1. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Data Mining: Association Analysis This lecture node is modified based on Lecture Notes for Chapter 6/7 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

2. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 2 Association Rule Mining l Given a set of transactions, find rules that will predict the occurrence of an item based on the occurrences of other items in the transaction. Market-Basket transactions Example of Association Rules {Diaper}  {Beer}, {Milk, Bread}  {Eggs,Coke}, {Beer, Bread}  {Milk},

3. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 3 Definition: Frequent Itemset l Itemset –A collection of one or more items  Example: {Milk, Bread, Diaper} –k-itemset  An itemset that contains k items l Support count (  ) –Frequency of occurrence of an itemset –E.g.  ({Milk, Bread,Diaper}) = 2 l Support –Fraction of transactions that contain an itemset –E.g. s({Milk, Bread, Diaper}) = 2/5 l Frequent Itemset –An itemset whose support is greater than or equal to a minsup threshold

4. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 4 Definition: Association Rule Example: l Association Rule –An implication expression of the form X  Y, where X and Y are itemsets –Example: {Milk, Diaper}  {Beer} l Rule Evaluation Metrics –Support (s)  Fraction of transactions that contain both X and Y –Confidence (c)  Measures how often items in Y appear in transactions that contain X

5. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 5 Association Rule Mining Task l Given a set of transactions T, the goal of association rule mining is to find all rules having –support ≥ minsup threshold –confidence ≥ minconf threshold l Brute-force approach: –List all possible association rules –Compute the support and confidence for each rule –Prune rules that fail the minsup and minconf thresholds  Computationally prohibitive!

6. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 6 Mining Association Rules Example of Rules: {Milk,Diaper}  {Beer} (s=0.4, c=0.67) {Milk,Beer}  {Diaper} (s=0.4, c=1.0) {Diaper,Beer}  {Milk} (s=0.4, c=0.67) {Beer}  {Milk,Diaper} (s=0.4, c=0.67) {Diaper}  {Milk,Beer} (s=0.4, c=0.5) {Milk}  {Diaper,Beer} (s=0.4, c=0.5) Observations: All the above rules are binary partitions of the same itemset: {Milk, Diaper, Beer} Rules originating from the same itemset have identical support but can have different confidence Thus, we may decouple the support and confidence requirements

7. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 7 Mining Association Rules l Two-step approach: 1.Frequent Itemset Generation – Generate all itemsets whose support  minsup 2.Rule Generation – Generate high confidence rules from each frequent itemset, where each rule is a binary partitioning of a frequent itemset l Frequent itemset generation is still computationally expensive

8. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 8 Frequent Itemset Generation Given d items, there are 2 d possible candidate itemsets

9. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 9 Frequent Itemset Generation l Brute-force approach: –Each itemset in the lattice is a candidate frequent itemset –Count the support of each candidate by scanning the database –Match each transaction against every candidate –Complexity ~ O(NMw) => Expensive since M = 2 d !!!

10. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 10 Reducing Number of Candidates l Apriori principle: –If an itemset is frequent, then all of its subsets must also be frequent l Apriori principle holds due to the following property of the support measure: –Support of an itemset never exceeds the support of its subsets –This is known as the anti-monotone property of support

11. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 11 Found to be Infrequent Illustrating Apriori Principle Pruned supersets

12. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 12 Illustrating Apriori Principle Items (1-itemsets) Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Triplets (3-itemsets) Minimum Support = 3 If every subset is considered, 6 C 1 + 6 C 2 + 6 C 3 = 41 With support-based pruning, 6 + 6 + 1 = 13

13. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 13 Apriori Algorithm l Let k=1 l Generate frequent itemsets of length 1 l Repeat until no new frequent itemsets are identified –Generate length (k+1) candidate itemsets from length k frequent itemsets  Let two k-itemsets be (X,Y) and (X, Z), where X is (k-1)-items, and Y and Z are 1-item.  The new (k+1)-itemset will be (X, Y, Z). –Prune candidate itemsets containing subsets of length k that are infrequent –Count the support of each candidate by scanning the DB –Eliminate candidates that are infrequent, leaving only those that are frequent

14. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 14 The Apriori Algorithm — Example Database D Scan D C1C1 L1L1 L2L2 C2C2 C2C2 C3C3 L3L3

15. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 15 Maximal Frequent Itemset Border Infrequent Itemsets Maximal Itemsets An itemset is maximal frequent if none of its immediate supersets is frequent

16. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 16 Closed Itemset l An itemset is closed if none of its immediate supersets has the same support as the itemset.

17. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 17 Maximal vs Closed Itemsets Transaction Ids! Not supported by any transactions

18. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 18 Maximal vs Closed Frequent Itemsets Minimum support = 2 # Closed = 9 # Maximal = 4 Closed and maximal Closed but not maximal

19. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 19 Maximal vs Closed Itemsets

20. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 20 Rule Generation l Given a frequent itemset L, find all non-empty subsets f  L such that f  L – f satisfies the minimum confidence requirement –If {A,B,C,D} is a frequent itemset, candidate rules: ABC  D, ABD  C, ACD  B, BCD  A, A  BCD,B  ACD,C  ABD, D  ABC AB  CD,AC  BD, AD  BC, BC  AD, BD  AC, CD  AB, l If |L| = k, then there are 2 k – 2 candidate association rules (ignoring L   and   L)

21. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 21 Rule Generation l How to efficiently generate rules from frequent itemsets? –In general, confidence does not have an anti- monotone property c(ABC  D) can be larger or smaller than c(AB  D) –But confidence of rules generated from the same itemset has an anti-monotone property –e.g., L = {A,B,C,D}: c(ABC  D)  c(AB  CD)  c(A  BCD)  Confidence is anti-monotone w.r.t. number of items on the RHS of the rule

22. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 22 Rule Generation for Apriori Algorithm Lattice of rules Pruned Rules Low Confidence Rule

23. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 23 Effect of Support Distribution l Many real data sets have skewed support distribution Support distribution of a retail data set

24. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 24 Effect of Support Distribution l How to set the appropriate minsup threshold? –If minsup is set too high, we could miss itemsets involving interesting rare items (e.g., expensive products) –If minsup is set too low, it is computationally expensive and the number of itemsets is very large l Using a single minimum support threshold may not be effective

25. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 25 Criticism to Support and Confidence l Example: –Among 5000 students  3000 play basketball  3750 eat cereal  2000 both play basket ball and eat cereal –play basketball  eat cereal [40%, 66.7%] is misleading because the overall percentage of students eating cereal is 75% which is higher than 66.7%. –play basketball  not eat cereal [20%, 33.3%] is far more accurate, although with lower support and confidence 2000/50002000/3000

26. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 26 Criticism to Support and Confidence (Cont.) l Example 2: –X and Y: positively correlated, –X and Z, negatively related –support and confidence of X=>Z dominates l We need a measure of dependent or correlated events l P(B|A)/P(B) is also called the lift of rule A => B

27. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 27 Other Interestingness Measures: Interest l Interest (correlation, lift) –taking both P(A) and P(B) in consideration –P(A^B)=P(B)*P(A), if A and B are independent events –A and B negatively correlated, if the value is less than 1; otherwise A and B positively correlated

28. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 28 Infrequent Patterns l Example-1: –The sale of DVDs and VCRs together is low, because people don’t buy both at the same time. –They are negative-correlated, and they are competing items. l Example-2: –If {Fire = Yes} is frequent but {Fire = Yes, Alarm = No} is infrequent, then the latter is an important infrequent pattern because it indicates faulty alarm systems.

29. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 29 Negative Patterns l Let I = {i 1, i 2, …} be a set of items. A negative item denote the absence of item i k from a given transaction. –For example, is a negative item whose value is 1 if a transaction does not contain coffee. l A negative itemset X is an itemset that has the following properties: where A is a set of positive items, is a set of negative items, s(X) ≥ minsup.

30. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 30 Negative Patterns l A negative association rule is an association rule that has the following properties –The rule is extracted from a negative itemset. –The support of the rule is greater than or equal to minsup. –The confidence of the rule is greater than or equal to minconf. l An example

31. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 31 Negatively Correlated Patterns l Let X = {x 1, x 2,...., x k } be a k-itemsets and P(X) be the probability that a transaction contains X. l The probability is itemset support s(X). l Negatively correlated itemset: An itemset X is negatively correlated if l An association rule X -> Y is negatively correlated if where X and Y are disjoint itemsets.

32. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 32 Negatively Correlated Patterns Because The condition for negative correlation can be stated below.

33. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 33 Comparisons l Comparisons among infrequent patterns, negative patterns, and negatively correlated patterns. l Many infrequent patterns have corresponding negative patterns. l Many negatively correlated patterns also have corresponding negative patterns. l The lower the support s(X U Y), the more negatively support the pattern is. l Negatively correlated patterns that are infrequent tend to be more interesting than negatively correlated patterns that are frequent.

34. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 34 Comparisons