Machine Learning. Learning non-linear functions
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1 Machine Learning /15 701/15-781, 781, Spring 2008 Decision Trees Eric Xing Lecture 6, February 4, 2008 Reading: Chap. 1.6, CB & Chap 3, TM Learning non-linear functions f: X Y X (vector of) continuous and/or discrete vars Y discrete vars Linear separator f might be non-linear function The XOR gate Speech recognition 1
2 10 A hypothesis for TaxFraud Input: a vector of attributes X=[Refund,MarSt,TaxInc] Output: Y= Cheating or t H as a procedure: Refund MarSt Single, Divorced Married Each internal node: test one attribute X i Each branch from a node: selects one value for X i Each leaf node: predict Y TaxInc < 80K > 80K YES Apply Model to Query Data Start from the root of tree. Query Data Refund Marital Cheat Refund Married 80K? Single, Divorced MarSt Married TaxInc < 80K > 80K YES 2
3 10 10 Apply Model to Test Data Query Data Refund Marital Cheat Refund Married 80K? Single, Divorced MarSt Married TaxInc < 80K > 80K YES Apply Model to Test Data Query Data Refund Marital Cheat Refund Married 80K? Single, Divorced MarSt Married TaxInc < 80K > 80K YES 3
4 10 10 Apply Model to Test Data Query Data Refund Marital Cheat Refund Married 80K? Single, Divorced MarSt Married TaxInc < 80K > 80K YES Apply Model to Test Data Query Data Refund Marital Cheat Refund Married 80K? Single, Divorced MarSt Married TaxInc < 80K > 80K YES 4
5 10 Apply Model to Test Data Query Data Refund Marital Cheat Refund Married 80K? Single, Divorced MarSt Married Assign Cheat to TaxInc < 80K > 80K YES A Tree to Predict C-Section Risk Learned from medical records of 1000 wonman Negative examples are C-sections 5
6 Expressiveness Decision trees can express any function of the input attributes. E.g., for Boolean functions, truth table row path to leaf: Trivially, there is a consistent decision tree for any training set with one path to leaf for each example (unless f nondeterministic in x) but it probably won't generalize to new examples Prefer to find more compact decision trees Hypothesis spaces How many distinct decision trees with n Boolean attributes? = number of Boolean functions = number of distinct truth tables with 2 n rows = 2 2n E.g., with 6 Boolean attributes, there are 18,446,744,073,709,551,616 trees 6
7 10 10 Hypothesis spaces How many distinct decision trees with n Boolean attributes? = number of Boolean functions = number of distinct truth tables with 2 n rows = 2 2n E.g., with 6 Boolean attributes, there are 18,446,744,073,709,551,616 trees How many purely conjunctive hypotheses (e.g., Hungry Rain)? Each attribute can be in (positive), in (negative), or out 3 n distinct conjunctive hypotheses More expressive hypothesis space increases chance that target function can be expressed increases number of hypotheses consistent with training set may get worse predictions Decision Tree Learning Tid Attrib1 Attrib2 Attrib3 Class 1 Large 125K 2 Medium 100K 3 Small 70K 4 Medium 120K 5 Large 95K 6 Medium 60K 7 Large 220K 8 Small 85K 9 Medium 75K Learn Model 10 Small 90K Tid Attrib1 Attrib2 Attrib3 Class 11 Small 55K? 12 Medium 80K? 13 Large 110K? 14 Small 95K? 15 Large 67K? Apply Model Decision Tree 7
8 10 10 Example of a Decision Tree categorical Tid Refund Marital categorical continuous Cheat class Splitting Attributes 1 Single 125K 2 Married 100K 3 Single 70K 4 Married 120K 5 Divorced 95K 6 Married 60K 7 Divorced 220K 8 Single 85K 9 Married 75K 10 Single 90K Refund MarSt Single, Divorced TaxInc < 80K > 80K YES Married Training Data Model: Decision Tree Another Example of Decision Tree categorical Tid Refund Marital categorical continuous 1 Single 125K 2 Married 100K 3 Single 70K 4 Married 120K Cheat 5 Divorced 95K 6 Married 60K 7 Divorced 220K 8 Single 85K 9 Married 75K 10 Single 90K class Married MarSt Single, Divorced Refund TaxInc < 80K > 80K YES There could be more than one tree that fits the same data! Training Data 8
