Algorithms for Classification:

1. Algorithms for Classification: Notes by Gregory Piatetsky

2. Classification • Task: Given a set of pre-classified examples, build a model or classifier to classify new cases. • Supervised learning: classes are known for the examples used to build the classifier. • A classifier can be a set of rules, a decision tree, a neural network, etc. • Typical applications: credit approval, direct marketing, fraud detection, medical diagnosis, …..

3. Simplicity first • Simple algorithms often work very well! • There are many kinds of simple structure, eg: • One attribute does all the work • All attributes contribute equally & independently • A weighted linear combination might do • Instance-based: use a few prototypes • Use simple logical rules • Success of method depends on the domain witten&eibe

4. Inferring rudimentary rules • 1R: learns a 1-level decision tree • I.e., rules that all test one particular attribute • Basic version • One branch for each value • Each branch assigns most frequent class • Error rate: proportion of instances that don’t belong to the majority class of their corresponding branch • Choose attribute with lowest error rate (assumes nominal attributes) witten&eibe

5. Bayesian (Statistical) modeling • “Opposite” of 1R: use all the attributes • Two assumptions: Attributes are • equally important • statistically independent (given the class value) • I.e., knowing the value of one attribute says nothing about the value of another(if the class is known) • Independence assumption is almost never correct! • But … this scheme works well in practice witten&eibe

6. Bayes’s rule • Probability of event H given evidence E : • A priori probability of H : • Probability of event before evidence is seen • A posteriori probability of H : • Probability of event after evidence is seen From Bayes “Essay towards solving a problem in the doctrine of chances” (1763) Thomas Bayes Born: 1702 in London, England Died: 1761 in Tunbridge Wells, Kent, England witten&eibe

7. Probabilities for weather data witten&eibe

8. Probabilities for Weather data • A new day: witten&eibe

9. Weather data example Evidence E Probability of class “yes” witten&eibe

10. The “zero-frequency problem” • What if an attribute value doesn’t occur with every class value?(e.g. “Humidity = high” for class “yes”) • Probability will be zero! • A posteriori probability will also be zero!(No matter how likely the other values are!) • This is problematic since it will wipe out all information in the other probabilities when they are multiplied. It is therefore often desirable to incorporate a small-sample correction in all probability estimates such that no probability is ever set to be exactly zero. • Remedy: add 1 to the count for every attribute value-class combination (Laplace estimator) • Result: probabilities will never be zero! (also: stabilizes probability estimates) witten&eibe

11. Probabilities for weather data witten&eibe

12. Probabilities for Weather data • A new day: witten&eibe

13. Weather data example Evidence E Probability of class “yes” witten&eibe

14. The “zero-frequency problem” • What if an attribute value doesn’t occur with every class value?(e.g. “Humidity = high” for class “yes”) • Probability will be zero! • A posteriori probability will also be zero!(No matter how likely the other values are!) • This is problematic since it will wipe out all information in the other probabilities when they are multiplied. It is therefore often desirable to incorporate a small-sample correction in all probability estimates such that no probability is ever set to be exactly zero. • Remedy: add 1 to the count for every attribute value-class combination (Laplace estimator) • Result: probabilities will never be zero! (also: stabilizes probability estimates) witten&eibe

15. *Modified probability estimates • E.G. For Outlook, Yes we had:Sunny=2/9, Overcast=4/9 and Rainy=3/9. • If we add the constant we get: Sunny= (2+1)/(9+3), Overcast=(4+1)/(9+3) and Rainy=(3+1)/(9+3). • In some cases adding a constant different from 1 might be more appropriate. • Weights don’t need to be equal(but they must sum to μ) Rainy Sunny Overcast witten&eibe

16. Missing values (one approach) • Training: instance is not included in frequency count for attribute value-class combination • Classification: attribute will be omitted from calculation • Example: witten&eibe

17. Numeric attributes • Discretization, or assume a distribution. • Usual assumption: a normal or Gaussian probability distribution (given the class) • The probability density function for the normal distribution is defined by two parameters: • Sample mean: • Standard deviation • Then the density function f(x) is: Karl Gauss, 1777-1855 German mathematician witten&eibe

18. Statistics forweather data • Example density value: witten&eibe

19. Classifying a new day • A new day: witten&eibe

20. Missing values in tuples (one approach) • Training: instance is not included in frequency count for attribute value-class combination • Classification: attribute will be omitted from calculation • Example: witten&eibe

21. Naïve Bayes: discussion • Naïve Bayes works surprisingly well (even if independence assumption is clearly violated) • Why? Because classification doesn’t require accurate probability estimates as long as maximum probability is assigned to correct class • However: adding too many redundant attributes will cause problems (e.g. identical attributes) • Note also: many numeric attributes are not normally distributed ( kernel density estimators) witten&eibe

22. Naïve Bayes Extensions • Improvements: • select best attributes (e.g. with greedy search) • often works as well or better with just a fraction of all attributes • Bayesian Networks witten&eibe

23. Summary • OneR – uses rules based on just one attribute • Naïve Bayes – use all attributes and Bayes rules to estimate probability of the class given an instance. • Simple methods frequently work well, but … • Complex methods can be better (as we will see)

24. A. Mining Queries, in SQL 1. Naïve Bayesian classifier is one of the few mining methods expressible in SQL: via the count aggregate Probability of yes/no outcome can be evaluated independently for each column. Using a count aggregate Probability of yes/no outcome can be evaluated independently for each column. Using a count aggregate Select count(*) from TrainSetgroup by Outlook, Play Repeat this statement for each column—could be hundreds. Different queries for different number of columns—a generic query for all NBC would be preferable.

25. Dec The input table is verticalized into B… e.g. use a DB2 a table function A Training: various count on B produces this NBC: B C After training: Prediction on tuples just like Cabove (w.o. Dec). Joining C with the NBC extracts counts and that allow us to estimate probabilities that that are then multiplied.

26. BNC in SQL • Verticalization: for better simplicity and genericity • A Table functions can help. • Simple counts can produce the classifier • If input is also verticalized, the NBC table and the test tuples will have a similar form (but no ALL column in test tuples) • joins followed by multiplications and comparisons will produce the correct decision. • Do we have a multiplication aggregate in SQL? • Any other solution?

27. Classification: Decision Trees

28. Outline • Top-Down Decision Tree Construction • Choosing the Splitting Attribute • Information Gain and Gain Ratio

29. DECISION TREE • An internal node is a test on an attribute. • A branch represents an outcome of the test, e.g., Color=red. • A leaf node represents a class label or class label distribution. • At each node, one attribute is chosen to split training examples into distinct classes as much as possible • A new case is classified by following a matching path to a leaf node.

30. Weather Data: Play or not Play? Note: Outlook is the Forecast, no relation to Microsoft email program

31. Example Tree for “Play?” Outlook sunny rain overcast Yes Humidity Windy high normal false true No Yes No Yes

32. Building Decision Tree [Q93] • Top-down tree construction • At start, all training examples are at the root. • Partition the examples recursively by choosing one attribute each time. • Bottom-up tree pruning • Remove subtrees or branches, in a bottom-up manner, to improve the estimated accuracy on new cases.

33. Choosing the Splitting Attribute • At each node, available attributes are evaluated on the basis of separating the classes of the training examples. A Goodness function is used for this purpose. • Typical goodness functions: • information gain (ID3/C4.5) • information gain ratio • gini index witten&eibe

34. Which attribute to select? witten&eibe

35. A criterion for attribute selection • Which is the best attribute? • The one which will result in the smallest tree • Heuristic: choose the attribute that produces the “purest” nodes Popular impurity (disuniformity) criteria: • Gini Index • Information gain • Strategy: choose attribute that results in greatest information gain witten&eibe

36. *CART Splitting Criteria: Gini Index • If a data set T contains examples from n classes, gini index, gini(T) is defined as: where pj is the relative frequency of class j in T. gini(T) is minimized if the classes in T are skewed.

37. *Gini Index After splitting T into two subsets T1 and T2 with sizes N1 and N2, N= N1+N2, the gini index of the split data is defined as: ginisplit (T)= N1/N gini(T1) + N2/N gini(T2) • The attribute providing smallest ginisplit(T)is chosen to split the node.

38. Information Gain or Entropy • Information gain increases with the average purity of the subsets that an attribute produces • Information is measured in bits • Given a probability distribution, the info required to predict an event is the distribution’s entropy • Entropy gives the information required in bits (this can involve fractions of bits!) • Formula for computing the entropy: witten&eibe

39. Information Gain or Entropy • Assume that we have a set of tuples which belongs to class 1 or 2 with respective probability p1 and p2. • The additional information we provide when we say that a specific tuple belongs to one of these two classes is: • When the classes become unbalanced and either p1 or p2 become zero then no additional information is needed. witten&eibe

40. *Claude Shannon “Father of information theory” Born: 30 April 1916 Died: 23 February 2001 Claude Shannon, who has died aged 84, perhaps more than anyone laid the groundwork for today’s digital revolution. His exposition of information theory, stating that all information could be represented mathematically as a succession of noughts and ones, facilitated the digital manipulation of data without which today’s information society would be unthinkable. Shannon’s master’s thesis, obtained in 1940 at MIT, demonstrated that problem solving could be achieved by manipulating the symbols 0 and 1 in a process that could be carried out automatically with electrical circuitry. That dissertation has been hailed as one of the most significant master’s theses of the 20th century. Eight years later, Shannon published another landmark paper, A Mathematical Theory of Communication, generally taken as his most important scientific contribution. Shannon applied the same radical approach to cryptography research, in which he later became a consultant to the US government. Many of Shannon’s pioneering insights were developed before they could be applied in practical form. He was truly a remarkable man, yet unknown to most of the world. witten&eibe

41. Example: attribute “Outlook”, 1 witten&eibe

42. Example: attribute “Outlook”, 2 • “Outlook” = “Sunny”: • “Outlook” = “Overcast”: • “Outlook” = “Rainy”: • Expected information for attribute: Note: log(0) is not defined, but we evaluate 0*log(0) as zero witten&eibe

43. Computing the information gain • Information gain: (information before split) – (information after split) • Compute for attribute “Humidity” witten&eibe

44. Example: attribute “Humidity” • “Humidity” = “High”: • “Humidity” = “Normal”: • Expected information for attribute: • Information Gain:

45. Computing the information gain • Information gain: (information before split) – (information after split) • Information gain for attributes from weather data: witten&eibe

46. Continuing to split witten&eibe

47. The final decision tree • Note: not all leaves need to be pure; sometimes identical instances have different classes  Splitting stops when data can’t be split any further witten&eibe

48. Highly-branching attributes • Problematic: attributes with a large number of values (extreme case: ID code) • Subsets are more likely to be pure if there is a large number of values • Information gain is biased towards choosing attributes with a large number of values • This may result in overfitting (selection of an attribute that is non-optimal for prediction) witten&eibe

49. Weather Data with ID code

50. Split for ID Code Attribute Entropy of split = 0 (since each leaf node is “pure”, having only one case. Information gain is maximal for ID code witten&eibe