lecture 7 classification itcs 6163. chapter 7. classification and prediction what is classification?...

Lecture 7 Classification

ITCS 6163

Chapter 7. Classification and Prediction

• What is classification? What is prediction?• Classification by decision tree induction• Bayesian Classification• Other Classification Methods (SVM)• Classification accuracy • Prediction• Summary

Classification problem• Given:

– Tuples each assigned a class level.

• Develop a model for each class– Example:– Good creditor : (age in [25,40]) AND (income > 50K)

AND (status = MARRIED)

• Applications:– Credit approval (good, bad)– Store locations (good, fair, poor)– Emergency situations (emergency, non-emergency)

• Classification: – predicts categorical class labels– classifies data (constructs a model) based on the training set and

the values (class labels) in a classifying attribute and uses it in classifying new data

• Prediction: – models continuous-valued functions, i.e., predicts unknown or

missing values

• Typical Applications– credit approval– target marketing– medical diagnosis– treatment effectiveness analysis

Classification vs. Prediction

Classification—A Two-Step Process

• Model construction: describing a set of predetermined classes– Each tuple/sample is assumed to belong to a predefined class, as

determined by the class label attribute

– The set of tuples used for model construction: training set

– The model is represented as classification rules, decision trees, or mathematical formulae

• Model usage: for classifying future or unknown objects– Estimate accuracy of the model

• The known label of test sample is compared with the classified result from the model

• Accuracy rate is the percentage of test set samples that are correctly classified by the model

• Test set is independent of training set, otherwise over-fitting will occur

Supervised vs. Unsupervised Learning

• Supervised learning (classification)

– Supervision: The training data (observations,

measurements, etc.) are accompanied by labels indicating

the class of the observations

– New data is classified based on the training set

• Unsupervised learning (clustering)

– The class labels of training data is unknown

– Given a set of measurements, observations, etc. with the

aim of establishing the existence of classes or clusters in

the data


• What is classification? What is prediction?• Classification by decision tree induction• Bayesian Classification• Other Classification Methods• Classification accuracy• Prediction• Summary

Classification by Decision Tree Induction

• Decision tree – A flow-chart-like tree structure

– Internal node denotes a test on an attribute

– Branch represents an outcome of the test

– Leaf nodes represent class labels or class distribution

• Decision tree generation consists of two phases– Tree construction

• At start, all the training examples are at the root

• Partition examples recursively based on selected attributes

– Tree pruning

• Identify and remove branches that reflect noise or outliers

• Use of decision tree: Classifying an unknown sample– Test the attribute values of the sample against the decision tree

Training Dataset

age income student credit_rating buys_computer<=30 high no fair no<=30 high no excellent no30…40 high no fair yes>40 medium no fair yes>40 low yes fair yes>40 low yes excellent no31…40 low yes excellent yes<=30 medium no fair no<=30 low yes fair yes>40 medium yes fair yes<=30 medium yes excellent yes31…40 medium no excellent yes31…40 high yes fair yes>40 medium no excellent no

This follows an example from Quinlan’s ID3

Output: A Decision Tree for “buys_computer”

age?

overcast

student? credit rating?

no yes fairexcellent

<=30 >40

no noyes yes

yes

30..40

Algorithm for Decision Tree Induction

• Basic algorithm (a greedy algorithm)– Tree is constructed in a top-down recursive divide-and-conquer manner– At start, all the training examples are at the root– Attributes are categorical (if continuous-valued, they are discretized in

advance)– Examples are partitioned recursively based on selected attributes– Test attributes are selected on the basis of a heuristic or statistical

measure (e.g., information gain)

• Conditions for stopping partitioning– All samples for a given node belong to the same class– There are no remaining attributes for further partitioning – majority

voting is employed for classifying the leaf– There are no samples left

Decision trees

Salary Education Class

10000 HS R

40000 C A

15000 C R

75000 G A

18000 G A

Training setSalary < 20000

N Y

AEducation = G

NY

A R

Decision trees

• Pros:– Fast.– Rules easy to interpret.– High dimensional data

• Cons:– No correlations– Axis-parallel cuts.

