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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_set8/3/2019 Chap4 Basic Classification
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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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Il lustrating Classif ication Task
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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Examples of Classif ication Task
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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Decision Tree Classif ication Task
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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Decision Tree Classif ication Task
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.
q What about this split?
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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Decision Tree Based Classif icationqAdvantages:
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
irrelevant to the classification task
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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: