Classification: Basic Concepts and Decision Trees

1. Classification: Basic Concepts and Decision Trees

2. A programming task

3. 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.

4. Illustrating Classification Task

5. Examples of Classification Task • Predicting tumor cells as benign or malignant • Classifying credit card transactions as legitimate or fraudulent • Classifying secondary structures of protein as alpha-helix, beta-sheet, or random coil • Categorizing news stories as finance, weather, entertainment, sports, etc

6. Classification Using Distance • Place items in class to which they are “closest”. • Must determine distance between an item and a class. • Classes represented by • Centroid: Central value. • Medoid: Representative point. • Individual points • Algorithm: KNN

7. Classification Techniques • Decision Tree based Methods • Rule-based Methods • Memory based reasoning • Neural Networks • Naïve Bayes and Bayesian Belief Networks • Support Vector Machines

8. A first example Database of 20,000 images of handwritten digits, each labeled by a human [28 x 28 greyscale; pixel values 0-255; labels 0-9] Use these to learn a classifier which will label digit-images automatically…

9. Learning Algorithm The learning problem Input space X = {0,1,…,255}784 Output space Y = {0,1,…,9} Training set (x1, y1), …, (xm, ym) m = 20000 Classifier f: X ! Y To measure how good f is: use a test set [Our test set: 100 instances of each digit.]

10. A possible strategy Input space X = {0,1,…,255}784 Output space Y = {0,1,…,9} Treat each image as a point in 784-dimensional Euclidean space To classify a new image: find its nearest neighbor in the database (training set) and return that label f = entire training set + search engine

11. K Nearest Neighbor (KNN): • Training set includes classes. • Examine K items near item to be classified. • New item placed in class with the most number of close items. • O(q) for each tuple to be classified. (Here q is the size of the training set.)

12. KNN

13. Nearest neighbor Image to label Nearest neighbor Overall: error rate = 6% (on test set) Question: what is the error rate for random guessing?

14. What does it get wrong? Who knows… but here’s a hypothesis: Each digit corresponds to some connected region of R784. Some of the regions come close to each other; problems occur at these boundaries. R1 R2 e.g. a random point in this ball has only a 70% chance of being in R2

15. Nearest neighbor: pros and cons Pros Simple Flexible Excellent performance on a wide range of tasks Cons Algorithmic: time consuming – with n training points in Rd, time to label a new point is O(nd) Statistical: memorization, not learning! no insight into the domain would prefer a compact classifier

16. Prototype selection A possible fix: instead of the entire training set, just keep a “representative sample” Voronoi cells “Decision boundary”

17. Prototype selection A possible fix: instead of the entire training set, just keep a “representative sample” Voronoi cells “Decision boundary”

18. How to pick prototypes? They needn’t be actual data points Idea 2: one prototype per class: mean of training points Examples: Error = 23%

19. Postscript: learning models Batch learning On-line learning Training data See a new point x Learning Algorithm predict label See y Classifier f Update classifer test

20. categorical categorical continuous class Example of a Decision Tree Splitting Attributes Refund Yes No NO MarSt Married Single, Divorced TaxInc NO < 80K > 80K YES NO Model: Decision Tree Training Data

21. NO Another Example of Decision Tree categorical categorical continuous class Single, Divorced MarSt Married NO Refund No Yes TaxInc < 80K > 80K YES NO There could be more than one tree that fits the same data!

22. Decision Tree Classification Task Decision Tree

23. Refund Yes No NO MarSt Married Single, Divorced TaxInc NO < 80K > 80K YES NO Apply Model to Test Data Test Data Start from the root of tree.

24. Refund Yes No NO MarSt Married Single, Divorced TaxInc NO < 80K > 80K YES NO Apply Model to Test Data Test Data

25. Apply Model to Test Data Test Data Refund Yes No NO MarSt Married Single, Divorced TaxInc NO < 80K > 80K YES NO

26. Apply Model to Test Data Test Data Refund Yes No NO MarSt Married Single, Divorced TaxInc NO < 80K > 80K YES NO

27. Apply Model to Test Data Test Data Refund Yes No NO MarSt Married Single, Divorced TaxInc NO < 80K > 80K YES NO

28. Apply Model to Test Data Test Data Refund Yes No NO MarSt Assign Cheat to “No” Married Single, Divorced TaxInc NO < 80K > 80K YES NO

29. Decision Tree Classification Task Decision Tree

30. Decision Tree Induction • Many Algorithms: • Hunt’s Algorithm (one of the earliest) • CART • ID3, C4.5 • SLIQ,SPRINT

31. 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. Dt ?

32. Refund Refund Yes No Yes No Don’t Cheat Marital Status Don’t Cheat Marital Status Single, Divorced Refund Married Married Single, Divorced Yes No Don’t Cheat Taxable Income Cheat Don’t Cheat Don’t Cheat Don’t Cheat < 80K >= 80K Don’t Cheat Cheat Hunt’s Algorithm Don’t Cheat

33. 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

34. 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

35. How to Specify Test Condition? • Depends on attribute types • Nominal • Ordinal • Continuous • Depends on number of ways to split • 2-way split • Multi-way split

36. CarType Family Luxury Sports CarType CarType {Family, Luxury} {Sports, Luxury} {Family} {Sports} Splitting Based on Nominal Attributes • Multi-way split: Use as many partitions as distinct values. • Binary split: Divides values into two subsets. Need to find optimal partitioning. OR

37. Size Small Large Medium Size Size Size {Medium, Large} {Small, Medium} {Small, Large} {Medium} {Large} {Small} Splitting Based on Ordinal Attributes • Multi-way split: Use as many partitions as distinct values. • Binary split: Divides values into two subsets. Need to find optimal partitioning. • What about this split? OR

38. 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

39. Splitting Based on Continuous Attributes

40. 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

41. How to determine the Best Split Before Splitting: 10 records of class 0, 10 records of class 1 Which test condition is the best?

42. Greedy approach: Nodes with homogeneous class distribution are preferred Need a measure of node impurity: How to determine the Best Split Non-homogeneous, High degree of impurity Homogeneous, Low degree of impurity

43. Measures of Node Impurity • Gini Index • Entropy • Misclassification error

44. M0 M2 M3 M4 M1 M12 M34 How to Find the Best Split Before Splitting: A? B? Yes No Yes No Node N1 Node N2 Node N3 Node N4 Gain = M0 – M12 vs M0 – M34

45. Measure of Impurity: GINI • Gini Index for a given node t : (NOTE: p( j | t) is the relative frequency of class j at node t). • Maximum (1 - 1/nc) 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

46. Examples for computing GINI P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0 P(C1) = 1/6 P(C2) = 5/6 Gini = 1 – (1/6)2 – (5/6)2 = 0.278 P(C1) = 2/6 P(C2) = 4/6 Gini = 1 – (2/6)2 – (4/6)2 = 0.444

47. Splitting Based on GINI • Used in CART, SLIQ, SPRINT. • When a node p is split into k partitions (children), the quality of split is computed as, where, ni = number of records at child i, n = number of records at node p.

48. Binary Attributes: Computing GINI Index • Splits into two partitions • Effect of Weighing partitions: • Larger and Purer Partitions are sought for. B? Yes No Node N1 Node N2 Gini(N1) = 1 – (5/6)2 – (2/6)2= 0.194 Gini(N2) = 1 – (1/6)2 – (4/6)2= 0.528 Gini(Children) = 7/12 * 0.194 + 5/12 * 0.528= 0.333

49. 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)

50. 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.