Decision trees(cont.)• Machine learning:

– ID3 (Quinlan86)– C4.5 (Quinlan93 )– CART (Breiman, Friedman, Olshen, Stone,

Classification and Regression Trees 1984)• Database:

– SLIQ (Metha, Agrawal and Rissanen, EDBT96)– SPRINT (Shafer, Agrawal, Metha, VLDB96)– Rainforest (Gherke, Ramakrishnan, Ghanti VLDB98)

Decision trees

• Finding the best tree is NP-Hard• We look at non-backtracking algorithms (never

look back at a previous decision)• Assume we have a test with n outcomes that

partitions T into subsets T1, T2,…, Tn

If the test is to be evaluated without exploring subsequent dimensions of the Ti’s, the only information available for guidance is the distribution of classes in T and its subsets.

Decision tree algorithms

• Building phase:– Recursively split nodes using best splitting

attribute and value for node

• Pruning phase:– Smaller (yet imperfect) tree achieves better

prediction accuracy.– Prune leaf nodes recursively to avoid over-

fitting.

Predictor variables (attributes)

• Numerically ordered: values are ordered and they can be represented in real line. ( E.g., salary.)

• Categorical: takes values from a finite set not having any natural ordering. (E.g., color.)

• Ordinal: takes values from a finite set whose values posses a clear ordering, but the distances between them are unknown. (E.g., preference scale: good, fair, bad.)

Binary Splits

Recursive (binary) partitioning– Univariate split on numerically ordered or

ordinal X

X <= c– on categorical X X A– Linear combination on numerical

ai Xi <= c

c and A are chosen to maximize separation.

Some probability...S = cases

freq(Ci,S) = # cases in S that belong to Ci

Gain entropic meassure:

Prob(“this case belongs to Ci”) = freq(Ci,S)/|S|

Information conveyed:

-log (freq(Ci,S)/|S|)

Entropy = expected information =

- (freq(Ci,S)/|S|) log (freq(Ci,S)/|S|) = info(S)

Gain

Test X:

infoX (T) = |Ti|/T info(Ti)

gain(X) = info (T) - infoX(T)

Example

Info(T) (9 play, 5 don’t) info(T) = -9/14log(9/14)- 5/14log(5/14) = 0.94 (bits)

Test: outlook

infoOutlook =

5/14 (-2/5 log(2/5)-3/5 log(3/5))+

4/14 (-4/4 log(4/4)) +

5/14 (-3/5 log(3/5) - 2/5 log(2/5))gainOutlook = 0.94-0.64= 0.3

= 0.64 (bits)

Test Windy

infowindy=

7/14(-4/7log(4/7)-3/7 log(3/7))

+7/14(-5/7log(5/7)-2/7log(2/(7))

= 0.278gainWindy = 0.94-0.278= 0.662

Windy is a better test

Outlook Temp Humidity Windy Classsunny 75 70 Y Playsunny 80 90 Y Don'tsunny 85 85 N Don'tsunny 72 95 N Don'tsunny 69 70 N Playovercast 72 90 Y Playovercast 83 78 N Playovercast 64 65 Y Playovercast 81 75 N Playrain 71 80 Y Don'train 65 70 Y Don'train 75 80 Y Playrain 68 80 N Playrain 70 96 N Play

Problem with Gain

Strong bias towards test with many outcomes.

Example: Z = Name

|Ti| = 1 (each name unique)

infoZ (T) = 1/|T| (- 1/N log (1/N)) 0

Maximal gain!! (but useless division--- overfitting--)

Split

Split-info (X) = - |Ti|/|T| log (|Ti|/|T|)

gain-ratio(X) = gain(X)/split-info(X)

Gain <= log(k)

Split <= log(n)

ratio small

Extracting Classification Rules from Trees

• Represent the knowledge in the form of IF-THEN rules• One rule is created for each path from the root to a leaf• Each attribute-value pair along a path forms a conjunction• The leaf node holds the class prediction• Rules are easier for humans to understand• Example

IF age = “<=30” AND student = “no” THEN buys_computer = “no”

IF age = “<=30” AND student = “yes” THEN buys_computer = “yes”

IF age = “31…40” THEN buys_computer = “yes”

IF age = “>40” AND credit_rating = “excellent” THEN buys_computer = “yes”

IF age = “<=30” AND credit_rating = “fair” THEN buys_computer = “no”

OVERFITTING

• Decision trees can grow so long that there is a leaf for each training example.

