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Data Mining Classification This lecture node is modified based on Lecture Notes for Chapter 4/5 of Introduction to Data Mining by Tan, Steinbach, Kumar, and slides from Jiawei Han for the book of Data Mining – Concepts and Techniqies by Jiawei Han and Micheline Kamber. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Classification: Definition Given a collection of records (training set ) – Each record contains a set of attributes, one of the attributes is the class. Find a model for class attribute as a function of the values of other attributes. Goal: previously unseen records should be assigned a class as accurately as possible. – A test set is used to determine the accuracy of the model. Usually, the given data set is divided into training and test sets, with training set used to build the model and test set used to validate it. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Classification Techniques Decision Tree Naïve Bayes kNN Classification © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Decision Tree Classification Task Tid Attrib1 Attrib2 Attrib3 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 Tree Induction algorithm Class Induction Learn Model Model 10 Training Set Tid Attrib1 Attrib2 Attrib3 11 No Small 55K ? 12 Yes Medium 80K ? 13 Yes Large 110K ? 14 No Small 95K ? 15 No Large 67K ? Apply Model Class Decision Tree Deduction 10 Test Set © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Example of a Decision Tree Tid Refund Marital Status Taxable Income Cheat 1 Yes Single 125K No 2 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K Splitting Attributes Refund Yes No NO MarSt Single, Divorced TaxInc < 80K NO Married NO > 80K YES 10 Model: Decision Tree Training Data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Another Example of Decision Tree MarSt Tid Refund Married Marital Status Taxable Income Cheat 1 Yes Single 125K No 2 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K NO Single, Divorced Refund No Yes NO TaxInc < 80K > 80K NO YES There could be more than one tree that fits the same data! 10 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Decision Tree Classification Task Tid Attrib1 Attrib2 Attrib3 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 Tree Induction algorithm Class Induction Learn Model Model 10 Training Set Tid Attrib1 Attrib2 Attrib3 11 No Small 55K ? 12 Yes Medium 80K ? 13 Yes Large 110K ? 14 No Small 95K ? 15 No Large 67K ? Apply Model Class Decision Tree Deduction 10 Test Set © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Apply Model to Test Data Test Data Start from the root of tree. Refund Yes Refund Marital Status Taxable Income Cheat No 80K Married ? 10 No NO MarSt Single, Divorced TaxInc < 80K NO © Tan,Steinbach, Kumar Married NO > 80K YES Introduction to Data Mining 4/18/2004 ‹#› Apply Model to Test Data Test Data Refund Yes Refund Marital Status Taxable Income Cheat No 80K Married ? 10 No NO MarSt Single, Divorced TaxInc < 80K NO © Tan,Steinbach, Kumar Married NO > 80K YES Introduction to Data Mining 4/18/2004 ‹#› Apply Model to Test Data Test Data Refund Yes Refund Marital Status Taxable Income Cheat No 80K Married ? 10 No NO MarSt Single, Divorced TaxInc < 80K NO © Tan,Steinbach, Kumar Married NO > 80K YES Introduction to Data Mining 4/18/2004 ‹#› Apply Model to Test Data Test Data Refund Yes Refund Marital Status Taxable Income Cheat No 80K Married ? 10 No NO MarSt Single, Divorced TaxInc < 80K NO © Tan,Steinbach, Kumar Married NO > 80K YES Introduction to Data Mining 4/18/2004 ‹#› Apply Model to Test Data Test Data Refund Yes Refund Marital Status Taxable Income Cheat No 80K Married ? 10 No NO MarSt Single, Divorced TaxInc < 80K NO © Tan,Steinbach, Kumar Married NO > 80K YES Introduction to Data Mining 4/18/2004 ‹#› Apply Model to Test Data Test Data Refund Yes Refund Marital Status Taxable Income Cheat No 80K Married ? 10 No NO MarSt Single, Divorced TaxInc < 80K NO © Tan,Steinbach, Kumar Married Assign Cheat to “No” NO > 80K YES Introduction to Data Mining 4/18/2004 ‹#› Performance Metrics PREDICTED CLASS Class=Yes ACTUAL CLASS Class=No Class=Yes a (TP) b (FN) Class=No c (FP) d (TN) a+d TP + TN = Accuracy = a + b + c + d TP + TN + FP + FN © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› General Structure of Hunt’s Algorithm Let Dt be the set of training records that reach a node t General Procedure: – If Dt contains records that belong the same class yt, then t is a leaf node labeled as yt – If Dt is an empty set, then t is a leaf node labeled by the default class, yd – If Dt contains records that belong to more than one class, use an attribute test to split the data into smaller subsets. Recursively apply the procedure to each subset. © Tan,Steinbach, Kumar Introduction to Data Mining Tid Refund Marital Status Taxable Income Cheat 1 Yes Single 125K No 2 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K 10 Dt ? 