1.  2003, G.Tecuci, Learning Agents Laboratory 1 Learning Agents Laboratory Computer Science Department George Mason University Prof. Gheorghe Tecuci 4. Inductive Learning from Examples: Decision Tree Learning 4. Inductive Learning from Examples: Decision Tree Learning

2.  2003, G.Tecuci, Learning Agents Laboratory 2 Overview The basic ID3 learning algorithm Discussion and refinement of the ID3 method Applicability of the decision tree learning The decision tree learning problem Recommended reading Exercises

3.  2003, G.Tecuci, Learning Agents Laboratory 3 The decision tree learning problem Given language of instances: feature value vectors language of generalizations: decision trees a set of positive examples (E1,..., En) of a concept a set of negative examples (C1,..., Cm) of the same concept learning bias: preference for shorter decision trees Determine a concept description in the form of a decision tree which is a generalization of the positive examples that does not cover any of the negative examples

4.  2003, G.Tecuci, Learning Agents Laboratory 4 Examples Illustration heighthaireyesclass shortblondblue + tallblondbrown - tallredblue + shortdarkblue - talldarkblue - tallblondblue + talldarkbrown - shortblondbrown - Feature vector representation of examples That is, there is a fixed set of attributes, each attribute taking values from a specified set. Decision tree concept

5.  2003, G.Tecuci, Learning Agents Laboratory 5 What is the logical expression represented by the decision tree? Decision tree concept: hair dark red blond eyes blue brown - - + + Disjunction of conjunctions (one conjunct per path to a + node): (hair = red)  [(hair = blond) & (eyes = blue)] Which is the concept represented by this decision tree?

6.  2003, G.Tecuci, Learning Agents Laboratory 6 Feature-value representation If the training set (i.e. the set of positive and negative examples from which the tree is learned) contains a positive example and a negative example that have identical values for each attribute, it is impossible to differentiate between the instances with reference only to the given attributes. In such a case the attributes are inadequate for the training set and for the induction task. Is the feature value representation adequate?

7.  2003, G.Tecuci, Learning Agents Laboratory 7 Feature-value representation (cont.) The problem is that there are many such correct decision trees and the task of induction is to construct a decision tree that correctly classifies not only instances from the training set but other (unseen) instances as well. When could a decision tree be built? If the attributes are adequate, it is always possible to construct a decision tree that correctly classifies each instance in the training set. So what is the difficulty in learning a decision tree?

8.  2003, G.Tecuci, Learning Agents Laboratory 8 Overview The basic ID3 learning algorithm Discussion and refinement of the ID3 method Applicability of the decision tree learning The decision tree learning problem Recommended reading Exercises

9.  2003, G.Tecuci, Learning Agents Laboratory 9 The basic ID3 learning algorithm Let C be the set of training examples If all the examples in C are positive then create a node with label + If all the examples in C are negative then create a node with label - If there is no attribute left then create a node with the same label as the majority of examples in C Otherwise: - partition the examples into subsets C1, C2,..., Ck according to the values of A. - apply the algorithm recursively to each of the sets Ci which is not empty - for each Ci which is empty create a node with the same label as the majority of examples in C the node - select the best attribute A and create a decision node, where v 1, v 2,..., v k are the values of A:... A v 1 v 2 v k

10.  2003, G.Tecuci, Learning Agents Laboratory 10 Features selection: information theory Let us consider a set S containing objects from n classes S1,..., Sn, so that the probability of an object to belong to a class Si is pi. According to the information theory, the amount of information needed to identify the class of one particular member of S is: Ii = - log 2 pi Intuitively, Ii represents the number of questions required to identify the class Si of a given element in S. The average amount of information needed to identify the class of an element in S is: - ∑ pi log 2 pi

11.  2003, G.Tecuci, Learning Agents Laboratory 11 Discussion Consider the following letters: A B C D E F G H Think of one of them (call it, the secret letter). How many questions need to be asked in order to find the secret letter?