• Extremes:– Overfitted: “Whatever I haven’t seen can’t be

classified”– Too General: “If it is green, it is a tree”

Avoid Overfitting in Classification

• The generated tree may overfit the training data – Too many branches, some may reflect anomalies due to

noise or outliers– Result is in poor accuracy for unseen samples

• Two approaches to avoid overfitting – Prepruning: Halt tree construction early—do not split a

node if this would result in the goodness measure falling below a threshold

• Difficult to choose an appropriate threshold– Postpruning: Remove branches from a “fully grown” tree

—get a sequence of progressively pruned trees• Use a set of data different from the training data to

decide which is the “best pruned tree”

Approaches to Determine the Final Tree Size

• Separate training (2/3) and testing (1/3) sets

• Use cross validation, e.g., 10-fold cross validation

• Use all the data for training

– but apply a statistical test (e.g., chi-square) to estimate whether expanding or pruning a node may improve the entire distribution

• Use minimum description length (MDL) principle:

– halting growth of the tree when the encoding is minimized

Enhancements to basic decision tree induction

• Allow for continuous-valued attributes– Dynamically define new discrete-valued attributes that

partition the continuous attribute value into a discrete set of intervals

• Handle missing attribute values– Assign the most common value of the attribute

– Assign probability to each of the possible values

• Attribute construction– Create new attributes based on existing ones that are

sparsely represented

– This reduces fragmentation, repetition, and replication

Classification in Large Databases

• Classification—a classical problem extensively studied by statisticians and machine learning researchers

• Scalability: Classifying data sets with millions of examples and hundreds of attributes with reasonable speed

• Why decision tree induction in data mining?

– relatively faster learning speed (than other classification methods)

– convertible to simple and easy to understand classification rules

– can use SQL queries for accessing databases

– comparable classification accuracy with other methods

Scalable Decision Tree Induction Methods in Data Mining Studies

• SLIQ (EDBT’96 — Mehta et al.)

– builds an index for each attribute and only class list and the current attribute list reside in memory

• SPRINT (VLDB’96 — J. Shafer et al.)

– constructs an attribute list data structure

• PUBLIC (VLDB’98 — Rastogi & Shim)

– integrates tree splitting and tree pruning: stop growing the tree earlier

• RainForest (VLDB’98 — Gehrke, Ramakrishnan & Ganti)

– separates the scalability aspects from the criteria that determine the quality of the tree

– builds an AVC-list (attribute, value, class label)

SPRINT

For large data sets.

Age Car Type Risk23 Family H17 Sports H43 Sports H68 Family L32 Truck L20 Family H

Age < 25

H Car = Sports

H L

Gini Index (IBM IntelligentMiner)

• If a data set T contains examples from n classes, gini index, gini(T) is defined as

where pj is the relative frequency of class j in T.• If a data set T is split into two subsets T1 and T2 with sizes N1 and

N2 respectively, the gini index of the split data contains examples from n classes, the gini index gini(T) is defined as

• The attribute provides the smallest ginisplit(T) is chosen to split the node (need to enumerate all possible splitting points for each attribute).

n

jp jTgini

1

21)(

)()()( 22

11 Tgini

NN

TginiNNTginisplit

SPRINT

Partition (S)

if all points of S are in the same class

return;

else

for each attribute A do

evaluate_splits on A;

use best split to partition into S1,S2;

Partition(S1);Partition(S2);

SPRINT Data Structures

Age Risk Tuple17 H 120 H 523 H 032 L 443 H 368 L 2

Car Type Risk TupleFamily H 0Sports H 1Sports H 2Family L 3Truck L 4

Family H 5

Age Car Type Risk23 Family H17 Sports H43 Sports H68 Family L32 Truck L20 Family H

