1. Data Science Algorithms: The Basic Methods
Decision Trees WFH: Data Mining, Chapter 4.3
Rodney Nielsen
Many of these slides were adapted from: I. H. Witten, E. Frank and M. A. Hall

2. Algorithms: The Basic Methods
Inferring rudimentary rules
Naïve Bayes, probabilistic model
Constructing decision trees
Constructing rules
Association rule learning
Linear models
Instance-based learning
Clustering

3. Decision Trees

4. Constructing Decision Trees
Strategy: top down Recursive divide-and-conquer fashion
First: select attribute for root node Create branch for each possible attribute value
Then: split instances into subsets One for each branch extending from the node
Finally: repeat recursively for each branch, using only instances that reach the branch
Stop if all instances have the same class

5. DT: Algorithm
Learning: Depth-first greedy search through the state space
Top down, recursive, divide and conquer
Select attribute for node Create branch for each value
Split instances into subsets One for each branch
Repeat recursively, using only instances that reach the branch
Stop if all instances have same class
This example assumes numeric features
Consider weather example

6. DT: Algorithm Classification:
Run through tree according to attribute values
Note: this example assumes each xi is numeric
x1<α1 x
x2<α2
x3<α3
x4<α4
This example assumes numeric features
Consider weather example

7. DT: ID3 Learning Algorithm
ID3(trainingData, attributes)
recursive function
If (attributes=) OR (all/most trainingData is in one class)
return leaf node predicting the majority class
x* f best attribute to split on
Nd f Create decision node splitting on x*
For each possible value, vk, of x*
addChild(ID3(trainingData subset with x*=vk , attributes – x*))
(assumes k-way split on categorical ftr)
return Nd
Student Q: Since Divide and Conquer Decision Trees are recursive, would they be preferred for massive amounts of data?

8. DT: Attribute Selection
Evaluate each attribute
Use heuristic choice (generally based on statistics or information theory) x1 x2 x3 - - + + - + - + + - - + + - - - - + - - + - + + + + - + + - - + + - - - + - -

9. DT: Inductive Bias Inductive bias Small vs. Large Trees Occam’s Razor
Student Q: Is it better to have a more extensive (larger) decision tree, or a less extensive (smaller) decision tree? Why do you think so?

10. Which Attribute to Select?
64% yes
36% no
60%
100%
60%
57%
86%
75%
50%
50%
75%
67%

11. Which Attribute to Select?
64% yes
36% no
60%
100%
60%
57%
86%
75%
50%
50%
75%
67%

12. Criterion for Attribute Selection
Which is the best attribute?
Heuristic: choose the attribute that produces the “purest” nodes
Want to get the smallest tree
Popular impurity criterion: information gain
Information gain increases as the purity of a subset/leaf increases
Strategy: choose attribute that gives greatest information gain
Student Q: Why should we seek purity in our decision nodes?

13. Computing Information Gain
Measure information in bits
Given a probability distribution, the information required to predict/specify an event is the distribution’s entropy
Entropy gives the information required in bits (can involve fractions of bits!)
Formula for computing the entropy:
entropy(p1, p2,…, pn) = - p1 log p1 - p2 log p2 - … - pn log pn
Specifying the outcome of a fair coin flip:
entropy(0.5,0.5) = -0.5 log log 0.5 = 1.0
Specifying the outcome from a roll of a die:
entropy(1/6,1/6,1/6,1/6,1/6,1/6) = 6 * (-1/6 log 1/6) = 2.58

14. Entropy
H(X) =
Entropy(X)
[0,N]
[N/2,N/2]
[N,0]

15. Example: Attribute Outlook
Outlook = Sunny :
Outlook = Overcast :
Outlook = Rainy :
Expected information for attribute:
Info([2,3]) = entropy(2/5,3/5) = -2/5 log(2/5) – 3/5 log(3/5) = bits
Note: this is normally
undefined.
Info([4,0]) = entropy(1,0) = -1 log(1) - 0 log(0) = 0.0 bits
Info([3,2]) = entropy(3/5,2/5) = - 3/5 log(3/5) – 2/5 log(2/5) = bits
Info([3,2],[4,0],[3,2]) = (5/14) x (4/14) x 0 + (5/14) x = bits

16. Computing Information Gain
Information gain: unknown info before splitting - unknown info after splitting
Information gain for attributes from weather data:
gain(Outlook ) = info([9,5]) - info([2,3],[4,0],[3,2])
= = bits
gain(Outlook) = bits
gain(Temperature) = bits
gain(Humidity) = bits
gain(Windy) = bits

