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# Chap4 Basic Classification

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Data MiningClassif ication:It is the process of discovering new patterns from large data sets involving methods

from statistics and artificial intelligence but also database management. In contrast to

machine learning, the emphasis lies on the discovery ofpreviously unknown patterns as

opposed to generalizing known patterns to new data.

Introduction to Data Mining

by

Tan, Steinbach, Kumar

http://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Database_managementhttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Database_managementhttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Data_set
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Classif ication: Definit ionq

Given a collection of records (training set) Each record contains a set ofattributes, one of the

attributes is the class.

q Find a model for class attribute as a function

of the values of other attributes.q Goal: previously unseen records should be

assigned a class as accurately as possible.A test setis used to determine the accuracy of the

model. Usually, the given data set is divided intotraining and test sets, with training set used to buildthe model and test set used to validate it.

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Apply

Model

Induction

Deduction

Learn

Model

Model

Tid Attrib1 Attrib2 Attrib3 Class

1 Yes

Large 125K No

2

No Medium 100K No

3

No Small 70K No

4

Yes

Medium 120K No

5 No Large 95K

Yes

6

No Medium 60K

No

7 Yes

Large 220K

No

8

No Small 85K

Yes

9

No Medium 75K No

10 No Small 90K

Yes10

Tid Attrib1 Attrib2 Attrib3 Class

11 No Small 55K ?

12 Yes

Medium 80K

?

13 Yes

Large 110K ?

14 No Small 95K

?

15 No Large 67K

?10

Test Set

Learning

algorithm

Training Set

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q Classifying credit card transactions

as legitimate or fraudulent

q Classifying secondary structures of protein

as alpha-helix, beta-sheet, or random

coil

q Categorizing news stories as finance,

weather, entertainment, sports, etc

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Classif ication Techniquesq

Decision Tree based Methodsq Rule-based Methods

q Memory based reasoning

q Neural Networks

q Nave Bayes and Bayesian Belief Networks

q Support Vector Machines

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Example of a Decision Tree

Tid Refund Marital

Status

Taxable

Income Cheat

1 Yes Single 125K No

2 No Married 100K No

3 No Single 70K No

4 Yes Married 120K No

5 No Divorced 95K Yes

6 No Married 60K No

7 Yes Divorced 220K No

8 No Single 85K Yes

9 No Married 75K No

10 No Single 90K Yes10

categ

oric

al

categ

oric

al

contin

uous

class

Training Data Model: Decision Tree

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Example of a Decision Tree

Tid Refund Marital

Status

Taxable

Income Cheat

1 Yes Single 125K No

2 No Married 100K No

3 No Single 70K No

4 Yes Married 120K No

5 No Divorced 95K Yes

6 No Married 60K No

7 Yes Divorced 220K No

8 No Single 85K Yes

9 No Married 75K No

10 No Single 90K Yes10

categ

oric

al

categ

oric

al

contin

uous

class

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Splitting Attributes

Training Data Model: Decision Tree

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Another Example of Decision Tree

Tid Refund Marital

Status

Taxable

Income Cheat

1 Yes Single 125K No

2 No Married 100K No

3 No Single 70K No

4 Yes Married 120K No

5 No Divorced 95K Yes

6 No Married 60K No

7 Yes Divorced 220K No

8 No Single 85K Yes

9 No Married 75K No

10 No Single 90K Yes10

cate

gorical

cate

gorical

continuo

us

clas

sMarSt

Refund

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle,

Divorced

< 80K > 80K

There could be more than one tree that

fits the same data!

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Apply

Model

Induction

Deduction

Learn

Model

Model

Tid Attrib1 Attrib2 Attrib3 Class

1 Yes Large 125K No

2 No Medium 100K No

3 No Small 70K No

4 Yes Medium 120K No

5 No Large 95K Yes

6 No Medium 60K No

7 Yes Large 220K No

8 No Small 85K Yes

9 No Medium 75K No

10 No Small 90K Yes10

Tid Attrib1 Attrib2 Attrib3 Class

11 No Small 55K ?

12 Yes Medium 80K ?

13 Yes Large 110K ?

14 No Small 95K ?

15 No Large 67K ?10

Test Set

Tree

Inductionalgorithm

Training Set

Decision

Tree

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Apply Model to Test Data

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Refund Marital

Status

Taxable

Income Cheat

No Married 80K ?10

Test Data

Start from the root of tree.