9 Top-Down Induction of DT Tree Induction Greedy strategy. Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting 9
10 Tree Induction Greedy strategy. Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting How to Specify Test Condition? Depends on attribute types minal Ordinal Continuous Depends on number of ways to split 2-way split Multi-way split 10
11 Splitting Based on minal Attributes Multi-way split: Use as many partitions as distinct values. CarType Family Sports Luxury Binary split: Divides values into two subsets. Need to find optimal partitioning. {Sports, Luxury} CarType {Family} OR {Family, Luxury} CarType {Sports} Splitting Based on Ordinal Attributes Multi-way split: Use as many partitions as distinct values. Size Small Medium Large Binary split: Divides values into two subsets. Need to find optimal partitioning. {Small, Medium} Size {Large} OR {Medium, Large} Size {Small} What about this split? {Small, Large} Size {Medium} 11
12 Splitting Based on Continuous Attributes Different ways of handling Discretization to form an ordinal categorical attribute Static discretize once at the beginning Dynamic ranges can be found by equal interval bucketing, equal frequency bucketing (percentiles), or clustering. Binary Decision: (A < v) or (A v) consider all possible splits and finds the best cut can be more compute intensive Splitting Based on Continuous Attributes 12
13 Tree Induction Greedy strategy. Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting How to determine the Best Split Idea: a good attribute splits the examples into subsets that are (ideally) "all positive" or "all negative" Homogeneous, Low degree of impurity n-homogeneous, High degree of impurity Greedy approach: des with homogeneous class distribution are preferred Need a measure of node impurity: 13
14 How to compare attribute? Entropy Entropy H(X) of a random variable X H(X) is the expected number of bits needed to encode a randomly drawn value of X (under most efficient code) Why? Information theory: Most efficient code assigns -log 2 P(X=i) bits to encode the message X=I, So, expected number of bits to code one random X is: How to compare attribute? Conditional Entropy Specific conditional entropy H(X Y=v) of X given Y=v : NX H(X y = j) = P (x = i y = j)log 2 P (x = i y = j) i=1 Conditional entropy H(X Y) of X given Y : X H(X Y )= P (y = j)log 2 H(X y = j) j Val(y) Mututal information (aka information gain) of X and Y : I(X; Y ) = H(X) H(X Y )=H(Y) H(Y X) = H(X)+H(Y ) H(X, Y ) 14
15 Sample Entropy S is a sample of training examples p + is the proportion of positive examples in S p - is the proportion of negative examples in S Entropy measure the impurity of S Examples for computing Entropy C1 0 C2 6 P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Entropy = 0 log 0 1 log 1 = 0 0 = 0 C1 1 C2 5 P(C1) = 1/6 P(C2) = 5/6 Entropy = (1/6) log 2 (1/6) (5/6) log 2 (1/6) = 0.65 C1 2 C2 4 P(C1) = 2/6 P(C2) = 4/6 Entropy = (2/6) log 2 (2/6) (4/6) log 2 (4/6) =
16 Information Gain Information Gain: k ni GAIN = Entropy( p) Entropy( i) split i= 1 n Parent de, p is split into k partitions; n i is number of records in partition i Measures Reduction in Entropy achieved because of the split. Choose the split that achieves most reduction (maximizes GAIN) Used in ID3 and C4.5 Disadvantage: Tends to prefer splits that result in large number of partitions, each being small but pure. Gain(S,A) = mutual information between A and target class variable over sample S Splitting Based on INFO... Gain Ratio: GainRATIO GAIN Split k = split i= SplitINFO Parent de, p is split into k partitions n i is the number of records in partition i n n i i SplitINFO = log 1 n n Adjusts Information Gain by the entropy of the partitioning (SplitINFO). Higher entropy partitioning (large number of small partitions) is penalized! Used in C4.5 Designed to overcome the disadvantage of Information Gain 16
