CISC 4631 Data Mining Lecture 04: Decision Trees Theses slides are based on the slides by Tan,...

CISC 4631Data Mining

Lecture 04:• Decision Trees

Theses slides are based on the slides by • Tan, Steinbach and Kumar (textbook authors)• Eamonn Koegh (UC Riverside)• Raymond Mooney (UT Austin)

Classification: Definition• Given a collection of records (training set )

– Each record contains a set of attributes, one of the attributes is the class.

• Find a model for class attribute as a function of the values of other attributes.

• Goal: previously unseen records should be assigned a class as accurately as possible.– A test set is used to determine the accuracy of the

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

Illustrating Classification Task

Induction

Deduction

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 Yes 10

11 No Small 55K ?

12 Yes Medium 80K ?

13 Yes Large 110K ?

14 No Small 95K ?

15 No Large 67K ? 10

Test Set

Learningalgorithm

Training Set

Classification Techniques

• Decision Tree based Methods• Rule-based Methods• Memory based reasoning• Neural Networks• Naïve Bayes and Bayesian Belief Networks• Support Vector Machines

Example of a Decision Tree

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

categoric

contin

Refund

TaxInc

Yes No

Married Single, Divorced

< 80K > 80K

Splitting Attributes

Training Data Model: Decision Tree

Another Example of Decision Tree

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

categoric

contin

Refund

TaxInc

Yes No

Married Single,

Divorced

< 80K > 80K

There could be more than one tree that fits the same data!

Decision Tree Classification Task

Induction

Deduction

1 Yes Large 125K No

2 No Medium 100K No

3 No Small 70K 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

11 No Small 55K ?

12 Yes Medium 80K ?

13 Yes Large 110K ?

14 No Small 95K ?

Test Set

TreeInductionalgorithm

Training Set

Decision Tree

Apply Model to Test Data

Refund

TaxInc

Yes No

< 80K > 80K

Refund Marital Status

Taxable Income Cheat

No Married 80K ? 10

Test DataStart from the root of tree.

Refund

TaxInc

Yes No

< 80K > 80K

No Married 80K ? 10

Test Data

Refund

TaxInc

Yes No

< 80K > 80K

No Married 80K ? 10

Test Data

Refund

TaxInc

Yes No

< 80K > 80K

No Married 80K ? 10

Test Data

Refund

TaxInc

Yes No

< 80K > 80K

No Married 80K ? 10

Test Data

Refund

TaxInc

Yes No

< 80K > 80K

No Married 80K ? 10

Test Data

Assign Cheat to “No”

Decision Tree Terminology

Decision Tree Classification Task

Induction

Deduction

1 Yes Large 125K No

2 No Medium 100K No

3 No Small 70K 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

11 No Small 55K ?

12 Yes Medium 80K ?

13 Yes Large 110K ?

14 No Small 95K ?

Test Set

TreeInductionalgorithm

Training Set

Decision Tree

Decision Tree Induction

• Many Algorithms:– Hunt’s Algorithm (one of the earliest)– CART– ID3, C4.5– SLIQ,SPRINT

• John Ross Quinlan is a computer science researcher in data mining and decision theory. He has contributed extensively to the development of decision tree algorithms, including inventing the canonical C4.5 and ID3 algorithms.

Decision Tree Classifier

Ross Quinlan

1 2 3 4 5 6 7 8 9 10

123456789

Abdomen Length

Abdomen Length > 7.1?

no yes

KatydidAntenna Length > 6.0?

no yes

KatydidGrasshopper

Grasshopper

Antennae shorter than body?

Cricket

Foretiba has ears?

Katydids Camel Cricket

3 Tarsi?

Decision trees predate computers

Decision tree is a classifier in the form of a tree structure– Decision node: specifies a test on a single attribute– Leaf node: indicates the value of the target attribute – Arc/edge: split of one attribute– Path: a disjunction of test to make the final decision

Decision trees classify instances or examples by starting at the root of the tree and moving through it until a leaf node.

Definition

• 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

Decision Tree Classification

Decision Tree Representation

• Each internal node tests an attribute• Each branch corresponds to attribute value• Each leaf node assigns a classification

outlook

sunny overcast rain

yeshumidity wind

high normal strong weak

yes yesno no

• 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 can be 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

How do we construct the decision tree?