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Apply Model to Test Data

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Refund Marital

Status

Taxable

Income Cheat

No Married 80K ?10

Test Data

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Apply Model to Test Data

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Refund Marital

Status

Taxable

Income Cheat

No Married 80K ?10

Test Data

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Apply Model to Test Data

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Refund Marital

Status

Taxable

Income Cheat

No Married 80K ?10

Test Data

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Apply Model to Test Data

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Refund Marital

Status

Taxable

Income Cheat

No Married 80K ?10

Test Data

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Apply Model to Test Data

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

Refund Marital

Status

Taxable

Income Cheat

No Married 80K ?10

Test Data

Assign Cheat to No

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Apply

Model

Induction

Deduction

Learn

Model

Model

TidAttrib1 Attrib2 Attrib3 Class

1 Yes Large 125K No

2 No Medium 100K No

3 No Small 70K No

4 Yes Medium 120K No

5 No Large 95K Yes

6 No Medium 60K No

7 Yes Large 220K No

8 No Small 85K Yes

9 No Medium 75K No

10 No Small 90K Yes10

Tid Attrib1 Attrib2 Attrib3 Class

11 No Small 55K ?

12 Yes Medium 80K ?

13 Yes Large 110K ?

14 No Small 95K ?

15 No Large 67K ?10

Test Set

Tree

Induction

algorithm

Training Set

Decision

Tree

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Decision Tree Inductionq Many Algorithms:

Hunts Algorithm (one of the earliest)

CART

ID3, C4.5

SLIQ,SPRINT

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Tree Inductionq Greedy strategy.

Split the records based on an attribute test

that optimizes certain criterion.

q Issues

Determine how to split the recordsHow to specify the attribute test condition?

How to determine the best split?

Determine when to stop splitting

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Tree Inductionq Greedy strategy.

Split the records based on an attribute test

that optimizes certain criterion.

q Issues

Determine how to split the recordsHow to specify the attribute test condition?

How to determine the best split?

Determine when to stop splitting

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How to Specify Test Condition?q Depends on attribute types

Nominal

Ordinal

Continuous

q Depends on number of ways to split

2-way split

Multi-way split

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Splitt ing Based on Nominal Attributesq

Multi-way split: Use as many partitions as distinctvalues.

q Binary split: Divides values into two subsets.

Need to find optimal partitioning.

CarTypeFamily

Sports

Luxury

CarType{Family,

Luxury} {Sports}

CarType{Sports,

Luxury} {Family}OR

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q Multi-way split: Use as many partitions as distinct

values.

q Binary split: Divides values into two subsets.

Need to find optimal partitioning.

Splitt ing Based on Ordinal Attributes

SizeSmall

Medium

Large

Size{Medium,

Large} {Small}

Size{Small,

Medium} {Large}OR

Size{Small,

Large} {Medium}

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Attributesq Different ways of handling

Discretization to form an ordinal categorical

attribute Static discretize once at the beginning

Dynamic ranges can be found by equal intervalbucketing, equal frequency bucketing

(percentiles), or clustering.

Binary Decision: (A < v) or (A v) consider all possible splits and finds the best cut

can be more compute intensive

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Attributes

Taxable

Income

> 80K?

Yes No

Taxable

Income?

(i) Binary split (ii) Multi-way split

< 10K

[10K,25K) [25K,50K) [50K,80K)

> 80K

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Tree Inductionq Greedy strategy.

Split the records based on an attribute test

that optimizes certain criterion.

q Issues

Determine how to split the recordsHow to specify the attribute test condition?

How to determine the best split?

Determine when to stop splitting

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How to determine the Best Split

Own

Car?

