Classification

— Chapter 4—

1
Outlines

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
2
Supervised vs. Unsupervised Learning

◼ Supervised learning (classification)

◼ Supervision: The training data (observations,
measurements, etc.) are accompanied by labels indicating
the class of the observations
◼ New data is classified based on the training set
◼ Unsupervised learning (clustering)
◼ The class labels of training data is unknown
◼ Given a set of measurements, observations, etc. with the
aim of establishing the existence of classes or clusters in
the data
3
Prediction Problems: Classification vs.
Numeric Prediction
◼ Classification
◼ predicts categorical class labels (discrete or nominal)

◼ classifies data (constructs a model) based on the training

set and the values (class labels) in a classifying attribute
and uses it in classifying new data
◼ Numeric Prediction
◼ models continuous-valued functions, i.e., predicts
unknown or missing values
◼ Typical applications
◼ Credit/loan approval:

◼ Medical diagnosis: if a tumor is cancerous or benign

◼ Fraud detection: if a transaction is fraudulent

◼ Web page categorization: which category it is

4
Classification—A Two-Step Process
◼ Model construction: describing a set of predetermined classes
◼ Each tuple/sample is assumed to belong to a predefined class, as

determined by the class label attribute

◼ The set of tuples used for model construction is training set

◼ The model is represented as classification rules, decision trees, or

mathematical formulae
◼ Model usage: for classifying future or unknown objects
◼ Estimate accuracy of the model

◼ The known label of test sample is compared with the classified

result from the model

◼ Accuracy rate is the percentage of test set samples that are

correctly classified by the model

◼ Test set is independent of training set (otherwise overfitting)

◼ If the accuracy is acceptable, use the model to classify new data

◼ Note: If the test set is used to select models, it is called validation (test) set
5
Process (1): Model Construction

Classification
Algorithms
Training
Data

NAME RANK YEARS TENURED Classifier

M ike A ssistant P rof 3 no (Model)
M ary A ssistant P rof 7 yes
B ill P rofessor 2 yes
Jim A ssociate P rof 7 yes
IF rank = ‘professor’
D ave A ssistant P rof 6 no
OR years > 6
A nne A ssociate P rof 3 no
THEN tenured = ‘yes’
6
Process (2): Using the Model in Prediction

Classifier

Testing
Data Unseen Data

(Jeff, Professor, 4)
NAME RANK YEARS TENURED
T om A ssistant P rof 2 no Tenured?
M erlisa A ssociate P rof 7 no
G eorge P rofessor 5 yes
Joseph A ssistant P rof 7 yes
7
Chapter 4. Classification: Basic Concepts

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
8
formation
Decision Tree Induction: An Example
age income student credit_rating buys_computer
<=30 high no fair no
❑ Training data set: Buys_computer <=30 high no excellent no
❑ Resulting tree: 31…40 high no fair yes
>40 medium no fair yes
>40 low yes fair yes
>40 low yes excellent no
31…40 low yes excellent yes
age? <=30 medium no fair no
<=30 low yes fair yes
>40 medium yes fair yes
<=30 medium yes excellent yes
<=30 overcast
31..40 >40 31…40 medium no excellent yes
31…40 high yes fair yes
>40 medium no excellent no

student? yes credit rating?

no yes excellent fair

no yes yes
9
Algorithm for Decision Tree Induction
◼ Basic algorithm (a greedy algorithm)
◼ Tree is constructed in a top-down recursive divide-and-

conquer manner
◼ At start, all the training examples are at the root

◼ Attributes are categorical (if continuous-valued, they are

discretized in advance)
◼ Examples are partitioned recursively based on selected

attributes
◼ Test attributes are selected on the basis of a heuristic or

statistical measure (e.g., information gain)

◼ Conditions for stopping partitioning
◼ All samples for a given node belong to the same class

◼ There are no remaining attributes for further partitioning –

majority voting is employed for classifying the leaf

◼ There are no samples left
10
Brief Review of Entropy
◼

m=2

11
Brief Review of Entropy
age income student credit_rating buys_computer
◼ Example: <=30 high no fair no
<=30 high no excellent no
◼ Training data set: Buys_computer 31…40 high no fair yes
>40 medium no fair yes
◼ Find entropy of feature age, H(age) >40 low yes fair yes
>40 low yes excellent no
31…40 low yes excellent yes
<=30 medium no fair no
<=30 low yes fair yes
>40 medium yes fair yes
<=30 medium yes excellent yes
31…40 medium no excellent yes
9 9 5 5
31…40 high yes fair yes
fo( D) = I (9,5) = − log 2 ( )− log 2 ( ) =0.940 >40 medium no excellent no
14 14 14 14

