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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}, Implication means co-occurrence, not causality! 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 ‹#› Another Example Customer buys both Customer buys diaper Customer buys beer Transaction ID Items Bought Let minimum support 50%, and minimum confidence 50%, we 2000 A,B,C have 1000 A,C – A C (50%, 66.6%) 4000 A,D – C A (50%, 100%) 5000 B,E,F © Tan,Steinbach, Kumar 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 ‹#› 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) 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 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 if their first k-1 items are identical 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 ‹#› 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 ‹#› 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 – – Dimensionality (number of items) of the data set – – more space is needed to store support count of each item if number of frequent items also increases, both computation and I/O costs may also increase Size of database – lowering support threshold results in more frequent itemsets this may increase number of candidates and max length of frequent itemsets 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 – This may increase max length of frequent itemsets and traversals of hash tree (number of subsets in a transaction increases with its width) © 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 Solution: use multiple minimum support, so a larger minsup for frequent items (e.g., milk, bread) and a smaller minsup for rare items (e.g., diamond) © 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 ‹#› 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 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 Traversal of Itemset Lattice – General-to-specific vs Specific-to-general Frequent itemset border null null .. .. .. .. {a1,a2,...,an} {a1,a2,...,an} (a) General-to-specific © Tan,Steinbach, Kumar Frequent itemset border .. .. Frequent itemset border (b) Specific-to-general Introduction to Data Mining null {a1,a2,...,an} (c) Bidirectional 4/18/2004 ‹#› Alternative Methods for Frequent Itemset Generation Traversal of Itemset Lattice – Equivalent Classes null A AB ABC B AC AD ABD ACD null C BC BD D CD A AB BCD AC ABC B C BC AD ABD D BD CD ACD BCD ABCD ABCD (a) Prefix tree © Tan,Steinbach, Kumar Introduction to Data Mining (b) Suffix tree 4/18/2004 ‹#› Alternative Methods for Frequent Itemset Generation Traversal of Itemset Lattice – Breadth-first vs Depth-first (b) Depth first (a) Breadth first © 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 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 Transaction Database null B:3 A:7 B:5 C:1 C:3 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 C:1 Conditional Pattern base for D: P = {(A:1,B:1,C:1), (A:1,B:1), (A:1,C:1), (A:1), (B:1,C:1)} D:1 Recursively apply FPgrowth on P null A:7 B:5 B:1 C:1 C:3 D:1 D:1 Frequent Itemsets found (with sup > 1): AD, BD, CD, ABD, ACD, BCD D:1 D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Tree Projection null Set enumeration tree: A Possible Extension: E(A) = {B,C,D,E} B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE Possible Extension: E(ABC) = {D,E} ABCD ABCE ABDE ACDE BCDE ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Tree Projection Items are listed in lexicographic order Each node P stores the following information: – – – – Itemset for node P List of possible lexicographic extensions of P: E(P) Pointer to projected database of its ancestor node Bitvector containing information about which transactions in the projected database contain the itemset © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Projected Database Original Database: 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} Projected Database for node A: TID 1 2 3 4 5 6 7 8 9 10 Items {B} {} {C,D,E} {D,E} {B,C} {B,C,D} {} {B,C} {B,D} {} For each transaction T, projected transaction at node A is T E(A) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› ECLAT For each item, store a list of transaction ids (tids) Horizontal Data Layout TID 1 2 3 4 5 6 7 8 9 10 © Tan,Steinbach, Kumar 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 Vertical Data Layout A 1 4 5 6 7 8 9 B 1 2 5 7 8 10 C 2 3 4 8 9 D 2 4 5 9 E 1 3 6 TID-list Introduction to Data Mining 4/18/2004 ‹#› ECLAT