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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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    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:

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