Data Mining Classification: Basic Concepts, Decision Trees, and Model Evaluation

1
Data Mining Classification Basic Concepts,
Decision Trees, and Model Evaluation

• Lecture Notes for Chapter 4
• Introduction to Data Mining
• by
• Tan, Steinbach, Kumar

2
Classification Definition

• Given a collection of records (training set )
• Each record contains a set of attributes, one of
  the attributes is the class.
• Find a model for class attribute as a function
  of the values of other attributes.
• Goal previously unseen records should be
  assigned a class as accurately as possible.
• A test set is used to determine the accuracy of
  the model. Usually, the given data set is divided
  into training and test sets, with training set
  used to build the model and test set used to
  validate it.

3
Illustrating Classification Task
4
Examples of Classification Task

• Predicting tumor cells as benign or malignant
• Classifying credit card transactions as
  legitimate or fraudulent
• Classifying secondary structures of protein as
  alpha-helix, beta-sheet, or random coil
• Categorizing news stories as finance, weather,
  entertainment, sports, etc

5
Classification Techniques

• Decision Tree based Methods
• Rule-based Methods
• Memory based reasoning
• Neural Networks
• Naïve Bayes and Bayesian Belief Networks
• Support Vector Machines

6
Example of a Decision Tree
Splitting Attributes
Refund
Yes
No
MarSt
NO
Married
Single, Divorced
TaxInc
NO
lt 80K
gt 80K
YES
NO
Model Decision Tree
Training Data
7
Another Example of Decision Tree
categorical
categorical
continuous
class
Single, Divorced
MarSt
Married
Refund
NO
No
Yes
TaxInc
lt 80K
gt 80K
YES
NO
There could be more than one tree that fits the
same data!
8
Decision Tree Classification Task
Decision Tree
9
Apply Model to Test Data
Test Data
Start from the root of tree.
10
Apply Model to Test Data
Test Data
11
Apply Model to Test Data
Test Data
Refund
Yes
No
MarSt
NO
Married
Single, Divorced
TaxInc
NO
lt 80K
gt 80K
YES
NO
12
Apply Model to Test Data
Test Data
Refund
Yes
No
MarSt
NO
Married
Single, Divorced
TaxInc
NO
lt 80K
gt 80K
YES
NO
13
Apply Model to Test Data
Test Data
Refund
Yes
No
MarSt
NO
Married
Single, Divorced
TaxInc
NO
lt 80K
gt 80K
YES
NO
14
Apply Model to Test Data
Test Data
Refund
Yes
No
MarSt
NO
Assign Cheat to No
Married
Single, Divorced
TaxInc
NO
lt 80K
gt 80K
YES
NO
15
Decision Tree Classification Task
Decision Tree
16
Decision Tree Induction

• Many Algorithms
• Hunts Algorithm (one of the earliest)
• CART
• ID3, C4.5
• SLIQ,SPRINT

17
General Structure of Hunts Algorithm

• Let Dt be the set of training records that reach
  a node t
• General Procedure
• If Dt contains records that belong the same class
  yt, then t is a leaf node labeled as yt
• If Dt is an empty set, then t is a leaf node
  labeled by the default class, yd
• If Dt contains records that belong to more than
  one class, use an attribute test to split the
  data into smaller subsets. Recursively apply the
  procedure to each subset.

Dt
?
18
Hunts Algorithm
Dont Cheat
19
Tree Induction

• Greedy strategy.
• Split the records based on an attribute test that
  optimizes certain criterion.
• Issues
• Determine how to split the records
• How to specify the attribute test condition?
• How to determine the best split?
• Determine when to stop splitting

20
Tree Induction

• Greedy strategy.
• Split the records based on an attribute test that
  optimizes certain criterion.
• Issues
• Determine how to split the records
• How to specify the attribute test condition?
• How to determine the best split?
• Determine when to stop splitting

21
How to Specify Test Condition?

• Depends on attribute types
• Nominal
• Ordinal
• Continuous
• Depends on number of ways to split
• 2-way split
• Multi-way split

22
Splitting Based on Nominal Attributes

• Multi-way split Use as many partitions as
  distinct values.
• Binary split Divides values into two subsets.
  Need to find optimal partitioning.

