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Data Mining Association Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 6 Introduction to Data Mining by Tan, Steinbach, Kumar © 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 ‹#› Computational Complexity Given d unique items: – Total number of itemsets = 2d – Total number of possible association rules: d d k R k j 3 2 1 d 1 d k k 1 j 1 d d 1 If d=6, R = 602 rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Frequent Itemset Generation Strategies Reduce the number of candidates (M) – Complete search: M=2d – Use pruning techniques to reduce M Reduce the number of transactions (N) – Reduce size of N as the size of itemset increases Reduce the number of comparisons (NM) – Use efficient data structures to store the candidates or transactions – No need to match every candidate against every transaction © 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) Minimum Support = 3 Itemset {Bread,Milk} {Bread,Beer} {Bread,Diaper} {Milk,Beer} {Milk,Diaper} {Beer,Diaper} Count 3 2 3 2 3 3 Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Triplets (3-itemsets) Itemset {Bread,Milk,Diaper} © Tan,Steinbach, Kumar Introduction to Data Mining Count 3 4/18/2004 ‹#› Apriori Algorithm Method: – 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 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 Kumar 3 5} © Tan,Steinbach, Scan D {1 3} {2 3} {2 5} {3 5} 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 L3 itemset sup {2 3 5} 2 Introduction to Data Mining {1 {1 {2 {2 {3 3} 5} 3} 5} 5} 4/18/2004 ‹#› Reducing Number of Comparisons Candidate counting: – Scan the database of transactions to determine the support of each candidate itemset – To reduce the number of comparisons, store the candidates in a hash structure Instead of matching each transaction against every candidate, match it against candidates contained in the hashed buckets Transactions N TID 1 2 3 4 5 Hash Structure Items Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke k Buckets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Reducing Number of Comparisons © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Generate Hash Tree Suppose you have 15 candidate itemsets of length 3: {1 4 5}, {1 2 4}, {4 5 7}, {1 2 5}, {4 5 8}, {1 5 9}, {1 3 6}, {2 3 4}, {5 6 7}, {3 4 5}, {3 5 6}, {3 5 7}, {6 8 9}, {3 6 7}, {3 6 8} You need: • Hash function • Max leaf size: max number of itemsets stored in a leaf node (if number of candidate itemsets exceeds max leaf size, split the node) Hash function 3,6,9 1,4,7 234 567 345 136 145 2,5,8 124 457 © Tan,Steinbach, Kumar 125 458 Introduction to Data Mining 356 357 689 367 368 159 4/18/2004 ‹#› Association Rule Discovery: Hash tree Hash Function 1,4,7 Candidate Hash Tree 3,6,9 2,5,8 234 567 145 136 345 Hash on 1, 4 or 7 124 457 © Tan,Steinbach, Kumar 125 458 159 Introduction to Data Mining 356 357 689 367 368 4/18/2004 ‹#› Association Rule Discovery: Hash tree Hash Function 1,4,7 Candidate Hash Tree 3,6,9 2,5,8 234 567 145 136 345 Hash on 2, 5 or 8 124 457 © Tan,Steinbach, Kumar 125 458 159 Introduction to Data Mining 356 357 689 367 368 4/18/2004 ‹#› Association Rule Discovery: Hash tree Hash Function 1,4,7 Candidate Hash Tree 3,6,9 2,5,8 234 567 145 136 345 Hash on 3, 6 or 9 124 457 © Tan,Steinbach, Kumar 125 458 159 Introduction to Data Mining 356 357 689 367 368 4/18/2004 ‹#› Subset Operation Given a transaction t, what are the possible subsets of size 3? Transaction, t 1 2 3 5 6 Level 1 1 2 3 5 6 2 3 5 6 3 5 6 Level 2 12 3 5 6 13 5 6 123 125 126 135 136 Level 3 © Tan,Steinbach, Kumar 15 6 156 23 5 6 235 236 25 6 256 35 6 356 Subsets of 3 items Introduction to Data Mining 4/18/2004 ‹#› Subset Operation Using Hash Tree Hash Function 1 2 3 5 6 transaction 1+ 2356 2+ 356 1,4,7 3+ 56 3,6,9 2,5,8 234 567 145 136 345 124 457 125 458 © Tan,Steinbach, Kumar 159 356 357 689 Introduction to Data Mining 367 368 4/18/2004 ‹#› Subset Operation Using Hash Tree Hash Function 1 2 3 5 6 transaction 1+ 2356 2+ 356 12+ 356 1,4,7 3+ 56 3,6,9 2,5,8 13+ 56 234 567 15+ 6 145 136 345 124 457 © Tan,Steinbach, Kumar 125 458 159 Introduction to