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Decision Trees and Decision Tree Learning Philipp Kärger Outline: 1. Decision Trees 2. Decision Tree Learning 1. ID3 Algorithm 2. Which attribute to split on? 3. Some examples 3. Overfitting 4. Where to use Decision Trees? Decision tree representation for PlayTennis Outlook Sunny Overcast Humidity High No Rain Yes Normal Yes Wind Strong No Weak Yes Decision tree representation for PlayTennis Outlook Attribute Sunny Overcast Humidity High No Rain Yes Normal Yes Wind Strong No Weak Yes Decision tree representation for PlayTennis Outlook Value Sunny Overcast Humidity High No Rain Yes Normal Yes Wind Strong No Weak Yes Decision tree representation for PlayTennis Outlook Classification Sunny Overcast Humidity High No Rain Yes Normal Yes Wind Strong No Weak Yes PlayTennis: Other representations • Logical expression for PlayTennis=Yes: – (Outlook=Sunny Humidity=Normal) (Outlook=Overcast) (Outlook=Rain Wind=Weak) • If-then rules – IF Outlook=Sunny Humidity=Normal THEN PlayTennis=Yes – IF Outlook=Overcast THEN PlayTennis=Yes – IF Outlook=Rain Wind=Weak THEN PlayTennis=Yes – IF Outlook=Sunny Humidity=High THEN PlayTennis=No – IF Outlook=Rain Wind=Strong THEN PlayTennis=Yes Decision Trees - Summary • a model of a part of the world • allows us to classify instances (by performing a sequence of tests) • allows us to predict classes of (unseen) instances • understandable by humans (unlike many other representations) Decision Tree Learning • Goal: Learn from known instances how to classify unseen instances • by means of building and exploiting a Decision Tree • supervised or unsupervised learning? Application: classification of medical patients by their disease Classification Task seen patients Attrib2 Tid Attrib1 Attrib3 Class 1 Yes Large 125K No 2 No Medium 100K No 3 No Small 70K No 4 Yes Medium 120K No 5 No Large 95K Yes 6 No Medium 60K No 7 Yes Large 220K No 8 No Small 85K Yes 9 No Medium 75K No 10 No Small 90K Yes Learning algorithm Induction Learn Model Decision Model Tree 10 Training Set unseen patients Attrib2 Tid Attrib1 Attrib3 Class 11 No Small 55K ? 12 Yes Medium 80K ? 13 Yes Large 110K ? 14 No Small 95K ? 15 No Large 67K ? 10 Test Set Apply Model check Deduction attributes of an unseen patient rules telling which attributes of the patient indicates a disease Basic algorithm: ID3 (simplified) ID3 = Iterative Dichotomiser 3 - given a goal class to build the tree for A1=weather A2=day happy sun odd yes rain odd no rain even no - create a root node for the tree sun even - if all examples from the test set belong to the same goal class C rain odd then label the root with C sun even - else – select the ‘most informative’ attribute A – split the training set according to the values V1..Vn of A – recursively build the resulting subtrees T1 … Tn – generate decision tree T: A v1 ... yes no yes Humidity vn Low High T1 ... Tn No Yes • lessons learned: – there is always more than one decision tree – finding the “best” one is NP complete – all the known algorithms use heuristics • finding the right attribute A to split on is tricky Search heuristics in ID3 • Which attribute should we split on? • Need a heuristic – Some function gives big numbers for “good” splits • Want to get to “pure” sets • How can we measure “pure”? odd even sunny rain Measuring Information: Entropy • The average amount of information I needed to classify an object is given by the entropy measure p(c) = probability of class Cc (sum over all classes) • For a two-class problem: entropy p(c) • What is the entropy of the set of happy/unhappy days? A1=weather A2=day happy sun odd yes odd rain odd no even rain even no sun even yes rain odd no sun even yes sunny rain Residual Information • After applying attribute A, S is partitioned into subsets according to values v of A • Ires represents the amount of information still needed to classify an instance • Ires is equal to weighted sum of the amounts of information for the subsets p(c|v) = probability that an instance belongs to class C given that it belongs to v = I(v) • What is Ires(A) if I split for “weather” and if I split for “day”? A1=weather A2=day happy sun odd yes rain odd no rain even no sun even yes rain odd no sun even yes Ires(weather) = 0 Ires(day) = 1 odd even sunny rain Information Gain: = the amount of information I rule out by splitting on attribute A: Gain(A) = I – Ires(A) = information in the current set minus the residual information after splitting The most ‘informative’ attribute is the one that minimizes Ires, i.e., maximizes the Gain Triangles and Squares # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Color green green yellow red red red green green yellow red green yellow yellow red Attribute Outline dashed dashed dashed dashed solid solid solid dashed solid solid solid dashed solid dashed Shape Dot no yes no no no yes no no yes no yes yes no yes triange triange square square square triange square triange square square square square square triange Data Set: A set of classified objects . . . . . . Entropy . . . . . • 5 triangles • 9 squares • class probabilities . • entropy of the data set Entropy reduction by data set partitioning . . . . . . . . red Color? green . yellow . . . residual information . . . . . . . . red Color? green yellow . . . . Information Gain . . . . . . . . red Color? green yellow . . . . Information Gain of The Attribute • Attributes – Gain(Color) = 0.246 – Gain(Outline) = 0.151 – Gain(Dot) = 0.048 • Heuristics: attribute with the highest gain is chosen • This heuristics is local (local minimization of impurity) . . . . . . . . red Color? green . . yellow . . Gain(Outline) = 0.971 – 0 = 0.971 bits Gain(Dot) = 0.971 – 0.951 = 0.020 bits . . . . . . . . red Gain(Outline) = 0.971 – 0.951 = 0.020 bits Gain(Dot) = 0.971 – 0 = 0.971 bits Color? green . . yellow . . solid Outline? dashed . . . . . . . . . . red . yes Dot? Color? . no green . . yellow . . solid Outline? dashed . . Decision Tree . . . . . . Color red Dot yes triangle yellow green square no square Outline dashed triangle solid square A Defect of Ires • Ires favors attributes with many values • Such attribute splits S to many subsets, and if these are small, they will tend to be pure anyway • One way to rectify this is through a corrected measure of information gain ratio. A1=weather A2=day happy sun 17.1.08 yes rain 18.1.08 no rain 19.1.08 no sun 20.1.08 yes sun 21.1.08 yes Information Gain Ratio • I(A) is amount of information needed to determine the value of an attribute A • Information gain ratio Information Gain Ratio . . . . . . . . red Color? green yellow . . . . Information Gain and Information Gain Ratio A |v(A)| Gain(A) GainRatio(A) Color 3 0.247 0.156 Outline 2 0.152 0.152 Dot 2 0.048 0.049 Overfitting (Example) Overfitting Overfitting Underfitting: when model is too simple, both training and test errors are large Notes on Overfitting • Overfitting results in decision trees that are more complex than necessary • Training error no longer provides a good estimate of how well the tree will perform on previously unseen records How to Address Overfitting Idea: prune the tree so that it is not too specific Two possibilities: Pre-Pruning - prune while building the tree Post-Pruning - prune after building the tree How to Address Overfitting • Pre-Pruning (Early Stopping Rule) – Stop the algorithm before it becomes a fullygrown tree – More restrictive stopping conditions: • Stop if number of instances is less than some userspecified threshold • Stop if expanding the current node does not improve impurity measures (e.g., information gain). – Not successful in practice How to Address Overfitting… • Post-pruning – Grow decision tree to its entirety – Trim the nodes of the decision tree in a bottom-up fashion – If generalization error improves after trimming, replace sub-tree by a leaf node. – Class label of leaf node is determined from majority class of instances in the sub-tree Occam’s Razor • Given two models of similar generalization errors, one should prefer the simpler model over the more complex model • For complex models, there is a greater chance that it was fitted accidentally by errors in data • Therefore, one should prefer less complex models in general When to use Decision Tree Learning? Appropriate problems for decision tree learning • Classification problems • Characteristics: – – – – instances described by attribute-value pairs target function has discrete output values training data may be noisy training data may contain missing attribute values Strengths • • • • can generate understandable rules perform classification without much computation can handle continuous and categorical variables provide a clear indication of which fields are most important for prediction or classification Weakness • Not suitable for prediction of continuous attribute. • Perform poorly with many class and small data. • Computationally expensive to train. – At each node, each candidate splitting field must be sorted before its best split can be found. – In some algorithms, combinations of fields are used and a search must be made for optimal combining weights. – Pruning algorithms can also be expensive since many potential sub-trees must be formed and compared