Top-Down Decision Tree Induction

• Main loop:1. A the “best” decision attribute for next node2. Assign A as decision attribute for node3. For each value of A, create new descendant of node4. Sort training examples to leaf nodes5. If training examples perfectly classified,

Then STOP, Else iterate over new leaf nodes

Tree Induction

• Greedy strategy.– Split the records based on an attribute test that optimizes

certain criterion.

• Issues– Determine how to split the records

• How to specify the attribute test condition?• How to determine the best split?

– Determine when to stop splitting

How To Split Records• Random Split

– The tree can grow huge – These trees are hard to understand. – Larger trees are typically less accurate than smaller trees.

• Principled Criterion– Selection of an attribute to test at each node - choosing the most useful attribute

for classifying examples. – How?– Information gain

• measures how well a given attribute separates the training examples according to their target classification

• This measure is used to select among the candidate attributes at each step while growing the tree

Tree Induction

• Greedy strategy:– Split the records based on an attribute test that optimizes

certain criterion:– Hunt’s algorithm: recursively partition training records into

successively purer subsets. How to measure purity/impurity

• Entropy and information gain (covered in the lectures slides)• Gini (covered in the textbook)• Classification error

How to determine the Best Split

OwnCar?

C0: 6C1: 4

C0: 4C1: 6

C0: 1C1: 3

C0: 8C1: 0

C0: 1C1: 7

CarType?

C0: 1C1: 0

C0: 0C1: 1

StudentID?

Yes No Family

Sports

Luxury c1c10

C0: 0C1: 1

Before Splitting: 10 records of class 0, 10 records of class 1

Which test condition is the best?

Why is student id a bad feature to use?

Gender

How to determine the Best Split

• Greedy approach: – Nodes with homogeneous class distribution are preferred

• Need a measure of node impurity:

C0: 5C1: 5

C0: 9C1: 1

Non-homogeneous,

High degree of impurity

Homogeneous,

Low degree of impurity

Picking a Good Split Feature• Goal is to have the resulting tree be as small as possible, per Occam’s

razor.• Finding a minimal decision tree (nodes, leaves, or depth) is an NP-hard

optimization problem.• Top-down divide-and-conquer method does a greedy search for a simple

tree but does not guarantee to find the smallest.– General lesson in Machine Learning and Data Mining: “Greed is good.”

• Want to pick a feature that creates subsets of examples that are relatively “pure” in a single class so they are “closer” to being leaf nodes.

• There are a variety of heuristics for picking a good test, a popular one is based on information gain that originated with the ID3 system of Quinlan (1979).

R. Mooney, UT Austin

Information Theory• Think of playing "20 questions": I am thinking of an integer between 1 and

1,000 -- what is it? What is the first question you would ask?• What question will you ask?• Why?

• Entropy measures how much more information you need before you can identify the integer.

• Initially, there are 1000 possible values, which we assume are equally likely.

• What is the maximum number of question you need to ask?

31R. Mooney, UT Austin

Entropy• Entropy (disorder, impurity) of a set of examples, S, relative to a binary

classification is:

where p1 is the fraction of positive examples in S and p0 is the fraction of negatives.

• If all examples are in one category, entropy is zero (we define 0log(0)=0)• If examples are equally mixed (p1=p0=0.5), entropy is a maximum of 1.• Entropy can be viewed as the number of bits required on average to encode

the class of an example in S where data compression (e.g. Huffman coding) is used to give shorter codes to more likely cases.

• For multi-class problems with c categories, entropy generalizes to:

)(log)(log)( 020121 ppppSEntropy

iii ppSEntropy

12 )(log)(

Entropy Plot for Binary Classification

• The entropy is 0 if the outcome is certain. • The entropy is maximum if we have no knowledge of the system

(or any outcome is equally possible).

Entropy of a 2-class problem with regard to

the portion of one of the two groups

Information Gain• Is the expected reduction in entropy caused by partitioning the examples

according to this attribute. • is the number of bits saved when encoding the target value of an arbitrary

member of S, by knowing the value of attribute A.

