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 2 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 3 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 Support (s) Measures how often items in Y appear in transactions that contain X Tan,Steinbach, Kumar Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 4 Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke Example: Fraction of transactions that contain both X and Y Confidence (c) TID {Milk, Diaper} Beer (Milk , Diaper, Beer) 2 s 0.4 |T| 5 (Milk, Diaper, Beer) 2 c

0.67 (Milk , Diaper) 3 Introduction to Data Mining 4/18/2004 4 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 5 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 6 Mining Association Rules Example of Rules: TID Items 1 Bread, Milk 2 3 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 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 7 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 8 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 9 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 10 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 11 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 12 Illustrating Apriori Principle null A Found to be Infrequent B D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE

ADE BCD BCE BDE CDE ABCD ABCE Pruned supersets Tan,Steinbach, Kumar C Introduction to Data Mining ABDE ACDE BCDE ABCDE 4/18/2004 13 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} Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Triplets (3-itemsets) If every subset is considered, 6 C1 + 6C2 + 6C3 = 41 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 14 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 15 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 L2 itemset sup C2 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

16 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 17 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 159 356 357 689 4/18/2004 367 368 18 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 19 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 20 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 21 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 22 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 23 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 24 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 25 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 26 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 27 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 28 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 29 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 Prune rule D=>ABC if its subset AD=>BC does not have high confidence Tan,Steinbach, Kumar Introduction to Data Mining D=>ABC 4/18/2004 30 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 31 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 32 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 1 1 1 1 1 1 1 1 1 1 2 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 3 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 4 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 5 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 6

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 7 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 8 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 9 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 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 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 10 Number of frequent itemsets 3 k 10 k 1 Need a compact representation Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 33 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 Infrequent Itemsets Tan,Steinbach, Kumar ABCE ABDE ABCD E Introduction to Data Mining ACDE BCDE Border 4/18/2004 34 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 35 Maximal vs Closed Itemsets TID Items 1

ABC 2 ABCD 3 BCE 4 ACDE 5 DE Transaction Ids null 124 123 A 12 124 AB 12 24 AC ABC B AE 24 ABD ABE 2 2 ACD 3

BD 4 345 D BC 2 4 ACE BE ADE BCD E 24 3 CD BCE 34 CE 45 4 BDE 4 ABCD ABCE Not supported by any transactions Tan,Steinbach, Kumar 245 C 123 4 AD

2 1234 ABDE ACDE BCDE ABCDE Introduction to Data Mining 4/18/2004 36 DE CDE Maximal vs Closed Frequent Itemsets Minimum support = 2 124 123 A 12 124 AB 12 ABC 24 AC AD ABD ABE 2 1234 B AE

345 D 2 BC 3 BD 4 ACD 245 C 123 4 24 2 Closed but not maximal null 2 4 ACE BE ADE BCD E 24 3 CD BCE Closed and maximal 34 CE

BDE 45 4 DE CDE 4 ABCD ABCE ABDE ACDE BCDE # Closed = 9 # Maximal = 4 ABCDE Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 37 Maximal vs Closed Itemsets Frequent Itemsets Closed Frequent Itemsets Maximal Frequent Itemsets Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 38 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 39 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 40 Alternative Methods for Frequent Itemset Generation Traversal of Itemset Lattice Breadth-first vs Depth-first (a) Breadth first Tan,Steinbach, Kumar (b) Depth first Introduction to Data Mining 4/18/2004 41 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 42 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

43 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: A:1 null B:1 B:1 C:1 D:1 Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 44 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 A B C D E Pointer 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 45 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 D:1 Frequent Itemsets found (with sup > 1): AD, BD, CD, ABD, ACD, BCD D:1 Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 46 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 47 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 48 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 49 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

50 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 51 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 52 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 53 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 54 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 55 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 56 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 57 Lift and Interest (Another Example) Example 2: X and Y: positively correlated, X and Z, negatively related support and confidence of X=>Z dominates 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% Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 58 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 59 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 60 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 61 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 Ex. 1. sum(S.price) v is anti-monotone Ex. 2. range(S.profit) 15 is anti-monotone 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 TID Transaction 10 a, b, c, d, f 20 b, c, d, f, g, h 40 c, e, f, g Item Price Profit a

60 40 b 20 0 c 80 -20 d 30 10 e 70 -30 f 100 30 g 50 20 h 40 -10 4/18/2004 62 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 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 63 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 If an itemset afb violates C So It

does afbh, afb* becomes anti-monotone! Tan,Steinbach, Kumar Introduction to Data Mining TID Transaction 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 64 Strongly Convertible Constraints avg(X) 25 is convertible anti-monotone w.r.t. item value descending order R: 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: 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 65