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Data Mining Association Analysis: Basic Concepts and Algorithms Association Rule Mining l Given a set of transactions, find rules that will predict the occurrence of an item based on the occurrences of other items in the transaction Lecture Notes for Chapter 6 Market-Basket transactions Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Definition: Frequent Itemset l TID Items 1 2 3 4 5 Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke © Tan,Steinbach, Kumar l u l An itemset that contains k items – Frequency of occurrence of an itemset – E.g. σ({Milk, Bread,Diaper}) = 2 l Support TID Items 1 Bread, Milk 2 3 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke 4 5 Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke l Rule Evaluation Metrics u u Measures how often items in Y appear in transactions that contain X 4/18/2004 3 Association Rule Mining Task Given a set of transactions T, the goal of association rule mining is to find all rules having © Tan,Steinbach, Kumar {Milk , Diaper } ⇒ Beer σ (Milk, Diaper,Beer) 2 = = 0.67 σ (Milk, Diaper) 3 4/18/2004 4 4/18/2004 Example of Rules: Items 1 2 3 4 5 Bread, Milk 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: – List all possible association rules – Compute the support and confidence for each rule – Prune rules that fail the minsup and minconf thresholds ⇒ Computationally prohibitive! Introduction to Data Mining Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke Introduction to Data Mining TID Brute-force approach: © Tan,Steinbach, Kumar Bread, Milk 2 3 4 5 Mining Association Rules – support = minsup threshold – confidence = minconf threshold l Items 1 σ ( Milk , Diaper, Beer) 2 s= = = 0.4 |T| 5 c= – An itemset whose support is greater than or equal to a minsup threshold Introduction to Data Mining TID Example: Fraction of transactions that contain both X and Y – Confidence (c) – E.g. s({Milk, Bread, Diaper}) = 2/5 l 2 – Support (s) Frequent Itemset © Tan,Steinbach, Kumar 4/18/2004 – Example: {Milk, Diaper} → {Beer} – Fraction of transactions that contain an itemset l Introduction to Data Mining – An implication expression of the form X → Y, where X and Y are itemsets Example: {Milk, Bread, Diaper} Support count (σ) Implication means co-occurrence, not causality! Association Rule – A collection of one or more items – k-itemset {Diaper} → {Beer}, {Milk, Bread} → {Eggs,Coke}, {Beer, Bread} → {Milk}, Definition: Association Rule Itemset u Example of Association Rules • 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 5 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 6 Mining Association Rules Frequent Itemset Generation null l Two-step approach: 1. Frequent Itemset Generation – A 2. Rule Generation – l B C D E Generate all itemsets whose support ≥ minsup AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE 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 ABCD ABCE ABDE ACDE BCDE Given d items, there are 2d possible candidate itemsets ABCDE © Tan,Steinbach, Kumar l Introduction to Data Mining 4/18/2004 7 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 Frequent Itemset Generation Computational Complexity Brute-force approach: l Given d unique items: – Total number of itemsets = 2d – Total number of possible association rules: – Each itemset in the lattice is a candidate frequent itemset – Count the support of each candidate by scanning the database d d − k R = ∑ × ∑ k j = 3 − 2 +1 Transactions TID 1 2 3 4 5 8 Items Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke d −1 d −k k =1 j =1 d d +1 If d=6, R = 602 rules – Match each transaction against every candidate – Complexity ~ O(NMw) => Expensive since M = 2d !!! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 9 Frequent Itemset Generation Strategies l Reduce the number of candidates (M) © Tan,Steinbach, Kumar l Reduce the number of transactions (N) l – Use efficient data structures to store the candidates or transactions – No need to match every candidate against every transaction Introduction to Data Mining Apriori principle: 4/18/2004 Apriori principle holds due to the following property of the support measure: ∀X , Y : ( X ⊆ Y ) ⇒ s ( X ) ≥ s (Y ) Reduce the number of comparisons (NM) © Tan,Steinbach, Kumar 10 – If an itemset is frequent, then all of its subsets must also be frequent – Reduce size of N as the size of itemset increases – Used by DHP and vertical-based mining algorithms l 4/18/2004 Reducing Number of Candidates – Complete search: M=2d – Use pruning techniques to reduce M l Introduction to Data Mining 11 – 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 Illustrating Apriori Principle Item Bread Coke Milk Beer Diaper Eggs Found to be Infrequent Items (1-itemsets) Itemset {Bread,Milk} {Bread,Beer} {Bread,Diaper} {Milk,Beer} {Milk,Diaper} {Beer,Diaper} Minimum Support = 3 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 13 Apriori Algorithm © Tan,Steinbach, Kumar Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Item s e t {Bread,Milk,Diaper} Introduction to Data Mining Count 3 4/18/2004 14 Reducing Number of Comparisons l Method: 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 – Let k=1 – Generate frequent itemsets of length 1 – Repeat until no new frequent itemsets are identified Instead of matching each transaction against every candidate, match it against candidates contained in the hashed buckets u u Generate length (k+1) candidate itemsets from length k frequent itemsets u Prune candidate itemsets containing subsets of length k that are infrequent u Count the support of each candidate by scanning the DB u Eliminate candidates that are infrequent, leaving only those that are frequent © Tan,Steinbach, Kumar Count 3 2 3 2 3 3 Triplets (3-itemsets) If every subset is considered, 6C + 6C + 6C = 41 1 2 3 With support-based pruning, 6 + 6 + 1 = 13 Pruned supersets l Count 4 2 4 3 4 1 Introduction to Data Mining 4/18/2004 15 Generate Hash Tree Transactions 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 16 Association Rule Discovery: 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 1,4,7 • Hash function Candidate Hash Tree 3,6,9 2,5,8 • 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 234 567 145 136 345 356 357 689 367 368 Hash on 1, 4 or 7 124 457 159 4/18/2004 17 © Tan,Steinbach, Kumar 125 458 159 Introduction to Data Mining 356 357 689 367 368 4/18/2004 18 Association Rule Discovery: Hash tree Hash Function 1,4,7 Association Rule Discovery: Hash tree Hash Function Candidate Hash Tree 3,6,9 1,4,7 2,5,8 Candidate Hash Tree 3,6,9 2,5,8 234 567 145 234 567 136 145 345 Hash on 2, 5 or 8 124 457 159 125 458 © Tan,Steinbach, Kumar 356 357 689 Introduction to Data Mining 367 368 136 345 124 457 4/18/2004 356 357 689 Hash on 3, 6 or 9 19 Subset Operation 159 125 458 © Tan,Steinbach, Kumar Introduction to Data Mining 367 368 4/18/2004 20 Subset Operation Using Hash Tree Given a transaction t, what are the possible subsets of size 3? 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 21 Subset Operation Using Hash Tree 1+ 2356 159 © Tan,Steinbach, Kumar 2+ 356 4/18/2004 22 Subset Operation Using Hash Tree 1,4,7 3+ 56 367 368 Introduction to Data Mining Hash Function 1 2 3 5 6 transaction 12+ 356 125 458 356 357 689 1+ 2356 3,6,9 2+ 356 12+ 356 2,5,8 Hash Function 1 2 3 5 6 transaction 1,4,7 3+ 56 13+ 56 3,6,9 2,5,8 13+ 56 234 567 15+ 6 145 136 145 345 124 457 © Tan,Steinbach, Kumar 125 458 234 567 15+ 6 159 Introduction to Data Mining 356 357 689 136 367 368 345 124 457 4/18/2004 23 © 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 24 Factors Affecting Complexity l Choice of minimum support threshold – – l 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 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 – l l lowering support threshold results in more frequent itemsets this may increase number of candidates and max length of frequent itemsets Dimensionality (number of items) of the data set – – l Compact Representation 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 