12
Attribute Selection Measure:
Information Gain (ID3/C4.5)
◼ Select the attribute with the highest information gain
◼ Let pi be the probability that an arbitrary tuple in D belongs to
class Ci, estimated by |Ci, D|/|D|
◼ Ci, D is the set of tuples of class Ci in D
◼ |Ci,D| denote the number of tuples in Ci,D,
◼ Expected information (entropy) needed to classify a tuple in D:
m
Info( D) = − pi log 2 ( pi )
i =1
◼ Information needed (after using A to split D into v partitions) to
v | D |
classify D:
InfoA ( D) =   Info( D j )
j

j =1 | D |

◼ Information gained by branching on attribute A

Gain(A) = Info(D) − InfoA(D)
13
Attribute Selection: Information Gain
◼ Example: Induction of a decision tree using information
gain.
◼ Table below presents a training set, D, of class-labeled
tuples randomly selected from the AllElectronics customer
database. (In this example, each attribute is discrete
valued. Continuous-valued attributes have been
generalized.
age income student credit_rating buys_computer
<=30 high no fair no
<=30 high no excellent no
31…40 high no fair yes
>40 medium no fair yes
>40 low yes fair yes
>40 low yes excellent no
31…40 low yes excellent yes age pi ni I(pi, ni)
<=30 medium no fair no
<=30 low yes fair yes <=30 2 3 0.971
>40
<=30
medium
medium
yes fair
yes excellent
yes
yes 31…40 4 0 0
31…40 medium no excellent yes
31…40 high yes fair yes >40 3 2 0.971
>40 medium no excellent no 14
Attribute Selection: Information Gain
 Class P: buys_computer = “yes” There are nine tuples of class yes (Class
 Class N: buys_computer = “no” P ) and five tuples of class no (Class N)
 To find the splitting criterion for these tuples, we must compute the
information gain of each attribute.
 We first use above Eq. to compute the expected information needed to
classify a tuple in D:
9 9 5 5
Info( D) = I (9,5) = − log 2 ( )− log 2 ( ) =0.940
14 14 14 14

The expected information needed to classify a tuple in D if the tuples are

partitioned according to age is 5
5 4
I (2,3) means “age <=30” has 5 out of
Infoage ( D ) = I ( 2,3) + I ( 4,0) 14
14 14 14 samples, with 2 yes’es and 3
5 no’s.
+ I (3,2) = 0.694
14
age pi ni I(pi, ni)
<=30 2 3 0.971
31…40 4 0 0
>40 3 2 0.971
15
Attribute Selection: Information Gain
age income student credit_rating buys_computer
Hence <=30
<=30
high
high
no
no
fair
excellent
no
no
Gain(age) = Info( D) − Infoage ( D) = 0.246 31…40 high no fair yes
>40 medium no fair yes
Similarly, >40
>40
low
low
yes
yes
fair
excellent
yes
no
31…40
Gain(income) = 0.029
low yes excellent yes
<=30 medium no fair no
<=30 low yes fair yes

Gain( student ) = 0.151

>40 medium yes fair yes
<=30 medium yes excellent yes
31…40 medium no excellent yes

Gain(credit _ rating ) = 0.048 31…40

>40
high
medium
yes
no
fair
excellent
yes
no

▪ Because age has the highest information gain among the attributes, it is selected
as the splitting attribute.

• The tuples are then partitioned accordingly, as shown in Figure shown in next
slide.
•
• Notice that the tuples falling into the partition for age = middle aged all belong
to the same class.

• Because they all belong to class “yes,” a leaf should therefore be created at the
end of this branch and labeled “yes.”
• The final decision tree returned by the algorithm was shown earlier 16
Attribute Selection: Information Gain
Figure: The attribute age becomes the splitting attribute at the root node of the
decision tree

17
Attribute Selection: Information Gain
◼ Figure: Each leaf node represents a class (either buys computer = yes or buys
computer = no).
◼ Notice because the tuples falling into the partition for age = middle aged all
belong to the same class, “yes,”
◼ A leaf was created at the end of this branch and labeled “yes.”

18
Computing Information-Gain for
Continuous-Valued Attributes
◼ Let attribute A be a continuous-valued attribute
◼ For example, suppose that instead of the discretized version of age from the
example, we have the raw values for this attribute.
◼ Must determine the best split point for A
◼ Sort the value A in increasing order
◼ Typically, the midpoint between each pair of adjacent values is
considered as a possible split point
◼ (ai+ai+1)/2 is the midpoint between the values of ai and ai+1
◼ For each possible split-point for A, we evaluate
◼ where the number of partitions is two, that is, v = 2 (or j = 1,2)
◼ The point with the minimum expected information requirement for A is
selected as the split-point for A
◼ Split:
◼ D1 is the set of tuples in D satisfying A ≤ split-point, and D2 is the set of tuples in D
satisfying A > split-point 19
Gain Ratio for Attribute Selection (C4.5)
◼ Information gain measure is biased towards attributes with a large
number of values
◼ C4.5 (a successor of ID3) uses gain ratio to overcome the problem
(normalization to information gain) using a “split information” value
defined analogously with Info(D) as
v | Dj | | Dj |
SplitInfoA ( D) = −  log 2 ( )
j =1 |D| |D|

◼ This value represents the potential information generated by

splitting the training data set, D, into v partitions, corresponding to
the v outcomes of a test on attribute A.
◼ Note that, for each outcome, it considers the number of tuples
having that outcome with respect to the total number of tuples in D.
◼ It differs from information gain, which measures the information
with respect to classification that is acquired 20
Gain Ratio for Attribute Selection (C4.5)
◼ The gain ratio is defined as
◼ GainRatio(A) = Gain(A)/SplitInfo(A)

◼ Ex.

