1. Data Mining Decision Tree Induction

2. Classification Techniques
Linear Models Support Vector Machines
Decision Tree based Methods
Rule-based Methods
Memory based reasoning
Neural Networks
Naïve Bayes and Bayesian Belief Networks
Support Vector Machines

3. Example of a Decision Tree
categorical
continuous
class
Splitting Attributes
Refund
Yes No NO
MarSt
Single, Divorced
Married
TaxInc NO
< 80K
> 80K NO
YES
Training Data
Model: Decision Tree

4. Another Decision Tree Example
categorical
categorical
continuous
class
MarSt
Single, Divorced
Married NO
Refund No
Yes NO
TaxInc
< 80K
> 80K NO
YES
More than one tree may perfectly fit the data

5. Decision Tree Classification Task

6. Apply Model to Test Data
Start from the root of tree.
Refund
MarSt
TaxInc
YES NO
Yes No
Married
Single, Divorced
< 80K
> 80K

7. Apply Model to Test Data
Refund
MarSt
TaxInc
YES NO
Yes No
Married
Single, Divorced
< 80K
> 80K

8. Apply Model to Test Data
Refund
Yes No NO
MarSt
Single, Divorced
Married
TaxInc NO
< 80K
> 80K NO
YES

9. Apply Model to Test Data
Refund
Yes No NO
MarSt
Single, Divorced
Married
TaxInc NO
< 80K
> 80K NO
YES

10. Apply Model to Test Data
Refund
Yes No NO
MarSt
Single, Divorced
Married
TaxInc NO
< 80K
> 80K NO
YES

11. Apply Model to Test Data
Refund
Yes No NO
MarSt
Married
Assign Cheat to “No”
Single, Divorced
TaxInc NO
< 80K
> 80K NO
YES

12. Decision Tree Terminology

13. 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. 

14. Decision Tree Classifier10 1 2 3 4 5 6 7 8 9
Ross Quinlan
Abdomen Length > 7.1?
Antenna Length no
yes
Antenna Length > 6.0?
Katydid no
yes
Grasshopper
Katydid
Abdomen Length

15. Decision trees predate computers
Antennae shorter than body?
Yes No
3 Tarsi?
Grasshopper
Yes No
Foretiba has ears?
Yes No
Cricket
Decision trees predate computers
Katydids
Camel Cricket

16. Definition
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.

17. Decision Tree Classification
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
This can also be called supervised segmentation
This emphasizes that we are segmenting the instance space
Tree pruning
Identify and remove branches that reflect noise or outliers

18. 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
humidity
yes
wind
high
normal
strong
weak no
yes no
yes

19. How do we Construct a Decision Tree?
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
Examples are partitioned recursively based on selected attributes.
Test attributes are selected on the basis of a heuristic or statistical measure (e.g., info. gain)
Why do we call this a greedy algorithm?
Because it makes locally optimal decisions (at each node).

20. When Do we Stop Partitioning?
All samples for a node belong to same class
No remaining attributes
majority voting used to assign class
No samples left

21. How to Pick Locally Optimal Split
Hunt’s algorithm: recursively partition training records into successively purer subsets.
How to measure purity/impurity?
Entropy and associated information gain
Gini
Classification error rate
Never used in practice but good for understanding and simple exercises

22. How to Determine Best Split
Before Splitting: 10 records of class 0, records of class 1
Which test condition is the best?
Why is student id a bad feature to use?

23. How to Determine Best Split
Greedy approach:
Nodes with homogeneous class distribution are preferred
Need a measure of node impurity:
Non-homogeneous,
High degree of impurity
Homogeneous,
Low degree of impurity

24. Information Theory
Think of playing "20 questions": I am thinking of an integer between 1 and 1, 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?

25. 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 fraction of negatives.
If all examples are in one category, entropy is zero (we define 0log(0)=0)
If examples are equally mixed (p1=p0=0.5), entropy is a maximum of 1.
For multi-class problems with c categories, entropy generalizes to:

26. Entropy 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

27. Information Gain in Decision Tree Induction
Is the expected reduction in entropy caused by partitioning the examples according to this attribute.
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
The summation in the above formula is a bit misleading since when doing
the summation we weight each entropy by the fraction of total examples in
the particular child set. This applies to GINI and error rate also.

