Data Mining Classification: Alternative Techniques Lecture Notes for Chapter 5 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar.

Data Mining Classification: Alternative

Techniques

Lecture Notes for Chapter 5

Introduction to Data Miningby

Tan, Steinbach, Kumar

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1


Rule Based Method: See new lecture notes


Instance-Based Classifiers

Atr1 ……... AtrN ClassA

B

B

C

A

C

B

Set of Stored Cases

Atr1 ……... AtrN

Unseen Case

• Store the training records

• Use training records to predict the class label of unseen cases


Instance Based Classifiers

Examples:

– Rote-learner Memorizes entire training data and performs classification only if attributes of record match one of the training examples exactly

– Nearest neighbor Uses k “closest” points (nearest neighbors) for performing classification


Nearest Neighbor Classifiers

Basic idea:

– If it walks like a duck, quacks like a duck, then it’s probably a duck

Training Records

Test Record

Compute Distance

Choose k of the “nearest” records


Nearest-Neighbor Classifiers

Requires three things

– The set of stored records

– Distance Metric to compute distance between records

– The value of k, the number of nearest neighbors to retrieve

To classify an unknown record:

– Compute distance to other training records

– Identify k nearest neighbors

– Use class labels of nearest neighbors to determine the class label of unknown record (e.g., by taking majority vote)

Unknown record


Definition of Nearest Neighbor

X X X

(a) 1-nearest neighbor (b) 2-nearest neighbor (c) 3-nearest neighbor

K-nearest neighbors of a record x are data points that have the k smallest distance to x


1 nearest-neighbor

Voronoi Diagram


Nearest Neighbor Classification

Compute distance between two points:

– Euclidean distance

Determine the class from nearest neighbor list

– take the majority vote of class labels among the k-nearest neighbors

– Weigh the vote according to distance weight factor, w = 1/d2

i ii

qpqpd 2)(),(


Nearest Neighbor Classification…

Choosing the value of k:– If k is too small, sensitive to noise points

– If k is too large, neighborhood may include points from other classes

X



Scaling issues

– Attributes may have to be scaled to prevent distance measures from being dominated by one of the attributes

– Example: height of a person may vary from 1.5m to 1.8m weight of a person may vary from 90lb to 300lb income of a person may vary from $10K to $1M



Problem with Euclidean measure:

– High dimensional data curse of dimensionality

– Can produce counter-intuitive results

1 1 1 1 1 1 1 1 1 1 1 0

0 1 1 1 1 1 1 1 1 1 1 1

1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1vs

d = 1.4142 d = 1.4142

Solution: Normalize the vectors to unit length


Nearest neighbor Classification…

k-NN classifiers are lazy learners

– It does not build models explicitly

– Unlike eager learners such as decision tree induction and rule-based systems

– Classifying unknown records are relatively expensive


Example: PEBLS

PEBLS: Parallel Examplar-Based Learning System (Cost & Salzberg)

– Works with both continuous and nominal featuresFor nominal features, distance between two nominal values is computed using modified value difference metric (MVDM)

– Each record is assigned a weight factor

– Number of nearest neighbor, k = 1


Example: PEBLS

ClassMarital Status

Single Married Divorced

Yes 2 0 1

No 2 4 1

i

ii

n

n

n

nVVd

2

2

1

121 ),(

Distance between nominal attribute values:

d(Single,Married)

= | 2/4 – 0/4 | + | 2/4 – 4/4 | = 1

d(Single,Divorced)

= | 2/4 – 1/2 | + | 2/4 – 1/2 | = 0

d(Married,Divorced)

= | 0/4 – 1/2 | + | 4/4 – 1/2 | = 1

d(Refund=Yes,Refund=No)

= | 0/3 – 3/7 | + | 3/3 – 4/7 | = 6/7

Tid Refund MaritalStatus

TaxableIncome 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

ClassRefund

Yes No

Yes 0 3

No 3 4


Example: PEBLS

d

iiiYX YXdwwYX

1

2),(),(

Tid Refund Marital Status

Taxable Income Cheat

X Yes Single 125K No

Y No Married 100K No 10

Distance between record X and record Y:

where:

correctly predicts X timesofNumber

predictionfor used is X timesofNumber Xw

wX 1 if X makes accurate prediction most of the time

wX > 1 if X is not reliable for making predictions

Note: When X is a test data,We should have a weightWx=1, since for the test data,we do not have the class label.

