1. Decision Trees 10-601 Recitation 1/17/08 Mary McGlohon mmcgloho+10601@cs.cmu.edu

2. Announcements HW 1 out- DTs and basic probability Due Mon, Jan 28 at start of class Matlab High-level language, specialized for matrices Built-in plotting software, lots of math libraries On campus lab machines Interest in tutorial? Smiley Award Plug

4. AttendClass? Represent this tree as logical expression. AttendClass = Yes If Raining = False OR Material = New AND Before10am = False OR Is10601 = Yes Raining Is10601 Yes TrueFalse TrueFalse Yes Material NewOld Before10 TrueFalse Yes No Yes TrueFalse No Represent as a logical expression.

5. AttendClass? Represent this tree as logical expression. AttendClass = Yes If Raining = False OR Material = New AND Before10am = False OR Is10601 = Yes Raining Is10601 Yes TrueFalse TrueFalse Yes Material NewOld Before10 TrueFalse Yes No Yes TrueFalse No Represent as a logical expression. AttendClass = Yes if: (Raining = False) OR (Is10601 = True) OR (Material = New AND Before10 =False)

6. Split decisions There are other trees logically equivalent. How do we know which one to use?

7. Split decisions There are other trees logically equivalent. How do we know which one to use? Depends on what is important to us.

8. Information Gain Classically we rely on “information gain”, which uses the principle that we want to use the least number of bits, on average, to get our idea across. Suppose I want to send a weather forecast with 4 possible outcomes: Rain, Sun, Snow, and Tornado. 4 outcomes = 2 bits. In Pittsburgh there’s Rain 90% of the time, Snow 5%, Sun 4.9%, and Tornado.01%. So if you assign Rain to a 1-bit message, you rarely send >1 bit.

9. Entropy

10. RainIs10601Before10MaterialAttend ++-New+ +-+ + +-Old- +-- --+ -+ -+ -+ Set S has 6 positive, 2 negative examples. H(S) = -.75 log2(.75) -.25 log2(.25) =

11. Conditional Entropy “The average number of bits it would take to encode a message Y, given knowledge of X”

12. Conditional Entropy RainIs10601Before10MaterialAttend ++-New+ +-+ + +-Old- +-- --+ -+ -+ -+ H(Attend | Rain) = H(Attend | Rain=T)*P(Rain=T) + H(Attend|Rain=F)*P(Rain=F)

13. Conditional Entropy RainIs10601Before10MaterialAttend ++-New+ +-+ + +-Old- +-- --+ -+ -+ -+ H(Attend | Rain) = H(Attend | Rain=T)*P(Rain=T) + H(Attend|Rain=F)*P(Rain=F)= 1 * 0.5 + 0 * 0.5 = 0.5 Entropy of this set = 1 Entropy of this set = 0

14. Information Gain “How much conditioning on attribute A increases our knowledge (decreases entropy) of S. IG(S,A) = H(S) - H(S|A)

15. Information Gain IG(Attend,Rain) = H(Attend) - H(Attend|Rain)=.8113 -.5 =.3113 RainIs10601Before10MaterialAttend ++-New+ +-+ + +-Old- +-- --+ -+ -+ -+

16. What about this? Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo For some dataset, could we ever build this DT?

17. What about this? Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo For some dataset, could we ever build this DT? What if you were taking 20 classes, and it rains 90% of the time?

18. What about this? Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo For some dataset, could we ever build this DT? What if you were taking 20 classes, and it rains 90% of the time? If most information is gained from Material or Before10, we won’t ever need to traverse to 10-601. So even a bigger tree (node-wise) may be “simpler”, for some sets of data.

19. Node-based pruning Until further pruning is harmful, For each node n in trained tree T, Let Tn’ be T without n (and descendents). Assign removed node to be “best choice” under that traversal. Record error of Tn’ on validation set. Let T= Tk’ where Tk’ is pruned tree with best performance on validation set.

20. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo For each node, record performance on validation set of tree without node. Suppose our initial tree has 0.7 accurate performance on validation.

21. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo For each node, record performance on validation set of tree without node. Suppose our initial tree has 0.7 accurate performance on validation. Let’s test this node...

22. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo For each node, record performance on validation set of tree without node. Suppose our initial tree has 0.7 accurate performance on validation. Text Suppose that most examples where Material=New and Before10=True are “Yes”. Our new subtree has “Yes” here. Yes

23. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo For each node, record performance on validation set of tree without node. Suppose our initial tree has 0.7 accurate performance on validation. Text Suppose that most examples where Material=New and Before10=True are “Yes”. Our new subtree has “Yes” here. Yes Now, test this tree!

24. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo For each node, record performance on validation set of tree without node. Suppose our initial tree has 0.7 accurate performance on validation. Text Suppose that most examples where Material=New and Before10=True are “Yes”. Our new subtree has “Yes” here. Yes Now, test this tree!

25. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo For each node, record performance on validation set of tree without node. Suppose our initial tree has 0.7 accurate performance on validation. Text Suppose that most examples where Material=New and Before10=True are “Yes”. Our new subtree has “Yes” here. Yes Suppose we get accuracy of 0.73 on this pruned tree. Repeat the test procedure by removing a different node from the original tree...

26. Node-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo Try this tree (with a different node pruned)...

27. Node-based pruning Raining Material Before10 NewOld TrueFalse Yes TrueFalse No Raining Is10601 Yes TrueFalse TrueFalse YesNo Try this tree (with a different node pruned)... Now, test this tree and record its accuracy.