Information Gain in Decision Tree Induction

• Assume that using attribute A, a current set will be partitioned into some number of child sets

• The encoding information that would be gained by branching on A

)()()( setschildallEsetCurrentEAGain

Note: entropy is at its minimum if the collection of objects is completely uniform

Examples for Computing Entropy

C1 0 C2 6

C1 2 C2 4

C1 1 C2 5

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

Entropy = – 0 log2 0 – 1 log2 1 = – 0 – 0 = 0

P(C1) = 1/6 P(C2) = 5/6

Entropy = – (1/6) log2 (1/6) – (5/6) log2 (5/6) = 0.65

P(C1) = 2/6 P(C2) = 4/6

Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92

tjptjptEntropy )|(log)|()(2

C1 3 C2 3

P(C1) = 3/6=1/2 P(C2) = 3/6 = 1/2Entropy = – (1/2) log2 (1/2) – (1/2) log2 (1/2)

= -(1/2)(-1) – (1/2)(-1) = ½ + ½ = 1

NOTE: p( j | t) is computed as the relative frequency of class j at node t

How to Calculate log2x

• Many calculators only have a button for log10x and logex (note log typically means log10)

• You can calculate the log for any base b as follows:– logb(x) = logk(x) / logk(b)

– Thus log2(x) = log10(x) / log10(2)

– Since log10(2) = .301, just calculate the log base 10 and divide by .301 to get log base 2.

– You can use this for HW if needed

Splitting Based on INFO...

• 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.

splitiEntropy

pEntropyGAIN1

Continuous Attribute?(more on it later)

• Each non-leaf node is a test, its edge partitioning the attribute into subsets (easy for discrete attribute).

• For continuous attribute– Partition the continuous value of attribute A into a discrete set of

intervals– Create a new boolean attribute Ac , looking for a threshold c,

otherwisec

true A cA

How to choose c ?

Person Hair Length

Weight Age Class

Homer 0” 250 36 M

Marge 10” 150 34 F

Bart 2” 90 10 M

Lisa 6” 78 8 F

Maggie 4” 20 1 F

Abe 1” 170 70 M

Selma 8” 160 41 F

Otto 10” 180 38 M

Krusty 6” 200 45 M

Comic 8” 290 38 ?

Hair Length <= 5?yes no

Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911

Entropy(1F,3M) = -(1/4)log2(1/4) - (3/4)log2(3/4)

= 0.8113

Entropy(3F,2M) = -(3/5)log2(3/5) - (2/5)log

2(2/5)

= 0.9710

pSEntropy 22 loglog)(

Gain(Hair Length <= 5) = 0.9911 – (4/9 * 0.8113 + 5/9 * 0.9710 ) = 0.0911

Let us try splitting on Hair length

Weight <= 160?yes no

= 0.7219

2(4/4)

Gain(Weight <= 160) = 0.9911 – (5/9 * 0.7219 + 4/9 * 0 ) = 0.5900

Let us try splitting on Weight

age <= 40?yes no

2(2/3)

= 0.9183

Gain(Age <= 40) = 0.9911 – (6/9 * 1 + 3/9 * 0.9183 ) = 0.0183

Let us try splitting on Age

Weight <= 160?yes no

Hair Length <= 2?yes no

Of the 3 features we had, Weight was best. But while people who weigh over 160 are perfectly classified (as males), the under 160 people are not perfectly classified… So we simply recurse!

This time we find that we can split on Hair length, and we are done!

Weight <= 160?

yes no

Hair Length <= 2?

yes no

We don’t need to keep the data around, just the test conditions.

Male Female

How would these people be classified?

It is trivial to convert Decision Trees to rules… Weight <= 160?

yes no

Hair Length <= 2?

yes no

Male Female

Rules to Classify Males/Females

If Weight greater than 160, classify as MaleElseif Hair Length less than or equal to 2, classify as MaleElse classify as Female

Rules to Classify Males/Females

If Weight greater than 160, classify as MaleElseif Hair Length less than or equal to 2, classify as MaleElse classify as Female

Decision tree for a typical shared-care setting applying the system for the diagnosis of prostatic obstructions.

Once we have learned the decision tree, we don’t even need a computer!

This decision tree is attached to a medical machine, and is designed to help nurses make decisions about what type of doctor to call.

Wears green?

Yes No

The worked examples we have seen were performed on small datasets. However with small datasets there is a great danger of overfitting the data…

When you have few datapoints, there are many possible splitting rules that perfectly classify the data, but will not generalize to future datasets.

