Legal Analytics Course - Class 7 - Binary Classification with Decision Tree Learning - Professor Daniel Martin Katz + Professor Michael J Bommarito

Class 7Binary Classification & Decision Tree Learning

Legal Analytics

Professor Daniel Martin KatzProfessor Michael J Bommarito II

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< Binary Classification >

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http://scikit-learn.org/stable/tutorial/machine_learning_map/index.htmlaccess more at legalanalyticscourse.com

http://scikit-learn.org/stable/tutorial/machine_learning_map/index.html

http://scikit-learn.org/stable/tutorial/machine_learning_map/index.html

Classification to Predict Quantity

Classification to Predict CategoryRegression Methods

Trees, Forests, Knn, etc.


Adapted from Slides ByVictor Lavrenko and Nigel Goddard

@ University of EdinburghTake A Look These 12


72FemaleHuman

3FemaleHorse

36Male

Human

21Male

Human

67Male

Human

29FemaleHuman

54Male

Human

44Male

Human

50Male

Human

42FemaleHuman

6MaleDog

7FemaleHuman

Task = Determine Whether the Agents Will Obtain Employment?

Yes

Nof( )

Job?

Binary Classification (Supervised Learning)


Classification (Supervised Learning)Yes

Nof( )

Job?


Classification (Supervised Learning)

decision boundary

Yes

Nof( )

Job?

decision boundary


Multi Class Classification


https://www.youtube.com/watch?v=p5rTio1G4ys



Task = Determine Whether the Agents Will Obtain a Loan?

YesPerhapsf( )

Loan?

Multi Class Classification (Supervised Learning)

No


f( )


Loan?

YesPerhapsNo


f( )

Loan?

Yes


No

Maybe

YesPerhapsNo


Multiclass = Hyperplaneaccess more at legalanalyticscourse.com

Task = Determine the Age of the Respective Agents

f( )

Age?

Regression (Supervised Learning)

#


Generative vs.

Discriminant Models



Follow the video and take your

own notes

Intro to Decision Tree LearningClassification And Regression Tree (CART)


Decision Trees in Decision Theory

Decision Trees in Machine Learning≠


Uses a set of binary rules applied to calculate a target value

Used for classification (categorical variables) or regression (continuous variables)

Different algorithms are used to determine the“best” split at a node

Introduction to Decision Trees


“CART Approach” to Decision Trees

Classification And Regression Tree (CART)


https://www.youtube.com/watch?v=WOOTNBxbi8caccess more at legalanalyticscourse.com

https://www.youtube.com/watch?v=WOOTNBxbi8c

https://www.youtube.com/watch?v=WOOTNBxbi8c

http://www.r-bloggers.com/a-brief-tour-of-the-trees-and-forests/access more at legalanalyticscourse.com

http://www.r-bloggers.com/a-brief-tour-of-the-trees-and-forests/

http://www.r-bloggers.com/a-brief-tour-of-the-trees-and-forests/

http://www.r-bloggers.com/classification-tree-models/



https://www.youtube.com/watch?v=_RxqyvRK0Rw&list=PLD0F06AA0D2E8FFBAaccess more at legalanalyticscourse.com

https://www.youtube.com/watch?v=_RxqyvRK0Rw&list=PLD0F06AA0D2E8FFBA

https://www.youtube.com/watch?v=_RxqyvRK0Rw&list=PLD0F06AA0D2E8FFBA

Given Some Data:(X1, Y1), ... , (Xn, Yn)

Now We Have a New Set of X’sWe Want to Predict the Y


Form a Binary Tree thatMinimizes the Error in each leaf of the tree

CART (Classification & Regression Trees)


Observe the Correspondence Between the Data and Trees


10

1

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0010

Xi1

Xi2

0

Adapted from Example By Mathematical Monk

10

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Xi1

Xi2

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We want to build an approach which can lead to the proper classification (labeling) of new data points ( ) that are dropped into this space

10

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Xi1

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Xi1

Xi2

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L e t s B e g i n t o Partition the Space

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Xi1

Xi2

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1 2

1

2


L e t s B e g i n t o Partition the Space

split 1

(a)

10

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Xi1

Xi2

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1 2

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This Split Will Be Memorialized in the Tree

split 1

(a)

10

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Xi1

Xi2

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1 2

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2


We Ask the Question isXi1 > 1 ? - with a binary (yes or no) response

split 1

(a)

Xi1 > 1 ?

YesNo

10

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Xi1

Xi2

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1 2

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If No - then we are in zone (a) ... we tally the number of zeros and ones

Using Majority Rule do we assign a classification to this rule this leaf

split 1

(a)

Xi1 > 1 ?

YesNo

(0,5)Classify as 1

zone (a)

10

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0

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1 11

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1

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0010

Xi1

Xi2

0

1 2

1

2


Here we Classify as a 1 because (0,5) which is 0 zero’s and 5 one’s

split 1

(a)

Xi1 > 1 ?