OR
23
Splitting Based on Ordinal Attributes

• Multi-way split Use as many partitions as
  distinct values.
• Binary split Divides values into two subsets.
  Need to find optimal partitioning.
• What about this split?

OR
24
Splitting Based on Continuous Attributes

• Different ways of handling
• Discretization to form an ordinal categorical
  attribute
• Static discretize once at the beginning
• Dynamic ranges can be found by equal interval
  bucketing, equal frequency bucketing (percenti
  les), or clustering.
• Binary Decision (A lt v) or (A ? v)
• consider all possible splits and finds the best
  cut
• can be more compute intensive

25
Splitting Based on Continuous Attributes
26
Tree Induction

• Greedy strategy.
• Split the records based on an attribute test that
  optimizes certain criterion.
• Issues
• Determine how to split the records
• How to specify the attribute test condition?
• How to determine the best split?
• Determine when to stop splitting

27
How to determine the Best Split
Before Splitting 10 records of class 0, 10
records of class 1
Which test condition is the best?
28
How to determine the Best Split

• Greedy approach
• Nodes with homogeneous class distribution are
  preferred
• Need a measure of node impurity

Non-homogeneous, High degree of impurity
Homogeneous, Low degree of impurity
29
Measures of Node Impurity

• Gini Index
• Entropy
• Misclassification error

30
How to Find the Best Split
Before Splitting
A?
B?
Yes
No
Yes
No
Node N1
Node N2
Node N3
Node N4
Gain M0 M12 vs M0 M34
31
Measure of Impurity GINI

• Gini Index for a given node t
• (NOTE p( j t) is the relative frequency of
  class j at node t).
• Maximum (1 - 1/nc) when records are equally
  distributed among all classes, implying least
  interesting information
• Minimum (0.0) when all records belong to one
  class, implying most interesting information

32
Examples for computing GINI
P(C1) 0/6 0 P(C2) 6/6 1 Gini 1
P(C1)2 P(C2)2 1 0 1 0
P(C1) 1/6 P(C2) 5/6 Gini 1
(1/6)2 (5/6)2 0.278
P(C1) 2/6 P(C2) 4/6 Gini 1
(2/6)2 (4/6)2 0.444
33
Splitting Based on GINI

• Used in CART, SLIQ, SPRINT.
• When a node p is split into k partitions
  (children), the quality of split is computed as,
• where, ni number of records at child i,
• n number of records at node p.

34
Binary Attributes Computing GINI Index

• Splits into two partitions
• Effect of Weighing partitions
• Larger and Purer Partitions are sought for.

B?
Yes
No
Node N1
Node N2
Gini(N1) 1 (5/7)2 (2/7)2 0.408 Gini(N2)
1 (1/5)2 (4/5)2 0.320
Gini(Children) 7/12 0.408 5/12
0.320 0.371
35
Categorical Attributes Computing Gini Index

• For each distinct value, gather counts for each
  class in the dataset
• Use the count matrix to make decisions

Multi-way split
Two-way split (find best partition of values)
36
Continuous Attributes Computing Gini Index

• Use Binary Decisions based on one value
• Several Choices for the splitting value
• Number of possible splitting values Number of
  distinct values
• Each splitting value has a count matrix
  associated with it
• Class counts in each of the partitions, A lt v and
  A ? v
• Simple method to choose best v
• For each v, scan the database to gather count
  matrix and compute its Gini index
• Computationally Inefficient! Repetition of work.