28. Node-based pruning Raining Material Before10 NewOld TrueFalse Yes TrueFalse No Raining Is10601 Yes TrueFalse TrueFalse YesNo Try this tree (with a different node pruned)... Now, test this tree and record its accuracy. Once we test all possible prunings, modify our tree T with the pruning that has the best performance. Repeat the entire pruning selection procedure on new T, replacing T each time with the best performing pruned tree, until we no longer gain anything by pruning.

29. Rule-based pruning Raining Material Before10 Is10601 NewOld TrueFalse Yes TrueFalse TrueFalse YesNo Raining Is10601 Yes TrueFalse TrueFalse YesNo 1. Convert tree to rules, one for each leaf: IF Material=Old AND Raining = False THEN Attend = Yes IF Material=Old AND Raining=True AND Is601=True THEN Attend=Yes...

30. Rule-based pruning 2. Prune each rule. For instance, to prune this rule: IF Material=Old AND Raining = F THEN Attend = T Test potential rule without preconditions on validation set, compare to performance of original rule on set. IF Material=OLD THEN Attend=T IF Raining=F THEN Attend = T

31. Rule-based pruning Suppose we got the following accuracy for each rule: IF Material=Old AND Raining = F THEN Attend = T -- 0.6 IF Material=OLD THEN Attend=T -- 0.5 IF Raining=F THEN Attend = T -- 0.7

32. Rule-based pruning Suppose we got the following accuracy for each rule: IF Material=Old AND Raining = F THEN Attend = T -- 0.6 IF Material=OLD THEN Attend=T -- 0.5 IF Raining=F THEN Attend = T -- 0.7 Then, we would keep the best one and drop the others.

33. Rule-based pruning Repeat for next rule, comparing the original rule with each rule with one precondition removed. IF Material=Old AND Raining=T AND Is601=T then Attend=T If Material=Old AND Raining=T then Attend=T If Material=Old AND Is601=T then Attend=T If Raining=T and Is601=T then Attend=T

34. Rule-based pruning Repeat for next rule, comparing the original rule with each rule with one precondition removed. IF Material=Old AND Raining=T AND Is601=T then Attend=T-- 0.6 If Material=Old AND Raining=T then Attend=T-- 0.7 If Material=Old AND Is601=T then Attend=T-- 0.3 If Raining=T and Is601=T then Attend=T-- 0.65

35. Rule-based pruning Repeat for next rule, comparing the original rule with each rule with one precondition removed. IF Material=Old AND Raining=T AND Is601=T then Attend=T-- 0.6 If Material=Old AND Raining=T then Attend=T-- 0.7 If Material=Old AND Is601=T then Attend=T-- 0.3 If Raining=T and Is601=T then Attend=T-- 0.65 If a shorter rule works better, we may also choose to further prune on this step before moving on to next leaf. If Material=Old AND Raining=T then Attend=T-- 0.7 If Material=Old then Attend=T-- 0.3 If Raining = T then Attend = T-- 0.2

36. Rule-based pruning Repeat for next rule, comparing the original rule with each rule with one precondition removed. IF Material=Old AND Raining=T AND Is601=T then Attend=T-- 0.6 If Material=Old AND Raining=T then Attend=T-- 0.75 If Material=Old AND Is601=T then Attend=T-- 0.3 If Raining=T and Is601=T then Attend=T-- 0.65 If a shorter rule works better, we may also choose to further prune on this step before moving on to next leaf. If Material=Old AND Raining=T then Attend=T-- 0.75 If Material=Old then Attend=T-- 0.3 If Raining = T then Attend = T-- 0.2 Well, maybe not this time!

37. Rule-based pruning Once we have done the same pruning procedure for each rule in the tree.... 3. Order the ‘kept rules’ by their accuracy, and do all subsequent classification with that priority. -IF Material=Old AND Raining=T THEN Attend=T-- 0.75 -IF Raining=F THEN Attend = T -- 0.7 -....(and so on for other pruned rules)... (Note that you may wind up with a differently-structured DT than before, as discussed in class)

38. Adding randomness RainIs601Material Before1 0 Attend? TF???F Raining Is10601 Yes TrueFalse TrueFalse Yes Material NewOld Before10 No Yes TrueFalse No What if you didn’t know if you had new material? For instance, you wanted to classify this:

39. Adding randomness RainIs601Material Before1 0 Attend? TF???F Raining Is10601 Yes TrueFalse TrueFalse Yes Material NewOld Before10 No Yes TrueFalse No What if you didn’t know if you had new material? For instance, you wanted to classify this: where to go? You could look at training set, and see that when Rain=T an 10601=F, p fraction of the examples had new material. Then flip a p-biased coin and descend the appropriate branch. But that might not be the best idea. Why not?

40. Adding randomness Also, you may have missing data in the training set. RainIs601Material Before1 0 Attend? TF???FT There are also methods to deal with this using probability. “Well, 60% of the time when Rain and not 601, there’s new material (when we know there is new material). So we’ll just randomly select 60% of rainy, non-601 examples where we don’t know the material, to be old material. Raining Is10601 Yes TrueFalse TrueFalse Yes ?

41. Adventures in Probability That approach tends to work well. Still, we may have the following trouble. What if there aren’t very many training examples where Rain = True and 10601=False? Wouldn’t we still want to use examples where Rain=False to get the missing value? Well, it “depends”. Stay tuned for lecture next week!