Data Mining 356 357 689 367 368 4/18/2004 ‹#› Subset Operation Using Hash Tree Hash Function 1 2 3 5 6 transaction 1+ 2356 2+ 356 12+ 356 1,4,7 3+ 56 3,6,9 2,5,8 13+ 56 234 567 15+ 6 145 136 345 124 457 © Tan,Steinbach, Kumar 125 458 159 356 357 689 367 368 Match transaction against 11 out of 15 candidates Introduction to Data Mining 4/18/2004 ‹#› Factors Affecting Complexity Choice of minimum support threshold – lowering support threshold results in more frequent itemsets Dimensionality (number of items) of the data set – if number of frequent items also increases, both computation and I/O costs may also increase Size of database – since Apriori makes multiple passes, run time of algorithm may increase with number of transactions Average transaction width – transaction width increases with denser data sets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Compact Representation of Frequent Itemsets Some itemsets are redundant because they have identical support as their supersets TID A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 10 Number of frequent itemsets 3 k Need a compact representation 10 k 1 © Tan,Steinbach, Kumar 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 ‹#› Closed Itemsets TID Items 1 ABC 2 ABCD 3 BCE 4 ACDE 5 DE Closed Not closed 124 123 A 12 124 AB 12 24 AC AD ABC ABD ABE 2 AE 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 245 C 123 24 2 1234 B 4 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 Closed and maximal 34 CE 3 BCD 45 DE 4 BCE BDE CDE 4 2 ABCD ABCE ABDE ACDE BCDE # Closed = 9 # Maximal = 4 ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining 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 ‹#› Alternative Methods for Frequent Itemset Generation Representation of Database – horizontal vs vertical data layout Horizontal Data Layout TID 1 2 3 4 5 6 7 8 9 10 Items A,B,E B,C,D C,E A,C,D A,B,C,D A,E A,B A,B,C A,C,D B © Tan,Steinbach, Kumar Vertical Data Layout A 1 4 5 6 7 8 9 Introduction to Data Mining B 1 2 5 7 8 10 C 2 3 4 8 9 D 2 4 5 9 E 1 3 6 4/18/2004 ‹#› FP-growth Algorithm Use a compressed representation of the database using an FP-tree Once an FP-tree has been constructed, it uses a recursive divide-and-conquer approach to mine the frequent itemsets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› FP-tree construction null After reading TID=1: TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E} A:1 B:1 After reading TID=2: null A:1 B:1 B:1 C:1 D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› FP-tree construction After reading TID=3: TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E} © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› FP-Tree Construction TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E} Header table Item Pointer A B C D E © Tan,Steinbach, Kumar After reading TID=10: null B:1 A:7 B:5 C:1 C:1 D:1 D:1 C:3 D:1 D:1 D:1 E:1 E:1 E:1 Pointers are used to assist frequent itemset generation Introduction to Data Mining 4/18/2004 ‹#› FP-growth Algorithm Conditional Pattern base for D: P = {(A:7,B:5,C:3), (A:7,B:5), (A:7,C:1), (A:7), (B:1,C:1)} null A:7 B:5 B:1 C:1 C:3 D:1 D:1 C:1 Recursively apply FPgrowth on P D:1 Frequent Itemsets found (with sup > 1): A, AB, ABC D:1 D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Why Is FP-Growth Fast? FP-growth is faster than Apriori No candidate generation, no candidate test Use compact data structure Eliminate repeated database scan Basic operation is counting and FP-tree building © 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 ‹#› Rule Generation for Apriori Algorithm Candidate rule is generated by merging two rules that share the same prefix in the rule consequent CD=>AB BD=>AC join(CD=>AB,BD=>AC) would produce the candidate rule D => ABC D=>ABC Prune rule D=>ABC if its subset AD=>BC does not have high confidence © 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 ‹#› Multiple Minimum Support How to apply multiple minimum supports? – MS(i): minimum support for item i – e.g.: MS(Milk)=5%, MS(Coke) = 3%, MS(Broccoli)=0.1%, MS(Salmon)=0.5% – MS({Milk, Broccoli}) = min (MS(Milk), MS(Broccoli)) = 0.1% – Challenge: Support is no longer anti-monotone Suppose: Support(Milk, Coke) = 1.5% and Support(Milk, Coke, Broccoli) = 0.5% {Milk,Coke} is infrequent but {Milk,Coke,Broccoli} is frequent © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Multiple Minimum Support (Liu 1999) Order the items according to their minimum support (in ascending order) – e.g.: MS(Milk)=5%, MS(Coke) = 3%, MS(Broccoli)=0.1%, MS(Salmon)=0.5% – Ordering: Broccoli, Salmon, Coke, Milk Need to modify Apriori such that: – L1 : set of frequent items – F1 : set of items whose support is MS(1) where MS(1) is mini( MS(i) ) – C2 : candidate itemsets of size 2 is generated from F1 instead of L1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Multiple Minimum Support (Liu 1999) Modifications to Apriori: – In traditional Apriori, A candidate (k+1)-itemset is generated by merging two frequent itemsets of size k The candidate is pruned if it contains any infrequent subsets of size k – Pruning step has to be modified: Prune only if subset contains the first item e.g.: Candidate={Broccoli, Coke, Milk} (ordered according to minimum support) {Broccoli, Coke} and {Broccoli, Milk} are frequent but {Coke, Milk} is infrequent – Candidate is not pruned because {Coke,Milk} does not contain the first item, i.e., Broccoli. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Pattern Evaluation Association rule algorithms tend to produce too many rules – many of them are uninteresting or redundant – Redundant if {A,B,C} {D} and {A,B} {D} have same support & confidence Interestingness measures can be used to prune/rank the derived patterns In the original formulation of association rules, support & confidence are the only measures used © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Application of Interestingness Measure Interestingness Measures © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Computing Interestingness Measure Given a rule X Y, information needed to compute rule interestingness can be obtained from a contingency table Contingency table for X Y Y Y X f11 f10 f1+ X f01 f00 fo+ f+1 f+0 |T| f11: support of X and Y f10: support of X and Y f01: support of X and Y f00: support of X and Y Used to define various measures © Tan,Steinbach, Kumar support, confidence, lift, Gini, J-measure, etc. Introduction to Data Mining 4/18/2004 ‹#› Drawback of Confidence Coffee Coffee Tea 15 5 20 Tea 75 5 80 90 10 100 Association Rule: Tea Coffee Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9 Although confidence is high, rule is misleading P(Coffee|Tea) = 0.9375 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Correlation and statistical dependence Population of 1000 students – 600 students know how to swim (S) – 700 students know how to bike (B) – 420 students know how to swim and bike (S,B) – P(SB) = 420/1000 = 0.42 – P(S) P(B) = 0.6 0.7 = 0.42 If corrS,B = 1: independence If corrS,B > 1: positive correlation If corrS,B < 1: negative correlation © Tan,Steinbach, Kumar Introduction to Data Mining P( S B) corrS , B P( S ) P( B) 4/18/2004 ‹#› Statistical-based Measures Measures that take into account statistical dependence P(Y | X ) Lift P(Y ) P( X , Y ) Interest P( X ) P(Y ) © Tan,Steinbach, Kumar Introduction to Data Mining Confidence (XY) / Support (Y) Support (X, Y) / support(X)* support(Y) 4/18/2004 ‹#› Example: Lift/Interest Coffee Coffee Tea 15 5 20 Tea 75 5 80 90 10 100 Association Rule: Tea Coffee Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9 Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated) Interest = 0.15 / (0.9 * 0.2) = 0.8333 (< 1, therefore is negatively associated) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Drawback of Lift & Interest Y Y X 10 0 10 X 0 90 90 10 90 100 0.1 Lift 10 (0.1)(0.1) Y Y X 90 0 90 X 0 10 10 90 10 100 0.9 Lift 1.11 (0.9)(0.9) Statistical independence: If P(X,Y)=P(X)P(Y) => Lift = 1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› There are lots of measures proposed in the literature Some measures are good for certain applications, but not for others What criteria should we use to determine whether a measure is good or bad? What about Aprioristyle support based