12.  2003, G.Tecuci, Learning Agents Laboratory 12 Features selection: the best attribute Let us suppose that the decision tree has been built from a training set C consisting of p positive examples and n negative examples. The average amount of information needed to classify an instance from C is If attribute A with values {v1, v2,...,vk} is used for the root of the decision tree, it will partition C into {C1, C2,...,Ck}, where each Ci contains pi positive examples and ni negative examples. The expected information required to classify an instance in Ci is I(pi, ni). The expected amount of information required to classify an instance after the value of the attribute A is known is therefore: The information gained by branching on A is: gain(A) = I(p, n) - Ires(A)

13.  2003, G.Tecuci, Learning Agents Laboratory 13 Features selection: the heuristic The information gained by branching on A is: gain(A) = I(p, n) - Ires(A) Choose the attribute which leads to the greatest information gain. What would be a good heuristic? Why is this a heuristic and not a guaranteed method? Hint: What kind of search method for the best attribute does ID3 uses?

14.  2003, G.Tecuci, Learning Agents Laboratory 14 Features selection: the heuristic Why is this a heuristic and not a guaranteed method? Hint: Think of a situation where a is the best attribute, but the combination of “b and c” would actually be better than any of “a and b”, or “a and c”. That is, knowing b and c you can classify, but knowing only a and b (or only a and c) you cannot. This shows that the attributes may not be independent. How could we deal with this? Hint: Consider also combination of attributes, not only a, b, c, but also ab, bc, ca What is a problem with this approach?

15.  2003, G.Tecuci, Learning Agents Laboratory 15 Features selection The built tree depends of the heuristic used to select the attribute to test at each node. ID3 selects the 'most informative' attribute first. This criterion is based on concepts from information theory. Let us consider a set S containing objects from n classes S1,..., Sn, so that the probability of an object to belong to a class Si is pi. Then, according to the information theory, the amount of information needed to identify the class of one particular member of S is: Ii = - log 2 pi Intuitively, Ii represents the number of questions required to identify the class Si of a given element in S. Therefore, the average amount of information needed to identify the class of an element in S is: - ∑ pi log 2 pi Let us now consider the problem of determining the most informative attribute for building a decision tree. To classify an instance as being a positive or a negative example of a concept, a certain amount of information is needed. After we have learned the value of some attribute in the instance, we only need some remaining amount of information to classify the instance. This remaining amount is normally smaller than the initial amount, and is called the 'residual information'. The 'most informative' attribute is the one that minimizes the residual information. Let us suppose that the decision tree has been built from a training set C consisting of p positive examples and n negative examples. Then, according to the above formula, the average amount of information needed to classify an instance from C is If attribute A with values {v1, v2,...,vk} is used for the root of the decision tree, it will partition C into {C1, C2,...,Ck}, where Ci contains those examples in C that have value vi of A. Let Ci contain pi positive examples and ni negative examples. The expected information required to classify an instance in Ci is I(pi, ni). The expected amount of information required to classify an instance after the value of the attribute A is known is therefore: where the weight for the i-th branch is the proportion of the examples in C that belong to Ci. Therefore, the information gained by branching on A is: gain(A) = I(p, n) - Ires(A)

16.  2003, G.Tecuci, Learning Agents Laboratory 16 A good heuristic is to choose that attribute to branch which leads to the greatest information gain. Since I(p, n) is constant for all attributes, maximizing the gain is equivalent to minimizing Ires(A), which in turn is equivalent to minimizing the following expression: ID3 examines all candidate attributes and chooses A to maximize gain(A) (or minimize Ires(A)), forms the tree as above, and then uses the same process recursively to form decision trees for the residual subsets C1, C2,...,Ck. The intuition behind the information content criterion is the following one. We are interested in small trees. Therefore we need powerful attributes to discriminate between classes. Ideally, such a powerful attribute should divide the objects in C into subsets so that only one class is represented in each subset. Such a totally uniform subset (uniform with respect to the class) is said to be 'pure' and no additional information is needed to classify an instance from the subset. In a non-ideal situation, the set is not completely pure, but we want it to be as pure as possible. Thus we prefer the attributes that minimize the impurity of the resulting subsets. I(p,n) is a measure of impurity. i p log 2 + - i i n 2 + ( i - ) where is the number of positive examples in Ci p i is the number of negative examples in Ci if then the corresponding term in the sum is 0= 0 or n i p i p i p i n i n i n i n i p Σ