◼ From earlier example, we have Gain(income) = 0.029.

◼ gain_ratio(income) = 0.029/1.557 = 0.019

◼ The attribute with the maximum gain ratio is selected as the splitting
attribute

21
Gini Index (CART, IBM IntelligentMiner)
◼ If a data set D contains examples from n classes, gini index,
gini(D) is defined as n 2
gini( D) = 1−  p j
j =1
where pj is the relative frequency of class j in D
◼ If a data set D is split on A into two subsets D1 and D2, the gini
index gini(D) is defined as
|D1| |D |
gini A ( D) = gini( D1) + 2 gini( D 2)
|D| |D|
◼ Reduction in Impurity:
gini( A) = gini(D) − giniA (D)
◼ The attribute that provides the smallest ginisplit(D) (or the
largest reduction in impurity) is chosen to split the node (need
to enumerate all the possible splitting points for each attribute)
22
Computation of Gini Index
◼ Ex. D has 9 tuples in buys_computer = “yes” age income studentcredit_rating
buys_compu
and 5 in “no” <=30 high no fair no
2 2
 9   5  <=30 high no excellent no
gini( D ) = 1 −   −  = 0.459
 14   14  31…40 high no fair yes
◼ Suppose the attribute income partitions D into
10 in D1: {low, medium} and 4 in D2 >40 medium no fair yes
>40 low yes fair yes
 10   4
giniincome{low,medium} ( D) =  Gini( D1 ) +  Gini( D2 ) >40 low yes excellent no
 14   14 
31…40 low yes excellent yes
<=30 medium no fair no
<=30 low yes fair yes
>40 medium yes fair yes
Gini{low,high} is 0.458; Gini{medium,high} <=30 31…40
medium yes excellent yes
medium no excellent yes
is 0.450. Thus, split on the 31…40 high yes fair yes
{low,medium} (and {high}) since >40 medium no excellent no
it has the lowest Gini index

23
Comparing Attribute Selection Measures

◼ The three measures, in general, return good results but

◼ Information gain:
◼ biased towards multivalued attributes
◼ Gain ratio:
◼ tends to prefer unbalanced splits in which one partition is
much smaller than the others
◼ Gini index:
◼ biased to multivalued attributes
◼ has difficulty when # of classes is large
◼ tends to favor tests that result in equal-sized partitions
and purity in both partitions
24
Enhancements to Basic Decision Tree Induction

◼ Allow for continuous-valued attributes

◼ Dynamically define new discrete-valued attributes that
partition the continuous attribute value into a discrete set of
intervals
◼ Handle missing attribute values
◼ Assign the most common value of the attribute
◼ Assign probability to each of the possible values
◼ Attribute construction
◼ Create new attributes based on existing ones that are
sparsely represented
◼ This reduces fragmentation, repetition, and replication
25
Classification in Large Databases
◼ Why is decision tree induction popular?
◼ relatively faster learning speed (than other classification
methods)
◼ convertible to simple and easy to understand classification
rules
◼ can use SQL queries for accessing databases

◼ comparable classification accuracy with other methods

◼ RainForest (VLDB’98 — Gehrke, Ramakrishnan & Ganti)

◼ Builds an AVC-list (attribute, value, class label)

26
Scalability Framework for RainForest

◼ Separates the scalability aspects from the criteria that

determine the quality of the tree
◼ Builds an AVC-list: AVC (Attribute, Value, Class_label)
◼ AVC-set (of an attribute X )
◼ Projection of training dataset onto the attribute X and
class label where counts of individual class label are
aggregated
◼ AVC-group (of a node n )
◼ Set of AVC-sets of all predictor attributes at the node n

27
Rainforest: Training Set and Its AVC Sets

Training Examples AVC-set on Age AVC-set on income

age income studentcredit_rating
buys_computerAge Buy_Computer income Buy_Computer

<=30 high no fair no yes no

<=30 high no excellent no yes no
high 2 2
31…40 high no fair yes <=30 2 3
31..40 4 0 medium 4 2
>40 medium no fair yes
>40 low yes fair yes >40 3 2 low 3 1
>40 low yes excellent no
31…40 low yes excellent yes
AVC-set on
<=30 medium no fair no AVC-set on Student
credit_rating
<=30 low yes fair yes
student Buy_Computer
>40 medium yes fair yes Credit
Buy_Computer

<=30 medium yes excellent yes yes no rating yes no

31…40 medium no excellent yes yes 6 1 fair 6 2
31…40 high yes fair yes no 3 4 excellent 3 3
>40 medium no excellent no
28
Chapter 4. Classification: Basic Concepts