pruning? How does it affect these measures? Properties of A Good Measure Piatetsky-Shapiro: 3 properties a good measure M must satisfy: – M(A,B) = 0 if A and B are statistically independent – M(A,B) increase monotonically with P(A,B) when P(A) and P(B) remain unchanged – M(A,B) decreases monotonically with P(A) [or P(B)] when P(A,B) and P(B) [or P(A)] remain unchanged © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Comparing Different Measures 10 examples of contingency tables: Example f11 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 8123 8330 9481 3954 2886 1500 4000 4000 1720 61 Rankings of contingency tables using various measures: © Tan,Steinbach, Kumar Introduction to Data Mining f10 f01 f00 83 424 1370 2 622 1046 94 127 298 3080 5 2961 1363 1320 4431 2000 500 6000 2000 1000 3000 2000 2000 2000 7121 5 1154 2483 4 7452 4/18/2004 ‹#› Property under Variable Permutation B p r A A B q s B B A p q A r s Does M(A,B) = M(B,A)? Symmetric measures: support, lift, collective strength, cosine, Jaccard, etc Asymmetric measures: confidence, conviction, Laplace, J-measure, etc © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Property under Row/Column Scaling Grade-Gender Example (Mosteller, 1968): Male Female High 2 3 5 Low 1 4 5 3 7 10 Male Female High 4 30 34 Low 2 40 42 6 70 76 2x 10x Mosteller: Underlying association should be independent of the relative number of male and female students in the samples © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Property under Inversion Operation Transaction 1 . . . . . Transaction N A B C D E F 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 (a) © Tan,Steinbach, Kumar (b) Introduction to Data Mining (c) 4/18/2004 ‹#› Example: -Coefficient -coefficient is analogous to correlation coefficient for continuous variables Y Y X 60 10 70 X 10 20 30 70 30 100 60 * 20 10 *10 70 30 70 30 0.5238 f11 f 00 f 01 f10 f1 f 1 f 0 f 0 Y Y X 20 10 30 X 10 60 70 30 70 100 20 60 10 10 70 30 70 30 0.5238 Coefficient is the same for both tables © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Property under Null Addition A A B p r B q s A A B p r B q s+k Invariant measures: support, cosine, Jaccard, etc Non-invariant measures: correlation, Gini, mutual information, odds ratio, etc © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Different Measures have Different Properties Sym bol Measure Range P1 P2 P3 O1 O2 O3 O3' O4 Q Y M J G s c L V I IS PS F AV S Correlation Lambda Odds ratio Yule's Q Yule's Y Cohen's Mutual Information J-Measure Gini Index Support Confidence Laplace Conviction Interest IS (cosine) Piatetsky-Shapiro's Certainty factor Added value Collective strength Jaccard -1 … 0 … 1 0…1 0 … 1 … -1 … 0 … 1 -1 … 0 … 1 -1 … 0 … 1 0…1 0…1 0…1 0…1 0…1 0…1 0.5 … 1 … 0 … 1 … 0 .. 1 -0.25 … 0 … 0.25 -1 … 0 … 1 0.5 … 1 … 1 0 … 1 … 0 .. 1 Yes Yes Yes* Yes Yes Yes Yes Yes Yes No No No No Yes* No Yes Yes Yes No No Yes No Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No Yes Yes Yes Yes** Yes Yes Yes No No Yes Yes No No Yes Yes Yes No No No No No No No No No No No No No No No Yes No* Yes* Yes Yes No No* No No* No No No No No No Yes No No Yes* No Yes Yes Yes Yes Yes Yes Yes No Yes No No No Yes No No Yes Yes No Yes No No No No No No No No No No No Yes No No No Yes No No No No Yes 2 1 2 1 2 3 0 Yes 3 Introduction 3 to Data 3 3 Mining Yes Yes No No No No Klosgen's K © Tan,Steinbach, Kumar 4/18/2004 ‹#› No Support-based Pruning Most of the association rule mining algorithms use support measure to prune rules and itemsets Study effect of support pruning on correlation of itemsets – Generate 10000 random contingency tables – Compute support and pairwise correlation for each table – Apply support-based pruning and examine the tables that are removed © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Effect of Support-based Pruning All Itempairs 1000 900 800 700 600 500 400 300 200 100 2 3 4 5 6 7 8 9 0. 0. 0. 0. 0. 0. 0. 0. 1 1 0. 0 -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 Correlation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Effect of Support-based Pruning Support < 0.01 1 8 9 7 0. 0. 