25 l Need a compact representation © Tan,Steinbach, Kumar An itemset is maximal frequent if none of its immediate supersets is frequent l TID 1 2 3 4 5 Introduction to Data Mining 4/18/2004 27 2 ABCD 3 BCE 4 ACDE 5 DE Itemset {A} {B} {C} {D} {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} Items {A,B} {B,C,D} {A,B,C,D} {A,B,D} {A,B,C,D} © Tan,Steinbach, Kumar Maximal vs Closed Itemsets ABC 26 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 Border © Tan,Steinbach, Kumar Items 4/18/2004 An itemset is closed if none of its immediate supersets has the same support as the itemset Maximal Itemsets 1 k =1 Introduction to Data Mining Closed Itemset TID 10 Number of frequent itemsets Maximal Frequent Itemset Infrequent Itemsets 10 = 3× ∑ k l 123 Transaction Ids A 12 124 AB 12 24 AC AD ABD ABE B 245 C 123 4 AE 24 2 ABC 1234 2 3 BD 4 ACD CD 34 CE 45 3 BCD BDE ABCE ABDE 124 AB 12 CDE ABC 24 AC AD ABD ABE B AE ACDE 2 3 BD BE 2 4 ACE ADE E 24 CD Closed and maximal 34 CE 3 BCD 45 DE 4 BCE BDE CDE 4 ABCD BCDE 345 D BC 4 ACD 245 C 123 4 24 2 1234 2 4 ABCD DE 4 BCE 123 A 12 24 2 ADE 2 28 Closed but not maximal null 124 E BE 4 ACE Minimum support = 2 345 D BC 4/18/2004 Maximal vs Closed Frequent Itemsets null 124 Introduction to Data Mining ABCE ABDE ACDE BCDE # Closed = 9 # Maximal = 4 Not supported by any transactions © Tan,Steinbach, Kumar ABCDE ABCDE Introduction to Data Mining 4/18/2004 29 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 30 Maximal vs Closed Itemsets Alternative Methods for Frequent Itemset Generation l Traversal of Itemset Lattice – General-to-specific vs Specific-to-general © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 31 Alternative Methods for Frequent Itemset Generation l Traversal of Itemset Lattice © Tan,Steinbach, Kumar l Introduction to Data Mining 4/18/2004 33 Representation of Database 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 Traversal of Itemset Lattice © Tan,Steinbach, Kumar 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 35 Introduction to Data Mining 4/18/2004 34 FP-growth Algorithm l Use a compressed representation of the database using an FP-tree l Once an FP-tree has been constructed, it uses a recursive divide-and-conquer approach to mine the frequent itemsets – horizontal vs vertical data layout TID 1 2 3 4 5 6 7 8 9 10 32 – Breadth-first vs Depth-first Alternative Methods for Frequent Itemset Generation l 4/18/2004 Alternative Methods for Frequent Itemset Generation – Equivalent Classes © Tan,Steinbach, Kumar Introduction to Data Mining © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 36 FP-tree construction 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} TID 1 2 3 4 5 6 7 8 9 10 A:1 B:1 After reading TID=2: null A:1 C:1 D:1 © Tan,Steinbach, Kumar Introduction to Data Mining Transaction Database 4/18/2004 37 FP-growth Item A B C D E null B:3 A:7 B:5 Header table B:1 B:1 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} C:1 D:1 C:3 Pointer D:1 D:1 Pointers are used to assist frequent itemset generation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 A:7 C:3 38 Tree Projection null B:1 C:1 E:1 E:1 E:1 D:1 null Set enumeration tree: B:5 C:3 D:1 D:1 C:1 D:1 D: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)} Recursively apply FPgrowth on P Frequent Itemsets found (with sup > 1): AD, BD, CD, ACD, BCD D:1 D:1 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 39 Tree Projection Introduction to Data Mining 4/18/2004 40 Projected Database Items are listed in lexicographic order l Each node P stores the following information: l – – – – © Tan,Steinbach, Kumar Original Database: TID 1 2 3 4 5 6 7 8 9 10 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 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 41 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 42 ECLAT l 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 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 B 1 2 5 7 8 10 C 2 3 4 8 9 D 2 4 5 9 E 1 3 6 3 traversal