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
29
Bayesian Classification: Why?
◼ A statistical classifier: performs probabilistic prediction, i.e.,
predicts class membership probabilities
◼ Foundation: Based on Bayes’ Theorem.
◼ Performance: A simple Bayesian classifier, naïve Bayesian
classifier, has comparable performance with decision tree and
selected neural network classifiers
◼ Incremental: Each training example can incrementally
increase/decrease the probability that a hypothesis is correct —
prior knowledge can be combined with observed data
◼ Standard: Even when Bayesian methods are computationally
intractable, they can provide a standard of optimal decision
making against which other methods can be measured
30
Bayes’ Theorem: Basics
M
◼ Total probability Theorem: P(B) =  P(B | A )P( A )
i i
i =1

◼ Bayes’ Theorem: P( H | X) = P(X | H ) P( H ) = P(X | H ) P( H ) / P(X)

P(X)
◼ Let X be a data sample (“evidence”): class label is unknown
◼ Let H be a hypothesis that X belongs to class C
◼ Classification is to determine P(H|X), (i.e., posteriori probability): the
probability that the hypothesis holds given the observed data sample X
◼ P(H) (prior probability): the initial probability
◼ E.g., X will buy computer, regardless of age, income, …

◼ P(X): probability that sample data is observed

◼ P(X|H) (likelihood): the probability of observing the sample X, given that
the hypothesis holds
◼ E.g., Given that X will buy computer, the prob. that X is 31..40,

medium income
31
Prediction Based on Bayes’ Theorem
◼ Given training data X, posteriori probability of a hypothesis H,
P(H|X), follows the Bayes’ theorem

P( H | X) = P(X | H ) P(H ) = P(X | H ) P(H ) / P(X)

P(X)
◼ Informally, this can be viewed as
posteriori = likelihood x prior/evidence
◼ Predicts X belongs to Ci iff the probability P(Ci|X) is the highest
among all the P(Ck|X) for all the k classes
◼ Practical difficulty: It requires initial knowledge of many
probabilities, involving significant computational cost

32
Classification Is to Derive the Maximum Posteriori
◼ Let D be a training set of tuples and their associated class
labels, and each tuple is represented by an n-D attribute vector
X = (x1, x2, …, xn)
◼ Suppose there are m classes C1, C2, …, Cm.
◼ Classification is to derive the maximum posteriori, i.e., the
maximal P(Ci|X)
◼ This can be derived from Bayes’ theorem
P(X | C )P(C )
P(C | X) = i i
i P(X)
◼ Since P(X) is constant for all classes, only
P(C | X) = P(X | C )P(C )
i i i
needs to be maximized

33
Naïve Bayes Classifier
◼ A simplified assumption: attributes are conditionally
independent (i.e., no dependence relation between
attributes): n
P( X | C i) =  P( x | C i) = P( x | C i)  P( x | C i)  ... P( x | C i)
k 1 2 n
k =1
◼ This greatly reduces the computation cost: Only counts the
class distribution
◼ If Ak is categorical, P(xk|Ci) is the # of tuples in Ci having value xk
for Ak divided by |Ci, D| (# of tuples of Ci in D)
◼ If Ak is continous-valued, P(xk|Ci) is usually computed based on
Gaussian distribution with a mean μ and standard deviation σ
( x− )2
1 −
g ( x,  ,  ) = e 2 2
and P(xk|Ci) is 2 

P(X | C i) = g ( xk , Ci ,  Ci )
34
Naïve Bayes Classifier: Training Dataset
age income studentcredit_rating
buys_compu
<=30 high no fair no
Class: <=30 high no excellent no
C1:buys_computer = ‘yes’ 31…40 high no fair yes
C2:buys_computer = ‘no’ >40 medium no fair yes
>40 low yes fair yes
Data to be classified: >40 low yes excellent no
31…40 low yes excellent yes
X = (age <=30,
<=30 medium no fair no
Income = medium, <=30 low yes fair yes
Student = yes >40 medium yes fair yes
Credit_rating = Fair) <=30 medium yes excellent yes
31…40 medium no excellent yes
31…40 high yes fair yes
>40 medium no excellent no
35
Naïve Bayes Classifier: An Example age income studentcredit_rating
buys_comp
<=30 high no fair no
<=30 high no excellent no
31…40 high no fair yes

◼ P(Ci): P(buys_computer = “yes”) = 9/14 = 0.643 >40

>40
medium
low
no fair
yes fair
yes
yes
>40 low yes excellent no

P(buys_computer = “no”) = 5/14= 0.357 31…40

<=30
low
medium
yes excellent
no fair
yes
no
<=30 low yes fair yes
◼ Compute P(X|Ci) for each class >40
<=30
medium yes fair
medium yes excellent
yes
yes
31…40 medium no excellent yes
P(age = “<=30” | buys_computer = “yes”) = 2/9 = 0.222 31…40
>40
high
medium
yes fair
no excellent
yes
no

