2,
1,
,
1
)
(
:
Subject to
2
:
Minimize
1


















x
w
w
w
(61)



















r
i
i
i
i
i
r
i
i
i
r
i
i
P b
y
C
L
1
1
1
]
1
)
(
[
2
1




 x
w
w
w
(62)

Kuhn-Tucker conditions

From primal to dual
 As the linear separable case, we transform
the primal to a dual by setting to zero the
partial derivatives of the Lagrangian (62) with
respect to the primal variables (i.e., w, b
and i), and substituting the resulting
relations back into the Lagrangian.
 Ie.., we substitute Equations (63), (64) and
(65) into the primal Lagrangian (62).
 From Equation (65), C  i  i = 0, we can
deduce that i  C because i  0.

Dual
 The dual of (61) is
 Interestingly, i and its Lagrange multipliers i are
not in the dual. The objective function is identical to
that for the separable case.
 The only difference is the constraint i  C.

Find primal variable values
 The dual problem (72) can be solved numerically.
 The resulting i values are then used to compute w
and b. w is computed using Equation (63) and b is
computed using the Kuhn-Tucker complementarity
conditions (70) and (71).
 Since no values for i, we need to get around it.
 From Equations (65), (70) and (71), we observe that if 0 < i
< C then both i = 0 and yiw  xi + b – 1 + i = 0. Thus, we
can use any training data point for which 0 < i < C and
Equation (69) (with i = 0) to compute b.
.
0
1
1





 

j
r
i
i
i
i
i
y
y
b x
x
 (73)

(65), (70) and (71) in fact tell us more
 (74) shows a very important property of SVM.
 The solution is sparse in i. Many training data points are
outside the margin area and their i’s in the solution are 0.
 Only those data points that are on the margin (i.e., yi(w  xi
+ b) = 1, which are support vectors in the separable case),
inside the margin (i.e., i = C and yi(w  xi + b) < 1), or
errors are non-zero.
 Without this sparsity property, SVM would not be practical for
large data sets.

The final decision boundary
 The final decision boundary is (we note that many
i’s are 0)
 The decision rule for classification (testing) is the
same as the separable case, i.e.,
sign(w  x + b).
 Finally, we also need to determine the parameter C
in the objective function. It is normally chosen
through the use of a validation set or cross-
validation.
0
1









 

b
y
b
r
i
i
i
i x
x
x
w  (75)

How to deal with nonlinear separation?
 The SVM formulations require linear separation.
 Real-life data sets may need nonlinear separation.
 To deal with nonlinear separation, the same
formulation and techniques as for the linear case
are still used.
 We only transform the input data into another space
(usually of a much higher dimension) so that
 a linear decision boundary can separate positive and
negative examples in the transformed space,
 The transformed space is called the feature space.
The original data space is called the input space.

Space transformation
 The basic idea is to map the data in the input
space X to a feature space F via a nonlinear
mapping ,
 After the mapping, the original training data
set {(x1, y1), (x2, y2), …, (xr, yr)} becomes:
{((x1), y1), ((x2), y2), …, ((xr), yr)}
)
(
:
x
x 


F
X 
(76)
(77)

Geometric interpretation
 In this example, the transformed space is
also 2-D. But usually, the number of
dimensions in the feature space is much
higher than that in the input space

Optimization problem in (61) becomes

An example space transformation
 Suppose our input space is 2-dimensional,
and we choose the following transformation
(mapping) from 2-D to 3-D:
 The training example ((2, 3), -1) in the input
space is transformed to the following in the
feature space:
((4, 9, 8.5), -1)
)
2
,
,
(
)
,
( 2
1
2
2
2
1
2
1 x
x
x
x
x
x 

Problem with explicit transformation
 The potential problem with this explicit data
transformation and then applying the linear SVM is
that it may suffer from the curse of dimensionality.
 The number of dimensions in the feature space can
be huge with some useful transformations even with
reasonable numbers of attributes in the input space.
 This makes it computationally infeasible to handle.
 Fortunately, explicit transformation is not needed.

Kernel functions
 We notice that in the dual formulation both
 the construction of the optimal hyperplane (79) in F and
 the evaluation of the corresponding decision function (80)
only require dot products (x)  (z) and never the mapped
vector (x) in its explicit form. This is a crucial point.
 Thus, if we have a way to compute the dot product
(x)  (z) using the input vectors x and z directly,
 no need to know the feature vector (x) or even  itself.
 In SVM, this is done through the use of kernel
functions, denoted by K,
K(x, z) = (x)  (z) (82)

An example kernel function
 Polynomial kernel
K(x, z) = x  zd
 Let us compute the kernel with degree d = 2 in a 2-
dimensional space: x = (x1, x2) and z = (z1, z2).
 This shows that the kernel x  z2 is a dot product in
a transformed feature space
(83)
,
)
(
)
(
)
2
(
)
2
(
2
)
(
2
2
2
2
2
2
2
2
2
2
1
2
2
1
1
2
2
1
2
2
1
1
2
1
2
1
2
1
1
2
















z
x
z
x


z
z
,
z
,
z
x
x
,
x
,
x
z
x
z
x
z
x
z
x
z
x
z
x
(84)

Kernel trick
 The derivation in (84) is only for illustration
purposes.
 We do not need to find the mapping function.
 We can simply apply the kernel function
directly by
 replace all the dot products (x)  (z) in (79) and
(80) with the kernel function K(x, z) (e.g., the
polynomial kernel x  zd in (83)).
 This strategy is called the kernel trick.