Training set

Age Car

Attribute lists

SplitsAge Risk Tuple17 H 120 H 523 H 032 L 443 H 368 L 2

Age < 27.5

Car Type Risk TupleFamily H 0Sports H 1Family H 5

Car Type Risk TupleSports H 2Family L 3Truck L 4

Age Risk Tuple17 H 120 H 523 H 0

Age Risk Tuple32 L 443 H 268 L 3


Family H 5

Group1 Group2

Histograms

For continuous attributes

Associated with node (Cabove, Cbelow)

to process already processed

ExampleAge Risk Tuple

17 H 120 H 523 H 032 L 443 H 368 L 2

H LCb 0 0Ca 4 2

ginisplit0 = 0/6 gini(S1) + 6/6 gini(S2)

gini(S2) = 1 - [(4/6)2 +(2/6)2 ] = 0.444

ginisplit0 = 0.444

H LCb 1 0Ca 3 2

ginisplit1 = 1/6 gini(S1) +5/6 gini(S2)

gini(S1) = 1 - [(1/1) 2 ] = 0

gini(S2) = 1 - [(3/4)2 +(2/4)2 ] = 0.1875

ginisplit1= 0.156

H LCb 2 0Ca 2 2

ginisplit2 = 2/6 gini(S1) +4/6 gini(S2)

gini(S1) = 1 - [(2/2) 2 ] = 0

gini(S2) = 1 - [(2/4)2 +(2/4)2 ] = 0.5

ginisplit2= 0.333

H LCb 3 0Ca 1 2

ginisplit3 =3/6 gini(S1) +3/6 gini(S2)

gini(S1) = 1 - [(3/3) 2 ] = 0

gini(S2) = 1 - [(1/3)2 +(2/3)2 ] = 0.444

ginisplit3= 0.222

H LCb 3 1Ca 1 1


gini(S1) = 1 - [(3/4) 2 +(1/4) 2 ] = 0.375

gini(S2) = 1 - [(1/2)2 +(1/2)2 ] = 0.5ginisplit4= 0.416

H LCb 4 1Ca 0 1


gini(S1) = 1 - [(4/5) 2 +(1/5) 2 ] = 0.320

gini(S2) = 1 - [(1/1)2 ] = 0ginisplit5= 0.222

H LCb 4 2Ca 0 0


gini(S1) = 1 - [(4/6) 2 +(2/6) 2 ] = 0.320

ginisplit6= 0.444Age <= 18.5

Splitting categorical attributes

Single scan through the attribute list collecting counts on count matrix for each combination of class label + attribute value

ExampleCar Type Risk Tuple

Family H 0Sports H 1Sports H 2Family L 3Truck L 4

Family H 5

H LFamily 2 1Sports 2 0Truck 0 1

ginisplit(family)= 3/6 gini(S1) + 3/6 gini(S2)

gini(S1) = 1 - [(2/3)2 + (1/3)2] = 4/9

gini(S2) = 1- [(2/3)2 + (1/3)2] = 4/9

ginisplit(family)= 0.444 ginisplit((sports)= 2/6 gini(S1) + 4/6 gini(S2)

gini(S1) = 1 - [(2/2)2] = 0

gini(S2) = 1- [(2/4)2 + (2/4)2] = 0.5

ginisplit((sports) )= 0.333

ginisplit(truck)= 1/6 gini(S1) + 5/6 gini(S2)

gini(S1) = 1 - [(1/1)2] = 0

gini(S2) = 1- [(4/5)2 + (1/5)2] = 0.32ginisplit(truck) )= 0.266

Car Type = Truck

Example (2 attributes)

The winner is Age <= 18.5

Age Risk Tuple17 H 120 H 523 H 032 L 443 H 368 L 2


Family H 5

H

Y N

Age Risk Tuple20 H 523 H 032 L 443 H 368 L 2

Car Type Risk TupleFamily H 0

Sports H 2Family L 3Truck L 4

Family H 5

Performing the split

• Create 2 child nodes

• Split attribute lists for winning attribute

• For the remaining– Insert Tuple Ids in Hash Table (which child)– Scan lists of attributes and probe hash table

(may be too large and need several steps).

Drawbacks

• Large explosion of space (possibly tripling the size of database).

• Costly Hash-Join.


