1. The Major Data Mining Tasks Classification Clustering Associations Most of the other tasks (for example, outlier discovery or anomaly detection ) make heavy use of one or more of the above. So in this tutorial we will focus most of our energy on the above, starting with…

2. Grasshoppers Katydids The Classification Problem (informal definition) Given a collection of annotated data. In this case 5 instances Katydids of and five of Grasshoppers, decide what type of insect the unlabeled example is. Katydid or Grasshopper?

3. Thorax Length AbdomenLength AntennaeLength MandibleSize Spiracle Diameter Leg Length For any domain of interest, we can measure features Color {Green, Brown, Gray, Other} Has Wings?

4. Insect ID AbdomenLengthAntennaeLength Insect Class 12.75.5Grasshopper 28.09.1Katydid 30.94.7Grasshopper 41.13.1Grasshopper 55.48.5Katydid 62.91.9Grasshopper 76.16.6Katydid 80.51.0Grasshopper 98.36.6Katydid 108.14.7Katydids 115.17.0??????? We can store features in a database. My_Collection The classification problem can now be expressed as: Given a training database (My_Collection), predict the class label of a previously unseen instance Given a training database (My_Collection), predict the class label of a previously unseen instance previously unseen instance =

5. Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Grasshoppers Katydids Abdomen Length

6. Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Grasshoppers Katydids Abdomen Length We will also use this lager dataset as a motivating example… Each of these data objects are called… exemplars (training) examples instances tuples

7. We will return to the previous slide in two minutes. In the meantime, we are going to play a quick game. I am going to show you some classification problems which were shown to pigeons! Let us see if you are as smart as a pigeon! We will return to the previous slide in two minutes. In the meantime, we are going to play a quick game. I am going to show you some classification problems which were shown to pigeons! Let us see if you are as smart as a pigeon!

8. Examples of class A 3 4 1.5 5 6 8 2.5 5 Examples of class B 5 2.5 5 2 8 3 4.5 3 Pigeon Problem 1

9. Examples of class A 3 4 1.5 5 6 8 2.5 5 Examples of class B 5 2.5 5 2 8 3 4.5 3 8 1.5 4.5 7 What class is this object? What about this one, A or B? Pigeon Problem 1

10. Examples of class A 3 4 1.5 5 6 8 2.5 5 Examples of class B 5 2.5 5 2 8 3 4.5 3 8 1.5 This is a B! Pigeon Problem 1 Here is the rule. If the left bar is smaller than the right bar, it is an A, otherwise it is a B. Here is the rule. If the left bar is smaller than the right bar, it is an A, otherwise it is a B.

11. Examples of class A 4 5 6 3 Examples of class B 5 2.5 2 5 5 3 2.5 3 8 1.5 7 Even I know this one Pigeon Problem 2 Oh! This ones hard!

12. Examples of class A 4 5 6 3 Examples of class B 5 2.5 2 5 5 3 2.5 3 7 Pigeon Problem 2 So this one is an A. The rule is as follows, if the two bars are equal sizes, it is an A. Otherwise it is a B.

13. Examples of class A 4 1 5 6 3 3 7 Examples of class B 5 6 7 5 4 8 7 6 Pigeon Problem 3 This one is really hard! What is this, A or B? This one is really hard! What is this, A or B?

14. Examples of class A 4 1 5 6 3 3 7 Examples of class B 5 6 7 5 4 8 7 6 Pigeon Problem 3 It is a B! The rule is as follows, if the square of the sum of the two bars is less than or equal to 100, it is an A. Otherwise it is a B.

15. Why did we spend so much time with this game? Because we wanted to show that almost all classification problems have a geometric interpretation, check out the next 3 slides…

16. Examples of class A 3 4 1.5 5 6 8 2.5 5 Examples of class B 5 2.5 5 2 8 3 4.5 3 Pigeon Problem 1 Here is the rule again. If the left bar is smaller than the right bar, it is an A, otherwise it is a B. Here is the rule again. If the left bar is smaller than the right bar, it is an A, otherwise it is a B. Left Bar 10 12345678 9 1 2 3 4 5 6 7 8 9 Right Bar

17. Examples of class A 4 5 6 3 Examples of class B 5 2.5 2 5 5 3 2.5 3 Pigeon Problem 2 Left Bar 10 12345678 9 1 2 3 4 5 6 7 8 9 Right Bar Let me look it up… here it is.. the rule is, if the two bars are equal sizes, it is an A. Otherwise it is a B.

18. Examples of class A 4 1 5 6 3 3 7 Examples of class B 5 6 7 5 4 8 7 Pigeon Problem 3 Left Bar 100 1020304050607080 90 100 10 20 30 40 50 60 70 80 90 Right Bar The rule again: if the square of the sum of the two bars is less than or equal to 100, it is an A. Otherwise it is a B. The rule again: if the square of the sum of the two bars is less than or equal to 100, it is an A. Otherwise it is a B.

19. Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Grasshoppers Katydids Abdomen Length

20. Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Abdomen Length Katydids Grasshoppers previously unseen instance We can “project” the previously unseen instance into the same space as the database. We have now abstracted away the details of our particular problem. It will be much easier to talk about points in space. previously unseen instance We can “project” the previously unseen instance into the same space as the database. We have now abstracted away the details of our particular problem. It will be much easier to talk about points in space. 115.17.0??????? previously unseen instance =

21. Simple Linear Classifier previously unseen instance If previously unseen instance above the line then class is Katydid else class is Grasshopper Katydids Grasshoppers R.A. Fisher 1890-1962 10 123456789 1 2 3 4 5 6 7 8 9

22. The simple linear classifier is defined for higher dimensional spaces…

23. … we can visualize it as being an n-dimensional hyperplane

24. It is interesting to think about what would happen in this example if we did not have the 3 rd dimension…

25. We can no longer get perfect accuracy with the simple linear classifier… We could try to solve this problem by user a simple quadratic classifier or a simple cubic classifier.. However, as we will later see, this is probably a bad idea…

26. 10 12345678 9 1 2 3 4 5 6 7 8 9 100 1020304050607080 90 100 10 20 30 40 50 60 70 80 90 10 12345678 9 1 2 3 4 5 6 7 8 9 Which of the “Pigeon Problems” can be solved by the Simple Linear Classifier? 1)Perfect 2)Useless 3)Pretty Good Problems that can be solved by a linear classifier are call linearly separable.

27. A Famous Problem R. A. Fisher’s Iris Dataset. 3 classes 50 of each class The task is to classify Iris plants into one of 3 varieties using the Petal Length and Petal Width. Iris SetosaIris VersicolorIris Virginica Setosa Versicolor Virginica

28. Setosa Versicolor Virginica We can generalize the piecewise linear classifier to N classes, by fitting N-1 lines. In this case we first learned the line to (perfectly) discriminate between Setosa and Virginica/Versicolor, then we learned to approximately discriminate between Virginica and Versicolor. If petal width > 3.272 – (0.325 * petal length) then class = Virginica Elseif petal width…

29. Predictive accuracy Speed and scalability –time to construct the model –time to use the model –efficiency in disk-resident databases Robustness –handling noise, missing values and irrelevant features, streaming data Interpretability: –understanding and insight provided by the model We have now seen one classification algorithm, and we are about to see more. How should we compare them ?

30. Predictive Accuracy I How do we estimate the accuracy of our classifier? We can use K-fold cross validation Insect ID AbdomenLengthAntennaeLength Insect Class 12.75.5Grasshopper 28.09.1Katydid 30.94.7Grasshopper 41.13.1Grasshopper 55.48.5Katydid 62.91.9Grasshopper 76.16.6Katydid 80.51.0Grasshopper 98.36.6Katydid 108.14.7Katydids We divide the dataset into K equal sized sections. The algorithm is tested K times, each time leaving out one of the K section from building the classifier, but using it to test the classifier instead Accuracy = Number of correct classifications Number of instances in our database K = 5

31. Predictive Accuracy II Using K-fold cross validation is a good way to set any parameters we may need to adjust in (any) classifier. We can do K-fold cross validation for each possible setting, and choose the model with the highest accuracy. Where there is a tie, we choose the simpler model. Actually, we should probably penalize the more complex models, even if they are more accurate, since more complex models are more likely to overfit (discussed later). 10 123456789 1 2 3 4 5 6 7 8 9 123456789 1 2 3 4 5 6 7 8 9 123456789 1 2 3 4 5 6 7 8 9 Accuracy = 94%Accuracy = 100%

32. Predictive Accuracy III Accuracy = Number of correct classifications Number of instances in our database Accuracy is a single number, we may be better off looking at a confusion matrix. This gives us additional useful information… CatDogPig Cat10000 Dog9901 Pig45 10 Classified as a… True label is...

33. We need to consider the time and space requirements for the two distinct phases of classification : Time to construct the classifier In the case of the simpler linear classifier, the time taken to fit the line, this is linear in the number of instances. Time to use the model In the case of the simpler linear classifier, the time taken to test which side of the line the unlabeled instance is. This can be done in constant time. Speed and Scalability I As we shall see, some classification algorithms are very efficient in one aspect, and very poor in the other.

34. Speed and Scalability II For learning with small datasets, this is the whole picture However, for data mining with massive datasets, it is not so much the (main memory) time complexity that matters, rather it is how many times we have to scan the database. This is because for most data mining operations, disk access times completely dominate the CPU times. For data mining, researchers often report the number of times you must scan the database.

35. Robustness I We need to consider what happens when we have: Noise For example, a persons age could have been mistyped as 650 instead of 65, how does this effect our classifier? (This is important only for building the classifier, if the instance to be classified is noisy we can do nothing). Missing values 10 123456789 1 2 3 4 5 6 7 8 9 123456789 1 2 3 4 5 6 7 8 9 123456789 1 2 3 4 5 6 7 8 9 For example suppose we want to classify an insect, but we only know the abdomen length (X-axis), and not the antennae length (Y-axis), can we still classify the instance?

36. Robustness II We need to consider what happens when we have: Irrelevant features For example, suppose we want to classify people as either Suitable_Grad_Student Unsuitable_Grad_Student And it happens that scoring more than 5 on a particular test is a perfect indicator for this problem… 10 If we also use “hair_length” as a feature, how will this effect our classifier?

37. Robustness III We need to consider what happens when we have: Streaming data For many real world problems, we don’t have a single fixed dataset. Instead, the data continuously arrives, potentially forever… (stock market, weather data, sensor data etc) 10 Can our classifier handle streaming data?

38. Interpretability Some classifiers offer a bonus feature. The structure of the learned classifier tells use something about the domain. Height Weight As a trivial example, if we try to classify peoples health risks based on just their height and weight, we could gain the following insight (Based of the observation that a single linear classifier does not work well, but two linear classifiers do). There are two ways to be unhealthy, being obese and being too skinny.

39. Nearest Neighbor Classifier previously unseen instance If the nearest instance to the previously unseen instance is a Katydid class is Katydid else class is Grasshopper Katydids Grasshoppers Joe Hodges 1922-2000 Evelyn Fix 1904-1965 Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Abdomen Length

40. This division of space is called Dirichlet Tessellation (or Voronoi diagram, or Theissen regions). We can visualize the nearest neighbor algorithm in terms of a decision surface… Note the we don’t actually have to construct these surfaces, they are simply the implicit boundaries that divide the space into regions “belonging” to each instance.

41. The nearest neighbor algorithm is sensitive to outliers… The solution is to…

42. We can generalize the nearest neighbor algorithm to the K- nearest neighbor (KNN) algorithm. We measure the distance to the nearest K instances, and let them vote. K is typically chosen to be an odd number. K = 1K = 3

43. 10 123456789 Suppose the following is true, if an insects antenna is longer than 5.5 it is a Katydid, otherwise it is a Grasshopper. Using just the antenna length we get perfect classification! Suppose the following is true, if an insects antenna is longer than 5.5 it is a Katydid, otherwise it is a Grasshopper. Using just the antenna length we get perfect classification! The nearest neighbor algorithm is sensitive to irrelevant features… Training data 12345678910123456789 6 5 Suppose however, we add in an irrelevant feature, for example the insects mass. Using both the antenna length and the insects mass with the 1-NN algorithm we get the wrong classification! Suppose however, we add in an irrelevant feature, for example the insects mass. Using both the antenna length and the insects mass with the 1-NN algorithm we get the wrong classification!