Is it a kernel function?
 The question is: how do we know whether a
function is a kernel without performing the
derivation such as that in (84)? I.e,
 How do we know that a kernel function is indeed a
dot product in some feature space?
 This question is answered by a theorem
called the Mercer’s theorem, which we will
not discuss here.

Commonly used kernels
 It is clear that the idea of kernel generalizes the dot
product in the input space. This dot product is also
a kernel with the feature map being the identity

Some other issues in SVM
 SVM works only in a real-valued space. For a
categorical attribute, we need to convert its
categorical values to numeric values.
 SVM does only two-class classification. For multi-
class problems, some strategies can be applied, e.g.,
one-against-rest, and error-correcting output coding.
 The hyperplane produced by SVM is hard to
understand by human users. The matter is made
worse by kernels. Thus, SVM is commonly used in
applications that do not required human
understanding.

Road Map
 Basic concepts
 Rule induction
 Summary

k-Nearest Neighbor Classification (kNN)
 Unlike all the previous learning methods, kNN
does not build model from the training data.
 To classify a test instance d, define k-
neighborhood P as k nearest neighbors of d
 Count number n of training instances in P that
belong to class cj
 Estimate Pr(cj|d) as n/k
 No training is needed. Classification time is
linear in training set size for each test case.

kNNAlgorithm
 k is usually chosen empirically via a validation
set or cross-validation by trying a range of k
values.
 Distance function is crucial, but depends on
applications.

Example: k=6 (6NN)
Government
Science
Arts
A new point
Pr(science| )?

Discussions
 kNN can deal with complex and arbitrary
decision boundaries.
 Despite its simplicity, researchers have
shown that the classification accuracy of kNN
can be quite strong and in many cases as
accurate as those elaborated methods.
 kNN is slow at the classification time
 kNN does not produce an understandable
model

Road Map
 Basic concepts
 Rule induction
 Summary

Combining classifiers
 So far, we have only discussed individual
classifiers, i.e., how to build them and use
them.
 Can we combine multiple classifiers to
produce a better classifier?
 Yes, sometimes
 We discuss two main algorithms:
 Bagging
 Boosting

Bagging
 Breiman, 1996
 Bootstrap Aggregating = Bagging
 Application of bootstrap sampling
 Given: set D containing m training examples
 Create a sample S[i] of D by drawing m examples at
random with replacement from D
 S[i] of size m: expected to leave out 0.37 of examples
from D

Bagging (cont…)
 Training
 Create k bootstrap samples S[1], S[2], …, S[k]
 Build a distinct classifier on each S[i] to produce k
classifiers, using the same learning algorithm.
 Testing
 Classify each new instance by voting of the k
classifiers (equal weights)

Bagging Example
Original 1 2 3 4 5 6 7 8
Training set 1 2 7 8 3 7 6 3 1
Training set 2 7 8 5 6 4 2 7 1
Training set 3 3 6 2 7 5 6 2 2
Training set 4 4 5 1 4 6 4 3 8

Bagging (cont …)
 When does it help?
 When learner is unstable
 Small change to training set causes large change in the
output classifier
 True for decision trees, neural networks; not true for k-
nearest neighbor, naïve Bayesian, class association
rules
 Experimentally, bagging can help substantially for
unstable learners, may somewhat degrade results
for stable learners
Bagging Predictors, Leo Breiman, 1996

Boosting
 A family of methods:
 We only study AdaBoost (Freund & Schapire, 1996)
 Training
 Produce a sequence of classifiers (the same base
learner)
 Each classifier is dependent on the previous one,
and focuses on the previous one’s errors
 Examples that are incorrectly predicted in previous
classifiers are given higher weights
 Testing
 For a test case, the results of the series of
classifiers are combined to determine the final
class of the test case.

AdaBoost
Weighted
training set
(x1, y1, w1)
(x2, y2, w2)
…
(xn, yn, wn)
Non-negative weights
sum to 1
 Build a classifier ht
whose accuracy on
training set > ½
(better than random)
Change weights
called a weaker classifier

AdaBoost algorithm

Bagging, Boosting and C4.5
C4.5’s mean error
rate over the
10 cross-
validation.
Bagged C4.5
vs. C4.5.
Boosted C4.5
vs. C4.5.
Boosting vs.
Bagging

Does AdaBoost always work?
 The actual performance of boosting depends
on the data and the base learner.
 It requires the base learner to be unstable as
bagging.
 Boosting seems to be susceptible to noise.
 When the number of outliners is very large, the
emphasis placed on the hard examples can hurt
the performance.

Road Map
 Basic concepts
 Rule induction
 Summary

Summary
 Applications of supervised learning are in almost
any field or domain.
 We studied 8 classification techniques.
 There are still many other methods, e.g.,
 Bayesian networks
 Neural networks
 Genetic algorithms
 Fuzzy classification
This large number of methods also show the importance of
classification and its wide applicability.
 It remains to be an active research area.

CS583-supervised-learning.ppt

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