C0: 6

C1: 4

C0: 4

C1: 6

C0: 1

C1: 3

C0: 8

C1: 0

C0: 1

C1: 7

Car

Type?

C0: 1

C1: 0

C0: 1

C1: 0

C0: 0

C1: 1

Student

ID?

...

Yes No FamilySports

Luxury c1 c10

c20

C0: 0

C1: 1...

c11

Before Splitting: 10 records of class 0,

10 records of class 1

Which test condition is the best?

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How to determine the Best Splitq Greedy approach:

Nodes with homogeneous class distribution

are preferred

q Need a measure of node impurity:

C0: 5

C1: 5

C0: 9

C1: 1

Non-homogeneous,

High degree of impurity

Homogeneous,

Low degree of impurity

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Measures of Node Impurityq Gini Index

q Entropy

q Misclassification error

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Measure of Impurity: GINIq 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 interestinginformation

Minimum (0.0) when all records belong to one class,

implying most interesting information

=j

tjptGINI 2)]|([1)(

C1 0C2 6

Gini=0.000

C1 2C2 4

Gini=0.444

C1 3C2 3

Gini=0.500

C1 1C2 5

Gini=0.278

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Examples for computing GINI

C1 0C2 6

C1 2C2 4

C1 1C2 5

P(C1) = 0/6 = 0 P(C2) = 6/6 = 1

Gini = 1 P(C1)2 P(C2)2 = 1 0 1 = 0

=

j

tjptGINI 2)]|([1)(

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

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Splitt ing Based on GINIq Used in CART, SLIQ, SPRINT.

q 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.

==

k

i

i

split iGINIn

n

GINI 1 )(

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Indexq Splits into two partitions

q Effect of Weighing partitions:

Larger and Purer Partitions are sought for.

B?

Yes No

Node N1 Node N2

ParentC1 6C2 6

Gini = 0.500

N1 N2C1 5 1C2 2 4Gini=0.333

Gini(N1)

= 1 (5/7)2 (2/7)2

= 0.194

Gini(N2)

= 1 (1/5)2 (4/5)2

= 0.528

Gini(Children)= 7/12 * 0.194 +

5/12 * 0.528

= 0.333

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Index...q For efficient computation: for each attribute,

Sort the attribute on values

Linearly scan these values, each time updating the count matrixand computing gini index

Choose the split position that has the least gini index

Cheat No No No Yes Yes Yes No No No No

Taxable Income

60 70 75 85 90 95 100 120 125 220

55 65 72 80 87 92 97 110 122 172 230

Yes 0 3 0 3 0 3 0 3 1 2 2 1 3 0 3 0 3 0 3 0 3 0

No 0 7 1 6 2 5 3 4 3 4 3 4 3 4 4 3 5 2 6 1 7 0

Gini 0.420 0.400 0.375 0.343 0.417 0.400 0.300 0.343 0.375 0.400 0.420

Split Positions

Sorted Values

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Alternative Splitting Criteria based on INFOq 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

=j

tjptjptEntropy )|(log)|()(

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Examples for computing Entropy

C1 0C2 6

C1 2C2 4

C1 1C2 5

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

=j

tjptjptEntropy )|(log)|()(2

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Splitting Based on INFO...q 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.

=

=

k

i

i

splitiEntropy

n

npEntropyGAIN

1

)()(

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Errorq Classification error at a node t :

q 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

)|(max1)( tiPtErrori

=

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Examples for Computing Error

C1 0C2 6

C1 2C2 4

C1 1C2 5

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

)|(max1)( tiPtError i=

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Comparison among Splitt ing CriteriaFor a 2-class problem:

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Misclassif ication Error vs GiniA?

Yes No

Node N1 Node N2

P a r e n tC1 7C2 3G in i = 0 .42

N1 N2C1 3 4C2 0 3Gini=0.361

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 !!

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Tree Inductionq Greedy strategy.

Split the records based on an attribute test

that optimizes certain criterion.

q Issues

Determine how to split the recordsHow to specify the attribute test condition?