• What is classification? What is prediction?• Classification by decision tree induction• Bayesian Classification• Other methods (SVM)• Classification accuracy• Prediction• Summary

Bayesian Theorem

• Given training data D, posteriori probability of a hypothesis h, P(h|D) follows the Bayes theorem

• MAP (maximum posteriori) hypothesis

• Practical difficulty: require initial knowledge of many probabilities, significant computational cost

)()()|()|(

DPhPhDPDhP

.)()|(maxarg)|(maxarg hPhDPHh

DhPHhMAP

h

Naïve Bayes Classifier (I)

• A simplified assumption: attributes are conditionally independent:

• Greatly reduces the computation cost, only count the class distribution.

P V P Pj j i ji

n

C C v C( | ) ( ) ( | )

1

Naive Bayesian Classifier (II)• Given a training set, we can compute the probabilities

Outlook P N Humidity P Nsunny 2/9 3/5 high 3/9 4/5overcast 4/9 0 normal 6/9 1/5rain 3/9 2/5Tempreature Windyhot 2/9 2/5 true 3/9 3/5mild 4/9 2/5 false 6/9 2/5cool 3/9 1/5

3/9 x 3/9 x

ExampleE ={outlook = sunny, temp = [64,70], humidity= [65,70], windy = y} =

{E1,E2,E3,E4}

Pr[“Play”/E] = (Pr[E1/Play] x Pr[E2/Play] x Pr[E3/Play] x Pr[E4/Play] x Pr[Play]) / Pr[E] =(2/9x 4/9x

Outlook Temp Humidity Windy Classsunny 75 70 Y Playsunny 80 90 Y Don'tsunny 85 85 N Don'tsunny 72 95 N Don'tsunny 69 70 N Playovercast 72 90 Y Playovercast 83 78 N Playovercast 64 65 Y Playovercast 81 75 N Playrain 71 80 Y Don'train 65 70 Y Don'train 75 80 Y Playrain 68 80 N Playrain 70 96 N Play

9/14)/Pr[E]= 0.007/Pr[E]

Pr[“Don’t”/E] = (3/5 x 2/5 x 1/5 x 3/5 x 5/14)/Pr[E] = 0.010/Pr[E]

With E: Pr[“Play”/E] = 41 %, Pr[“Don’t”/E] = 59 %

Bayesian Belief Networks (I)FamilyHistory

LungCancer

PositiveXRay

Smoker

Emphysema

Dyspnea

LC

~LC

(FH, S) (FH, ~S)(~FH, S) (~FH, ~S)

0.8

0.2

0.5

0.5

0.7

0.3

0.1

0.9

Bayesian Belief Networks

The conditional probability table for the variable LungCancer

Bayesian Belief Networks (II)• Bayesian belief network allows a subset of the

variables conditionally independent

• A graphical model of causal relationships

• Several cases of learning Bayesian belief networks

– Given both network structure and all the variables: easy

– Given network structure but only some variables

– When the network structure is not known in advance

Another Example (Friedman & Goldzsmidt)

Variables : Burglary, Earthquake, Alarm, Neighbor call, Radio announcement.

Burglary and Earthquake are independent (P(BE) = P(B)*P(E))

Burglary and Radio announcement are independent given Earthquake (P(BR/E) = P(B/E)*P(R/E))

So, P(A,R,E,B)=P(A|R,E,B)*P(R|E,B)*P(E|B)*P(B)

can be reduced to:

P(A,R,E,B) = P(A|E,B)*P(R|E)*P(E)*P(B)

Example (cont.)

Earthquake Burglary

Radio announc.

Neigh. call

Alarm

Example (cont.)

E B P(A/EB) P(!A/EB)

E B 0.90 0.10

E !B 0.20 0.80 !E B 0.90 0.10 !E !B 0.01 0.99

Associated with each node is a set of conditional probability distributions. For example, the "Alarm" node might have thefollowing probability distribution


• What is classification? What is prediction?• Issues regarding classification and prediction• Classification by decision tree induction• Bayesian Classification• Other Methods• Classification accuracy• Prediction• Summary

Extending linear classification

Problem: all the algorithms we covered (plus many other ones) can only represent linear boundaries between classes

-> Age > 25Age <= 25 <-

Too simplistic for many real cases

Nonlinear class boundaries

Support vector machines (SVM)-- a misnomer, since they are algorithms, not machines--

Idea: use a non-linear mapping and transform the space into a new space.

Example:

x = w1a13 + w2 a1

2 a2 + w3 a1 a22 + w4 a2

3

SVMs

Based on an algorithm that finds a maximum marginal hyperplane (linear model).