37
Continuous Attributes Computing Gini Index...

• For efficient computation for each attribute,
• Sort the attribute on values
• Linearly scan these values, each time updating
  the count matrix and computing gini index
• Choose the split position that has the least gini
  index

38
Alternative Splitting Criteria based on INFO

• Entropy at a given node t
• (NOTE p( j t) is the relative frequency of
  class j at node t).
• Measures homogeneity of a node.
• Maximum (log nc) when records are equally
  distributed among all classes implying least
  information
• Minimum (0.0) when all records belong to one
  class, implying most information
• Entropy based computations are similar to the
  GINI index computations

39
Examples for computing Entropy
P(C1) 0/6 0 P(C2) 6/6 1 Entropy 0
log 0 1 log 1 0 0 0
P(C1) 1/6 P(C2) 5/6 Entropy
(1/6) log2 (1/6) (5/6) log2 (1/6) 0.65
P(C1) 2/6 P(C2) 4/6 Entropy
(2/6) log2 (2/6) (4/6) log2 (4/6) 0.92
40
Splitting Based on INFO...

• Information Gain
• Parent Node, p is split into k partitions
• ni is number of records in partition i
• Measures Reduction in Entropy achieved because of
  the split. Choose the split that achieves most
  reduction (maximizes GAIN)
• Used in ID3 and C4.5
• Disadvantage Tends to prefer splits that result
  in large number of partitions, each being small
  but pure.

41
Splitting Based on INFO...

• Gain Ratio
• Parent Node, p is split into k partitions
• ni is the number of records in partition i
• Adjusts Information Gain by the entropy of the
  partitioning (SplitINFO). Higher entropy
  partitioning (large number of small partitions)
  is penalized!
• Used in C4.5
• Designed to overcome the disadvantage of
  Information Gain

42
Splitting Criteria based on Classification Error

• Classification error at a node t
• Measures misclassification error made by a node.
• Maximum (1 - 1/nc) when records are equally
  distributed among all classes, implying least
  interesting information
• Minimum (0.0) when all records belong to one
  class, implying most interesting information

43
Examples for Computing Error
P(C1) 0/6 0 P(C2) 6/6 1 Error 1
max (0, 1) 1 1 0
P(C1) 1/6 P(C2) 5/6 Error 1 max
(1/6, 5/6) 1 5/6 1/6
P(C1) 2/6 P(C2) 4/6 Error 1 max
(2/6, 4/6) 1 4/6 1/3
44
Comparison among Splitting Criteria
For a 2-class problem
45
Misclassification Error vs Gini
A?
Yes
No
Node N1
Node N2
Gini(N1) 1 (3/3)2 (0/3)2 0 Gini(N2)
1 (4/7)2 (3/7)2 0.489
Gini(Children) 3/10 0 7/10 0.489
0.342 Gini improves !!
46
Tree Induction

• Greedy strategy.
• Split the records based on an attribute test that
  optimizes certain criterion.
• Issues
• Determine how to split the records
• How to specify the attribute test condition?
• How to determine the best split?
• Determine when to stop splitting

47
Stopping Criteria for Tree Induction

• Stop expanding a node when all the records belong
  to the same class
• Stop expanding a node when all the records have
  similar attribute values
• Early termination (pre-pruning)

48
Decision Tree Based Classification

• Advantages
• Inexpensive to construct
• Extremely fast at classifying unknown records
• Easy to interpret for small-sized trees
• Accuracy is comparable to other classification
  techniques for many simple data sets

49
Example C4.5

• Simple depth-first construction.
• Uses Information Gain
• Sorts Continuous Attributes at each node.
• Needs entire data to fit in memory.
• Unsuitable for Large Datasets.
• Needs out-of-core sorting.

50
Practical Issues of Classification

• Underfitting and Overfitting
• Missing Values
• Costs of Classification

51
Underfitting and Overfitting (Example)
500 circular and 500 triangular data
points. Circular points 0.5 ? sqrt(x12x22) ?
1 Triangular points sqrt(x12x22) gt 0.5
or sqrt(x12x22) lt 1
52
Underfitting and Overfitting
Overfitting
Underfitting when model is too simple, both
training and test errors are large
53
Overfitting due to Noise
Decision boundary is distorted by noise point
54
Overfitting due to Insufficient Examples
Lack of data points in the lower half of the
diagram makes it difficult to predict correctly
the class labels of that region - Insufficient
number of training records in the region causes
the decision tree to predict the test examples
using other training records that are irrelevant
to the classification task
55
Notes on Overfitting