17 10 Exercise categorical Tid Refund Marital categorical continuous 1 Single 125K 2 Married 100K 3 Single 70K 4 Married 120K Cheat 5 Divorced 95K 6 Married 60K 7 Divorced 220K 8 Single 85K 9 Married 75K 10 Single 90K class Training Data Tree Induction Greedy strategy. Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting 17
18 Stopping Criteria for Tree Induction Stop expanding a node when all the records belong to the same class Stop expanding a node when all the records have similar attribute values Early termination (to be discussed later) Decision Tree Based Classification Advantages: Inexpensive to construct Extremely fast at classifying unknown records Easy to interpret for small-sized trees Accuracy is comparable to other classification techniques for many simple data sets Example: C4.5 Simple depth-first construction. Uses Information Gain Sorts Continuous Attributes at each node. Needs entire data to fit in memory. Unsuitable for Large Datasets. Needs out-of-core sorting. You can download the software from: 18
19 Practical Issues of Classification Underfitting and Overfitting Missing Values Costs of Classification -- Later lectures Underfitting and Overfitting Overfitting Underfitting: when model is too simple, both training and test errors are large 19
20 tes 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 Which Tree Should We Output? Occam s razor: prefer the simplest hypothesis that fits the data 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 include model complexity when evaluating a model 20
21 Minimum Description Length (MDL) X y X 1 1 X 2 0 X 3 0 X 4 1 X n 1 A 0 A? B? B 1 B 2 C? 1 C 1 C B X y X 1? X 2? X 3? X 4? X n? Cost(Model,Data) = Cost(Data Model) + Cost(Model) Cost is the number of bits needed for encoding. Search for the least costly model. Cost(Data Model) encodes the misclassification errors. Cost(Model) uses node encoding (number of children) plus splitting condition encoding. How to Address Overfitting Pre-Pruning (Early Stopping Rule) Stop the algorithm before it becomes a fully-grown tree Typical stopping conditions for a node: Stop if all instances belong to the same class Stop if all the attribute values are the same More restrictive conditions: Stop if number of instances is less than some user-specified threshold Stop if class distribution of instances are independent of the available features (e.g., using χ 2 test) Stop if expanding the current node does not improve impurity measures (e.g., Gini or information gain). 21
22 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 Can use MDL for post-pruning Handling Missing Attribute Values Missing values affect decision tree construction in three different ways: Affects how impurity measures are computed Affects how to distribute instance with missing value to child nodes Affects how a test instance with missing value is classified 22
23 Computing Impurity Measure Tid Refund Marital Class 1 Single 125K 2 Married 100K 3 Single 70K 4 Married 120K 5 Divorced 95K 6 Married 60K 7 Divorced 220K 8 Single 85K 9 Married 75K 10? Single 90K Missing value Before Splitting: Entropy(Parent) = -0.3 log(0.3)-(0.7)log(0.7) = Class Class = = Refund= 0 3 Refund= 2 4 Refund=? 1 0 Split on Refund: Entropy(Refund=) = 0 Entropy(Refund=) = -(2/6)log(2/6) (4/6)log(4/6) = Entropy(Children) = 0.3 (0) (0.9183) = Gain = 0.9 ( ) = Distribute Instances Tid Refund Marital 1 Single 125K 2 Married 100K 3 Single 70K 4 Married 120K Class 5 Divorced 95K 6 Married 60K 7 Divorced 220K 8 Single 85K 9 Married 75K Class= 0 Class= 3 Refund Cheat= 2 Cheat= 4 Tid Refund Marital Refund Class 10? Single 90K Class= 0 + 3/9 Class= 3 Class= 2 + 6/9 Class= 4 Probability that Refund= is 3/9 Probability that Refund= is 6/9 Assign record to the left child with weight = 3/9 and to the right child with weight = 6/9 23
24 10 Classify Instances New record: Married Single Divorced Total Tid Refund Marital Class Class= ? 85K? Class= 6/ Refund Single, Divorced TaxInc MarSt < 80K > 80K YES Married Total Probability that Marital = Married is 3.67/6.67 Probability that Marital ={Single,Divorced} is 3/
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