44. How do we mitigate the nearest neighbor algorithms sensitivity to irrelevant features? Use more training instances Ask an expert what features are relevant to the task Use statistical tests to try to determine which features are useful Search over feature subsets (in the next slide we will see why this is hard)

45. Why searching over feature subsets is hard Suppose you have the following classification problem, with 100 features, where is happens that Features 1 and 2 (the X and Y below) give perfect classification, but all 98 of the other features are irrelevant… Using all 100 features will give poor results, but so will using only Feature 1, and so will using Feature 2! Of the 2 100 –1 possible subsets of the features, only one really works. Only Feature 1 Only Feature 2

46. 1234 3,42,41,42,31,31,2 2,3,41,3,41,2,41,2,3 1,2,3,4 Forward Selection Backward Elimination Bi-directional Search

47. The nearest neighbor algorithm is sensitive to the units of measurement X axis measured in centimeters Y axis measure in dollars The nearest neighbor to the pink unknown instance is red. X axis measured in millimeters Y axis measure in dollars The nearest neighbor to the pink unknown instance is blue. One solution is to normalize the units to pure numbers. Typically the features are Z-normalized to have a mean of zero and a standard deviation of one. X = (X – mean(X))/std(x)

48. We can speed up nearest neighbor algorithm by “throwing away” some data. This is called data editing. Note that this can sometimes improve accuracy! One possible approach. Delete all instances that are surrounded by members of their own class. We can also speed up classification with indexing

49. 10 123456789 1 2 3 4 5 6 7 8 9 Manhattan (p=1) Max (p=inf) Mahalanobis Weighted Euclidean Up to now we have assumed that the nearest neighbor algorithm uses the Euclidean Distance, however this need not be the case…

50. …In fact, we can use the nearest neighbor algorithm with any distance/similarity function IDNameClass 1GunopulosGreek 2PapadopoulosGreek 3KolliosGreek 4DardanosGreek 5 KeoghIrish 6GoughIrish 7GreenhaughIrish 8HadleighIrish For example, is “Faloutsos” Greek or Irish? We could compare the name “Faloutsos” to a database of names using string edit distance… edit_distance(Faloutsos, Keogh) = 8 edit_distance(Faloutsos, Gunopulos) = 6 Hopefully, the similarity of the name (particularly the suffix) to other Greek names would mean the nearest nearest neighbor is also a Greek name. Specialized distance measures exist for DNA strings, time series, images, graphs, videos, sets, fingerprints etc…

51. Advantages: –Simple to implement –Handles correlated features (Arbitrary class shapes) –Defined for any distance measure –Handles streaming data trivially Disadvantages: –Very sensitive to irrelevant features. –Slow classification time for large datasets –Works best for real valued datasets Advantages/Disadvantages of Nearest Neighbor

52. Decision Tree Classifier Ross Quinlan Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Abdomen Length Abdomen Length Abdomen Length > 7.1? no yes Katydid Antenna Length Antenna Length > 6.0? no yes Katydid Grasshopper

53. Antennae shorter than body? Cricket Foretiba has ears? KatydidsCamel Cricket Yes No 3 Tarsi? No Decision trees predate computers

54. Decision tree –A flow-chart-like tree structure –Internal node denotes a test on an attribute –Branch represents an outcome of the test –Leaf nodes represent class labels or class distribution Decision tree generation consists of two phases –Tree construction At start, all the training examples are at the root Partition examples recursively based on selected attributes –Tree pruning Identify and remove branches that reflect noise or outliers Use of decision tree: Classifying an unknown sample –Test the attribute values of the sample against the decision tree Decision Tree Classification

55. Basic algorithm (a greedy algorithm) –Tree is constructed in a top-down recursive divide-and-conquer manner –At start, all the training examples are at the root –Attributes are categorical (if continuous-valued, they can be discretized in advance) –Examples are partitioned recursively based on selected attributes. –Test attributes are selected on the basis of a heuristic or statistical measure (e.g., information gain) Conditions for stopping partitioning –All samples for a given node belong to the same class –There are no remaining attributes for further partitioning – majority voting is employed for classifying the leaf –There are no samples left How do we construct the decision tree?

56. Information Gain as A Splitting Criteria Select the attribute with the highest information gain ( information gain is the expected reduction in entropy ). Assume there are two classes, P and N –Let the set of examples S contain p elements of class P and n elements of class N –The amount of information, needed to decide if an arbitrary example in S belongs to P or N is defined as 0 log(0) is defined as 0

57. Information Gain in Decision Tree Induction Assume that using attribute A, a current set will be partitioned into some number of child sets The encoding information that would be gained by branching on A Note: entropy is at its minimum if the collection of objects is completely uniform

58. Person Hair Length WeightAgeClass Homer0”25036M Marge10”15034F Bart2”9010M Lisa6”788F Maggie4”201F Abe1”17070M Selma8”16041F Otto10”18038M Krusty6”20045M Comic8”29038?

59. Hair Length <= 5? yes no Entropy(4F,5M) = -(4/9)log 2 (4/9) - (5/9)log 2 (5/9) = 0.9911 Entropy(1F,3M) = -(1/4)log 2 (1/4) - (3/4)log 2 (3/4) = 0.8113 Entropy(3F,2M) = -(3/5)log 2 (3/5) - (2/5)log 2 (2/5) = 0.9710 Gain(Hair Length <= 5) = 0.9911 – (4/9 * 0.8113 + 5/9 * 0.9710 ) = 0.0911 Let us try splitting on Hair length

60. Weight <= 160? yes no Entropy(4F,5M) = -(4/9)log 2 (4/9) - (5/9)log 2 (5/9) = 0.9911 Entropy(4F,1M) = -(4/5)log 2 (4/5) - (1/5)log 2 (1/5) = 0.7219 Entropy(0F,4M) = -(0/4)log 2 (0/4) - (4/4)log 2 (4/4) = 0 Gain(Weight <= 160) = 0.9911 – (5/9 * 0.7219 + 4/9 * 0 ) = 0.5900 Let us try splitting on Weight

61. age <= 40? yes no Entropy(4F,5M) = -(4/9)log 2 (4/9) - (5/9)log 2 (5/9) = 0.9911 Entropy(3F,3M) = -(3/6)log 2 (3/6) - (3/6)log 2 (3/6) = 1 Entropy(1F,2M) = -(1/3)log 2 (1/3) - (2/3)log 2 (2/3) = 0.9183 Gain(Age <= 40) = 0.9911 – (6/9 * 1 + 3/9 * 0.9183 ) = 0.0183 Let us try splitting on Age

62. Weight <= 160? yes no Hair Length <= 2? yes no Of the 3 features we had, Weight was best. But while people who weigh over 160 are perfectly classified (as males), the under 160 people are not perfectly classified… So we simply recurse! This time we find that we can split on Hair length, and we are done!

