# Lecture 3: Decision Trees

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1 Lecture 3: Decision Trees Cognitive Systems - Machine Learning Part I: Basic Approaches of Concept Learning ID3, Information Gain, Overfitting, Pruning last change November 26, 2014 Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

2 Outline Outline Decision Trees ID3 Information Gain Overfitting Pruning Summary Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

3 Decision Tree Decision Tree Representation classification of instances by sorting them down the tree from the root to some leaf node node test of some attribute branch one of the possible values for the attribute decision trees represent a disjunction of conjunctions of constraints on the attribute values of instances i.e., ( ) ( )... equivalent to a set of if-then-rules each branch represents one if-then-rule if-part: conjunctions of attribute tests on the nodes then-part: classification of the branch Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

4 Decision Tree Decision Tree Representation This decision tree is equivalent to: if (Outlook = Sunny) (Humidity = Normal) then Yes; if (Outlook = Overcast) then Yes; if (Outlook = Rain) (Wind = Weak) then Yes; Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

5 Decision Tree Appropriate Problems Instances are represented by attribute-value pairs, e.g. (Temperature, Hot) Target function has discrete output values, e.g. yes or no (concept/classification learning) Disjunctive descriptions may be required Training data may contain errors Training data may contain missing attribute values last three points make Decision Tree Learning more attractive than CANDIDATE-ELIMINATION Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

6 ID3 ID3 learns decision trees by constructing them top-down employs a greedy search algorithm without backtracking through the space of all possible decision trees finds a short tree (wrt path length) but not necessarily the best decision tree key idea: selection of the next attribute according to a statistical measure all examples are considered at the same time (simultaneous covering) recursive application with reduction of selectable attributes until each training example can be classified unambiguously Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

7 ID3 Algorithm for Concept Learning ID3(Examples, Target attribute, Attributes) Create a Root for the tree ID3 If all examples are positive, Return single-node tree Root, with label = + If all examples are negative, Return single-node tree Root, with label = If Attributes is empty, Return single-node tree Root, with label = most common value of Target attribute in Examples Otherwise, Begin A attribute in Attributes that best classifies Examples Decision attribute for Root A For each possible value v i of A Add new branch below Root with A = v i Let Examples vi be the subset of Examples with v i for A If Examples vi is empty Then add a leaf node with label = most common value of Target attribute in Examples Else add ID3(Examples vi, Target Attribute, Attributes {A}) Return Root Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

8 The best classifier Information Gain central choice: Which attribute classifies the examples best? ID3 uses the information gain statistical measure that indicates how well a given attribute separates the training examples according to their target classification Gain(S, A) Entropy(S) }{{} original entropy of S S v S Entropy(S v) v values(a) }{{} relative entropy of S interpretation: denotes the reduction in entropy caused by partitioning S according to A alternative: number of saved yes/no questions (i.e., bits) attribute with max A Gain(S, A) is selected! Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

9 Information Gain Entropy statistical measure from information theory that characterizes (im-)purity of an arbitrary collection of examples S for binary classification: H(S) p log 2 p p log 2 p for n-ary classification: H(S) n i=1 p i log 2 p i interpretation: specification of the minimum number of bits of information needed to encode the classification of an arbitrary member of S alternative: number of yes/no questions Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

10 Entropy Information Gain minimum of H(S) for minimal impurity point distribution H(S) = 0 maximum of H(S) for maximal impurity uniform distribution for binary classification: H(S) = 1 for n-ary classification: H(S) = log 2 n Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

11 Illustrative Example Information Gain example days (Feature sky with values sunny, rainy is replaced by outlook with three values) Day Outlook Temp. Humidity Wind PlayTennis D1 Sunny Hot High Weak No D2 Sunny Hot High Strong No D3 Overcast Hot High Weak Yes D4 Rain Mild High Weak Yes D5 Rain Cool Normal Weak Yes D6 Rain Cool Normal Strong No D7 Overcast Cool Normal Strong Yes D8 Sunny Mild High Weak No D9 Sunny Cool Normal Weak Yes D10 Rain Mild Normal Weak Yes D11 Sunny Mild Normal Strong Yes D12 Overcast Mild High Strong Yes D13 Overcast Hot Normal Weak Yes D14 Rain Mild High Strong No Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

12 Illustrative Example Information Gain entropy of S S = {D1,..., D14} = [9+, 5 ] H(S) = 9 14 log log = information gain (e.g. Wind) S Weak = {D1, D3, D4, D5, D8, D9, D10, D13} = [6+, 2 ] S Strong = {D2, D6, D7, D11, D12, D14} = [3+, 3 ] Gain(S, Wind) = H(S) v Wind S v S H(S v) = H(S) 8 14 H(S Weak) 6 14 H(S Strong) = = Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