4/18/2004 ‹#› Hunt’s Algorithm Don’t Cheat Refund Yes No Don’t Cheat Don’t Cheat Refund Refund Yes Yes No No Tid Refund Marital Status Taxable Income Cheat 1 Yes Single 125K No 2 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K 10 Don’t Cheat Don’t Cheat Marital Status Single, Divorced Cheat Married Single, Divorced Don’t Cheat © Tan,Steinbach, Kumar Marital Status Married Don’t Cheat Taxable Income < 80K >= 80K Don’t Cheat Cheat Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› How to determine the Best Split Before Splitting: 10 records of class 0, 10 records of class 1 Own Car? Yes Car Type? No Family Student ID? Luxury c1 Sports C0: 6 C1: 4 C0: 4 C1: 6 C0: 1 C1: 3 C0: 8 C1: 0 C0: 1 C1: 7 C0: 1 C1: 0 ... c10 c11 C0: 1 C1: 0 C0: 0 C1: 1 c20 ... C0: 0 C1: 1 Which test condition is the best? © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› How to determine the Best Split Greedy approach: – Nodes with homogeneous class distribution are preferred Need a measure of node impurity: C0: 5 C1: 5 C0: 9 C1: 1 Non-homogeneous, Homogeneous, High degree of impurity Low degree of impurity © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› How to Find the Best Split Before Splitting: C0 C1 N00 N01 M0 A? B? Yes No Node N1 C0 C1 Node N2 N10 N11 C0 C1 N20 N21 M2 M1 Yes No Node N3 C0 C1 Node N4 N30 N31 C0 C1 M3 M12 N40 N41 M4 M34 Gain = M0 – M12 vs M0 – M34 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Measure of Impurity: GINI Gini Index for a given node t : GINI (t ) = 1 [ p( j | t )]2 j (NOTE: p( j | t) is the relative frequency of class j at node t). – Maximum when records are equally distributed among all classes, implying least interesting information – Minimum (0.0) when all records belong to one class, implying most interesting information C1 C2 0 6 Gini=0.000 © Tan,Steinbach, Kumar C1 C2 1 5 Gini=0.278 C1 C2 2 4 Gini=0.444 Introduction to Data Mining C1 C2 3 3 Gini=0.500 4/18/2004 ‹#› Examples for computing GINI GINI (t ) = 1 [ p( j | t )]2 j C1 C2 0 6 P(C1) = 0/6 = 0 C1 C2 1 5 P(C1) = 1/6 C1 C2 2 4 P(C1) = 2/6 © Tan,Steinbach, Kumar P(C2) = 6/6 = 1 Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0 P(C2) = 5/6 Gini = 1 – (1/6)2 – (5/6)2 = 0.278 P(C2) = 4/6 Gini = 1 – (2/6)2 – (4/6)2 = 0.444 Introduction to Data Mining 4/18/2004 ‹#› Splitting Based on GINI When a node p is split into k partitions (children), the quality of split is computed as, k ni GINI split = GINI (i ) i =1 n where, © Tan,Steinbach, Kumar ni = number of records at child i, n = number of records at node p. Introduction to Data Mining 4/18/2004 ‹#› Binary Attributes: Computing GINI Index Split into two partitions Effect of weighing partitions: – Larger and Purer Partitions are sought for. Parent B? Yes No C1 6 C2 6 Gini = 0.500 Gini(N1) = 1 – (5/7)2 – (2/7)2 = 0.408 Gini(N2) = 1 – (1/5)2 – (4/5)2 = 0.320 © Tan,Steinbach, Kumar Node N1 Node N2 N1 N2 C1 5 1 C2 2 4 Gini=0.371 Introduction to Data Mining Gini(Children) = 7/12 * 0.408 + 5/12 * 0.320 = 0.371 4/18/2004 ‹#› Categorical Attributes: Computing Gini Index For each distinct value, gather counts for each class in the dataset Use the count matrix to make decisions Multi-way split Two-way split (find best partition of values) CarType Family Sports Luxury C1 C2 Gini 1 4 2 1 0.393 © Tan,Steinbach, Kumar 1 1 C1 C2 Gini CarType {Sports, {Family} Luxury} 3 1 2 4 0.400 Introduction to Data Mining C1 C2 Gini CarType {Family, {Sports} Luxury} 2 2 1 5 0.419 4/18/2004 ‹#› Continuous Attributes: Computing Gini Index Use Binary Decisions based on one value Several Choices for the splitting value – Number of possible splitting values = Number of distinct values Each splitting value has a count matrix associated with it – Class counts in each of the partitions, A < v and A v Simple method to choose best v – For each v, scan the database to gather count matrix and compute its Gini index – Computationally Inefficient! Repetition of work. © Tan,Steinbach, Kumar Introduction to Data Mining Tid Refund Marital Status Taxable Income Cheat 1 Yes Single 125K No 2 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K 10 Taxable Income > 80K? Yes 4/18/2004 No ‹#› Alternative Splitting Criteria based on INFO Entropy at a given node