How to determine the best split? Determine when to stop splitting

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Stopping Criteria for Tree Inductionq Stop expanding a node when all the records

belong to the same class

q Stop expanding a node when all the records have

similar attribute values

q Early termination (to be discussed later)

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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

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Example: C4.5q Simple depth-first construction.

q Uses Information Gain

q Sorts Continuous Attributes at each node.

q Needs entire data to fit in memory.

q Unsuitable for Large Datasets.

Needs out-of-core sorting.

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Practical Issues of Classif icationq Underfitting and Overfitting

q Missing Values

q Costs of Classification

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(Example)

500 circular and 500

triangular data points.

Circular points:0.5 sqrt(x1

2+x22) 1

Triangular points:

sqrt(x12+x22) > 0.5 or

sqrt(x12+x2

2) < 1

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Underfitt ing and Overfitt ingOverfitting

Underfitting: when model is too simple, both training and test errors are large

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Overfitt ing due to Noise

Decision boundary is distorted by noise point

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Examples

Lack of data points in the lower half of the diagram makes it difficult topredict 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

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Notes on Overfitt ingq Overfitting results in decision trees that are more

complex than necessary

q Training error no longer provides a good estimate

of how well the tree will perform on previouslyunseen records

q

Need new ways for estimating errors

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Occams Razorq Given two models of similar generalization errors,

one should prefer the simpler model over the

more complex model

q For complex models, there is a greater chancethat it was fitted accidentally by errors in data

q

Therefore, one should include model complexitywhen evaluating a model

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How to Address Overfitt ingq 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).

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How to Address Overfitt ingq 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 Can use MDL for post-pruning

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Example of Post-Pruning

A?

A1

A2 A3

A4

Class = Yes 20

Class = No 10

Error = 10/30

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 8

Class = No 4

Class = Yes 3

Class = No 4

Class = Yes 4

Class = No 1

Class = Yes 5

Class = No 0

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Estimating Generalization Errorsq Re-substitution errors: error on training ( e(t) )

q Generalization errors: error on testing ( e(t))q 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

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Handling Missing Attribute Valuesq 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

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Model Evaluationq Metrics for Performance Evaluation

How to evaluate the performance of a model?

q Methods for Performance Evaluation

How to obtain reliable estimates?

q Methods for Model Comparison

How to compare the relative performance

among competing models?

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Model Evaluationq Metrics for Performance Evaluation

How to evaluate the performance of a model?

q Methods for Performance Evaluation

How to obtain reliable estimates?

q Methods for Model Comparison

How to compare the relative performance

among competing models?

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Metrics for Performance Evaluationq Focus on the predictive capability of a model

Rather than how fast it takes to classify or

build models, scalability, etc.

q Confusion Matrix:

PREDICTED CLASS

ACTUALCLASS

Class=Yes Class=No

Class=Yes a b

Class=No c d

a: TP (true positive)

b: FN (false negative)

c: FP (false positive)

d: TN (true negative)

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Metrics for Performance Evaluation

q Most widely-used metric:

PREDICTED CLASS

ACTUALCLASS

Class=Yes Class=No

Class=Yes a(TP)

b(FN)

Class=No c(FP)

d(TN)

FNFPTNTP

TNTP

dcba

da

++++

=+++

+=Accuracy

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Limitation of Accuracyq Consider a 2-class problem

Number of Class 0 examples = 9990

Number of Class 1 examples = 10

q 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

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Cost Matrix

PREDICTED CLASS

ACTUALCLASS

C(i|j) Class=Yes Class=No

Class=Yes C(Yes|Yes) C(No|Yes)

Class=No C(Yes|No) C(No|No)

C(i|j): Cost of misclassifying class j example as class i

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Computing Cost of Classif icationCost

MatrixPREDICTED CLASS

ACTUALCLASS

C(i|j) + -

+ -1 100

- 1 0

Model M1 PREDICTED CLASS

ACTUALCLASS

+ -

+ 150 40

- 60 250

Model M2 PREDICTED CLASS

ACTUALCLASS

+ -

+ 250 45

- 5 200

Accuracy = 80%

Cost = 3910

Accuracy = 90%

Cost = 4255

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Cost vs AccuracyCount PREDICTED CLASS