P(age = “<= 30” | buys_computer = “no”) = 3/5 = 0.6

P(income = “medium” | buys_computer = “yes”) = 4/9 = 0.444
P(income = “medium” | buys_computer = “no”) = 2/5 = 0.4
P(student = “yes” | buys_computer = “yes) = 6/9 = 0.667
P(student = “yes” | buys_computer = “no”) = 1/5 = 0.2
P(credit_rating = “fair” | buys_computer = “yes”) = 6/9 = 0.667
P(credit_rating = “fair” | buys_computer = “no”) = 2/5 = 0.4
◼ X = (age <= 30 , income = medium, student = yes, credit_rating = fair)
P(X|Ci) : P(X|buys_computer = “yes”) = 0.222 x 0.444 x 0.667 x 0.667 = 0.044
P(X|buys_computer = “no”) = 0.6 x 0.4 x 0.2 x 0.4 = 0.019
P(X|Ci)*P(Ci) : P(X|buys_computer = “yes”) * P(buys_computer = “yes”) = 0.028
P(X|buys_computer = “no”) * P(buys_computer = “no”) = 0.007
Therefore, X belongs to class (“buys_computer = yes”) 36
Naïve Bayes Classifier: Comments
◼ Advantages
◼ Easy to implement

◼ Good results obtained in most of the cases

◼ Disadvantages
◼ Assumption: class conditional independence, therefore loss
of accuracy
◼ Practically, dependencies exist among variables

◼ E.g., hospitals: patients: Profile: age, family history, etc.

Symptoms: fever, cough etc., Disease: lung cancer,

diabetes, etc.
◼ Dependencies among these cannot be modeled by Naïve

Bayes Classifier
◼ How to deal with these dependencies? Bayesian Belief Networks
37
Chapter 4. Classification: Basic Concepts

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
38
Using IF-THEN Rules for Classification
◼ Represent the knowledge in the form of IF-THEN rules
R: IF age = youth AND student = yes THEN buys_computer = yes
◼ Rule antecedent/precondition vs. rule consequent

◼ Assessment of a rule: coverage and accuracy

◼ ncovers = # of tuples covered by R

◼ ncorrect = # of tuples correctly classified by R

coverage(R) = ncovers /|D| /* D: training data set */

accuracy(R) = ncorrect / ncovers
◼ If more than one rule are triggered, need conflict resolution
◼ Size ordering: assign the highest priority to the triggering rules that has

the “toughest” requirement (i.e., with the most attribute tests)

◼ Class-based ordering: decreasing order of prevalence or misclassification

cost per class

◼ Rule-based ordering (decision list): rules are organized into one long

priority list, according to some measure of rule quality or by experts

39
Rule Extraction from a Decision Tree
◼ Rules are easier to understand than large age?

trees <=30 31..40 >40

◼ One rule is created for each path from the student? credit rating?
yes
root to a leaf
no yes excellent fair
◼ Each attribute-value pair along a path forms a yes
no yes
conjunction: the leaf holds the class
prediction
◼ Rules are mutually exclusive and exhaustive
◼ Example: Rule extraction from our buys_computer decision-tree
IF age = young AND student = no THEN buys_computer = no
IF age = young AND student = yes THEN buys_computer = yes
IF age = mid-age THEN buys_computer = yes
IF age = old AND credit_rating = excellent THEN buys_computer = no
IF age = old AND credit_rating = fair THEN buys_computer = yes
40
Chapter 4. Classification: Basic Concepts

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
41
Model Evaluation and Selection
◼ Evaluation metrics: How can we measure accuracy? Other
metrics to consider?
◼ Use validation test set of class-labeled tuples instead of
training set when assessing accuracy
◼ Methods for estimating a classifier’s accuracy:
◼ Holdout method, random subsampling
◼ Cross-validation
◼ Bootstrap
◼ Comparing classifiers:
◼ Confidence intervals
◼ Cost-benefit analysis and ROC Curves
42
Classifier Evaluation Metrics: Confusion
Matrix
Confusion Matrix:
Actual class\Predicted class C1 ¬ C1
C1 True Positives (TP) False Negatives (FN)
¬ C1 False Positives (FP) True Negatives (TN)

Example of Confusion Matrix:

Actual class\Predicted buy_computer buy_computer Total
class = yes = no
buy_computer = yes 6954 46 7000
buy_computer = no 412 2588 3000
Total 7366 2634 10000

◼ Given m classes, an entry, CMi,j in a confusion matrix indicates

# of tuples in class i that were labeled by the classifier as class j
◼ May have extra rows/columns to provide totals
43
Classifier Evaluation Metrics: Accuracy,
Error Rate, Sensitivity and Specificity
A\P C ¬C ◼ Class Imbalance Problem:
C TP FN P
◼ One class may be rare, e.g.
¬C FP TN N
fraud, or HIV-positive
P’ N’ All
◼ Significant majority of the

◼ Classifier Accuracy, or negative class and minority of

recognition rate: percentage of the positive class
test set tuples that are correctly ◼ Sensitivity: True Positive
classified recognition rate
Accuracy = (TP + TN)/All ◼ Sensitivity = TP/P

