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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 Association Rule Mining 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 TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke © Tan,Steinbach, Kumar Introduction to Data Mining Example of Association Rules {Diaper} {Beer}, {Milk, Bread} {Eggs,Coke}, {Beer, Bread} {Milk}, 4/18/2004 ‹#› Definition: Frequent Itemset Itemset – A collection of one or more items Example: {Milk, Bread, Diaper} – k-itemset An itemset that contains k items Support count () – Frequency of occurrence of an itemset – E.g. ({Milk, Bread,Diaper}) = 2 Support TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke – Fraction of transactions that contain an itemset – E.g. s({Milk, Bread, Diaper}) = 2/5 Frequent Itemset – An itemset whose support is greater than or equal to a minsup threshold © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Definition: Association Rule Association Rule – An implication expression of the form X Y, where X and Y are itemsets – Example: {Milk, Diaper} {Beer} Rule Evaluation Metrics TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke – Support (s) Example: Fraction of transactions that contain both X and Y {Milk , Diaper } Beer – Confidence (c) Measures how often items in Y appear in transactions that contain X © Tan,Steinbach, Kumar s (Milk, Diaper, Beer ) |T| 2 0.4 5 (Milk, Diaper, Beer ) 2 c 0.67 (Milk, Diaper ) 3 Introduction to Data Mining 4/18/2004 ‹#› Association Rule Mining Task Given a set of transactions T, the goal of association rule mining is to find all rules having – support ≥ minsup threshold – confidence ≥ minconf threshold 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! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mining Association Rules Example of Rules: TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke {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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mining Association Rules 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 Frequent itemset generation is still computationally expensive © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Frequent Itemset Generation null A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining BCDE Given d items, there are 2d possible candidate itemsets 4/18/2004 ‹#› Frequent Itemset Generation Brute-force approach: – Each itemset in the lattice is a candidate frequent itemset – Count the support of each candidate by scanning the database Transactions N TID 1 2 3 4 5 Items Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke List of Candidates M w – Match each transaction against every candidate – Complexity ~ O(NMw) => Expensive since M = 2d !!! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Reducing Number of Candidates Apriori principle: – If an itemset is frequent, then all of its subsets must also be frequent Apriori principle holds due to the following property of the support measure: X , Y : ( X Y ) s( X ) s(Y ) – Support of an itemset never exceeds the support of its subsets – This is known as the anti-monotone property of support © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Illustrating Apriori Principle null A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE Found to be Infrequent ABCD ABCE Pruned supersets © Tan,Steinbach, Kumar Introduction to Data Mining ABDE ACDE BCDE ABCDE 4/18/2004 ‹#› Illustrating Apriori Principle Item Bread Coke Milk Beer Diaper Eggs Count 4 2 4 3 4 1 Items (1-itemsets) Itemset {Bread,Milk} {Bread,Beer} {Bread,Diaper} {Milk,Beer} {Milk,Diaper} {Beer,Diaper} Minimum Support = 3 Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Triplets (3-itemsets) If every subset is considered, 6C + 6C + 6C = 41 1 2 3 With support-based pruning, 6 + 6 + 1 = 13 © Tan,Steinbach, Kumar Count 3 2 3 2 3 3 Introduction to Data Mining Itemset {Bread,Milk,Diaper} Count 3 4/18/2004 ‹#› Apriori Algorithm Let k=1 Generate frequent itemsets of length 1 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› The Apriori Algorithm — Example Database D TID 100 200 300 400 itemset sup. C1 {1} 2 {2} 3 Scan D {3} 3 {4} 1 {5} 3 Items 134 235 1235 25 C2 itemset sup L2 itemset sup 2 2 3 2 {1 {1 {1 {2 {2 {3 C3 itemset {2 3 5} Scan D {1 3} {2 3} {2 5} {3 5} © Tan,Steinbach, Kumar 2} 3} 5} 3} 5} 5} 1 2 1 2 3 2 L1 itemset sup. {1} {2} {3} {5} 2 3 3 3 C2 itemset {1 2} Scan D {1 {1 {2 {2 {3 3} 5} 3} 5} 5} L3 itemset sup {2 3 5} 2 Introduction to Data Mining 4/18/2004 ‹#› Maximal Frequent Itemset An itemset is maximal frequent if none of its immediate supersets is frequent null Maximal Itemsets A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE Infrequent Itemsets ABCD E © Tan,Steinbach, Kumar Introduction to Data Mining ACDE BCDE Border 4/18/2004 ‹#› Closed Itemset An itemset is closed if none of its immediate supersets has the same support as the itemset. TID 1 2 3 4 5 Items {A,B} {B,C,D} {A,B,C,D} {A,B,D} {A,B,C,D} © Tan,Steinbach, Kumar Itemset {A} {B} {C} {D} {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} Introduction to Data Mining Support 4 5 3 4 4 2 3 3 4 3 Itemset Support {A,B,C} 2 {A,B,D} 3 {A,C,D} 2 {B,C,D} 3 {A,B,C,D} 2 4/18/2004 ‹#› Maximal