For example, the rule “Wears green?” perfectly classifies the data, so does “Mothers name is Jacqueline?”, so does “Has blue shoes”…

MaleFemale

How to Find the Best Split: GINI

Yes No

Node N3 Node N4

Yes No

Node N1 Node N2

Before Splitting:

C0 N10 C1 N11

C0 N20 C1 N21

C0 N30 C1 N31

C0 N40 C1 N41

C0 N00 C1 N01

M1 M2 M3 M4

M12 M34Gain = M0 – M12 vs M0 – M34

Measure of Impurity: GINI (at node t)• Gini Index for a given node t with classes j

NOTE: p( j | t) is computed as the relative frequency of class j at node t

• Example: Two classes C1 & C2 and node t has 5 C1 and 5 C2 examples. Compute Gini(t)– 1 – [p(C1|t) + p(C2|t)] = 1 – [(5/10)2 + [(5/10)2 ]– 1 – [¼ + ¼] = ½.– Do you think this Gini value indicates a good split or bad

split? Is it an extreme value?

tjptGINI 2)]|([1)(

More on Gini• Worst Gini corresponds to probabilities of 1/nc, where nc is the

number of classes.– For 2-class problems the worst Gini will be ½

• How do we get the best Gini? Come up with an example for node t with 10 examples for classes C1 and C2– 10 C1 and 0 C2 – Now what is the Gini?

• 1 – [(10/10)2 + (0/10)2 = 1 – [1 + 0] = 0– So 0 is the best Gini

• So for 2-class problems:– Gini varies from 0 (best) to ½ (worst).

Some More Examples• Below we see the Gini values for 4 nodes with

different distributions. They are ordered from best to worst. See next slide for details– Note that thus far we are only computing GINI for one

node. We need to compute it for a split and then compute the change in Gini from the parent node.

C1 0C2 6

Gini=0.000

C1 2C2 4

Gini=0.444

C1 3C2 3

Gini=0.500

C1 1C2 5

Gini=0.278

Examples for computing GINI

C1 0 C2 6

C1 2 C2 4

C1 1 C2 5

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

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

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

Splitting Criteria based on Classification Error

• Classification error at a node t :

• 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

Examples for Computing Error

C1 0 C2 6

C1 2 C2 4

C1 1 C2 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)( tiPtErrori

Comparison among Splitting CriteriaFor a 2-class problem:

Discussion• Error rate is often the metric used to evaluate a

classifier (but not always)– So it seems reasonable to use error rate to determine the

best split– That is, why not just use a splitting metric that matches the

ultimate evaluation metric?– But this is wrong!

• The reason is related to the fact that decision trees use a greedy strategy, so we need to use a splitting metric that leads to globally better results

• The other metrics will empirically outperform error rate, although there is no proof for this.

DTs in practice...• Growing to purity is bad (overfitting)

x1: petal length

DTs in practice...

• Growing to purity is bad (overfitting)– Terminate growth early– Grow to purity, then prune back

x1: petal length

Not statisticallysupportable leaf

Remove split& merge leaves

Avoid Overfitting in Classification(more on overfitting later)

• 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”

Tree Induction

• Greedy strategy.– Split the records based on an attribute test that optimizes

certain criterion.

• Issues– Determine how to split the records

• How to specify the attribute test condition?• How to determine the best split?

– Determine when to stop splitting

How to Specify Test Condition?

• Depends on attribute types– Nominal– Ordinal– Continuous

• Depends on number of ways to split– 2-way split– Multi-way split

Splitting Based on Nominal Attributes

• Multi-way split: Use as many partitions as distinct values.

• Binary split: Divides values into two subsets. Need to find optimal partitioning.

CarTypeFamily

SportsLuxury

CarType{Family, Luxury} {Sports}

CarType{Sports, Luxury} {Family} OR

• Multi-way split: Use as many partitions as distinct values.

• Binary split: Divides values into two subsets. Need to find optimal partitioning.

• What about this split?

Splitting Based on Ordinal Attributes

SizeSmall

MediumLarge

Size{Medium,

Large} {Small}

Size{Small,

Medium} {Large}OR

Size{Small, Large} {Medium}

Splitting Based on Continuous Attributes

• Different ways of handling– Discretization to form an ordinal categorical attribute

• Static – discretize once at the beginning• Dynamic – ranges can be found by equal interval

bucketing, 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

Splitting Based on Continuous Attributes

TaxableIncome> 80K?

Yes No

TaxableIncome?

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

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

Data Fragmentation

• Number of instances gets smaller as you traverse down the tree

• Number of instances at the leaf nodes could be too small to make any statistically significant decision

Search Strategy

• Finding an optimal decision tree is NP-hard

• The algorithm presented so far uses a greedy, top-down, recursive partitioning strategy to induce a reasonable solution

Expressiveness• Decision tree provides expressive representation for learning

discrete-valued function– But they do not generalize well to certain types of Boolean

functions• Example: parity function:

– Class = 1 if there is an even number of Boolean attributes with truth value = True

– Class = 0 if there is an odd number of Boolean attributes with truth value = True

• For accurate modeling, must have a complete tree

• Not expressive enough for modeling continuous variables– Particularly when test condition involves only a single

attribute at-a-time

Decision Boundary

y < 0.33?