6 0. -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 Correlation 5 0 0. 0 4 50 0. 50 3 100 0. 100 2 150 0. 150 1 200 0. 200 0 250 1 250 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 300 -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 300 0. Support < 0.03 Correlation Support < 0.05 300 250 Small Support values decrease negatively correlated itemsets 200 150 100 50 Correlation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 1 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 Effect of Support-based Pruning Investigate how support-based pruning affects other measures Steps: – Generate 10000 contingency tables – Rank each table according to the different measures – Compute the pair-wise correlation between the measures © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Effect of Support-based Pruning Without Support Pruning (All Pairs) All Pairs (40.14%) Conviction Odds ratio Col Strength Correlation Interest PS CF Yule Y Reliability Kappa Klosgen Yule Q Confidence Laplace IS Support Jaccard Lambda Gini J-measure Mutual Info 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Red cells indicate correlation between pairs of measures > 0.85 40.14% pairs have correlation > 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 20 4/18/2004 21 ‹#› Effect of Support-based Pruning 0.5% support 50% 0.005 <= s upport <= 0.500 (61.45%) Interest Conviction Odds ratio Col Strength Laplace Confidence Correlation Klosgen Reliability PS Yule Q CF Yule Y Kappa IS Jaccard Support Lambda Gini J-measure Mutual Info 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 61.45% pairs have correlation > 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Effect of Support-based Pruning 0.5% support 30% 0.005 <= s upport <= 0.300 (76.42%) Support Interest Reliability Conviction Yule Q Odds ratio Confidence CF Yule Y Kappa Correlation Col Strength IS Jaccard Laplace PS Klosgen Lambda Mutual Info Gini J-measure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 76.42% pairs have correlation > 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› ? ? ? SIMPSON’S PARADOX How does it happen? © Tan,Steinbach, Kumar Introduction to Data Mining Pamela Leutwyler 4/18/2004 ‹#› A perfume company is testing two new scents Citrus And Orange blossom © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 28 single women volunteer to test these products © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 15 are Eagles cheerleaders 13 are members of Granny’s bingo club © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 13 women choose CITRUS 15 women choose ORANGE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 80% success rate 70% success rate 4 OF THE 5 CHEERLEADERS WHO USED CITRUS FOUND LOVE! 7 OF THE 10 CHEERLEADERS WHO USED ORANGE FOUND LOVE! CITRUS appears to be more effective for cheerleaders © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 2 OF THE 8 GRANNIES WHO USED CITRUS FOUND LOVE! 1 OF THE 5 GRANNIES WHO USED ORANGE FOUND LOVE! 80% success rate success rate CITRUS appears to be more effective for 70% grannies 25% success rate © Tan,Steinbach, Kumar Introduction to Data Mining 20% success rate 4/18/2004 ‹#› CITRUS is better for cheerleaders 80% success rate 70% success rate CITRUS is better for grannies 25% success rate © Tan,Steinbach, Kumar Introduction to Data Mining 20% success rate 4/18/2004 ‹#› © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 6 of the 13 women who used Citrus found love. 46% 8 of the 15 women who used Orange found love. 53% Overall Orange has a higher success rate © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› How can it happen that CITRUS works better for cheerleaders and CITRUS works better for grannies While ORANGE works better overall??? Where are most of the grannies? © Tan,Steinbach, Kumar Introduction to Data Mining Where are most of the cheerleaders? 4/18/2004 ‹#› © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Simpson’s paradox Contingency matrix for women Love Not-Love Citrus 6 7 13 Orange 8 7 15 14 14 28 Confidence (CitrusLove)=6/13= 46% Confidence (OrangeLove)=8/15= 53% © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›