17.  2003, G.Tecuci, Learning Agents Laboratory 17 Examples Illustration of the method heighthaireyesclass shortblondblue + tallblondbrown - tallredblue + shortdarkblue - talldarkblue - tallblondblue + talldarkbrown - shortblondbrown - 1.Find the attribute that maximizes the information gain: gain(A) = I(p, n) - I(3+, 5-) = -3/8log 2 3/8 – 5/8log 2 5/8 = 0.954434003 Height:short (1+, 2-)tall(2+, 3-) Gain(height) = 0.954434003 - 3/8*I(1+,2-) - 5/8*I(2+,3-) = = 0.954434003 – 3/8(-1/3log 2 1/3 - 2/3log 2 2/3) – 5/8(-2/5log 2 2/5 - 3/5log 2 3/5) = 0.003228944 Hair:blond(2+, 2-)red(1+, 0-)dark(0+, 3-) Gain(hair) = 0.954434003 – 4/8(-2/4log 2 2/4 – 2/4log 2 2/4) – 1/8(-1/1log 2 1/1-0) – -3/8(0-3/3log 2 3/3) = 0.954434003 – 0.5 = 0.454434003 Eyes:blue(3+, 2-)brown(0+, 3-) Gain(eyes) = 0.954434003 – 5/8(-3/5log 2 3/5 – 2/5log 2 2/5) -5/8(= = 0.954434003 - 0.606844122 = 0.347589881 “Hair” is the best attribute.

18.  2003, G.Tecuci, Learning Agents Laboratory 18 Examples Illustration of the method (cont.) heighthaireyesclass shortblondblue + tallblondbrown - tallredblue + shortdarkblue - talldarkblue - tallblondblue + talldarkbrown - shortblondbrown - 2. “Hair” is the best attribute. Build the tree using it.

19.  2003, G.Tecuci, Learning Agents Laboratory 19 Illustration of the method (cont.) 3. Select the best attribute for the set of examples: short, blond, blue: + tall, blond, brown: - tall, blond, blue: + short, blond, brown: - I(2+, 2-) = -2/4log 2 2/4 – 2/4log 2 2/4 = -log 2 1/2=1 Height:short (1+, 1-)tall(1+, 1-) Eyes:blue (2+, 0-)brown(0+, 2-) Gain(height) = 1 – 2/4*I(1+,1-) – 2/4*I(1+,1-) = 1 - I(1+,1-) = 1-1 = 0 Gain(eyes) = 1 – 2/4*I(2+,0-) – 2/4*I(0+,2-) = 1 – 0 – 0 = 1 “Eyes” is the best attribute.

20.  2003, G.Tecuci, Learning Agents Laboratory 20 Illustration of the method (cont.) 4. “Eyes” is the best attribute. Expand the tree using it:

21.  2003, G.Tecuci, Learning Agents Laboratory 21 Illustration of the method (cont.) 5. Build the decision tree: What induction hypothesis is made?

22.  2003, G.Tecuci, Learning Agents Laboratory 22 Overview The basic ID3 learning algorithm Discussion and refinement of the ID3 method Applicability of the decision tree learning The decision tree learning problem Recommended reading Exercises

23.  2003, G.Tecuci, Learning Agents Laboratory 23 How could we transform a tree into a set of rules? Answer: IF(hair = red) THENpositive example IF[(hair = blond) & (eyes = blue)] THENpositive example Why should we make such a transformation? Converting to rules improves understandability.

24.  2003, G.Tecuci, Learning Agents Laboratory 24 Learning from noisy data errors in the values of attributes (due to measurements or subjective judgments); errors of classifications of the instances (for instance a negative example that was considered a positive example). What errors could be found in an example (also called noise in data)? What are the effects of noise? How to change the ID3 algorithm to deal with noise?