approaches: l Advantage: very fast support counting Disadvantage: intermediate tid-lists may become too large for memory l 4/18/2004 43 © Tan,Steinbach, Kumar l Introduction to Data Mining 4/18/2004 44 Rule Generation l ACD →B, C →ABD, AD → BC, How to efficiently generate rules from frequent itemsets? – In general, confidence does not have an antimonotone property – If {A,B,C,D} is a frequent itemset, candidate rules: ABD →C, B →ACD, AC → BD, CD →AB, → AB 1 5 7 8 l TID-list Introduction to Data Mining ∧ B 1 2 5 7 8 10 – top-down, bottom-up and hybrid Given a frequent itemset L, find all non-empty subsets f ⊂ L such that f → L – f satisfies the minimum confidence requirement ABC →D, A →BCD, AB →CD, BD →AC, 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 Rule Generation l l c(ABC →D) can be larger or smaller than c(AB →D) BCD →A, D →ABC BC →AD, – But confidence of rules generated from the same itemset has an anti-monotone property – e.g., L = {A,B,C,D}: If |L| = k, then there are 2k – 2 candidate association rules (ignoring L → ∅ and ∅ → L) 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 u © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 45 Rule Generation for Apriori Algorithm © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 46 Rule Generation for Apriori Algorithm Lattice of rules l Low Confidence Rule Candidate rule is generated by merging two rules that share the same prefix in the rule consequent CD=>AB l join(CD=>AB,BD=>AC) would produce the candidate rule D => ABC l Prune rule D=>ABC if its subset AD=>BC does not have high confidence BD=>AC D=>ABC Pruned Rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 47 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 48 Effect of Support Distribution l Effect of Support Distribution Many real data sets have skewed support distribution l – 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 Support distribution of a retail data set l © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 49 Multiple Minimum Support l How to set the appropriate minsup threshold? Using a single minimum support threshold may not be effective © Tan,Steinbach, Kumar Introduction to Data Mining Item – 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% MS(I) Sup(I) A 0.10% 0.25% B 0.20% 0.26% C 0.30% 0.29% – Challenge: Support is no longer anti-monotone Suppose: D Support(Milk, Coke) = 1.5% and Support(Milk, Coke, Broccoli) = 0.5% 0.50% 0.05% E u 50 Multiple Minimum Support How to apply multiple minimum supports? u 4/18/2004 3% 4.20% AB ABC AC ABD AD ABE A B AE ACD BC ACE C D BD ADE BE BCD CD BCE CE BDE E {Milk,Coke} is infrequent but {Milk,Coke,Broccoli} is frequent DE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 51 Multiple Minimum Support Item MS(I) © Tan,Steinbach, Kumar AB ABC AC ABD l Sup(I) 0.10% 0.25% B 0.20% 0.26% C 0.30% 0.29% B CDE 4/18/2004 52 Multiple Minimum Support (Liu 1999) A A Introduction to Data Mining AD ABE AE ACD BC ACE BD ADE BE BCD CD BCE CE BDE 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 C D D l – 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 0.50% 0.05% E E 3% 4.20% DE © Tan,Steinbach, Kumar Introduction to Data Mining CDE 4/18/2004 Need to modify Apriori such that: 53 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 54 Multiple Minimum Support (Liu 1999) l Pattern Evaluation Modifications to Apriori: l – In traditional Apriori, A candidate (k+1)-itemset is generated by merging two frequent itemsets of size k u The candidate is pruned if it contains any infrequent subsets of size k u 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 – 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) u {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. u u © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 55 Knowledge Postprocessing l In the original formulation of association rules, support & confidence are the only measures used 4/18/2004 56 Given a rule X → Y, information needed to compute rule interestingness can be obtained from a