• Overfitting results in decision trees that are
  more complex than necessary
• Training error no longer provides a good estimate
  of how well the tree will perform on previously
  unseen records
• Need new ways for estimating errors

56
Estimating Generalization Errors

• Re-substitution errors error on training (? e(t)
  )
• Generalization errors error on testing (? e(t))
• Methods for estimating generalization errors
• Optimistic approach e(t) e(t)
• Pessimistic approach
• For each leaf node e(t) (e(t)0.5)
• Total errors e(T) e(T) N ? 0.5 (N number
  of leaf nodes)
• For a tree with 30 leaf nodes and 10 errors on
  training (out of 1000 instances)
  Training error 10/1000 1
• Generalization error (10
  30?0.5)/1000 2.5
• Reduced error pruning (REP)
• uses validation data set to estimate
  generalization error

57
Occams Razor

• Given two models of similar generalization
  errors, one should prefer the simpler model over
  the more complex model
• For complex models, there is a greater chance
  that it was fitted accidentally by errors in data
• Therefore, one should include model complexity
  when evaluating a model

58
Minimum Description Length (MDL)

• Cost(Model,Data) Cost(DataModel) Cost(Model)
• Cost is the number of bits needed for encoding.
• Search for the least costly model.
• Cost(DataModel) encodes the misclassification
  errors.
• Cost(Model) uses node encoding (number of
  children) plus splitting condition encoding.

59
How to Address Overfitting

• Pre-Pruning (Early Stopping Rule)
• Stop the algorithm before it becomes a
  fully-grown tree
• Typical stopping conditions for a node
• Stop if all instances belong to the same class
• Stop if all the attribute values are the same
• More restrictive conditions
• Stop if number of instances is less than some
  user-specified threshold
• Stop if class distribution of instances are
  independent of the available features (e.g.,
  using ? 2 test)
• Stop if expanding the current node does not
  improve impurity measures (e.g., Gini or
  information gain).

60
How to Address Overfitting

• Post-pruning
• Grow decision tree to its entirety
• Trim the nodes of the decision tree in a
  bottom-up fashion
• If generalization error improves after trimming,
  replace sub-tree by a leaf node.
• Class label of leaf node is determined from
  majority class of instances in the sub-tree

61
Example of Post-Pruning
Training Error (Before splitting)
10/30 Pessimistic error (10 0.5)/30
10.5/30 Training Error (After splitting)
9/30 Pessimistic error (After splitting) (9
4 ? 0.5)/30 11/30 PRUNE!
Class Yes 20
Class No 10
Error 10/30 Error 10/30
Class Yes 8
Class No 4
Class Yes 3
Class No 4
Class Yes 4
Class No 1
Class Yes 5
Class No 1
62
Examples of Post-pruning

• Optimistic error?
• Pessimistic error?
• Reduced error pruning?

Case 1
Dont prune for both cases
Dont prune case 1, prune case 2
Case 2
Depends on validation set
63
Handling Missing Attribute Values

• Missing values affect decision tree construction
  in three different ways
• Affects how impurity measures are computed
• Affects how to distribute instance with missing
  value to child nodes
• Affects how a test instance with missing value is
  classified

64
Computing Impurity Measure
Before Splitting Entropy(Parent) -0.3
log(0.3)-(0.7)log(0.7) 0.8813
Split on Refund Entropy(RefundYes) 0
Entropy(RefundNo) -(2/6)log(2/6)
(4/6)log(4/6) 0.9183 Entropy(Children)
0.3 (0) 0.6 (0.9183) 0.551 Gain 0.9 ?
(0.8813 0.551) 0.3303
Missing value
65
Distribute Instances
Refund
Yes
No
Probability that RefundYes is 3/9 Probability
that RefundNo is 6/9 Assign record to the left
child with weight 3/9 and to the right child
with weight 6/9
Refund
Yes
No
66
Classify Instances
New record
Married Single Divorced Total
ClassNo 3 1 0 4
ClassYes 0 16/9 1 2.67
Total 3 2.67 1 6.67
Refund
Yes
No
MarSt
NO
Single, Divorced
Married
Probability that Marital Status Married is
3/6.67 Probability that Marital Status
Single,Divorced is 3.67/6.67
TaxInc
NO
lt 80K
gt 80K
YES
NO
67
Search Strategy

