Section 18.3 Learning Decision Trees

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1 Section 18.3 Learning Decision Trees CS Artificial Intelligence Nilufer Onder Department of Computer Science Michigan Technological University

2 Outline Attribute-based representations Decision tree learning as a search problem A greedy algorithm

3 Decision trees A decision tree allows a classification of an object by testing its values for certain properties. An example is the 20 questions game. A player asks questions to an answerer and tries to guess the object that the answerer chose at the beginning of the game. The objective of decision tree learning is to learn a tree of questions which determines class membership at the leaf of each branch. Check out an online example at

4 Possible decision tree

5 Possible decision tree (cont d)

6 What might the original data look like?

7 The search problem This is an attribute-based representation where examples are described by attribute values (Boolean, discrete, continuous, etc.) Classification of examples is positive (T) or negative (F). Given a table of observable properties, search for a decision tree that correctly represents the data (for now, assume that the data is noise-free) is as small as possible What does the search tree look like?

8 Predicate as a decision tree

9 The training set

10 Possible decision tree

11 Smaller decision tree

12 Building the decision tree - getting started (1)

13 Getting started (2)

14 Getting started (3)

15 How to compute the probability of error (1)

16 How to compute the probability of error (2)

17 Assume it s A

18 Assume it s B

19 Assume it s C

20 Assume it s D

21 Assume it s E

22 Probability of error for each

23 Choice of second predicate

24 Choice of third predicate

25

26 The decision tree learning algorithm function Decision-Tree-Learning (examples, attributes, parent-examples ) returns a tree if examples is empty then return Plurality-Value(parent-examples) else if all examples have the same classification then return the classification else if attributes is empty then return Plurality-Value(examples) else A argmax a attributes Importance(a, examples) tree a new decision tree with root test A for each value v k of A do exs { e : e examples and e.a = v k } subtree Decision-Tree-Learning (exs, attributes-a, examples) add a branch to tree with label (A = v k ) and subtree subtree return tree

27 Notes on the algorithm Notice that the probability of error calculations boil down to summing up the minority numbers and dividing by the total number of examples in that category. This is due to fraction cancellations. Probability of error is: minority 1 + minority total number of examples in this category After an attribute is selected take only the examples that have the attribute as labelled on the branch.

28 What happens if there is noise in the training set? Consider a very small but inconsistent data set: A T F F classification T F T

29 Issues in learning decision trees If data for some attribute is missing and is hard to obtain, it might be possible to extrapolate or use unknown. If some attributes have continuous values, groupings might be used. If the data set is too large, one might use bagging to select a sample from the training set. Or, one can use boosting to assign a weight showing importance to each instance. Or, one can divide the sample set into subsets and train on one, and test on others.

30 How large is the hypothesis space? How many decision trees with n Boolean attributes? = number of Boolean functions = number of distinct truth tables with 2 n rows. = 2 2n

31 Using probability of error The probability of error is based on a measure of the quantity of information that is contained in the truth value of an observable attribute. It shows how predictable the classification is after getting information about an attribute. The lower the probability of error, the higher the predictability. The attribute with the minimal probability of error yields the maximum predictability. That is what we chose A at the root of the decision tree.

32 Using information theory Entropy gives information about unpredictability. The scale is to use 1 bit to answer a Boolean question with prior < 0.5, 0.5 >. This is least predictability (highest unpredicatability). Information answers questions: the more clueless we are about the answer initially, the more information is contained in the answer. i.e., we have a gain after getting an answer about attribute A. We select the attribute with the highest gain. Let p be the number of positive examples, and n the number of negative examples. Entropy(p, n) is defined as plog 2 p nlog 2 n

33 Information gain Gain(A) is the expected reduction on entropy after getting an answer on attribute A. Let p i be the number of positive examples when the answer to A is i, and n i be the number of negative examples when the answer to A is i. Assuming two possible answers, Gain(A) is defined as entropy(p, n) p 1 + n 1 p + n entropy(p 1, n 1 ) p 2 + n 2 p + n entropy(p 2, n 2 )

34 Example Assuming two possible answers, Gain(A) is defined as entropy(p, n) p 1 + n 1 p + n entropy(p 1, n 1 ) p 2 + n 2 p + n entropy(p 2, n 2 ) Initially there are 6 positive and 7 negative examples. Entropy(6,7) = There are 6 positive and 2 negative examples for A being true and 0 positive and 5 negative example for A being false. Therefore the gain is entropy(6, 2) 5 entropy(5, 0) = =

35 Example(cont d) The gain values are: A: B: C: D: E:

36 Summary Decision tree learning is a supervised learning paradigm. The hypothesis is a decision tree. The greedy algorithm uses information gain to decide which attribute should be placed at each node of the tree. Due to the greedy approach, the decision tree might not be optimal but the algorithm is fast. If the data set is complete and not noisy, then the learned decision tree will be accurate.

37 Sources for the slides AIMA textbook (3 rd edition) AIMA slides: Jean-Claude Latombe s CS121 slides latombe/cs121 (Accessed prior to 2009) Wikipedia article for Twenty Questions Questions (Accessed in March 2012)

P(A, B) = P(A B) = P(A) + P(B) - P(A B)

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Supervised learning can be done by choosing the hypothesis that is most probable given the data: = arg max ) = arg max

The learning problem is called realizable if the hypothesis space contains the true function; otherwise it is unrealizable On the other hand, in the name of better generalization ability it may be sensible

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