contingency table Contingency table for X Preprocessed Data Prod Prod uct Prod uct Prod uct Prod uct Prod uct Prod uct Prod uct Prod uct Prod uct uct Featur Featur e Featur e Featur e Featur e Featur e Featur e Featur e Featur e Featur e e Introduction to Data Mining Computing Interestingness Measure l Patterns Interestingness measures can be used to prune/rank the derived patterns © Tan,Steinbach, Kumar Application of Interestingness Measure Interestingness Measures l M ining Selected Data →Y Y Y X f11 f10 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 f1+ Preprocessing Data Used to define various measures u Selection © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 57 Drawback of Confidence © Tan,Steinbach, Kumar Coffee Tea 15 5 Tea 75 5 80 90 10 100 Introduction to Data Mining 4/18/2004 58 Statistical Independence l Coffee support, confidence, lift, Gini, J-measure, etc. 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) 20 Association Rule: Tea → Coffee – P(S∧B) = 420/1000 = 0.42 – P(S) × P(B) = 0.6 × 0.7 = 0.42 Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9 – P(S∧B) = P(S) × P(B) => Statistical independence – P(S∧B) > P(S) × P(B) => Positively correlated – P(S∧B) < P(S) × P(B) => Negatively correlated ⇒ Although confidence is high, rule is misleading ⇒ P(Coffee|Tea) = 0.9375 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 59 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 60 Statistical-based Measures l Example: Lift/Interest Measures that take into account statistical dependence P(Y | X ) Lift = P (Y ) P( X ,Y ) Interest = P ( X ) P (Y ) PS = P ( X , Y ) − P ( X ) P (Y ) φ − coefficien t = © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 80 90 10 100 20 Y Y Introduction to Data Mining 4/18/2004 62 Some measures are good for certain applications, but not for others Y 10 0 10 X 90 0 0 90 90 X 0 10 10 10 90 100 90 10 100 Lift = © Tan,Steinbach, Kumar There are lots of measures proposed in the literature 90 What criteria should we use to determine whether a measure is good or bad? 0.9 = 1.11 (0.9)(0.9) What about Aprioristyle support based pruning? How does it affect these measures? Statistical independence: If P(X,Y)=P(X)P(Y) => Lift = 1 Introduction to Data Mining 4/18/2004 63 Properties of A Good Measure l 5 ⇒ Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated) 61 X © Tan,Steinbach, Kumar 75 but P(Coffee) = 0.9 P ( X , Y ) − P ( X ) P (Y ) P( X )[1 − P ( X )]P (Y )[1 − P (Y )] 0.1 = 10 (0.1)(0.1) 5 Tea Confidence= P(Coffee|Tea) = 0.75 X Lift = Coffee 15 Association Rule: Tea → Coffee Drawback of Lift & Interest Y Coffee Tea Comparing Different Measures 10 examples of contingency tables: Piatetsky-Shapiro: 3 properties a good measure M must satisfy: Example f11 f10 f01 f00 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 8123 8330 9481 3954 2886 1500 4000 4000 1720 61 83 2 94 3080 1363 2000 2000 2000 7121 2483 424 622 127 5 1320 500 1000 2000 5 4 1370 1046 298 2961 4431 6000 3000 2000 1154 7452 – 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 Rankings of contingency tables using various measures: – 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 65 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 66 Property under Variable Permutation Property under Row/Column Scaling Grade-Gender Example (Mosteller, 1968): B p r A A B q s B B A p q A r s Male Female Male Female High 2 3 5 High 4 30 34 Low 1 4 5 Low 2 40 42 3 7 10 6 70 76 2x 10x Does M(A,B) = M(B,A)? Symmetric measures: u support, lift, collective strength, cosine, Jaccard, etc Asymmetric measures: u confidence, conviction, Laplace, J-measure, etc © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 67 . . . . . 