t: Entropy(t ) = p( j | t ) log p( j | t ) j (NOTE: p( j | t) is the relative frequency of class j at node t). – Measures homogeneity of a node. Maximum when records are equally distributed among all classes implying least information Minimum (0.0) when all records belong to one class, implying most information – Entropy based computations are similar to the GINI index computations © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Examples for computing Entropy Entropy(t ) = p( j | t ) log p( j | t ) j C1 C2 0 6 C1 C2 1 5 P(C1) = 1/6 C1 C2 2 4 P(C1) = 2/6 © Tan,Steinbach, Kumar P(C1) = 0/6 = 0 2 P(C2) = 6/6 = 1 Entropy = – 0 log 0 – 1 log 1 = – 0 – 0 = 0 P(C2) = 5/6 Entropy = – (1/6) log2 (1/6) – (5/6) log2 (1/6) = 0.65 P(C2) = 4/6 Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92 Introduction to Data Mining 4/18/2004 ‹#› Splitting Based on INFO... Information Gain: n GAIN = Entropy ( p) Entropy (i ) n k split i i =1 Parent Node, p is split into k partitions; ni 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) – Disadvantage: Tends to prefer splits that result in large number of partitions, each being small but pure. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Splitting Based on INFO... Gain Ratio: GAIN n n GainRATIO = SplitINFO = log SplitINFO n n Split split k i i i =1 Parent Node, p is split into k partitions ni is the number of records in partition i – Adjusts Information Gain by the entropy of the partitioning (SplitINFO). Higher entropy partitioning (large number of small partitions) is penalized! – Designed to overcome the disadvantage of Information Gain © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Splitting Criteria based on Classification Error Classification error at a node t : Error (t ) = 1 max P(i | t ) i Measures misclassification error made by a node. Maximum when records are equally distributed among all classes, implying least interesting information Minimum (0.0) when all records belong to one class, implying most interesting information © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Examples for Computing Error Error (t ) = 1 max P(i | t ) i C1 C2 0 6 C1 C2 1 5 P(C1) = 1/6 C1 C2 2 4 P(C1) = 2/6 © Tan,Steinbach, Kumar P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Error = 1 – max (0, 1) = 1 – 1 = 0 P(C2) = 5/6 Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6 P(C2) = 4/6 Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3 Introduction to Data Mining 4/18/2004 ‹#› Comparison among Splitting Criteria For a 2-class problem: © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Training Dataset This follows an example from Quinlan’s ID3 © Tan,Steinbach, Kumar age <=30 <=30 30…40 >40 >40 >40 31…40 <=30 <=30 >40 <=30 31…40 31…40 >40 income student credit_rating high no fair high no excellent high no fair medium no fair low yes fair low yes excellent low yes excellent medium no fair low yes fair medium yes fair medium yes excellent medium no excellent high yes fair medium no excellent Introduction to Data Mining buys_computer no no yes yes yes no yes no yes yes yes yes yes no 4/18/2004 ‹#› More on Information Gain Entropy(t ) = p( j | t ) log p( j | t ) j 2 S contains si tuples of class Ci for i = {1, …, m} information measures info required to classify any m arbitrary tuple si si I(s1 ,s2 ,...,sm ) = -å log2 s i=1 s entropy of attribute A with values {a1,a2,…,av} s1 j +... + smj E(A) = å I(s1 j ,..., smj ) s j=1 information gained by branching on attribute A v Gain(A) = I(s1,s2,...,sm )- E(A) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Attribute Selection by Information Gain Class P: buys_computer = “yes” Class N: buys_computer = “no” I(p, n) = I(9, 5) =0.940 Compute the entropy for age: age <=30 30…40 >40 pi 2 4 3 age income student <=30 high no <=30 high no 31…40 high no >40 medium no >40 low yes >40 low yes 31…40 low yes <=30 medium no <=30 low yes >40 medium yes <=30 medium yes 31…40 medium no © Tan,Steinbach, Kumar 31…40 high yes >40 medium no ni I(pi, ni) 3 0.971 0 0 2 0.971 5 4 I (2,3) + I ( 4,0) 14 14 5 + I (3,2) = 0.694 14 E ( age) = 5 I (2,3) means “age <=30” has 5 14 out of 14 samples, with 2 yes’es and 3 no’s. Hence Gain(age) = I ( p, n) E (age) = 0.246 credit_rating buys_computer fair no excellent no fair yes fair yes fair yes excellent no excellent yes fair no fair yes fair yes excellent yes excellent yes Introduction to Data Mining fair yes excellent no Similarly, Gain(income) = 0.029 Gain( student ) = 0.151 Gain(credit _ rating ) = 0.048 4/18/2004 ‹#› Output: A Decision Tree for “buys_computer” age? <=30 overcast 30..40 