63. Weight <= 160? yesno Hair Length <= 2? yes no We need don’t need to keep the data around, just the test conditions. Male Female How would these people be classified?

64. It is trivial to convert Decision Trees to rules… Weight <= 160? yesno Hair Length <= 2? yes no Male Female Rules to Classify Males/Females If Weight greater than 160, classify as Male Elseif Hair Length less than or equal to 2, classify as Male Else classify as Female Rules to Classify Males/Females If Weight greater than 160, classify as Male Elseif Hair Length less than or equal to 2, classify as Male Else classify as Female

65. Decision tree for a typical shared-care setting applying the system for the diagnosis of prostatic obstructions. Once we have learned the decision tree, we don’t even need a computer! This decision tree is attached to a medical machine, and is designed to help nurses make decisions about what type of doctor to call.

66. Wears green? Yes No The worked examples we have seen were performed on small datasets. However with small datasets there is a great danger of overfitting the data… When you have few datapoints, there are many possible splitting rules that perfectly classify the data, but will not generalize to future datasets. For example, the rule “Wears green?” perfectly classifies the data, so does “Mothers name is Jacqueline?”, so does “Has blue shoes”… Male Female

67. Avoid Overfitting in Classification The generated tree may overfit the training data –Too many branches, some may reflect anomalies due to noise or outliers –Result is in poor accuracy for unseen samples Two approaches to avoid overfitting –Prepruning: Halt tree construction early—do not split a node if this would result in the goodness measure falling below a threshold Difficult to choose an appropriate threshold –Postpruning: Remove branches from a “fully grown” tree—get a sequence of progressively pruned trees Use a set of data different from the training data to decide which is the “best pruned tree”

68. 10 12345678 9 1 2 3 4 5 6 7 8 9 12345678 9 1 2 3 4 5 6 7 8 9 Which of the “Pigeon Problems” can be solved by a Decision Tree? 1)Deep Bushy Tree 2)Useless 3)Deep Bushy Tree The Decision Tree has a hard time with correlated attributes ?

69. Advantages: –Easy to understand (Doctors love them!) –Easy to generate rules Disadvantages: –May suffer from overfitting. –Classifies by rectangular partitioning (so does not handle correlated features very well). –Can be quite large – pruning is necessary. –Does not handle streaming data easily Advantages/Disadvantages of Decision Trees

70. Naïve Bayes Classifier We will start off with a visual intuition, before looking at the math… Thomas Bayes 1702 - 1761

71. Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Grasshoppers Katydids Abdomen Length Remember this example? Let’s get lots more data…

72. Antenna Length 10 123456789 1 2 3 4 5 6 7 8 9 Katydids Grasshoppers With a lot of data, we can build a histogram. Let us just build one for “Antenna Length” for now…

73. We can leave the histograms as they are, or we can summarize them with two normal distributions. Let us us two normal distributions for ease of visualization in the following slides…

74. p(c j | d) = probability of class c j, given that we have observed d 3 Antennae length is 3 We want to classify an insect we have found. Its antennae are 3 units long. How can we classify it? We can just ask ourselves, give the distributions of antennae lengths we have seen, is it more probable that our insect is a Grasshopper or a Katydid. There is a formal way to discuss the most probable classification…

75. 10 2 P( Grasshopper | 3 ) = 10 / (10 + 2)= 0.833 P( Katydid | 3 ) = 2 / (10 + 2)= 0.166 3 Antennae length is 3 p(c j | d) = probability of class c j, given that we have observed d

76. 9 3 P( Grasshopper | 7 ) = 3 / (3 + 9)= 0.250 P( Katydid | 7 ) = 9 / (3 + 9)= 0.750 7 Antennae length is 7 p(c j | d) = probability of class c j, given that we have observed d

77. 6 6 P( Grasshopper | 5 ) = 6 / (6 + 6)= 0.500 P( Katydid | 5 ) = 6 / (6 + 6)= 0.500 5 Antennae length is 5 p(c j | d) = probability of class c j, given that we have observed d

78. Bayes Classifiers That was a visual intuition for a simple case of the Bayes classifier, also called: Idiot Bayes Naïve Bayes Simple Bayes We are about to see some of the mathematical formalisms, and more examples, but keep in mind the basic idea. previously unseen instance Find out the probability of the previously unseen instance belonging to each class, then simply pick the most probable class.

79. Bayes Classifiers Bayesian classifiers use Bayes theorem, which says p(c j | d ) = p(d | c j ) p(c j ) p(d) p(c j | d) = probability of instance d being in class c j, This is what we are trying to compute p(d | c j ) = probability of generating instance d given class c j, We can imagine that being in class c j, causes you to have feature d with some probability p(c j ) = probability of occurrence of class c j, This is just how frequent the class c j, is in our database p(d) = probability of instance d occurring This can actually be ignored, since it is the same for all classes

80. Assume that we have two classes malefemale c 1 = male, and c 2 = female. We have a person whose sex we do not know, say “drew” or d. malefemale malefemale Classifying drew as male or female is equivalent to asking is it more probable that drew is male or female, I.e which is greater p(male | drew) or p(female | drew) malemalemale p(male | drew) = p(drew | male ) p(male) p(drew) (Note: “Drew can be a male or female name”) What is the probability of being called “drew” given that you are a male? What is the probability of being a male? What is the probability of being named “drew”? (actually irrelevant, since it is that same for all classes) Drew Carey Drew Barrymore

81. p(c j | d) = p(d | c j ) p(c j ) p(d) Officer Drew NameSex DrewMale ClaudiaFemale DrewFemale Female AlbertoMale KarinFemale NinaFemale SergioMale This is Officer Drew (who arrested me in 1997). Is Officer Drew a Male or Female? Luckily, we have a small database with names and sex. We can use it to apply Bayes rule…

82. male p(male | drew) = 1/3 * 3/8 = 0.125 3/8 3/8 female p(female | drew) = 2/5 * 5/8 = 0.250 3/8 3/8 Officer Drew p(c j | d) = p(d | c j ) p(c j ) p(d) NameSex DrewMale ClaudiaFemale DrewFemale Female AlbertoMale KarinFemale NinaFemale SergioMale Female Officer Drew is more likely to be a Female.