◼ Error rate: 1 – accuracy, or ◼ Specificity: True Negative

Error rate = (FP + FN)/All recognition rate

◼ Specificity = TN/N

44
Classifier Evaluation Metrics:
Precision and Recall, and F-measures
◼ Precision: exactness – what % of tuples that the classifier
labeled as positive are actually positive

◼ Recall: completeness – what % of positive tuples did the

classifier label as positive?
◼ Perfect score is 1.0
◼ Inverse relationship between precision & recall
◼ F measure (F1 or F-score): harmonic mean of precision and
recall,

◼ Fß: weighted measure of precision and recall

◼ assigns ß times as much weight to recall as to precision

45
Classifier Evaluation Metrics: Example

Actual Class\Predicted class cancer = yes cancer = no Total Recognition(%)

cancer = yes 90 210 300 (sensitivity=
cancer = no 140 9560 9700 (specificity)=
Total 230 9770 10000 (accuracy)=

◼ Precision = Recall =

46
Classifier Evaluation Metrics: Example

Actual Class\Predicted class cancer = yes cancer = no Total Recognition(%)

cancer = yes 90 210 300 30.00 (sensitivity
cancer = no 140 9560 9700 98.56 (specificity)
Total 230 9770 10000 96.40 (accuracy)

◼ Precision = 90/230 = 39.13% Recall = 90/300 = 30.00%

47
Evaluating Classifier Accuracy:
Holdout & Cross-Validation Methods
◼ Holdout method
◼ Given data is randomly partitioned into two independent sets

◼ Training set (e.g., 2/3) for model construction

◼ Test set (e.g., 1/3) for accuracy estimation

◼ Random sampling: a variation of holdout

◼ Repeat holdout k times, accuracy = avg. of the accuracies

obtained
◼ Cross-validation (k-fold, where k = 10 is most popular)
◼ Randomly partition the data into k mutually exclusive subsets,
each approximately of equal size
◼ At i-th iteration, use Di as test set and others as training set

◼ Leave-one-out: k folds where k = # of tuples, for small sized data

◼ *Stratified cross-validation*: folds are stratified so that class

dist. in each fold is approx. the same as that in the initial data
48
Evaluating Classifier Accuracy: Bootstrap
◼ Bootstrap
◼ Works well with small data sets
◼ Samples the given training tuples uniformly with replacement
◼ i.e., each time a tuple is selected, it is equally likely to be selected
again and re-added to the training set
◼ Several bootstrap methods, and a common one is .632 boostrap
◼ A data set with d tuples is sampled d times, with replacement, resulting in
a training set of d samples. The data tuples that did not make it into the
training set end up forming the test set. About 63.2% of the original data
end up in the bootstrap, and the remaining 36.8% form the test set (since
(1 – 1/d)d ≈ e-1 = 0.368)
◼ Repeat the sampling procedure k times, overall accuracy of the model:

49
Estimating Confidence Intervals:
Classifier Models M1 vs. M2
◼ Suppose we have 2 classifiers, M1 and M2, which one is better?

◼ Use 10-fold cross-validation to obtain and

◼ These mean error rates are just estimates of error on the true
population of future data cases

◼ What if the difference between the 2 error rates is just

attributed to chance?

◼ Use a test of statistical significance

◼ Obtain confidence limits for our error estimates

50
Estimating Confidence Intervals:
Null Hypothesis
◼ Perform 10-fold cross-validation
◼ Assume samples follow a t distribution with k–1 degrees of
freedom (here, k=10)
◼ Use t-test (or Student’s t-test)
◼ Null Hypothesis: M1 & M2 are the same
◼ If we can reject null hypothesis, then
◼ we conclude that the difference between M1 & M2 is
statistically significant
◼ Chose model with lower error rate

51
Estimating Confidence Intervals: t-test

◼ If only 1 test set available: pairwise comparison

◼ For ith round of 10-fold cross-validation, the same cross
partitioning is used to obtain err(M1)i and err(M2)i
◼ Average over 10 rounds to get
and
◼ t-test computes t-statistic with k-1 degrees of
freedom:
where

◼ If two test sets available: use non-paired t-test

where

where k1 & k2 are # of cross-validation samples used for M1 & M2, resp.
52
Estimating Confidence Intervals:
Table for t-distribution

◼ Symmetric
◼ Significance level,
e.g., sig = 0.05 or
5% means M1 & M2
are significantly
different for 95% of
population
◼ Confidence limit, z
= sig/2

53
Estimating Confidence Intervals:
Statistical Significance
◼ Are M1 & M2 significantly different?
◼ Compute t. Select significance level (e.g. sig = 5%)

◼ Consult table for t-distribution: Find t value corresponding

to k-1 degrees of freedom (here, 9)

◼ t-distribution is symmetric: typically upper % points of

distribution shown → look up value for confidence limit

z=sig/2 (here, 0.025)
◼ If t > z or t < -z, then t value lies in rejection region:

◼ Reject null hypothesis that mean error rates of M1 & M2

are same
◼ Conclude: statistically significant difference between M1

& M2
◼ Otherwise, conclude that any difference is chance
54
Model Selection: ROC Curves
◼ ROC (Receiver Operating
Characteristics) curves: for visual
comparison of classification models
◼ Originated from signal detection theory
◼ Shows the trade-off between the true
positive rate and the false positive rate
◼ The area under the ROC curve is a ◼ Vertical axis
measure of the accuracy of the model represents the true
positive rate
◼ Rank the test tuples in decreasing ◼ Horizontal axis rep.
order: the one that is most likely to the false positive rate
belong to the positive class appears at ◼ The plot also shows a
the top of the list diagonal line
◼ The closer to the diagonal line (i.e., the ◼ A model with perfect
closer the area is to 0.5), the less accuracy will have an
accurate is the model area of 1.0
55
Issues Affecting Model Selection
◼ Accuracy
◼ classifier accuracy: predicting class label
◼ Speed
◼ time to construct the model (training time)
◼ time to use the model (classification/prediction time)
◼ Robustness: handling noise and missing values
◼ Scalability: efficiency in disk-resident databases
◼ Interpretability
◼ understanding and insight provided by the model
◼ Other measures, e.g., goodness of rules, such as decision tree
size or compactness of classification rules
56
Chapter 4. Classification: Basic Concepts

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
57
Ensemble Methods: Increasing the Accuracy

◼ Ensemble methods
◼ Use a combination of models to increase accuracy

◼ Combine a series of k learned models, M1, M2, …, Mk, with

the aim of creating an improved model M*

◼ Popular ensemble methods
◼ Bagging: averaging the prediction over a collection of

classifiers
◼ Boosting: weighted vote with a collection of classifiers

◼ Ensemble: combining a set of heterogeneous classifiers

58
Bagging: Boostrap Aggregation
◼ Analogy: Diagnosis based on multiple doctors’ majority vote
◼ Training
◼ Given a set D of d tuples, at each iteration i, a training set Di of d tuples

is sampled with replacement from D (i.e., bootstrap)

◼ A classifier model Mi is learned for each training set Di

◼ Classification: classify an unknown sample X

◼ Each classifier Mi returns its class prediction

◼ The bagged classifier M* counts the votes and assigns the class with the

most votes to X
◼ Prediction: can be applied to the prediction of continuous values by taking
the average value of each prediction for a given test tuple
◼ Accuracy
◼ Often significantly better than a single classifier derived from D

◼ For noise data: not considerably worse, more robust

◼ Proved improved accuracy in prediction

59
Boosting
◼ Analogy: Consult several doctors, based on a combination of
weighted diagnoses—weight assigned based on the previous
diagnosis accuracy
◼ How boosting works?
◼ Weights are assigned to each training tuple
◼ A series of k classifiers is iteratively learned
◼ After a classifier Mi is learned, the weights are updated to
allow the subsequent classifier, Mi+1, to pay more attention to
the training tuples that were misclassified by Mi
◼ The final M* combines the votes of each individual classifier,
where the weight of each classifier's vote is a function of its
accuracy
◼ Boosting algorithm can be extended for numeric prediction
◼ Comparing with bagging: Boosting tends to have greater accuracy,
but it also risks overfitting the model to misclassified data 60
Adaboost (Freund and Schapire, 1997)
◼ Given a set of d class-labeled tuples, (X1, y1), …, (Xd, yd)
◼ Initially, all the weights of tuples are set the same (1/d)
◼ Generate k classifiers in k rounds. At round i,
◼ Tuples from D are sampled (with replacement) to form a training set
Di of the same size
◼ Each tuple’s chance of being selected is based on its weight
◼ A classification model Mi is derived from Di
◼ Its error rate is calculated using Di as a test set
◼ If a tuple is misclassified, its weight is increased, o.w. it is decreased
◼ Error rate: err(Xj) is the misclassification error of tuple Xj. Classifier Mi
error rate is the sum of the weights of the misclassified tuples:
d
error ( M i ) =  w j  err ( X j )
j
◼ The weight of classifier Mi’s vote is 1 − error( M i )
log
error( M i )
61
Random Forest (Breiman 2001)
◼ Random Forest:
◼ Each classifier in the ensemble is a decision tree classifier and is

generated using a random selection of attributes at each node to

determine the split
◼ During classification, each tree votes and the most popular class is

returned
◼ Two Methods to construct Random Forest:
◼ Forest-RI (random input selection): Randomly select, at each node, F

attributes as candidates for the split at the node. The CART methodology
is used to grow the trees to maximum size
◼ Forest-RC (random linear combinations): Creates new attributes (or

features) that are a linear combination of the existing attributes

(reduces the correlation between individual classifiers)
◼ Comparable in accuracy to Adaboost, but more robust to errors and outliers
◼ Insensitive to the number of attributes selected for consideration at each
split, and faster than bagging or boosting
62
Classification of Class-Imbalanced Data Sets
◼ Class-imbalance problem: Rare positive example but numerous
negative ones, e.g., medical diagnosis, fraud, oil-spill, fault, etc.
◼ Traditional methods assume a balanced distribution of classes
and equal error costs: not suitable for class-imbalanced data
◼ Typical methods for imbalance data in 2-class classification:
◼ Oversampling: re-sampling of data from positive class