13 Illustrative Example Information Gain Which attribute is the best classifier? Gain (S, Humidity) = ( 14 7 ) ( 14 7 ) = Gain (S, Wind) = ( 14 8 ) ( 14 6 ) 1.0 = Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

14 Information Gain Illustrative Example informations gains for the four attributes: Gain(S, Outlook) = Gain(S, Humidity) = Gain(S, Wind) = Gain(S, Temperature) = Outlook is selected as best classifier and is therefore Root of the tree now branches are created below the root for each possible value because every example for which Outlook = Overcast is positive, this node becomes a leaf node with the classification Yes the other descendants are still ambiguous (H(S) 0) hence, the decision tree has to be further elaborated below these nodes Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

15 Illustrative Example Information Gain Which attribute should be tested here? S sunny = {D1, D2, D8, D9, D11} Gain(S sunny, Humidity) = ( 3 5 ) 0.0 ( 2 ) 0.0 = Gain(S sunny, Temperature) = ( 2 5 ) 0.0 ( 2 5 ) 1.0 ( 1 ) 0.0 = Gain(S sunny, Humidity) = ( 2 5 ) 1.0 ( 3 ) = Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

16 Illustrative Example Information Gain Resulting decision tree Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

17 Information Gain Hypothesis Space Search H complete space of finite discrete functions, relative to the available attributes (i.e. all possible decision trees) capabilities and limitations: returns just one single consistent hypothesis performs greedy search (i.e., max A Gain(S, A)) susceptible to the usual risks of hill-climbing without backtracking uses all training examples at each step simultaneous covering Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

18 Inductive Bias Inductive Bias As mentioned above, ID3 searches complete space of possible hypotheses (wrt instance space), but not completely Preference Bias Inductive Bias: Shorter trees are preferred to longer trees. Trees that place high information gain attributes close to the root are also preferred. Why prefer shorter hypotheses? Occam s Razor: Prefer the simplest hypothesis that fits the data! (aka W. Ockham) see Minimum Description Length Principle (Bayesian Learning) e.g., if there are two decision trees, one with 500 nodes and another with 5 nodes, the second one should be preferred better chance to avoid overfitting Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

19 Overfitting Overfitting Given a hypothesis space H, a hypothesis h H is said to overfit the training data if there exists some alternative hypothesis h H, such that h has smaller error than h over the training data, but h has smaller error than h over the entire distribution of instances. Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

20 Overfitting Example Overfitting Day Outlook Temp. Humidity Wind PlayTennis D1 Sunny Hot High Weak No D2 Sunny Hot High Strong No D3 Overcast Hot High Weak Yes D4 Rain Mild High Weak Yes D5 Rain Cool Normal Weak Yes D6 Rain Cool Normal Strong No D7 Overcast Cool Normal Strong Yes D8 Sunny Mild High Weak No D9 Sunny Cool Normal Weak Yes D10 Rain Mild Normal Weak Yes D11 Sunny Mild Normal Strong Yes D12 Overcast Mild High Strong Yes D13 Overcast Hot Normal Weak Yes D14 Rain Mild High Strong No D15 Sunny Hot Normal Strong No Wrong classification in last example (noise) Resulting tree is more complex and has different structure Tree still fits training set but wrong classification of unseen examples Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

21 Overfitting Overfitting reasons for overfitting: noise in the data number of training examples is too small to produce a representative sample of the target function how to avoid overfitting: stop the tree growing earlier, before it reaches the point where it perfectly classifies the training data, i.e. create a leaf and assign the most common concept allow overfitting and then post-prune the tree (more successful in practice!) how to determine the perfect tree size: separate validation set to evaluate utility of post-pruning apply statistical test to estimate whether expanding (or pruning) produces an improvement Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

22 Overfitting Training Set, Validation Set and Test Set Training Set used to form the learned hypothesis Validation Set separated from training set used to evaluate the accuracy of learned hypothesis over subsequent data (in particular) used to evaluate the impact of pruning this hypothesis Test Set separated from training set used only to evaluate the accuracy of learned hypothesis over subsequent data no more learning / adjustment of the parameters when applying the test set Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

23 Pruning Post-Pruning Prune the tree after it has been generated to avoid overfitting. Two approaches: 1 Reduced Error Pruning 2 Rule Post-Pruning Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

24 Pruning Reduced Error Pruning each of the decision nodes is considered to be a candidate for pruning pruning a decision node consists of removing the sub-tree rooted at the node, making it a leaf node and assigning the most common classification of the training examples affiliated with that node nodes are removed only if the resulting tree performs not worse than the original tree over the validation set pruning starts with the node whose removal most increases accuracy and continues until further pruning is harmful Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