83. Officer Drew IS a female! Officer Drew male p(male | drew) = 1/3 * 3/8 = 0.125 3/8 3/8 female p(female | drew) = 2/5 * 5/8 = 0.250 3/8 3/8

84. NameOver 170 CM EyeHair lengthSex DrewNoBlueShortMale ClaudiaYesBrownLongFemale DrewNoBlueLongFemale DrewNoBlueLongFemale AlbertoYesBrownShortMale KarinNoBlueLongFemale NinaYesBrownShortFemale SergioYesBlueLongMale p(c j | d) = p(d | c j ) p(c j ) p(d) So far we have only considered Bayes Classification when we have one attribute (the “antennae length”, or the “name”). But we may have many features. How do we use all the features?

85. To simplify the task, naïve Bayesian classifiers assume attributes have independent distributions, and thereby estimate p(d|c j ) = p(d 1 |c j ) * p(d 2 |c j ) * ….* p(d n |c j ) The probability of class c j generating instance d, equals…. The probability of class c j generating the observed value for feature 1, multiplied by.. The probability of class c j generating the observed value for feature 2, multiplied by..

86. To simplify the task, naïve Bayesian classifiers assume attributes have independent distributions, and thereby estimate p(d|c j ) = p(d 1 |c j ) * p(d 2 |c j ) * ….* p(d n |c j ) p( officer drew |c j ) = p(over_170 cm = yes|c j ) * p(eye =blue|c j ) * …. Officer Drew is blue-eyed, over 170 cm tall, and has long hair Female p( officer drew | Female) = 2/5 * 3/5 * …. Male p( officer drew | Male) = 2/3 * 2/3 * ….

87. p(d 1 |c j ) p(d 2 |c j ) p(d n |c j ) cjcj The Naive Bayes classifiers is often represented as this type of graph… Note the direction of the arrows, which state that each class causes certain features, with a certain probability …

88. Naïve Bayes is fast and space efficient We can look up all the probabilities with a single scan of the database and store them in a (small) table… SexOver190 cmMaleYes0.15 No0.85 FemaleYes0.01 No0.99 cjcj … p(d 1 |c j ) p(d 2 |c j ) p(d n |c j ) SexLong HairMaleYes0.05 No0.95 FemaleYes0.70 No0.30 SexMale Female

89. Naïve Bayes is NOT sensitive to irrelevant features... Suppose we are trying to classify a persons sex based on several features, including eye color. (Of course, eye color is completely irrelevant to a persons gender) Female p( Jessica | Female) = 9,000/10,000 * 9,975/10,000 * …. Male p( Jessica | Male) = 9,001/10,000 * 2/10,000 * …. p( Jessica |c j ) = p(eye = brown|c j ) * p( wears_dress = yes|c j ) * …. However, this assumes that we have good enough estimates of the probabilities, so the more data the better. Almost the same!

90. An obvious point. I have used a simple two class problem, and two possible values for each example, for my previous examples. However we can have an arbitrary number of classes, or feature values AnimalMass >10 kgCatYes0.15 No0.85 DogYes0.91 No0.09 PigYes0.99 No0.01 cjcj … p(d 1 |c j ) p(d 2 |c j ) p(d n |c j ) AnimalCat Dog Pig ColorCatBlack0.33 White0.23 Brown0.44 DogBlack0.97 White0.03 Brown0.90 PigBlack0.04 White0.01 Brown0.95

91. Naïve Bayesian Classifier p(d 1 |c j ) p(d 2 |c j ) p(d n |c j ) p(d|cj)p(d|cj) Problem! Naïve Bayes assumes independence of features… SexOver 6 foot MaleYes0.15 No0.85 FemaleYes0.01 No0.99 SexOver 200 pounds MaleYes0.11 No0.80 FemaleYes0.05 No0.95

92. Naïve Bayesian Classifier p(d 1 |c j ) p(d 2 |c j ) p(d n |c j ) p(d|cj)p(d|cj) Solution Consider the relationships between attributes… SexOver 6 foot MaleYes0.15 No0.85 FemaleYes0.01 No0.99 SexOver 200 pounds MaleYes and Over 6 foot0.11 No and Over 6 foot0.59 Yes and NOT Over 6 foot0.05 No and NOT Over 6 foot0.35 FemaleYes and Over 6 foot0.01

93. Naïve Bayesian Classifier p(d 1 |c j ) p(d 2 |c j ) p(d n |c j ) p(d|cj)p(d|cj) Solution Consider the relationships between attributes… But how do we find the set of connecting arcs??

94. 10 123456789 1 2 3 4 5 6 7 8 9 The Naïve Bayesian Classifier has a quadratic decision boundary

95. Dear SIR, I am Mr. John Coleman and my sister is Miss Rose Colemen, we are the children of late Chief Paul Colemen from Sierra Leone. I am writing you in absolute confidence primarily to seek your assistance to transfer our cash of twenty one Million Dollars ($21,000.000.00) now in the custody of a private Security trust firm in Europe the money is in trunk boxes deposited and declared as family valuables by my late father as a matter of fact the company does not know the content as money, although my father made them to under stand that the boxes belongs to his foreign partner. …

96. This mail is probably spam. The original message has been attached along with this report, so you can recognize or block similar unwanted mail in future. See http://spamassassin.org/tag/ for more details. Content analysis details: (12.20 points, 5 required) NIGERIAN_SUBJECT2 (1.4 points) Subject is indicative of a Nigerian spam FROM_ENDS_IN_NUMS (0.7 points) From: ends in numbers MIME_BOUND_MANY_HEX (2.9 points) Spam tool pattern in MIME boundary URGENT_BIZ (2.7 points) BODY: Contains urgent matter US_DOLLARS_3 (1.5 points) BODY: Nigerian scam key phrase ($NN,NNN,NNN.NN) DEAR_SOMETHING (1.8 points) BODY: Contains 'Dear (something)' BAYES_30 (1.6 points) BODY: Bayesian classifier says spam probability is 30 to 40% [score: 0.3728]