• Finding an optimal decision tree is NP-hard
• The algorithm presented so far uses a greedy,
  top-down, recursive partitioning strategy to
  induce a reasonable solution
• Other strategies?
• Bottom-up (CART)
• Bi-directional

68
Decision Boundary

• Border line between two neighboring regions of
  different classes is known as decision boundary
• Decision boundary is parallel to axes because
  test condition involves a single attribute
  at-a-time

69
Oblique Decision Trees

• Test condition may involve multiple attributes
• More expressive representation
• Finding optimal test condition is
  computationally expensive

70
Tree Replication

• Same subtree appears in multiple branches

71
Model Evaluation

• Metrics for Performance Evaluation
• How to evaluate the performance of a model?
• Methods for Performance Evaluation
• How to obtain reliable estimates?
• Methods for Model Comparison
• How to compare the relative performance among
  competing models?

72
Model Evaluation

• Metrics for Performance Evaluation
• How to evaluate the performance of a model?
• Methods for Performance Evaluation
• How to obtain reliable estimates?
• Methods for Model Comparison
• How to compare the relative performance among
  competing models?

73
Metrics for Performance Evaluation

• Focus on the predictive capability of a model
• Rather than how fast it takes to classify or
  build models, scalability, etc.
• Confusion Matrix

PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS ClassYes ClassNo
ACTUALCLASS ClassYes a b
ACTUALCLASS ClassNo c d
a TP (true positive) b FN (false negative) c
FP (false positive) d TN (true negative)
74
Metrics for Performance Evaluation

• Most widely-used metric

PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS ClassYes ClassNo
ACTUALCLASS ClassYes a(TP) b(FN)
ACTUALCLASS ClassNo c(FP) d(TN)
75
Limitation of Accuracy

• Consider a 2-class problem
• Number of Class 0 examples 9990
• Number of Class 1 examples 10
• If model predicts everything to be class 0,
  accuracy is 9990/10000 99.9
• Accuracy is misleading because model does not
  detect any class 1 example

76
Cost Matrix
PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS C(ij) ClassYes ClassNo
ACTUALCLASS ClassYes C(YesYes) C(NoYes)
ACTUALCLASS ClassNo C(YesNo) C(NoNo)
C(ij) Cost of misclassifying class j example as
class i
77
Computing Cost of Classification
Cost Matrix PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS C(ij) -
ACTUALCLASS -1 100
ACTUALCLASS - 1 0
Model M1 PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS -
ACTUALCLASS 150 40
ACTUALCLASS - 60 250
Model M2 PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS -
ACTUALCLASS 250 45
ACTUALCLASS - 5 200
Accuracy 80 Cost 3910
Accuracy 90 Cost 4255
78
Cost vs Accuracy
Count PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS ClassYes ClassNo
ACTUALCLASS ClassYes a b
ACTUALCLASS ClassNo c d
Cost PREDICTED CLASS PREDICTED CLASS PREDICTED CLASS
ACTUALCLASS ClassYes ClassNo
ACTUALCLASS ClassYes p q
ACTUALCLASS ClassNo q p
79
Cost-Sensitive Measures

• Precision is biased towards C(YesYes)
  C(YesNo)
• Recall is biased towards C(YesYes) C(NoYes)
• F-measure is biased towards all except C(NoNo)

80
Model Evaluation

• Metrics for Performance Evaluation
• How to evaluate the performance of a model?
• Methods for Performance Evaluation
• How to obtain reliable estimates?
• Methods for Model Comparison
• How to compare the relative performance among
  competing models?