17. DT: Information Gain
Decrease in entropy as a result of partitioning the data
Ex: X=[6+,7-], H(X)= -6/13 log26/13 -7/13 log27/13 = 0.996
InfoGain = 0.996
3/13(-1 log21 – 0 log20)
-10/13(-.4 log log2.6)
= 0.249
InfoGain = 0.996
6/13(-5/6 log25/6 -1/6 log21/6)
-7/13(-2/7 log22/7 -5/7 log25/7)
= 0.231
0.996 – 2/13(-1 log21 -0 log20)
4/13(-3/4 log23/4 -1/4 log21/4)
-7/13(-2/7 log22/7 -5/7 log25/7)
= 0.281 x1 - + x2 - + x3 - +

18. DT: ID3 Learning Algorithm
ID3(trainingData, attributes)
If (attributes=) OR (all trainingData is in one class)
return leaf node predicting the majority class
x* f best attribute to split on
Nd f Create decision node splitting on x*
attributesLeft f attributes – x* (assumes k-way split on categorical ftr)
For each possible value, vk, of x*
addChild(ID3(trainingData subset with x*=vk , attributesLeft))
return Nd

19. DT: Inductive Bias
Inductive bias
Small vs. Large Trees
Occam’s Razor

20. Continuing to Split gain(Temperature ) = 0.571 bits
gain(Windy ) = bits
gain(Humidity ) = bits

21. Final Decision Tree
Note: not all leaves need to be pure; sometimes identical instances have different classes
 ID3 Splitting stops when data can’t be split any further

22. Wishlist for a Purity Measure
Properties we require from a purity measure:
When node is pure, measure should be zero
When impurity is maximal (i.e. all classes equally likely), measure should be maximal
Measure should obey multistage property (i.e. decisions can be made in several stages)
Entropy is the only function that satisfies all three properties!

23. Highly-Branching Attributes
Problematic: attributes with a large number of values (extreme case: ID code)
Subsets are more likely to be pure if there is a large number of values
Information gain is biased towards choosing attributes with a large number of values
This may result in overfitting (selection of an attribute that performs well on training data, but is non-optimal for prediction)
Another problem: fragmentation

24. Weather Data with ID CodeN M L K J I H G F E D C B A
ID code No
True
High
Mild
Rainy
Yes
False
Normal
Hot
Overcast
Sunny
Cool
Play
Windy
Humidity
Temp.
Outlook

25. Tree Stump for ID Code Attribute
Information gain is maximal for ID code (namely bits)

26. Gain Ratio
Gain ratio: a modification of the information gain that reduces its bias
Gain ratio takes number and size of branches into account when choosing an attribute
It corrects the information gain by taking the intrinsic information of a split into account
Intrinsic information: entropy of distribution of instances into branches (i.e. how much info do we need to tell which branch an instance belongs to)

27. Computing the Gain Ratio
Example: intrinsic information for ID code
Value of attribute decreases as intrinsic information gets larger
Called Split Information
Gain Ratio = Information Gain / Split Information
Info([1,1…,1]) = -1/14 x log1/14 x 14

28. Gain Ratios for Weather Data
0.019
Gain ratio: 0.029/1.557
0.157
Gain ratio: 0.247/1.577
1.557
Split info: info([4,6,4])
1.577
Split info: info([5,4,5])
0.029
Gain:
0.247
Gain:
0.911
Info:
0.693
Temperature
Outlook
0.049
Gain ratio: 0.048/0.985
0.152
Gain ratio: 0.152/1
0.985
Split info: info([8,6])
1.000
Split info: info([7,7])
0.048
Gain:
Gain:
0.892
Info:
0.788
Windy
Humidity

29. More on Gain Ratio “Outlook” still comes out top
Problem with gain ratio: it may overcompensate
May choose an attribute just because its intrinsic information is very low
Standard fix: only consider attributes with greater than average information gain

30. DT: Hypothesis Space
Unrestricted hypothesis space + -

31. Discussion
Top-down induction of decision trees: ID3, algorithm developed by Ross Quinlan
Gain ratio just one modification of this basic algorithm
C4.5: deals with numeric attributes, missing values, noisy data
Similar approach: CART
There are many other attribute selection criteria! (But little difference in accuracy of result)