97. Advantages: –Fast to train (single scan). Fast to classify –Not sensitive to irrelevant features –Handles real and discrete data –Handles streaming data well Disadvantages: –Assumes independence of features Advantages/Disadvantages of Naïve Bayes

98. Summary of Classification We have seen 4 major classification techniques: Simple linear classifier, Nearest neighbor, Decision tree. There are other techniques: Neural Networks, Support Vector Machines, Genetic algorithms.. In general, there is no one best classifier for all problems. You have to consider what you hope to achieve, and the data itself… Let us now move on to the other classic problem of data mining and machine learning, Clustering…

99. Organizing data into classes such that there is high intra-class similarity low inter-class similarity Finding the class labels and the number of classes directly from the data (in contrast to classification). More informally, finding natural groupings among objects. What is Clustering? Also called unsupervised learning, sometimes called classification by statisticians and sorting by psychologists and segmentation by people in marketing

100. What is a natural grouping among these objects?

101. School Employees Simpson's Family MalesFemales Clustering is subjective What is a natural grouping among these objects?

102. What is Similarity? The quality or state of being similar; likeness; resemblance; as, a similarity of features. Similarity is hard to define, but… “We know it when we see it” The real meaning of similarity is a philosophical question. We will take a more pragmatic approach. Webster's Dictionary

103. Defining Distance Measures Definition: Let O 1 and O 2 be two objects from the universe of possible objects. The distance (dissimilarity) between O 1 and O 2 is a real number denoted by D(O 1,O 2 ) 0.233342.7 PeterPiotr

104. What properties should a distance measure have? D(A,B) = D(B,A) Symmetry D(A,A) = 0 Constancy of Self-Similarity D(A,B) = 0 IIf A= B Positivity (Separation) D(A,B)  D(A,C) + D(B,C) Triangular Inequality PeterPiotr 3 d('', '') = 0 d(s, '') = d('', s) = |s| -- i.e. length of s d(s1+ch1, s2+ch2) = min( d(s1, s2) + if ch1=ch2 then 0 else 1 fi, d(s1+ch1, s2) + 1, d(s1, s2+ch2) + 1 ) When we peek inside one of these black boxes, we see some function on two variables. These functions might very simple or very complex. In either case it is natural to ask, what properties should these functions have?

105. Intuitions behind desirable distance measure properties D(A,B) = D(B,A) Symmetry Otherwise you could claim “Alex looks like Bob, but Bob looks nothing like Alex.” D(A,A) = 0 Constancy of Self-Similarity Otherwise you could claim “Alex looks more like Bob, than Bob does.” D(A,B) = 0 IIf A=B Positivity (Separation) Otherwise there are objects in your world that are different, but you cannot tell apart. D(A,B)  D(A,C) + D(B,C) Triangular Inequality Otherwise you could claim “Alex is very like Bob, and Alex is very like Carl, but Bob is very unlike Carl.”

106. Two Types of Clustering Hierarchical Partitional algorithms: Construct various partitions and then evaluate them by some criterion (we will see an example called BIRCH) Hierarchical algorithms: Create a hierarchical decomposition of the set of objects using some criterion Partitional

107. Desirable Properties of a Clustering Algorithm Scalability (in terms of both time and space) Ability to deal with different data types Minimal requirements for domain knowledge to determine input parameters Able to deal with noise and outliers Insensitive to order of input records Incorporation of user-specified constraints Interpretability and usability

108. A Useful Tool for Summarizing Similarity Measurements In order to better appreciate and evaluate the examples given in the early part of this talk, we will now introduce the dendrogram. The similarity between two objects in a dendrogram is represented as the height of the lowest internal node they share.

109. (Bovine:0.69395, (Spider Monkey 0.390, (Gibbon:0.36079,(Orang:0.33636,(Gorilla:0.17147,(Chimp:0.19268, Human:0.11927):0.08386):0.06124):0.15057):0.54939); There is only one dataset that can be perfectly clustered using a hierarchy…

110. Business & Economy B2BFinanceShoppingJobs Aerospace Agriculture… Banking Bonds…Animals ApparelCareer Workspace Note that hierarchies are commonly used to organize information, for example in a web portal. Yahoo’s hierarchy is manually created, we will focus on automatic creation of hierarchies in data mining.

111. Pedro (Portuguese) Petros (Greek), Peter (English), Piotr (Polish), Peadar (Irish), Pierre (French), Peder (Danish), Peka (Hawaiian), Pietro (Italian), Piero (Italian Alternative), Petr (Czech), Pyotr (Russian) Cristovao (Portuguese) Christoph (German), Christophe (French), Cristobal (Spanish), Cristoforo (Italian), Kristoffer (Scandinavian), Krystof (Czech), Christopher (English) Miguel (Portuguese) Michalis (Greek), Michael (English), Mick (Irish!) A Demonstration of Hierarchical Clustering using String Edit Distance Piotr Pyotr Petros Pietro Pedro Pierre Piero Peter Peder Peka Peadar Michalis Michael Miguel Mick Cristovao Christopher Christophe Christoph Crisdean Cristobal Cristoforo Kristoffer Krystof

112. Piotr Pyotr Petros Pietro Pedro Pierre Piero Peter Peder Peka Peadar Pedro (Portuguese/Spanish) Petros (Greek), Peter (English), Piotr (Polish), Peadar (Irish), Pierre (French), Peder (Danish), Peka (Hawaiian), Pietro (Italian), Piero (Italian Alternative), Petr (Czech), Pyotr (Russian)

113. ANGUILLAAUSTRALIA St. Helena & Dependencies South Georgia & South Sandwich Islands U.K. Serbia & Montenegro (Yugoslavia) FRANCENIGERINDIAIRELANDBRAZIL Hierarchal clustering can sometimes show patterns that are meaningless or spurious For example, in this clustering, the tight grouping of Australia, Anguilla, St. Helena etc is meaningful, since all these countries are former UK colonies. However the tight grouping of Niger and India is completely spurious, there is no connection between the two.