Determine support of any k-itemset by intersecting tid-lists of two of its (k-1) subsets. A 1 4 5 6 7 8 9 B 1 2 5 7 8 10 AB 1 5 7 8 3 traversal approaches: – top-down, bottom-up and hybrid Advantage: very fast support counting Disadvantage: intermediate tid-lists may become too large for memory © 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 and an objective interestingness measure. – An object interestingness measure is domain independent. © 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 ‹#› Statistical Independence 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 – P(SB) = P(S) P(B) => Statistical independence – P(SB) > P(S) P(B) => Positively correlated – P(SB) < P(S) P(B) => Negatively correlated © Tan,Steinbach, Kumar Introduction to Data Mining 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 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) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Lift and Interest (Another Example) Example 2: – X and Y: positively correlated, – X and Z, negatively related – support and confidence of X=>Z dominates © Tan,Steinbach, Kumar 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 Rule Support Confidence X=>Y 25% 50% X=>Z 37.50% 75% Introduction to Data Mining 4/18/2004 ‹#› Lift and Interest (Another Example) Interest (lift) P( A B ) P( A) P( B) – A and B negatively correlated, if the value is less than 1; otherwise A and B positively correlated X11110000 Y11000000 Z01111111 © 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 ‹#› 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 ‹#› Constraint-Based Frequent Pattern Mining Classification of constraints based on their constraintpushing capabilities – Anti-monotonic: If constraint c is violated, its further mining can be terminated – Monotonic: If c is satisfied, no need to check c again – Convertible: c is not monotonic nor anti-monotonic, but it can be converted into it if items in the transaction can be properly ordered © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Anti-Monotonicity TDB (min_sup=2) A constraint C is antimonotone if the super pattern satisfies C, all of its sub-patterns do so too In other words, anti-monotonicity: If an itemset S violates the constraint, so does any of its superset TID Transaction 10 a, b, c, d, f 20 30 40 b, c, d, f, g, h a, c, d, e, f c, e, f, g Item Price Profit Ex. 1. sum(S.price) v is anti-monotone a 60 40 Ex. 2. range(S.profit) 15 is anti-monotone b 20 0 c 80 -20 d 30 10 e 70 -30 f 100 30 g 50 20 h 40 -10 – Itemset ab violates C – So does every superset of ab Ex. 3. sum(S.Price) v is not anti-monotone Ex. 4. support count is anti-monotone: core property used in Apriori © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Monotonicity TDB (min_sup=2) A constraint C is monotone if the pattern satisfies C, we do not need to check C in subsequent mining Alternatively, monotonicity: If an itemset S satisfies the constraint, so does any of its superset TID Transaction 10 a, b, c, d, f 20 b, c, d, f, g, h 30 a, c, d, e, f 40 c, e, f, g Item Price Profit a 60 40 b 20 0 c 80 -20 d 30 10 – Itemset ab satisfies C e 70 -30 – So does every superset of ab f 100 30 g 50 20 h 40 -10 Ex. 1. sum(S.Price) v is monotone Ex. 2. min(S.Price) v is monotone Ex. 3. C: range(S.profit) 15 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Converting “Tough” Constraints TDB (min_sup=2) Convert tough constraints into antimonotone or monotone by properly ordering items Examine C: avg(S.profit) 25 – Order items in value-descending order <a, f, g, d, b, h, c, e> – If an itemset afb violates C So It does afbh, afb* becomes anti-monotone! © Tan,Steinbach, Kumar Introduction to Data Mining TID Transaction 10 a, b, c, d, f 20 b, c, d, f, g, h 30 a, c, d, e, f 40 c, e, f, g Item Profit a b c d e f g h 40 0 -20 10 -30 30 20 -10 4/18/2004 ‹#› Strongly Convertible Constraints avg(X) 25 is convertible anti-monotone w.r.t. item value descending order R: <a, f, g, d, b, h, c, e> – If an itemset af violates a constraint C, so does every itemset with af as prefix, such as afd avg(X) 25 is convertible monotone w.r.t. item value ascending order R-1: <e, c, h, b, d, g, f, a> – If an itemset d satisfies a constraint C, so does itemsets df and dfa, which having d as a prefix Item Profit a 40 b 0 c -20 d 10 e -30 f 30 g 20 h -10 Thus, avg(X) 25 is strongly convertible © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›