81
Methods for Performance Evaluation

• How to obtain a reliable estimate of performance?
• Performance of a model may depend on other
  factors besides the learning algorithm
• Class distribution
• Cost of misclassification
• Size of training and test sets

82
Learning Curve

• Learning curve shows how accuracy changes with
  varying sample size
• Requires a sampling schedule for creating
  learning curve
• Arithmetic sampling(Langley, et al)
• Geometric sampling(Provost et al)
• Effect of small sample size
• Bias in the estimate
• Variance of estimate

83
Methods of Estimation

• Holdout
• Reserve 2/3 for training and 1/3 for testing
• Random subsampling
• Repeated holdout
• Cross validation
• Partition data into k disjoint subsets
• k-fold train on k-1 partitions, test on the
  remaining one
• Leave-one-out kn
• Bootstrap
• Sampling with replacement

84
Model Evaluation

• Metrics for Performance Evaluation
• How to evaluate the performance of a model?
• Methods for Performance Evaluation
• How to obtain reliable estimates?
• Methods for Model Comparison
• How to compare the relative performance among
  competing models?

85
ROC (Receiver Operating Characteristic)

• Developed in 1950s for signal detection theory to
  analyze noisy signals
• Characterize the trade-off between positive hits
  and false alarms
• ROC curve plots TPRate (on the y-axis) against
  FPRate (on the x-axis)
• Performance of each classifier represented as a
  point on the ROC curve
• changing the threshold of algorithm, sample
  distribution or cost matrix changes the location
  of the point

86
ROC Curve

• (TPRate,FPRate)
• (0,0) declare everything to be
  negative class
• (1,1) declare everything to be positive
  class
• (1,0) ideal
• Diagonal line
• Random guessing
• Below diagonal line
• prediction is opposite of the true class

87
Using ROC for Model Comparison

• No model consistently outperform the other
• M1 is better for small FPR
• M2 is better for large FPR
• Area Under the ROC curve
• Ideal
• Area 1
• Random guess
• Area 0.5

88
How to Construct an ROC curve

• Use classifier that produces posterior
  probability for each test instance P(A)
• Sort the instances according to P(A) in
  decreasing order
• Apply threshold at each unique value of P(A)
• Count the number of TP, FP, TN, FN at each
  threshold
• TP rate, TPR TP/(TPFN)
• FP rate, FPR FP/(FP TN)

Instance P(A) True Class
1 0.95
2 0.93
3 0.87 -
4 0.85 -
5 0.85 -
6 0.85
7 0.76 -
8 0.53
9 0.43 -
10 0.25
89
How to construct an ROC curve
Threshold gt
ROC Curve
90
Rule-Based Classifier

• Classify records by using a collection of
  ifthen rules
• Rule (Condition) ? y
• where
• Condition is a conjunctions of attributes
• y is the class label
• LHS rule antecedent or condition
• RHS rule consequent
• Examples of classification rules
• (Blood TypeWarm) ? (Lay EggsYes) ? Birds
• (Taxable Income lt 50K) ? (RefundYes) ? EvadeNo

91
Rule-based Classifier (Example)

• R1 (Give Birth no) ? (Can Fly yes) ? Birds
• R2 (Give Birth no) ? (Live in Water yes) ?
  Fishes
• R3 (Give Birth yes) ? (Blood Type warm) ?
  Mammals
• R4 (Give Birth no) ? (Can Fly no) ? Reptiles
• R5 (Live in Water sometimes) ? Amphibians

92
Application of Rule-Based Classifier

• A rule r covers an instance x if the attributes
  of the instance satisfy the condition of the rule

R1 (Give Birth no) ? (Can Fly yes) ?
Birds R2 (Give Birth no) ? (Live in Water
yes) ? Fishes R3 (Give Birth yes) ? (Blood
Type warm) ? Mammals R4 (Give Birth no) ?
(Can Fly no) ? Reptiles R5 (Live in Water
sometimes) ? Amphibians
The rule R1 covers a hawk gt Bird The rule R3
covers the grizzly bear gt Mammal
93
Rule Coverage and Accuracy

• Coverage of a rule
• Fraction of records that satisfy the antecedent
  of a rule
• Accuracy of a rule
• Fraction of records that satisfy both the
  antecedent and consequent of a rule