28. Examples for Computing Entropy
NOTE: p( j | t) is computed as the relative frequency of class j at node t
P(C1) = 0/6 = P(C2) = 6/6 = 1
Entropy = – 0 log2 0 – 1 log2 1 = – 0 – 0 = 0
P(C1) = 1/ P(C2) = 5/6
Entropy = – (1/6) log2 (1/6) – (5/6) log2 (5/6) = 0.65
P(C1) = 2/ P(C2) = 4/6
Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92
P(C1) = 3/6=1/ P(C2) = 3/6 = 1/2
Entropy = – (1/2) log2 (1/2) – (1/2) log2 (1/2)
= -(1/2)(-1) – (1/2)(-1) = ½ + ½ = 1

29. How to Calculate log2x
Many calculators only have a button for log10x and logex (“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

30. Splitting Based on INFO...
Information Gain:
Parent Node, p is split into k partitions;
ni is number of records in partition i
Uses a weighted average of the child nodes, where weight is based on number of examples
Used in ID3 and C4.5 decision tree learners
WEKA’s J48 is a Java version of C4.5
Disadvantage: Tends to prefer splits that result in large number of partitions, each being small but pure.

31. How Split on Continuous Attributes?
For continuous attributes
Partition the continuous value of attribute A into a discrete set of intervals
Create a new boolean attribute Ac , looking for a threshold c
One method is to try all possible splits
How to choose c ?

32. Person Homer 0” 250 36 M Marge 10” 150 34 F Bart 2” 90 10 Lisa 6” 78 8
Hair Length
Weight
Age
Class
Homer 0”
250 36 M
Marge
10”
150 34 F
Bart 2” 90 10
Lisa 6” 78 8
Maggie 4” 20 1
Abe 1”
170 70
Selma 8”
160 41
Otto
180 38
Krusty
200 45
Comic 8”
290 38 ?

33. Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9)=
yes no
Hair Length <= 5?
Let us try splitting on Hair length
Entropy(1F,3M) = -(1/4)log2(1/4) - (3/4)log2(3/4) =
Entropy(3F,2M) = -(3/5)log2(3/5) - (2/5)log2(2/5) =
Gain(Hair Length <= 5) = – (4/9 * /9 * ) =

34. Gain(Weight <= 160) = 0.9911 – (5/9 * 0.7219 + 4/9 * 0 ) = 0.5900
Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) =
yes no
Weight <= 160?
Let us try splitting on Weight
Entropy(4F,1M) = -(4/5)log2(4/5) - (1/5)log2(1/5) =
Entropy(0F,4M) = -(0/4)log2(0/4) - (4/4)log2(4/4)
= 0
Gain(Weight <= 160) = – (5/9 * /9 * 0 ) =

35. Gain(Age <= 40) = 0.9911 – (6/9 * 1 + 3/9 * 0.9183 ) = 0.0183
Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) =
yes no
age <= 40?
Let us try splitting on Age
Entropy(3F,3M) = -(3/6)log2(3/6) - (3/6)log2(3/6)
= 1
Entropy(1F,2M) = -(1/3)log2(1/3) - (2/3)log2(2/3) =
Gain(Age <= 40) = – (6/9 * 1 + 3/9 * ) =

36. This time we find that we can split on Hair length, and we are done!
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!
yes no
Weight <= 160?
This time we find that we can split on Hair length, and we are done!
yes no
Hair Length <= 2?

37. Male Male Female Weight <= 160?
We don’t need to keep the data around, just the test conditions.
Weight <= 160?
yes no
How would these people be classified?
Hair Length <= 2?
Male
yes no
Male
Female

38. Male Male Female It is trivial to convert Decision Trees to rules…
Weight <= 160?
yes no
Hair Length <= 2?
Male
yes no
Male
Female
Rules to Classify Males/Females
If Weight greater than 160, classify as Male
Elseif Hair Length less than or equal to 2, classify as Male
Else classify as Female
Note: could avoid use of “elseif” by specifying all test conditions from root to corresponding leaf.