114. ANGUILLAAUSTRALIA St. Helena & Dependencies South Georgia & South Sandwich Islands U.K. Serbia & Montenegro (Yugoslavia) FRANCENIGERINDIAIRELANDBRAZIL The flag of Niger is orange over white over green, with an orange disc on the central white stripe, symbolizing the sun. The orange stands the Sahara desert, which borders Niger to the north. Green stands for the grassy plains of the south and west and for the River Niger which sustains them. It also stands for fraternity and hope. White generally symbolizes purity and hope. The Indian flag is a horizontal tricolor in equal proportion of deep saffron on the top, white in the middle and dark green at the bottom. In the center of the white band, there is a wheel in navy blue to indicate the Dharma Chakra, the wheel of law in the Sarnath Lion Capital. This center symbol or the 'CHAKRA' is a symbol dating back to 2nd century BC. The saffron stands for courage and sacrifice; the white, for purity and truth; the green for growth and auspiciousness.

115. We can look at the dendrogram to determine the “correct” number of clusters. In this case, the two highly separated subtrees are highly suggestive of two clusters. (Things are rarely this clear cut, unfortunately)

116. Outlier One potential use of a dendrogram is to detect outliers The single isolated branch is suggestive of a data point that is very different to all others

117. (How-to) Hierarchical Clustering The number of dendrograms with n leafs = (2n -3)!/[(2 (n -2) ) (n -2)!] Number Number of Possible of LeafsDendrograms 213 415 5105...… 10 34,459,425 Since we cannot test all possible trees we will have to heuristic search of all possible trees. We could do this.. Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. Top-Down (divisive): Starting with all the data in a single cluster, consider every possible way to divide the cluster into two. Choose the best division and recursively operate on both sides.

118. 08877 0244 033 01 0 D(, ) = 8 D(, ) = 1 We begin with a distance matrix which contains the distances between every pair of objects in our database.

119. Bottom-Up (agglomerative): Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best

120. Bottom-Up (agglomerative): Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best Consider all possible merges… … Choose the best

121. Bottom-Up (agglomerative): Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best Consider all possible merges… … Choose the best Consider all possible merges… Choose the best …

122. Bottom-Up (agglomerative): Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best Consider all possible merges… … Choose the best Consider all possible merges… Choose the best …

123. We know how to measure the distance between two objects, but defining the distance between an object and a cluster, or defining the distance between two clusters is non obvious. Single linkage (nearest neighbor): Single linkage (nearest neighbor): In this method the distance between two clusters is determined by the distance of the two closest objects (nearest neighbors) in the different clusters. Complete linkage (furthest neighbor): Complete linkage (furthest neighbor): In this method, the distances between clusters are determined by the greatest distance between any two objects in the different clusters (i.e., by the "furthest neighbors"). Group average linkage Group average linkage : In this method, the distance between two clusters is calculated as the average distance between all pairs of objects in the two different clusters. Wards Linkage Wards Linkage : In this method, we try to minimize the variance of the merged clusters

124. Average linkageWards linkage Single linkage

125. Summary of Hierarchal Clustering Methods No need to specify the number of clusters in advance. Hierarchal nature maps nicely onto human intuition for some domains They do not scale well: time complexity of at least O(n 2 ), where n is the number of total objects. Like any heuristic search algorithms, local optima are a problem. Interpretation of results is (very) subjective.

126. Up to this point we have simply assumed that we can measure similarity, but How do we measure similarity? 0.233342.7 PeterPiotr

127. A generic technique for measuring similarity To measure the similarity between two objects, transform one of the objects into the other, and measure how much effort it took. The measure of effort becomes the distance measure. The distance between Patty and Selma. Change dress color, 1 point Change earring shape, 1 point Change hair part, 1 point D(Patty,Selma) = 3 The distance between Marge and Selma. Change dress color, 1 point Add earrings, 1 point Decrease height, 1 point Take up smoking, 1 point Lose weight, 1 point D(Marge,Selma) = 5 This is called the “edit distance” or the “transformation distance”

128. Peter Piter Pioter Piotr Substitution (i for e) Insertion (o) Deletion (e) Edit Distance Example It is possible to transform any string Q into string C, using only Substitution, Insertion and Deletion. Assume that each of these operators has a cost associated with it. The similarity between two strings can be defined as the cost of the cheapest transformation from Q to C. Note that for now we have ignored the issue of how we can find this cheapest transformation How similar are the names “Peter” and “Piotr”? Assume the following cost function Substitution1 Unit Insertion1 Unit Deletion1 Unit D( Peter,Piotr ) is 3 Piotr Pyotr Petros Pietro Pedro Pierre Piero Peter

129. Partitional Clustering Nonhierarchical, each instance is placed in exactly one of K nonoverlapping clusters. Since only one set of clusters is output, the user normally has to input the desired number of clusters K.

130. Squared Error 10 123456789 1 2 3 4 5 6 7 8 9 Objective Function

131. Algorithm k-means 1. Decide on a value for k. 2. Initialize the k cluster centers (randomly, if necessary). 3. Decide the class memberships of the N objects by assigning them to the nearest cluster center. 4. Re-estimate the k cluster centers, by assuming the memberships found above are correct. 5. If none of the N objects changed membership in the last iteration, exit. Otherwise goto 3.

132. 0 1 2 3 4 5 012345 K-means Clustering: Step 1 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3

133. 0 1 2 3 4 5 012345 K-means Clustering: Step 2 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3

134. 0 1 2 3 4 5 012345 K-means Clustering: Step 3 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3

135. 0 1 2 3 4 5 012345 K-means Clustering: Step 4 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3

136. K-means Clustering: Step 5 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3

137. Comments on the K-Means Method Strength –Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. –Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms Weakness –Applicable only when mean is defined, then what about categorical data? –Need to specify k, the number of clusters, in advance –Unable to handle noisy data and outliers –Not suitable to discover clusters with non-convex shapes

138. The K-Medoids Clustering Method Find representative objects, called medoids, in clusters PAM (Partitioning Around Medoids, 1987) –starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering –PAM works effectively for small data sets, but does not scale well for large data sets

139. EM Algorithm Initialize K cluster centers Iterate between two steps –Expectation step: assign points to clusters –Maximation step: estimate model parameters