25.  2003, G.Tecuci, Learning Agents Laboratory 25 How to deal with noise? The algorithm must be able to work with inadequate attributes, because noise can cause even the most comprehensive set of attributes to appear inadequate. The algorithm must be able to decide that testing further attributes will not improve the predictive accuracy of the decision tree. For instance, it should refrain from increasing the complexity of the decision tree to accommodate a single noise-generated special case. Noise may cause the attributes to become inadequate. Noise may lead to decision trees of spurious complexity (overfitting). What are the effects of noise? How to change the ID3 algorithm to deal with noise?

26.  2003, G.Tecuci, Learning Agents Laboratory 26 How to deal with an inadequate attribute set? A collection C of instances may contain representatives of both classes, yet further testing of C may be ruled out, either because the attributes are inadequate and unable to distinguish among the instances in C, or because each attribute has been judged to be irrelevant to the class of instances in C. In this situation it is necessary to produce a leaf labeled with a class information, but the instances in C are not all of the same class. (inadequacy due to noise) What class to assign a leaf node that contains both + and - examples?

27.  2003, G.Tecuci, Learning Agents Laboratory 27 What class to assign a leaf node that contains both + and - examples? Approaches: 1. The notion of class could be generalized from a binary value (0 for negative examples and 1 for positive examples) to a number in the interval [0; 1]. In such a case, a class of 0.8 would be interpreted as 'belonging to class P with probability 0.8'. 2. Opt for the more numerous class, i.e. assign the leaf to class P if p>n, to class N if p<n, and to either if p=n. The first approach minimizes the sum of the squares of the errors over objects in C. The second approach minimizes the sum of the absolute errors over objects in C. If the aim is to minimize expected error, the second approach might be anticipated to be superior.

28.  2003, G.Tecuci, Learning Agents Laboratory 28 How to avoid overfitting the data? One says that a hypothesis overfits the training examples if some other hypothesis that fits the training examples less well actually performs better over the entire distribution of instances. Stop growing the tree before it overfits; Allow the tree to overfit and then prune it. How to determine the correct size of the tree? Use a testing set of examples to compare the likely errors of various trees. How to avoid overfitting?

29.  2003, G.Tecuci, Learning Agents Laboratory 29 Rule post pruning to avoid overfitting the data? Infer a decision tree Convert the tree into a set of rules Prune (generalize) the rules by removing antecedents as long as this improves their accuracy Sort the rules by their accuracy and use this order in classification Rule post pruning algorithm Compare tree pruning with rule post pruning. Rule post pruning is more general. We can remove an attribute from the top of the tree without removing all the attributes that follow.

30.  2003, G.Tecuci, Learning Agents Laboratory 30 How to use continuous attributes? Transform a continuous attribute into a discrete one. Give an example of such a transformation.

31.  2003, G.Tecuci, Learning Agents Laboratory 31 How to deal with missing attribute values? Estimate the value from the values of the other examples. How? Assign the value that is most common for the training examples at that node. Assign a probability to each of the values. How does this affect the algorithm? Consider fractional examples.

32.  2003, G.Tecuci, Learning Agents Laboratory 32 Comparison with the candidate elimination algorithm Generalization language ID3 – disjunctions of conjunctions CE – conjunctions ID3 – all in the same time (can deal with noise and missing values) CE – one at a time (can determine the most informative example) Use of examples ID3: hill climbing (may not find the concept but only an approximation) CE: exhaustive search Search strategy ID3 – preference bias (Occam’s razor) CE – representation bias Bias

33.  2003, G.Tecuci, Learning Agents Laboratory 33 Overview The basic ID3 learning algorithm Discussion and refinement of the ID3 method Applicability of the decision tree learning The decision tree learning problem Recommended reading Exercises

34.  2003, G.Tecuci, Learning Agents Laboratory 34 What problems are appropriate for decision tree learning? Problems for which: Instances can be represented by attribute-value pairs Disjunctive descriptions may be required to represent the learned concept Training data may contain errors Training data may contain missing attribute values

35.  2003, G.Tecuci, Learning Agents Laboratory 35 What practical applications could you envision? Classify: - Patients by their disease; - Equipment malfunctions by their cause; - Loan applicants by their likelihood to default on payments.

