1. Data Mining CSCI 307, Spring 2019 Lecture 15
Constructing Trees

2. 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):
measure([2,3,4]) = measure([2,7]) + (7/9) x measure([3,4])
This decision is made in two stages
Make the first decision
Then decide on the second case
Entropy is the only function that satisfies all three properties

3. Example: attribute Outlook
Yes No .
Sunny
Overcast
Rainy
Example: attribute Outlook
Outlook = Sunny : Outlook = Overcast : Outlook = Rainy :

4. Example: attribute Outlook
Outlook = Sunny : Outlook = Overcast : Outlook = Rainy : Expected information for the attribute:
info([2,3]) = bits
info([4,0]) = 0 bits
info([3,2]) = bits
info([2,3],[4,0],[3,2]) =

5. Computing Information Gain
information before splitting – information after splitting
We've calculated the information BEFORE splitting and we've calculated the information AFTER the split for the Outlook attribute. So we can calculate the information gain for Outlook.
gain(Outlook ) =

6. attribute: Temperature
Yes No
Hot
Mild
Cool
attribute: Temperature
Temperature = Hot : Temperature = Mild : Temperature = Cool :
info([2,2]) = entropy(2/4, 2/4)
= −2/4 log2(2/4) − 2/4 log2(2/4)= 1 bit
info([4,2]) = entropy(4/6, 2/6)
= −2/3 log(2/3) − 1/3 log(1/3)= bits
info([3,1]) = entropy(3/4, 1/4)
= −3/4 log(3/4) − 1/4 log(1/4) = bits

7. attribute: Temperature
Temperature = Hot :
Temperature = Mild :
Temperature = Cool :
Expected information for the attribute:
info([2,2]) = 1 bit
info([4,2]) = bits
info([3,1]) = bits
Average information value.
(Use the number of instances that go down each branch.)
info([2,2],[4,2],[3,1]) =
gain(Temperature )
= info([9,5]) – info([2,2],[4,2],[3,1]) =

8. attribute: Humidity Humidity = High : Humidity = Normal :
Yes No
High
Normal 6 1
attribute: Humidity
Humidity = High : Humidity = Normal :
info([3,4]) = entropy(3/7, 4/7)
= −3/7 log2(3/7) − 4/7 log2(4/7)
= bits
info([6,1]) = entropy(6/7, 1/7)
= −6/7 log(6/7) − 1/7 log(1/7)
= bits

9. attribute: Humidity = 0.788 bits Humidity = High : Humidity = Normal :
Expected information for the attribute:
info([3,4]) = bits
info([6,1]) = bits
Average information value.
(Use the number of instances that go down each branch.)
info([3,4],[6,1])
= 7/14 x /14 x 0.592
= bits
gain(Humidity )
= info([9,5]) – info([3,4],[6,1])
= – 0.788
= bits

10. attribute: Windy Windy = False : Windy = True :
Yes No
False
True
attribute: Windy
Windy = False : Windy = True :
info([6,2]) = entropy(6/8, 2/8)
= −6/8 log2(6/8) − 2/8 log2(2/8)
= bits
info([3,3]) = entropy(3/6, 3/6)
= −3/6 log(3/6) − 3/6 log(3/6)
= 1 bit

11. attribute: Windy = 0.892 bits Windy = False : Windy = True :
Expected information for the attribute:
info([6,2]) = bits
info([3,3]) = 1 bit
Average information value.
(Use the number of instances that go down each branch.)
info([6,2],[3,3])
= 8/14 x /14 x 1
= bits
gain(Windy )
= info([9,5]) – info([6,2],[3,3])
= – 0.892
= bits

12. Which Attribute to Select as Root?
For all the attributes from the weather data:
gain(Outlook ) = bits
gain(Temperature ) = bits
gain(Humidity )= bits
gain(Windy ) = bits
Outlook is the way to go ... it's the root.

13. Continuing to Split yes
Now, determine the gain for EACH of Outlook's branches, sunny, overcast, and rainy.
For the sunny branch we know at this point the entropy is 0.971; it is our "before" split information as we calculate our gain from here.
yes
The rainy branch entropy is also 0.971; use it is our "before" split information as we calculate our gain from here on down.
Splitting stops when we can't split any further; that is the case with the value overcast.
We don't need to consider Outlook further.

14. Continuing the Split at Sunny
Now, must determine the gain for EACH of Outlook's branches. For the sunny branch we know the "before" split entropy is 0.971

15. Find Subroot for Sunny humidity = high: info([0,3]) = entropy(0,1) =
humidity = normal:
info([2,0]) = entropy(1,0)
info([0,3],[2,0]) =
gain(Humidity ) = info([2,3]) – info([0,3],[2,0])

16. Find Subroot for Sunny (continued)
windy = false:
info([1,2]) = entropy(1/3,2/3) = −⅓ log(⅓) − ⅔log(⅔)
= bits
windy = true:
info([1,1]) = entropy(1/2,1/2) = −½ log(½) − ½ log(½)
= 1 bit
info([0,3],[2,0]) = 3/5 x /5 x 1 = bits
gain(Windy ) = info([2,3]) – info([1,2],[1,1])
= = 0.020

17. Find Subroot for Sunny (continued)
temperature = hot: info([0,2]) = entropy(0,1)
= −0 log(0) − 1 log(1) = 0 bits
temperature = mild: info([1,1]) = entropy(1/2,1/2)
= −½ log(½) − ½ log(½) = 1 bit
temperature = cool: info([1,0]) = entropy(1,0)
= −1 log(1) − 0 log(0) = 0 bits
info([0,2],[1,1],[1,0]) = 2/5 x /5 x /5 x 0
= = 0.4 bits
gain(Temperature ) = info([2,3]) – info([0,2],[1,1],[1,0]) = – 0.4 = bits

18. Finish the Split at Sunny
gain(Humidity )= bits
gain(Temperature ) = bits
gain(Windy ) = bits

19. Possible Splits at Rainy
No need to actually do the calculations because windy is pure

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

21. 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 is non-optimal for prediction)
Another problem: fragmentation

22. Tree Stump for ID code Attribute
This seems like a bad idea for a split.
Entropy of split:
info(ID code) = info([0,1]) + (info([0,1]) + (info([1,0])
(info([1,0]) + info([0,1]) = 0 bits
Information gain is maximal for ID code (namely bits, i.e. the before split information)

23. 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 information do we need to tell which branch an instance belongs to)

24. Computing the Gain Ratio
Example: intrinsic information for ID code
info([1,1,...,1]) = 14 x (-1/14 x log(1/14)) = 3.807bits
Value of attribute decreases as intrinsic information gets larger
Definition of gain ratio:
Example:
gain_ratio(attribute) =
gain(attribute)
intrinsic_info(attribute)
gain_ratio(ID code) = = 0.246
0.940 bits
3.807 bits