YesNo

(0,5)Classify as 1

zone (a)

10

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0

0

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1 11

00

1

1

1

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0010

Xi1

Xi2

0

1 2

1

2


Using a Similar Approach Lets Begin to Fill in the Rest of the Tree

split 1

(a)

Xi1 > 1 ?

YesNo

(0,5)Classify as 1

zone (a)

10

1

11

0

0

0

0

0

1

1 11

00

1

1

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0010

Xi1

Xi2

0

1 2

1

2


split 1

(a)

Xi1 > 1 ?

YesNo

(0,5)Classify as 1

zone (a) Xi2 > 1.45 ?

No Yes

split 2

10

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0010

Xi1

Xi2

0split 1

split 2

split 3

1 2 2.2

1

2 Xi1 > 1 ?

(0,5) Xi2 > 1.45 ?

(4,1)(2,3)

Classify as 1

Classify as 1 Classify as 0

(a)

zone (a)

1.45YesNo


No

(b)

(c)

zone (b) zone (c)

YesNo

Yes

Xi1 > 2 ?

10

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1

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0010

Xi1

Xi2

0split 1

split 2

split 3

split 4

1 2 2.2

1

2 Xi1 > 1 ?

(0,5) Xi2 > 1.45 ?

Xi1 > 2.2 ?

(1,4)(5,0)(4,1)(2,3)

Classify as 1


(a)

zone (a)

1.45YesNo


No

(b)

(c)

(d)

(e)

zone (b) zone (c)

YesNo YesNo

Yes

zone (d)Classify as 0 Classify as 1

zone (e)

Xi1 > 2 ?

Okay Lets Add Back the ( ) which are new items

to be classified

For simplicity sake there is one in each zone

We Will Use the Tree Becausethe Tree Is Our Prediction Machine

10

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1

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1

1

0010

Xi1

Xi2

0split 1

split 2

split 3

split 4

1 2 2.2

1

2 Xi1 > 1 ?

(0,5) Xi2 > 1.45 ?

Xi1 > 2.2 ?

(1,4)(5,0)(4,1)(2,3)

Classify as 1


(a)

zone (a)

1.45YesNo


No

(b)

(c)

(d)

(e)

zone (b) zone (c)

YesNo YesNo

Yes


zone (e)

Xi1 > 2 ?

10

1

11

0

0

0

0

0

1

1 11

00

1

1

1

1

0010

Xi1

Xi2

0split 1

split 2

split 3

split 4

1 2 2.2

1

2 Xi1 > 1 ?

(0,5) Xi2 > 1.45 ?

Xi1 > 2.2 ?

(1,4)(5,0)(4,1)(2,3)

Classify as 1


(a)

zone (a)

1.45YesNo


No

(b)

(c)

(d)

(e)

zone (b) zone (c)

Yes No YesNo

Yes


zone (e)

1

1

1

0 1

0

Xi1 > 2 ?

10

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0

0

0

0

1

1 11

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1

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Xi1

Xi2

0

1 2

1

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3

00

00

1

1

1

1

11 10

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0

11 11

1 1

00

1

1 1

0

A B C

D

E

F

G

How about this one?

In this simple example, we eyeballed the 2D space, partitioned

it and stopped after 4 Splits


Most Real Problems are Not So Simple ...


Real problems are n-dimensional (not 2D)

(1)


For real problems, you need to select criteria

(or a criterion) for deciding where to

partition (split) the data

(2)


For real problems you must develop a stopping condition

or pursue recursive partitioning of the space

(3)


Solutions to these 3 Problems are among the core questions in

algorithm selection / development


From an Algorithmic Perspective - The Task is to Develop a

Method to Partition the Trees


Must Do So Without Knowing the Specific Contours of the Data / Problem in Question


So How Do We Traverse Through

The Data?


Optimal Partitioning of Trees is NP-Complete


“Although any given solution to an NP-complete problem can be verified quickly (in polynomial time), there is no known efficient way to locate a solution in the first place; indeed, the most notable characteristic of NP-complete problems is that no fast solution to them is known. That is, the time required to solve the problem using any currently known algorithm increases very quickly as the size of the problem grows”

key implication is that one cannot in advance determine

the “optimal tree”


Breiman, et al (1984) uses a Greedy Optimization Method


Greedy Optimization Method is used to calculate the MLE

(maximum-likelihood estimation)


Greedy is a Heuristic “makes the locally optimal choice at each stage with the hope of finding a global optimum. In many problems, a greedy strategy does not in general produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a global optimal solution in a reasonable time.”


More on Trees (and Forests)Next Time ...


Legal AnalyticsClass 7 - Binary Classification with Decision Tree Learning

daniel martin katz

blog | ComputationalLegalStudies

corp | LexPredict

michael j bommarito

twitter | @computational

blog | ComputationalLegalStudies

corp | LexPredict

twitter | @mjbommar

more content available at legalanalyticscourse.com

site | danielmartinkatz.com site | bommaritollc.com

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Legal Analytics Course - Class 7 - Binary Classification with Decision Tree Learning - Professor Daniel Martin Katz + Professor Michael J Bommarito