◼ Under-sampling: randomly eliminate tuples from negative

class
◼ Threshold-moving: moves the decision threshold, t, so that
the rare class tuples are easier to classify, and hence, less
chance of costly false negative errors
◼ Ensemble techniques: Ensemble multiple classifiers
introduced above
◼ Still difficult for class imbalance problem on multiclass tasks
63
Chapter 4. Classification: Basic Concepts

◼ Classification: Basic Concepts

◼ Decision Tree Induction
◼ Bayes Classification Methods
◼ Rule-Based Classification
◼ Model Evaluation and Selection
◼ Techniques to Improve Classification Accuracy:
Ensemble Methods
◼ Summary
64
Summary (I)
◼ Classification is a form of data analysis that extracts models
describing important data classes.
◼ Effective and scalable methods have been developed for decision
tree induction, Naive Bayesian classification, rule-based
classification, and many other classification methods.
◼ Evaluation metrics include: accuracy, sensitivity, specificity,
precision, recall, F measure, and Fß measure.
◼ Stratified k-fold cross-validation is recommended for accuracy
estimation. Bagging and boosting can be used to increase overall
accuracy by learning and combining a series of individual models.

65
Summary (II)

◼ Significance tests and ROC curves are useful for model selection.
◼ There have been numerous comparisons of the different
classification methods; the matter remains a research topic
◼ No single method has been found to be superior over all others
for all data sets
◼ Issues such as accuracy, training time, robustness, scalability,
and interpretability must be considered and can involve trade-
offs, further complicating the quest for an overall superior
method

66
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Generation Computer Systems, 13, 1997
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an application to boosting. J. Computer and System Sciences, 1997.
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construction of large datasets. VLDB’98.
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Construction. SIGMOD'99.
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Mining, Inference, and Prediction. Springer-Verlag, 2001.
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Class-Association Rules, ICDM'01.
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and training time of thirty-three old and new classification algorithms. Machine
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mining. EDBT'96.
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Disciplinary Survey, Data Mining and Knowledge Discovery 2(4): 345-389, 1998
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70
Issues: Evaluating Classification Methods
◼ Accuracy
◼ classifier accuracy: predicting class label

◼ predictor accuracy: guessing value of predicted attributes

◼ Speed
◼ time to construct the model (training time)

◼ time to use the model (classification/prediction time)

◼ Robustness: handling noise and missing values

◼ Scalability: efficiency in disk-resident databases
◼ Interpretability
◼ understanding and insight provided by the model

◼ Other measures, e.g., goodness of rules, such as decision tree

size or compactness of classification rules

71
Predictor Error Measures

◼ Measure predictor accuracy: measure how far off the predicted value is from
the actual known value
◼ Loss function: measures the error betw. yi and the predicted value yi’
◼ Absolute error: | yi – yi’|
◼ Squared error: (yi – yi’)2
◼ Test error (generalization error):
d
the average loss over the test set
d

◼ Mean absolute error:  | y − yMean

i =1
'|i squared error:
i ( y − y ')
i =1
i i
2

d d
d

◼ Relative absolute error:

d

 | y −Relative
y '|
i squared error:
i
 ( yi − yi ' ) 2
i =1
i =1
d d
| y
i =1
i −y|
(y
i =1
i − y)2
The mean squared-error exaggerates the presence of outliers
Popularly use (square) root mean-square error, similarly, root relative
squared error
72
Scalable Decision Tree Induction Methods

◼ SLIQ (EDBT’96 — Mehta et al.)

◼ Builds an index for each attribute and only class list and the

current attribute list reside in memory

◼ SPRINT (VLDB’96 — J. Shafer et al.)
◼ Constructs an attribute list data structure

◼ PUBLIC (VLDB’98 — Rastogi & Shim)

◼ Integrates tree splitting and tree pruning: stop growing the

tree earlier
◼ RainForest (VLDB’98 — Gehrke, Ramakrishnan & Ganti)
◼ Builds an AVC-list (attribute, value, class label)

◼ BOAT (PODS’99 — Gehrke, Ganti, Ramakrishnan & Loh)

◼ Uses bootstrapping to create several small samples

73
Data Cube-Based Decision-Tree Induction
◼ Integration of generalization with decision-tree induction
(Kamber et al.’97)
◼ Classification at primitive concept levels
◼ E.g., precise temperature, humidity, outlook, etc.
◼ Low-level concepts, scattered classes, bushy classification-
trees
◼ Semantic interpretation problems
◼ Cube-based multi-level classification
◼ Relevance analysis at multi-levels
◼ Information-gain analysis with dimension + level
74