(StatusSingle) ? No Coverage 40,
Accuracy 50
94
How does Rule-based Classifier Work?
R1 (Give Birth no) ? (Can Fly yes) ?
Birds R2 (Give Birth no) ? (Live in Water
yes) ? Fishes R3 (Give Birth yes) ? (Blood
Type warm) ? Mammals R4 (Give Birth no) ?
(Can Fly no) ? Reptiles R5 (Live in Water
sometimes) ? Amphibians
A lemur triggers rule R3, so it is classified as
a mammal A turtle triggers both R4 and R5 A
dogfish shark triggers none of the rules
95
Characteristics of Rule-Based Classifier

• Mutually exclusive rules
• Classifier contains mutually exclusive rules if
  the rules are independent of each other
• Every record is covered by at most one rule
• Exhaustive rules
• Classifier has exhaustive coverage if it accounts
  for every possible combination of attribute
  values
• Each record is covered by at least one rule

96
From Decision Trees To Rules
Rules are mutually exclusive and exhaustive Rule
set contains as much information as the tree
97
Rules Can Be Simplified
Initial Rule (RefundNo) ?
(StatusMarried) ? No Simplified Rule
(StatusMarried) ? No
98
Effect of Rule Simplification

• Rules are no longer mutually exclusive
• A record may trigger more than one rule
• Solution?
• Ordered rule set
• Unordered rule set use voting schemes
• Rules are no longer exhaustive
• A record may not trigger any rules
• Solution?
• Use a default class

99
Ordered Rule Set

• Rules are rank ordered according to their
  priority
• An ordered rule set is known as a decision list
• When a test record is presented to the classifier
  
• It is assigned to the class label of the highest
  ranked rule it has triggered
• If none of the rules fired, it is assigned to the
  default class

R1 (Give Birth no) ? (Can Fly yes) ?
Birds R2 (Give Birth no) ? (Live in Water
yes) ? Fishes R3 (Give Birth yes) ? (Blood
Type warm) ? Mammals R4 (Give Birth no) ?
(Can Fly no) ? Reptiles R5 (Live in Water
sometimes) ? Amphibians
100
Rule Ordering Schemes

• Rule-based ordering
• Individual rules are ranked based on their
  quality
• Class-based ordering
• Rules that belong to the same class appear
  together

101
Building Classification Rules

• Direct Method
• Extract rules directly from data
• e.g. RIPPER, CN2, Holtes 1R
• Indirect Method
• Extract rules from other classification models
  (e.g. decision trees, neural networks, etc).
• e.g C4.5rules

102
Direct Method Sequential Covering

1. Start from an empty rule
2. Grow a rule using the Learn-One-Rule function
3. Remove training records covered by the rule
4. Repeat Step (2) and (3) until stopping criterion
   is met

103
Example of Sequential Covering
104
Example of Sequential Covering
105
Aspects of Sequential Covering

• Rule Growing
• Instance Elimination
• Rule Evaluation
• Stopping Criterion
• Rule Pruning

106
Rule Growing

• Two common strategies

107
Rule Growing (Examples)

• CN2 Algorithm
• Start from an empty conjunct
• Add conjuncts that minimizes the entropy measure
  A, A,B,
• Determine the rule consequent by taking majority
  class of instances covered by the rule
• RIPPER Algorithm
• Start from an empty rule gt class
• Add conjuncts that maximizes FOILs information
  gain measure
• R0 gt class (initial rule)
• R1 A gt class (rule after adding conjunct)
• Gain(R0, R1) t log (p1/(p1n1)) log
  (p0/(p0 n0))
• where t number of positive instances covered
  by both R0 and R1
• p0 number of positive instances covered by R0
• n0 number of negative instances covered by R0
• p1 number of positive instances covered by R1
• n1 number of negative instances covered by R1

108
Instance Elimination

• Why do we need to eliminate instances?
• Otherwise, the next rule is identical to previous
  rule
• Why do we remove positive instances?
• Ensure that the next rule is different
• Why do we remove negative instances?
• Prevent underestimating accuracy of rule
• Compare rules R2 and R3 in the diagram