For a training data Y, WY is largewhen Y is an outlier.


Bayes Classifier

A probabilistic framework for solving classification problems

Conditional Probability:

Bayes theorem:

)()()|(

)|(AP

CPCAPACP

)(),(

)|(

)(),(

)|(

CPCAP

CAP

APCAP

ACP


Example of Bayes Theorem

Given: – A doctor knows that meningitis causes stiff neck 50% of the

time

– Prior probability of any patient having meningitis is 1/50,000

– Prior probability of any patient having stiff neck is 1/20

If a patient has stiff neck, what’s the probability he/she has meningitis?

0002.020/150000/15.0

)()()|(

)|( SP

MPMSPSMP


Bayesian Classifiers

Consider each attribute and class label as random variables

Given a record with attributes (A1, A2,…,An)

– Goal is to predict class C

– Specifically, we want to find the value of C that maximizes P(C| A1, A2,…,An )

Can we estimate P(C| A1, A2,…,An ) directly from data?


Bayesian Classifiers

Approach:

– compute the posterior probability P(C | A1, A2, …, An) for all values of C using the Bayes theorem

– Choose value of C that maximizes P(C | A1, A2, …, An)

– Equivalent to choosing value of C that maximizes P(A1, A2, …, An|C) P(C)

How to estimate P(A1, A2, …, An | C )?

)()()|(

)|(21

21

21

n

n

n AAAPCPCAAAP

AAACP


Naïve Bayes Classifier

Assume independence among attributes Ai when class is

given:

– P(A1, A2, …, An |C) = P(A1| Cj) P(A2| Cj)… P(An| Cj)

– Can estimate P(Ai| Cj) for all Ai and Cj.

– New point is classified to Cj if P(Cj) P(Ai| Cj) is maximal.


How to Estimate Probabilities from Data?

Class: P(C) = Nc/N– e.g., P(No) = 7/10,

P(Yes) = 3/10

For discrete attributes:

P(Ai | Ck) = |Aik|/ Nc

– where |Aik| is number of instances having attribute Ai and belongs to class Ck

– Examples:

P(Status=Married|No) = 4/7P(Refund=Yes|Yes)=0

k



For continuous attributes: – Discretize the range into bins

one ordinal attribute per bin violates independence assumption

– Two-way split: (A < v) or (A > v) choose only one of the two splits as new attribute

– Probability density estimation: Assume attribute follows a normal distribution Use data to estimate parameters of distribution (e.g., mean and standard deviation) Once probability distribution is known, can use it to estimate the conditional probability P(Ai|c)

k



Normal distribution:

– One for each (Ai,ci) pair

For (Income, Class=No):

– If Class=No sample mean = 110 sample variance = 2975

2

2

2

)(

221

)|( ij

ijiA

ij

jiecAP

0072.0)54.54(2

1)|120( )2975(2

)110120( 2

eNoIncomeP


Example of Naïve Bayes Classifier

120K)IncomeMarried,No,Refund( X

P(X|Class=No) = P(Refund=No|Class=No) P(Married| Class=No) P(Income=120K|

Class=No) = 4/7 4/7 0.0072 = 0.0024

P(X|Class=Yes) = P(Refund=No| Class=Yes) P(Married| Class=Yes) P(Income=120K| Class=Yes)

= 1 0 1.2 10-9 = 0

Since P(X|No)P(No) > P(X|Yes)P(Yes)

Therefore P(No|X) > P(Yes|X) => Class = No

Given a Test Record:


Naïve Bayes Classifier

If one of the conditional probability is zero, then the entire expression becomes zero

Probability estimation:

mN

mpNCAP

cN

NCAP

N

NCAP

c

ici

c

ici

c

ici

)|(:estimate-m

1)|(:Laplace

)|( :Originalc: number of values for attribute A (eg, for Outlook, c=3)

p: prior probability set by user

m: parameter set by user

Note: when Nc =0, P(Ai|C)=p.