vs Closed Itemsets TID Items 1 ABC 2 ABCD 3 BCE 4 ACDE 5 DE 124 123 A 12 124 AB 12 24 AC ABE 2 245 C 123 4 AE 24 ABD 1234 B AD 2 ABC 2 3 BD 4 ACD 345 D BC BE 2 4 ACE ADE E 24 CD 34 CE 3 BCD 45 ABCE ABDE ACDE BDE CDE BCDE ABCDE Introduction to Data Mining DE 4 BCE 4 ABCD Not supported by any transactions © Tan,Steinbach, Kumar Transaction Ids! null 4/18/2004 ‹#› Maximal vs Closed Frequent Itemsets Minimum support = 2 124 123 A 12 124 AB 12 ABC 24 AC AD ABD ABE 1234 B AE 345 D 2 3 BC BD 4 ACD 245 C 123 4 24 2 Closed but not maximal null 24 BE 2 4 ACE E ADE CD ABCD ABCE ABDE ACDE # Closed = 9 # Maximal = 4 © Tan,Steinbach, Kumar 34 CE 3 BCD ABCDE Introduction to Data Mining 45 DE 4 BCE 4 2 Closed and maximal BCDE BDE CDE TID Items 1 ABC 2 ABCD 3 BCE 4 ACDE 5 DE 4/18/2004 ‹#› Maximal vs Closed Itemsets Frequent Itemsets Closed Frequent Itemsets Maximal Frequent Itemsets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Rule Generation 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, A BCD, AB CD, BD AC, ABD C, B ACD, AC BD, CD AB, ACD B, C ABD, AD BC, BCD A, D ABC BC AD, If |L| = k, then there are 2k – 2 candidate association rules (ignoring L and L) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Rule Generation How to efficiently generate rules from frequent itemsets? – In general, confidence does not have an antimonotone 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Rule Generation for Apriori Algorithm Lattice of rules Low Confidence Rule CD=>AB ABCD=>{ } BCD=>A ACD=>B BD=>AC D=>ABC BC=>AD C=>ABD ABD=>C AD=>BC B=>ACD ABC=>D AC=>BD AB=>CD A=>BCD Pruned Rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Effect of Support Distribution Many real data sets have skewed support distribution Support distribution of a retail data set © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Effect of Support Distribution 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 Using a single minimum support threshold may not be effective © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Criticism to Support and Confidence Example: 2000/5000 2000/3000 – 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 basketball not basketball sum(row) cereal 2000 1750 3750 not cereal 1000 250 1250 sum(col.) 3000 2000 5000 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Criticism to Support and Confidence (Cont.) Example 2: X 1 1 1 1 0 0 0 0 Y 1 1 0 0 0 0 0 0 Z 0 1 1 1 1 1 1 1 – X and Y: positively correlated, – X and Z, negatively related – support and confidence of X=>Z dominates We need a measure of dependent or correlated events corrA, B P( A B) P( A) P( B) Rule Support Confidence X=>Y 25% 50% X=>Z 37.50% 75% P(B|A)/P(B) is also called the lift of rule A => B © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Other Interestingness Measures: Interest Interest (correlation, lift) P( A B) P( A) P( B) – 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 X 1 1 1 1 0 0 0 0 Y 1 1 0 0 0 0 0 0 Z 0 1 1 1 1 1 1 1 © Tan,Steinbach, Kumar Itemset Support Interest X,Y X,Z Y,Z 25% 37.50% 12.50% 2 0.9 0.57 Introduction to Data Mining 4/18/2004 ‹#› Infrequent Patterns 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. 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Negative Patterns Let I = {i1, i2, …} be a set of items. A negative item ik denote the absence of item ik from a given transaction. – For example, coffee is a negative item whose value is 1 if a transaction does not contain coffee. A negative itemset X is an itemset that has the following properties: X A B where A is a set of positive items, B is a set of negative items, | B | 1 s(X) ≥ minsup. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Negative Patterns 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. An example © Tan,Steinbach, Kumar tea coffee Introduction to Data Mining 4/18/2004 ‹#› Negatively Correlated Patterns Let X = {x1, x2, ...., xk} be a k-itemsets and P(X) be the probability that a transaction contains X. The probability is itemset support s(X). Negatively correlated itemset: An itemset X is negatively correlated if k s( X ) s( x j ) s( x1 ) s( x2 ) ... s( xk ) j 1 An association rule X -> Y is negatively correlated if s( X Y ) s( X ) s(Y ) where X and Y are disjoint itemsets. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Negatively Correlated Patterns Because s ( X Y ) s ( X ) s (Y ) s ( X Y ) [ s ( X Y ) s ( X Y )][ s ( X Y ) s ( X Y )] s ( X Y )[1 s ( X Y ) s ( X Y ) s ( X Y )] s ( X Y )s ( X Y ) s( X Y )s( X Y ) s( X Y )s( X Y ) The condition for negative correlation can be stated below. s( X Y ) s( X Y ) s( X Y ) s( X Y ) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Comparisons Comparisons among infrequent patterns, negative patterns, and negatively correlated patterns. Many infrequent patterns have corresponding negative patterns. Many negatively correlated patterns also have corresponding negative patterns. The lower the support s(X U Y), the more negatively support the pattern is. Negatively correlated patterns that are infrequent tend to be more interesting than negatively correlated patterns that are frequent. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Comparisons © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›