109
Rule Evaluation

• Metrics
• Accuracy
• Laplace
• M-estimate

n Number of instances covered by rule nc
Number of positive instances covered by rule k
Number of classes p Prior probability
110
Stopping Criterion and Rule Pruning

• Stopping criterion
• Compute the gain
• If gain is not significant, discard the new rule
• Rule Pruning
• Similar to post-pruning of decision trees
• Reduced Error Pruning
• Remove one of the conjuncts in the rule
• Compare error rate on validation set before and
  after pruning
• If error improves, prune the conjunct

111
Summary of Direct Method

• Grow a single rule
• Remove Instances from rule
• Prune the rule (if necessary)
• Add rule to Current Rule Set
• Repeat

112
Direct Method RIPPER

• For 2-class problem, choose one of the classes as
  positive class, and the other as negative class
• Learn rules for positive class
• Negative class will be default class
• For multi-class problem
• Order the classes according to increasing class
  prevalence (fraction of instances that belong to
  a particular class)
• Learn the rule set for smallest class first,
  treat the rest as negative class
• Repeat with next smallest class as positive class

113
Direct Method RIPPER

• Growing a rule
• Start from empty rule
• Add conjuncts as long as they improve FOILs
  information gain
• Stop when rule no longer covers negative examples
• Prune the rule immediately using incremental
  reduced error pruning
• Measure for pruning v (p-n)/(pn)
• p number of positive examples covered by the
  rule in the validation set
• n number of negative examples covered by the
  rule in the validation set
• Pruning method delete any final sequence of
  conditions that maximizes v

114
Direct Method RIPPER

• Building a Rule Set
• Use sequential covering algorithm
• Finds the best rule that covers the current set
  of positive examples
• Eliminate both positive and negative examples
  covered by the rule
• Each time a rule is added to the rule set,
  compute the new description length
• stop adding new rules when the new description
  length is d bits longer than the smallest
  description length obtained so far

115
Direct Method RIPPER

• Optimize the rule set
• For each rule r in the rule set R
• Consider 2 alternative rules
• Replacement rule (r) grow new rule from scratch
• Revised rule(r) add conjuncts to extend the
  rule r
• Compare the rule set for r against the rule set
  for r and r
• Choose rule set that minimizes MDL principle
• Repeat rule generation and rule optimization for
  the remaining positive examples

116
Indirect Methods
117
Indirect Method C4.5rules

• Extract rules from an unpruned decision tree
• For each rule, r A ? y,
• consider an alternative rule r A ? y where A
  is obtained by removing one of the conjuncts in A
• Compare the pessimistic error rate for r against
  all rs
• Prune if one of the rs has lower pessimistic
  error rate
• Repeat until we can no longer improve
  generalization error

118
Indirect Method C4.5rules

• Instead of ordering the rules, order subsets of
  rules (class ordering)
• Each subset is a collection of rules with the
  same rule consequent (class)
• Compute description length of each subset
• Description length L(error) g L(model)
• g is a parameter that takes into account the
  presence of redundant attributes in a rule set
  (default value 0.5)

119
Example
120
C4.5 versus C4.5rules versus RIPPER
C4.5rules (Give BirthNo, Can FlyYes) ?
Birds (Give BirthNo, Live in WaterYes) ?
Fishes (Give BirthYes) ? Mammals (Give BirthNo,
Can FlyNo, Live in WaterNo) ? Reptiles ( ) ?
Amphibians
RIPPER (Live in WaterYes) ? Fishes (Have
LegsNo) ? Reptiles (Give BirthNo, Can FlyNo,
Live In WaterNo) ? Reptiles (Can FlyYes,Give
BirthNo) ? Birds () ? Mammals
121
C4.5 versus C4.5rules versus RIPPER
C4.5 and C4.5rules
RIPPER
122
Advantages of Rule-Based Classifiers

• As highly expressive as decision trees
• Easy to interpret
• Easy to generate
• Can classify new instances rapidly
• Performance comparable to decision trees