: 0 : 3

: 4 : 0

y < 0.47?

: 4 : 0

: 0 : 4

x < 0.43?

No Yes No

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

• Border line between two neighboring regions of different classes is known as decision boundary

• Decision boundary is parallel to axes because test condition involves a single attribute at-a-time

Oblique Decision Trees

x + y < 1

Class = + Class =

• Test condition may involve multiple attributes

• More expressive representation

• Finding optimal test condition is computationally expensive

Vertical/Horizontal Boundaries

500 circular and 500 triangular data points.

Circular points:

0.5 sqrt(x12+x2

Triangular points:

sqrt(x12+x2

2) > 0.5 or

sqrt(x12+x2

2) < 1

Tree ReplicationP

• Same subtree appears in multiple branches

Model Evaluation

• Metrics for Performance Evaluation– How to evaluate the performance of a model?

• Methods for Performance Evaluation– How to obtain reliable estimates?

1 2 3 4 5 6 7 8 9 10

123456789

10 20 30 40 50 60 70 80 90 100

102030405060708090

1 2 3 4 5 6 7 8 9 10

123456789

Which of the “Problems” can be solved by a Decision Tree?

1) Deep Bushy Tree2) Useless3) Deep Bushy Tree

The Decision Tree has a hard time with correlated attributes ?

• Advantages:– Easy to understand (Doctors love them!) – Easy to generate rules

• Disadvantages:– May suffer from overfitting.– Classifies by rectangular partitioning (so does not

handle correlated features very well).– Can be quite large – pruning is necessary.– Does not handle streaming data easily

Advantages/Disadvantages of Decision Trees

CISC 4631 Data Mining Lecture 04: Decision Trees Theses slides are based on the slides by Tan,...

Documents

Speaker 5 eamonn cashell

1 CISC 4631 Data Mining Lecture 09: Association Rule Mining Theses slides are based on the slides by Tan, Steinbach and Kumar (textbook authors) Prof

WheelHorse Comando 6 1-4631

CS 170 INTRO TO ARTIFICIAL INTELLIGENCE CS 170 INTRO TO ARTIFICIAL INTELLIGENCE Dr Eamonn Keogh eamonn@cs.uci.edu eamonn/ Office:

ClassificationContinued Dr Eamonn Keogh Computer Science & Engineering Department University of California - Riverside Riverside,CA 92521 eamonn@cs.ucr.edu

CISC 4631 Data Mining - Fordhamstorm.cis.fordham.edu/~gweiss/classes/cisc6950/slides/09 Clustering... · CISC 4631 Data Mining ... Cabletron-Sys-DOWN,CISCO -DOWN,HP -DOWN, ... 4 Baker-Hughes-UP,Dresser-Inds-UP,Halliburton

Eamonn Kelly, Architecture Portfolio Samples

Phone: (928) -641 4631 WELCOME!

Classification Dr Eamonn Keogh Computer Science & Engineering Department University of California - Riverside Riverside,CA 92521 eamonn@cs.ucr.edu Who

Slides at eamonn/public/cs170guest.ppt Eamonn Keogh eamonn@cs.ucr.edu

EAMONN BYRNE LANDSCAPE ARCHITECTUREd284f45nftegze.cloudfront.net/eblawebsite/EBLA_WEB... · 2014-01-24 · Eamonn Byrne Landscape Architecture (EBLA) is a consultancy providing services

Eamonn FingletonEamonn Fingleton

CISC 4631 Data Mining Lecture 08: Clustering Theses slides are based on the slides by Tan, Steinbach and Kumar (textbook authors) Eamonn Koegh (UC Riverside)

Eamonn Kneeland, Serzen 26

SAX: a Novel Symbolic Representation of Time Series Authors Jessica Lin Eamonn Keogh Li Wei Stefano Lonardi Presenter Arif Bin Hossain Slides incorporate

Eamonn de valera

Indexing Time Series Based on original slides by Prof. Dimitrios Gunopulos and Prof. Christos Faloutsos with some slides from tutorials by Prof. Eamonn

European Platform. Eamonn Power, TSSG

Eamonn Slattery

1 CISC 4631 Data Mining Lecture 03: Nearest Neighbor Learning Theses slides are based on the slides by Tan, Steinbach and Kumar (textbook authors) Prof