39. 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.
Decision tree for a typical shared-care setting applying the system for the diagnosis of prostatic obstructions.

40. The worked examples we have seen were performed on small datasets
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.
Yes No
Wears green?
Female
Male
For example, the rule “Wears green?” perfectly classifies the data, so does “Mothers name is Jacqueline?”, so does “Has blue shoes”…

41. GINI is Another Measure of Impurity
Gini for a given node t with classes j
NOTE: p( j | t) is again computed as relative frequency of class j at node t
Compute best split by computing the partition that yields the lowest GINI where we again take the weighted average of the children’s GINI
Best GINI = 0.0
Worst GINI = 0.5

42. 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. This is ½ for 2-class problems
Minimum (0.0) when all records belong to one class, implying most interesting information

43. Examples for Computing Error
P(C1) = 0/6 = P(C2) = 6/6 = 1
Error = 1 – max (0, 1) = 1 – 1 = 0
Equivalently, predict majority class and determine fraction of errors
P(C1) = 1/ P(C2) = 5/6
Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6
P(C1) = 2/ P(C2) = 4/6
Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3

44. Complete Example using Error Rate
Initial sample has 3 C1 and 15 C2
Based on one 3-way split you get the 3 child nodes to the left
What is the decrease in error rate?
What is the error rate initially?
What is it afterwards?
As usual you need to take the weighted average (but there is a shortcut)

45. Error Rate Example Continued
Error rate before: 3/18
Error rate after:
Shortcut:
Number of errors =
Out of 18 examples
Error rate = 3/18
Weighted average method:
6/18 x 0 + 6/18 x 1/6 + 6/18 x 2/6
Simplifies to 1/18 + 2/18 = 3/18

46. Comparison among Splitting Criteria
For a 2-class problem:

47. 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.

48. 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

49. 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.
CarType
Family
Sports
Luxury
CarType
{Sports, Luxury}
{Family}
CarType
{Family, Luxury}
{Sports} OR

50. Splitting Based on Ordinal Attributes
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?
Size
Small
Medium
Large
Size
{Small, Medium}
{Large}
Size
{Medium, Large}
{Small} OR
Size
{Small, Large}
{Medium}

51. 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

52. Splitting Based on Continuous Attributes

53. 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 statistically significant decision
Decision trees can suffer from data fragmentation
Especially true if there are many features and not too many examples
True or False: All classification methods may suffer data fragmentation.
False: not logistic regression or instance-based learning. Only applies to divide-and-conquer methods

54. Expressiveness
Expressiveness relates to flexibility of the classifier in forming decision boundaries
Linear models are not that expressive since they can only form linear boundaries
Decision tree models can form rectangular regions
Which is more expressive and why?
Decision trees because they can form many regions, but DTs do have the limitation of only forming axis-parallel boundaries.
Decision tree do not generalize well to certain types of functions (like parity which depends on all features)
For accurate modeling, must have a complete trees
Not expressive enough for modeling continuous variables especially when more than one variable at a time is involved

55. Decision Boundary
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

56. Oblique Decision Trees
x + y < 1
Class = +
Class =
This special type of decision tree avoids some weaknesses and increases the expressiveness of decision trees
This is not what we mean when we refer to decision trees (e.g., on an exam)

57. Tree Replication
This can be viewed as a weakness of decision trees, but this is really a minor issue

58. Pros and Cons of Decision Trees
Advantages:
Easy to understand
Can get a global view of what is going on and also explain individual decisions
Can generate rules from them
Fast to build and apply
Can handle redundant and irrelevant features and missing values
Disadvantages:
Limited expressive power
May suffer from overfitting and validation set may be necessary to avoid overfitting

59. More to Come on Decision Trees
We have covered most of the essential aspects of decision trees except pruning
We will cover pruning next and, more generally, overfitting avoidance
We will also cover evaluation, which applies to decision trees but also to all predictive models