36.  2003, G.Tecuci, Learning Agents Laboratory 36 Which are the main features of decision tree learning? May employ a large number of examples. Discovers efficient classification trees that are theoretically justified. Learns disjunctive concepts. Is limited to attribute-value representations. Has a non incremental nature (there are however also incremental versions that are less efficient). The tree representation is not very understandable. The method is limited to learning classification rules. The method was successfully applied to complex real world problems.

37.  2003, G.Tecuci, Learning Agents Laboratory 37 Overview The basic ID3 learning algorithm Discussion and refinement of the ID3 method Applicability of the decision tree learning The decision tree learning problem Recommended reading Exercises

38.  2003, G.Tecuci, Learning Agents Laboratory 38 Exercise foodmediumtypeclass herbivorelandharmlessmammal+deer (e1) carnivorelandharmfulmammal-lion (c1) omnivorouswaterharmlessfish+goldfish (e2) herbivoreamphibiousharmlessamphibian-frog (c2) omnivorousairharmlessbird-parrot (c3) carnivorelandharmfulreptile+cobra (e3) carnivorelandharmlessreptile-lizard (c4) omnivorouslandmoodymammal+bear (e4) Build two different decision trees corresponding to the examples and counterexamples from the following table. Indicate the concept represented by each decision tree. Apply the ID3 algorithm to build the decision tree corresponding to the examples and counterexamples from the above table.

39.  2003, G.Tecuci, Learning Agents Laboratory 39 Exercise shapesizeclass balllarge+e1 bricksmall-c1 cubelarge-c2 ballsmall+e2 a) You will be required to learn this concept by applying two different learning methods, the Induction of Decision Trees method, and the Versions Space (candidate elimination) method. Do you expect to learn the same concept with each method or different concepts? Explain in detail your prediction (You will need to consider various aspects like the instance space, the hypothesis space, and the method of learning). b) Learn the concept represented by the above examples by applying: - the Induction of Decision Trees method; - the Versions Space method. c) Explain the results obtained in b) and compare them with your predictions. d) Which will be the results of learning with the above two methods if only the first three examples are available? Consider the following positive and negative examples of a concept and the following background knowledge

40.  2003, G.Tecuci, Learning Agents Laboratory 40 Exercise workstationsoftwareprinterclass maclcmacwritelaserwriter+e1 sunframe-makerlaserwriter+e2 hpaccountinglaserjet-c1 sgispreadsheetlaserwriter-c2 macIImicrosoft-wordproprinter+e3 a) Build two decision trees corresponding to the above examples. Indicate the concept represented by each decision tree. In principle, how many different decision trees could you build? b) Learn the concept represented by the above examples by applying the Versions Space method. Which is the learned concept if only the first four examples are available? c) Compare and justify the obtained results. Consider the following positive and negative examples of a concept and the following background knowledge

41.  2003, G.Tecuci, Learning Agents Laboratory 41 Exercise True of false: If decision tree D2 is an elaboration of D1 (according to ID3), then D1 is more general than D2.

42.  2003, G.Tecuci, Learning Agents Laboratory 42 Recommended reading Mitchell T.M., Machine Learning, Chapter 3: Decision tree learning, pp. 52 -80, McGraw Hill, 1997. Quinlan J.R., Induction of decision trees, in Machine Learning Journal, 1:81-106. Also in Shavlik J. and Dietterich T. (eds), Readings in Machine Learning, Morgan Kaufmann, 1990. Barr A., Cohen P., and Feigenbaum E.(eds), The Handbook of Artificial Intelligence, vol III, pp.406-410, Morgan Kaufmann, 1982. Elwyn Edwards, Information Transmission, Chapter 4: Uncertainty, pp. 28- 39, Chapman and Hall, 1964.