143. Iteration 1 The cluster means are randomly assigned

144. Iteration 2

145. Iteration 5

146. Iteration 25

147. Nearest Neighbor Clustering Not to be confused with Nearest Neighbor Classification Items are iteratively merged into the existing clusters that are closest. Incremental Threshold, t, used to determine if items are added to existing clusters or a new cluster is created. What happens if the data is streaming…

148. 10 Threshold t t 1 2

149. 10 New data point arrives… It is within the threshold for cluster 1, so add it to the cluster, and update cluster center. 1 2 3

150. 10 New data point arrives… It is not within the threshold for cluster 1, so create a new cluster, and so on.. 1 2 3 4 Algorithm is highly order dependent… It is difficult to determine t in advance…

151. Partitional Clustering Algorithms Clustering algorithms have been designed to handle very large datasets E.g. the Birch algorithm Main idea: use an in-memory R-tree to store points that are being clustered Insert points one at a time into the R-tree, merging a new point with an existing cluster if is less than some  distance away If there are more leaf nodes than fit in memory, merge existing clusters that are close to each other At the end of first pass we get a large number of clusters at the leaves of the R-tree  Merge clusters to reduce the number of clusters

152. Partitional Clustering Algorithms The Birch algorithm R10 R11 R12 R1 R2 R3R4 R5 R6R7 R8 R9 Data nodes containing points R10R11 R12 We need to specify the number of clusters in advance, I have chosen 2

153. Partitional Clustering Algorithms The Birch algorithm R10 R11 R12 {R1,R2} R3R4 R5 R6R7 R8 R9 Data nodes containing points R10R11 R12

154. Partitional Clustering Algorithms The Birch algorithm R10R11 R12

155. How can we tell the right number of clusters? In general, this is a unsolved problem. However there are many approximate methods. In the next few slides we will see an example. For our example, we will use the familiar katydid/grasshopper dataset. However, in this case we are imagining that we do NOT know the class labels. We are only clustering on the X and Y axis values.

156. 12345678910 When k = 1, the objective function is 873.0

157. 12345678910 When k = 2, the objective function is 173.1

158. 12345678910 When k = 3, the objective function is 133.6

159. 0.00E+00 1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 9.00E+02 1.00E+03 123456 We can plot the objective function values for k equals 1 to 6… The abrupt change at k = 2, is highly suggestive of two clusters in the data. This technique for determining the number of clusters is known as “knee finding” or “elbow finding”. Note that the results are not always as clear cut as in this toy example k Objective Function

160. Association Rules Association Rules (market basket analysis) Retail shops are often interested in associations between different items that people buy. Someone who buys bread is quite likely also to buy milk A person who bought the book Database System Concepts is quite likely also to buy the book Operating System Concepts. Associations information can be used in several ways. E.g. when a customer buys a particular book, an online shop may suggest associated books. Association rules: bread  milk DB-Concepts, OS-Concepts  Networks Left hand side: antecedent, right hand side: consequent An association rule must have an associated population; the population consists of a set of instances E.g. each transaction (sale) at a shop is an instance, and the set of all transactions is the population

161. Association Rule Definitions Set of items: I={I 1,I 2,…,I m } Transactions: D={t 1,t 2, …, t n }, t j  I Itemset: {I i1,I i2, …, I ik }  I Support of an itemset: Percentage of transactions which contain that itemset. Large (Frequent) itemset: Itemset whose number of occurrences is above a threshold.

162. Association Rules Example I = { Beer, Bread, Jelly, Milk, PeanutButter} Support of {Bread,PeanutButter} is 60%

163. Association Rule Definitions Association Rule (AR): implication X  Y where X,Y  I and X  Y = the null set ; Support of AR (s) X  Y: Percentage of transactions that contain X  Y Confidence of AR (  ) X  Y: Ratio of number of transactions that contain X  Y to the number that contain X

164. Association Rules Example

165. Of 5 transactions, 3 involve both Bread and PeanutButter, 3/5 = 60% Of the 4 transactions that involve Bread, 3 of them also involve PeanutButter 3/4 = 75%

166. Association Rule Problem Given a set of items I={I 1,I 2,…,I m } and a database of transactions D={t 1,t 2, …, t n } where t i ={I i1,I i2, …, I ik } and I ij  I, the Association Rule Problem is to identify all association rules X  Y with a minimum support and confidence (supplied by user). NOTE: Support of X  Y is same as support of X  Y.

167. Association Rule Algorithm (Basic Idea) 1.Find Large Itemsets. 2.Generate rules from frequent itemsets. This is the simple naïve algorithm, better algorithms exist.

168. Association Rule Algorithm We are generally only interested in association rules with reasonably high support (e.g. support of 2% or greater) Naïve algorithm 1. Consider all possible sets of relevant items. 2. For each set find its support (i.e. count how many transactions purchase all items in the set). Large itemsets: sets with sufficiently high support Use large itemsets to generate association rules. From itemset A generate the rule A - {b}  b for each b  A. Support of rule = support (A). Confidence of rule = support (A ) / support (A - {b})

169. From itemset A generate the rule A - {b}  b for each b  A. Support of rule = support (A). Confidence of rule = support (A ) / support (A - {b}) Lets say itemset A = {Bread, Butter, Milk} Then A - {b}  b for each b  A includes 3 possibilities {Bread, Butter}  Milk {Bread, Milk}  Butter {Butter, Milk}  Bread

170. Apriori Algorithm Large Itemset Property: Any subset of a large itemset is large. Contrapositive: If an itemset is not large, none of its supersets are large.

171. Large Itemset Property If B is not frequent, then none of the supersets of B can be frequent. If {ACD} is frequent, then all subsets of {ACD} ({AC}, {AD}, {CD}) must be frequent.

172. Large Itemset Property If B is not frequent, then none of the supersets of B can be frequent. If {ACD} is frequent, then all subsets of {ACD} ({AC}, {AD}, {CD}) must be frequent.

173. Conclusions We have learned about the 3 major data mining/machine learning algorithms. Almost all data mining research is in these 3 areas, or is a minor extension of one or more of them. For further study, I recommend. Proceedings of SIGKDD, IEEE ICDM, SIAM SDM Data Mining: Concepts and Techniques (Jiawei Han and Micheline Kamber) Data Mining: Introductory and Advanced Topics (Margaret Dunham)