student? yes >40 credit rating? no yes excellent fair no yes no yes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Classification Rules Represent the knowledge in the form of IF-THEN rules One rule is created for each path from the root to a leaf Each attribute-value pair along a path forms a conjunction The leaf node holds the class prediction Rules are easier for humans to understand Example IF age = “<=30” AND student = “no” THEN buys_computer = “no” IF age = “<=30” AND student = “yes” THEN buys_computer = “yes” IF age = “31…40” THEN buys_computer = “yes” IF age = “>40” AND credit_rating = “excellent” THEN buys_computer = “yes” IF age = “<=30” AND credit_rating = “fair” THEN buys_computer = “no” © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Decision Boundary 1 0.9 x < 0.43? 0.8 0.7 Yes No y 0.6 y < 0.33? y < 0.47? 0.5 0.4 Yes 0.3 0.2 :4 :0 0.1 No Yes :0 :4 :0 :3 No :4 :0 0 0 0.1 0.2 0.3 0.4 0.5 x 0.6 0.7 0.8 0.9 1 • Border line between two neighboring regions of different classes is known as decision boundary • Decision boundary is parallel to axes because test condition involves a single attribute at-a-time © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Oblique Decision Trees x+y<1 Class = + Class = • Test condition may involve multiple attributes • More expressive representation • Finding optimal test condition is computationally expensive © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Model Underfitting/Overfitting The errors of a classification model are divided into two types – Training errors: the number of misclassification errors committed on training records. – Generalization errors: the expected error of the model on previously unseen records. A good model must have both errors low. Model underfitting: both type of errors are large when the decision tree is too small. Model overfitting: training error is small but generalization error is large, when the decision tree is too large. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Underfitting and Overfitting (Example) 500 circular and 500 triangular data points. Circular points: 0.5 sqrt(x12+x22) 1 Triangular points: sqrt(x12+x22) > 0.5 or sqrt(x12+x22) < 1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Underfitting and Overfitting Overfitting Underfitting: when model is too simple, both training and test errors are large © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› An Example: Training Dataset © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› An Example: Testing Dataset © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› An Example: Two Models, M1 and M2 0% training error 30% testing error 20% training error 10% testing error (human, warm-blooded, yes, no, no) (dolphin, warm-blooded, yes, no, no) (spiny anteater, warm-blooded, no, yes, yes) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Incorporating Model Complexity (2) Training errors: error on training ( e(t)) Generalization errors: error on testing ( e’(t)) Methods for estimating generalization errors: – Optimistic approach: e’(t) = e(t) – Pessimistic approach: For each leaf node: e’(t) = (e(t)+0.5) Total errors: e’(T) = e(T) + N 0.5 (N: number of leaf nodes) For a tree with 30 leaf nodes and 10 errors on training (out of 1000 instances): Training error = 10/1000 = 1% Generalization error = (10 + 300.5)/1000 = 2.5% © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› An Example: Pessimistic Error Estimate The generalization error of TL is (4 + 7 * 0.5)/24 = 0.3125, and the generalization error of TR is (6 + 4 * 0.5)/24 = 0.3333, where the penalty term is 0.5. Based on pessimistic error estimate, the TL should be used. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› How to Address Overfitting (1) 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 chi-squared test) Stop if expanding the current node does not improve impurity measures (e.g., Gini or information gain). © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› How to Address Overfitting (2) 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 (Minimum Description Length) for post-pruning © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Example of Post-Pruning Training Error (Before splitting) = 10/30 Class = Yes 20 Pessimistic error = (10 + 0.5)/30 = 10.5/30 Class = No 10 Training Error (After splitting) = 9/30 Pessimistic error (After splitting) Error = 10/30 = (9 + 4 0.5)/30 = 11/30 PRUNE! A? A1 A4 A3 A2 Class = Yes 8 Class = Yes 3 Class = Yes 4 Class = Yes 5 Class = No 4 Class = No 4 Class = No 1 Class = No 1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›