Example of Naïve Bayes Classifier

Name Give Birth Can Fly Live in Water Have Legs Class

human yes no no yes mammalspython no no no no non-mammalssalmon no no yes no non-mammalswhale yes no yes no mammalsfrog no no sometimes yes non-mammalskomodo no no no yes non-mammalsbat yes yes no yes mammalspigeon no yes no yes non-mammalscat yes no no yes mammalsleopard shark yes no yes no non-mammalsturtle no no sometimes yes non-mammalspenguin no no sometimes yes non-mammalsporcupine yes no no yes mammalseel no no yes no non-mammalssalamander no no sometimes yes non-mammalsgila monster no no no yes non-mammalsplatypus no no no yes mammalsowl no yes no yes non-mammalsdolphin yes no yes no mammalseagle no yes no yes non-mammals

Give Birth Can Fly Live in Water Have Legs Class

yes no yes no ?

0027.02013

004.0)()|(

021.0207

06.0)()|(

0042.0134

133

1310

131

)|(

06.072

72

76

76

)|(

NPNAP

MPMAP

NAP

MAP

A: attributes

M: mammals

N: non-mammals

P(A|M)P(M) > P(A|N)P(N)

=> Mammals


Naïve Bayes (Summary)

Robust to isolated noise points

Handle missing values by ignoring the instance during probability estimate calculations

Robust to irrelevant attributes

Independence assumption may not hold for some attributes– Use other techniques such as Bayesian Belief

Networks (BBN)


Artificial Neural Networks (ANN)

X1 X2 X3 Y1 0 0 01 0 1 11 1 0 11 1 1 10 0 1 00 1 0 00 1 1 10 0 0 0

X1

X2

X3

Y

Black box

Output

Input

Output Y is 1 if at least two of the three inputs are equal to 1.



X1 X2 X3 Y1 0 0 01 0 1 11 1 0 11 1 1 10 0 1 00 1 0 00 1 1 10 0 0 0

X1

X2

X3

Y

Black box

0.3

0.3

0.3 t=0.4

Outputnode

Inputnodes

otherwise0

trueis if1)( where

)04.03.03.03.0( 321

zzI

XXXIY



Model is an assembly of inter-connected nodes and weighted links

Output node sums up each of its input value according to the weights of its links

Compare output node against some threshold t

X1

X2

X3

Y

Black box

w1

t

Outputnode

Inputnodes

w2

w3

)( tXwIYi

ii Perceptron Model

)( tXwsignYi

ii

or


General Structure of ANN

Activationfunction

g(Si )Si Oi

I1

I2

I3

wi1

wi2

wi3

Oi

Neuron iInput Output

threshold, t

InputLayer

HiddenLayer

OutputLayer

x1 x2 x3 x4 x5

y

Training ANN means learning the weights of the neurons


Algorithm for learning ANN

Initialize the weights (w0, w1, …, wk)

Adjust the weights in such a way that the output of ANN is consistent with class labels of training examples

– Objective function:

– Find the weights wi’s that minimize the above objective function e.g., backpropagation algorithm (see lecture notes)

2),( i

iii XwfYE


More on Neural Networks: see additional notes


Support Vector Machines

Find a linear hyperplane (decision boundary) that will separate the data



One Possible Solution

B1



Another possible solution

B2



Other possible solutions

B2



Which one is better? B1 or B2? How do you define better?

B1

B2



Find hyperplane maximizes the margin => B1 is better than B2

B1

B2

b11

b12

b21

b22

margin



B1

b11

b12

0 bxw

1 bxw 1 bxw

1bxw if1

1bxw if1)(

xf Margin =2/||W||



We want to maximize:

– Which is equivalent to minimizing:

– But subjected to the following constraints:

This is a constrained optimization problem– Numerical approaches to solve it (e.g., quadratic programming)

2||||

2

w

1bxw if1

1bxw if1)(

i

i

ixf

2

||||)(

2wwL



What if the problem is not linearly separable?



What if the problem is not linearly separable?

– Introduce slack variables Need to minimize:

Subject to:

ii

ii

1bxw if1

-1bxw if1)(

ixf

N

i

kiC

wwL

1

2

2

||||)(


Nonlinear Support Vector Machines

What if decision boundary is not linear?


Nonlinear Support Vector Machines

Transform data into higher dimensional space


Ensemble Methods:

Construct a set of classifiers from the training data

Predict class label of previously unseen records by aggregating predictions made by multiple classifiers

See additional Notes

Data Mining Classification: Alternative Techniques Lecture Notes for Chapter 5 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar.

Documents

kumar tan

kumar introduction

data mining classification

nearest records slide

data points

nearest neighbor classifiers

nearestneighbor classifiers

classification slide