Convex hull: (tightest enclosing polygon)

Maximum margin hyperplane

Support vectors

Shortest line connecting the hulls

SVMs (cont.)• We have assumed that the two classes are linearly separable, so their convex hulls cannot overlap.

• The maximum margin hyperplane (MMH) is the one that is as far away as possible from both convex hulls. It is orthogonal to the shortest line connecting the hulls.

• The instances closest to the MMH (minimum distance to the line) are called support vectors (SV). (At least one for each class, often more.)

– Given the SVs, we can easily construct the MLH.

– All other training points can be deleted without any effect on the MMH

SVMs (cont.)

A hyperplane that separates the two classes can be written as:

x = w0 + w1a1 + w2 a2

for a two-attribute case.

However, the equation that defines the MMH, can be defined in terms of the SVs. Write the class value y of a training instance (point) as 1 (yes) or -1 (no). Then the MMH is:

x = b + i yi a(i). a i SVs

yi = class value of the point a(i); b and i are numerical values to be determined; a is a test point.

SVMs (cont.)

So, now…

Use the training values to determine b and i for

x = b + i yi a(i). a i SVs

Standard optimization problem: constrained quadratic optimization

(off-the-shelf software packages to solve this: Fletcher, Practical Methods of Optimization, 1987)

Dot product


• What is classification? What is prediction?• Issues regarding classification and prediction• Classification by decision tree induction• Bayesian Classification• Other Methods• Classification accuracy• Prediction• Summary

Classification Accuracy: Estimating Error Rates

• Partition: Training-and-testing

– use two independent data sets, e.g., training set (2/3), test set(1/3)

– used for data set with large number of samples

• Cross-validation

– divide the data set into k subsamples

– use k-1 subsamples as training data and one sub-sample as test data --- k-fold cross-validation

– for data set with moderate size

• Bootstrapping (leave-one-out)

– for small size data

Boosting and Bagging

• Boosting increases classification accuracy– Applicable to decision trees or Bayesian

classifier

• Learn a series of classifiers, where each classifier in the series pays more attention to the examples misclassified by its predecessor

• Boosting requires only linear time and constant space


• What is classification? What is prediction?• Issues regarding classification and prediction• Classification by decision tree induction• Bayesian Classification• Classification accuracy• Prediction• Summary

What Is Prediction?

• Prediction is similar to classification– First, construct a model

– Second, use model to predict unknown value

• Major method for prediction is regression

– Linear and multiple regression

– Non-linear regression

• Prediction is different from classification– Classification refers to predict categorical class label

– Prediction models continuous-valued functions

• Predictive modeling: Predict data values or construct generalized linear models based on the database data.

• One can only predict value ranges or category distributions• Method outline:

– Minimal generalization– Attribute relevance analysis– Generalized linear model construction– Prediction

• Determine the major factors which influence the prediction– Data relevance analysis: uncertainty measurement, entropy analysis,

expert judgement, etc.• Multi-level prediction: drill-down and roll-up analysis

Predictive Modeling in Databases

• Linear regression: Y = + X– Two parameters , and specify the line and are to be

estimated by using the data at hand.– using the least squares criterion to the known values of Y1,

Y2, …, X1, X2, ….

• Multiple regression: Y = b0 + b1 X1 + b2 X2.– Many nonlinear functions can be transformed into the above.

• Log-linear models:– The multi-way table of joint probabilities is approximated

by a product of lower-order tables.– Probability: p(a, b, c, d) = ab acad bcd

Regress Analysis and Log-Linear Models in Prediction


• What is classification? What is prediction?• Issues regarding classification and prediction• Classification by decision tree induction• Bayesian Classification• Other Classification Methods• Classification accuracy • Prediction• Summary

Summary

• Classification is an extensively studied problem (mainly in

statistics, machine learning & neural networks)

• Classification is probably one of the most widely used data

mining techniques with a lot of extensions

• Scalability is still an important issue for database applications:

thus combining classification with database techniques should

be a promising topic

• Research directions: classification of non-relational data, e.g.,

text, spatial, multimedia, etc..

lecture 7 classification itcs 6163. chapter 7. classification and prediction what is classification?...

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