ACTUALCLASS

Class=Yes Class=No

Class=Yes a b

Class=No c d

Cost PREDICTED CLASS

ACTUALCLASS

Class=Yes Class=No

Class=Yes p q

Class=No q p

N = a + b + c + d

Accuracy = (a + d)/N

Cost = p (a + d) + q (b + c)

= p (a + d) + q (N a d)

= q N (q p)(a + d)

= N [q (q-p) Accuracy]

Accuracy is proportional to cost if

1. C(Yes|No)=C(No|Yes) = q2. C(Yes|Yes)=C(No|No) = p

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Cost-Sensitive Measures

cba

a

pr

rp

ba

aca

a

++=+=

+=

+

=

2

22

(F)measure-F

(r)Recall

(p)Precision

q Precision is biased towards C(Yes|Yes) & C(Yes|No)

q Recall is biased towards C(Yes|Yes) & C(No|Yes)

q F-measure is biased towards all except C(No|No)

dwcwbwaw

dwaw

4321

41AccuracyWeighted+++

+=

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Model Evaluationq Metrics for Performance Evaluation

How to evaluate the performance of a model?

q Methods for Performance Evaluation

How to obtain reliable estimates?

q Methods for Model Comparison

How to compare the relative performanceamong competing models?

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Methods for Performance Evaluationq How to obtain a reliable estimate of performance?

q Performance of a model may depend on other

factors besides the learning algorithm:

Class distribution

Cost of misclassification

Size of training and test sets

Learning Curve

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q Learning curve shows

how accuracy changeswith varying sample size

q Requires a sampling

schedule for creating

learning curve:

q Arithmetic sampling(Langley, et al)

q Geometric sampling

(Provost et al)

Effect of small sample size:

- Bias in the estimate

- Variance of estimate

Methods of Estimation

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q Holdout

Reserve 2/3 for training and 1/3 for testingq Random subsampling

Repeated holdout

q Cross validation

Partition data into k disjoint subsets k-fold: train on k-1 partitions, test on the remaining one

Leave-one-out: k=n

q Stratified sampling

oversampling vs undersamplingq Bootstrap

Sampling with replacement

Model Evaluation

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q Metrics for Performance Evaluation

How to evaluate the performance of a model?

q Methods for Performance Evaluation

How to obtain reliable estimates?

q Methods for Model Comparison

How to compare the relative performanceamong competing models?

Characteristic)

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q Developed in 1950s for signal detection theory to

analyze noisy signals Characterize the trade-off between positive

hits and false alarms

q ROC curve plots TP (on the y-axis) against FP(on the x-axis)

q Performance of each classifier represented as apoint on the ROC curve

changing the threshold of algorithm, sampledistribution or cost matrix changes the locationof the point

ROC Curve

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At threshold t:

TP=0.5, FN=0.5, FP=0.12, FN=0.88

- 1-dimensional data set containing 2 classes (positive and negative)

- any points located at x > t is classified as positive

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Using ROC for Model Comparison

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q No model consistently

outperform the otherq M1 is better for

small FPR

q M2 is better for

large FPR

qArea Under the ROC

curve

q Ideal: Area = 1

q Random guess:

Area = 0.5

How to Construct an ROC curve

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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 +

Use classifier that produces

posterior probability for eachtest 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/(TP+FN)

FP rate, FPR = FP/(FP + TN)

How to construct an ROC curve

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Class + - + - - - + - + +

P0.25 0.43 0.53 0.76 0.85 0.85 0.85 0.87 0.93 0.95 1.00

TP 5 4 4 3 3 3 3 2 2 1 0

FP 5 5 4 4 3 2 1 1 0 0 0

TN 0 0 1 1 2 3 4 4 5 5 5

FN 0 1 1 2 2 2 2 3 3 4 5

TPR 1 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.2 0

FPR 1 1 0.8 0.8 0.6 0.4 0.2 0.2 0 0 0

Threshold >=

ROC Curve:

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