25 Reduced Error Pruning Pruning effect of reduced error pruning: any node added to coincidental regularities in the training set is likely to be pruned the stronger the pruning (less number of nodes), the better is the fitting to the test set the validation set used for pruning is distinct from both the training and test sets Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

26 Rule Post-Pruning Pruning rule post-pruning involves the following steps: 1 Infer the decision tree from the training set (Overfitting allowed!) 2 Convert the tree into a set of rules 3 Prune each rule by removing any preconditions that result in improving its estimated accuracy 4 Sort the pruned rules by their estimated accuracy one method to estimate rule accuracy is to use a separate validation set pruning rules is more precise than pruning the tree itself Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

27 Pruning Alternatives to Information Gain natural bias in information gain favors attributes with many values over those with few values e.g. attribute Date very large number of values (e.g. March 21, 2005) inserted in the above example, it would have the highest information gain, because it perfectly separates the training data but the classification of unseen examples would be impossible alternative measure: GainRatio GainRatio(S, A) = Gain(S,A) SplitInformation(S,A) SplitInformation(S, A) n S i i=1 log S i S 2 S SplitInformation(S, A) is sensitive to how broadly and uniformly A splits S (entropy of S with respect to the values of A) GainRatio penalizes attributes such as Date Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

28 Real-Valued Attributes Pruning Decision tree learning with real valued attributes is possible Discretization of data by split in two ranges (ID3) estimating number of best discriminating ranges (CAL5) One attribute must be allowed to occur more than one time on a path in the decision tree! Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

29 Pruning General ML-Methods Determining the Generalization Error Simple method: Randomly select part of the data from the training set for a test set Unbiased estimate of error because hypothesis is chosen independently of test cases But: the estimated error may still vary from the true error! Estimate the confidence interval in which the true error lies with a certain probability (see Mitchell, chap. 5) Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

30 General ML-Methods Cross Validation Pruning For many learning algorithms, it is useful to provide an additional validation set (e.g. for parameter fitting) k-fold cross validation: partition training set with m examples into k disjoint subsets of size m k run k times with a different subset as validation set each times (using the combined other subsets as training set) Calculate the mean of your estimates over the k runs Last run with complete training set and parameters fixed to the estimates Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

31 Compare Learners Pruning Which learner obtains better results in some domain? Compare whether generalization error of one is significantly lower than of the other Use similar procedure to k-fold cross-validation to obtain data for inference statistical comparison (see Mitchell, chap. 5) Source: Michael Wurtz eecs349michaelwurtz/ Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

32 General ML-Methods Bagging and Boosting Pruning Bagging (bootstrap aggregation, Breimann, 1996): Calculate M classifiers (e.g. decision trees) over different bootstrap samples Prediction by majority vote Boosting (Freund & Schapire, 1996) Additionally introduce weights for each classifier which are iteratively adjusted (due to classification failure/success) see diploma thesis of Jörg Mennicke: Classifier Learning for Imbalanced Data with Varying Misclassification Costs - A Comparison of knn, SVM, and Decision Tree Learning (2006) Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

33 Pruning The Problem of Imbalanced Data In realistic settings occurrence of different classes might be imbalanced (e.g. cancer screening, quality control) Undersampling (remove examples for the over-represented class), oversampling (clone data for the under-represented class) Estimate is worse for the class which occurs more seldom, this might be the class with higher misclassification costs (e.g. decide no cancer if true class is cancer) Instead of over-/, undersampling, introduce different costs for misclassification s and calculate weighted error measure! see e.g.: Tom Hecker and Jörg Mennicke, Diagnosing Cancerous Abnormalities with Decision Tree Learning, Student Project in cooperation with Fraunhofer IIS, Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

34 Summary Summary Decision tree learning is a practical and intuitively understandable method for concept learning Symbolic (rule-based) representation of hypotheses Able to learn disjunctive, discrete-valued concepts Noise in the data is allowed ID3 is a simultaneous covering algorithm based on information gain that performs a greedy top-down search through the space of possible decision trees The inductive bias of DT-algorithm is that short trees are preferred (Ockham s Razor) Overfitting is an important issue and can be reduced by pruning To estimate the generalization error of a hypothesis, typically k-fold cross validation is used Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

35 Learning Terminology Summary ID3 Supervised Learning Approaches: Concept / Classification symbolic inductive Learning Strategy: Data: Target Values: unsupervised learning Policy Learning statistical / neuronal network analytical learning from examples categorial/metric features concept/class Ute Schmid (CogSys, WIAI) ML Decision Trees November 26, / 35

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