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) l 4/18/2004 A A A A B p r B q s+k support, cosine, Jaccard, etc correlation, Gini, mutual information, odds ratio, etc © Tan,Steinbach, Kumar Introduction to Data Mining Y Y Y Y X 60 10 70 X 20 10 X 10 20 30 X 10 60 70 70 30 100 30 70 100 30 0.2 − 0.3 × 0.3 0.7 × 0.3 × 0.7 × 0.3 = 0.5238 φ= © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 70 Different Measures have Different Properties Non-invariant measures: u 68 φ Coefficient is the same for both tables 69 Invariant measures: u 4/18/2004 φ-coefficient is analogous to correlation coefficient for continuous variables 0.6 − 0.7 × 0.7 0.7 × 0.3 × 0.7 × 0.3 = 0.5238 Property under Null Addition B q s Introduction to Data Mining φ= (c) Introduction to Data Mining B p r © Tan,Steinbach, Kumar Example: φ-Coefficient Property under Inversion Operation Transaction 1 Mosteller: Underlying association should be independent of the relative number of male and female students in the samples 4/18/2004 71 Symbol Measure Range P1 P2 P3 O1 O2 O3 O3' Φ λ α Q Y κ M J G s c L V I IS PS F AV S ζ Correlation -1 … 0 … 1 Yes Yes Yes Yes No Yes Yes No Lambda 0…1 Yes No No Yes No No* Yes No O4 Odds ratio 0…1…∞ Yes* Yes Yes Yes Yes Yes* Yes No Yule's Q -1 … 0 … 1 Yes Yes Yes Yes Yes Yes Yes No Yule's Y -1 … 0 … 1 Yes Yes Yes Yes Yes Yes Yes Cohen's -1 … 0 … 1 Yes Yes Yes Yes No No Yes No Mutual Information 0…1 Yes Yes Yes Yes No No* Yes No J-Measure 0…1 Yes No No No No No No No Gini Index 0…1 Yes No No No No No* Yes Support 0…1 No Yes No Yes No No No No Confidence 0…1 No Yes No Yes No No No Yes No No Laplace 0…1 No Yes No Yes No No No No Conviction 0.5 … 1 … ∞ No Yes No Yes** No No Yes No Interest 0…1…∞ Yes* Yes Yes Yes No No No No IS (cosine) 0 .. 1 No Yes Yes Yes No No No Yes Piatetsky-Shapiro's -0.25 … 0 … 0.25 Yes Yes Yes Yes No Yes Yes No No Yes No Certainty factor -1 … 0 … 1 Yes Yes Yes No No Added value 0.5 … 1 … 1 Yes Yes Yes No No No No Collective strength 0…1…∞ No Yes Yes Yes No Yes* Yes No 0 .. 1 No Yes Yes Yes No No No Yes Yes Yes No No No 4/18/2004 No Jaccard Klosgen's K © Tan,Steinbach, Kumar 1 2 − 1 2 − 3 − Κ 0 Κ Yes 3 3 to Data 3 3 Mining Introduction 2 No 72 No Support-based Pruning Effect of Support-based Pruning Most of the association rule mining algorithms use support measure to prune rules and itemsets l All Itempairs 1000 900 800 Study effect of support pruning on correlation of itemsets 700 600 500 – Generate 10000 random contingency tables – Compute support and pairwise correlation for each table – Apply support-based pruning and examine the tables that are removed 400 300 200 100 -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 1 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 l Correlation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 73 © Tan,Steinbach, Kumar Effect of Support-based Pruning Support < 0.01 74 Effect of Support-based Pruning 0.7 0.8 1 0.6 0.9 0.5 0.8 1 0.4 0.9 0.3 0.6 0.7 0.2 0.5 0 0.1 0.4 Correlation 0.2 0.3 -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 1 0.9 0.8 0 0.7 50 0 0.5 0.6 100 50 0.3 0.4 150 100 0 0.1 0.2 150 -0.1 200 -0 .3 -0.2 250 200 -0.5 -0.4 300 250 -0.7 -0.6 4/18/2004 Support < 0.03 300 -1 -0 .9 -0.8 Introduction to Data Mining l Investigate how support-based pruning affects other measures l Steps: – Generate 10000 contingency tables – Rank each table according to the different measures – Compute the pair-wise correlation between the measures Correlation Support < 0.05 300 250 Support-based pruning eliminates mostly negatively correlated itemsets 200 150 100 50 0 0.1 -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 75 © Tan,Steinbach, Kumar Effect of Support-based Pruning 4/18/2004 76 Effect of Support-based Pruning Without Support Pruning (All Pairs) u Introduction to Data Mining 0.5% ≤ support ≤ 50% u 0.005 <= support <= 0.500 (61.45%) All Pairs (40.14%) 1 Conviction Odds ratio Conviction 0.9 0.9 Odds ratio Col Strength Correlation 1 Interest 0.8 Col Strength 0.7 Confidence 0.8 Laplace Interest Jaccard Kappa Klosgen 0.7 Correlation 0.6 0.6 Klosgen 0.5 Reliability 0.4 Yule Q Jaccard PS CF Yule Y Reliability PS 0.5 0.4 CF Yule Q 0.3 Laplace 0.3 Yule Y Confidence 0.2 Kappa 0.1 Jaccard 0.2 IS IS Support 0.1 Support Jaccard 0 -1 Lambda Gini -0.8 -0.6 -0.4 -0.2 0 0.2 Correlation 0.4 0.6 0.8 1 0 -1 Lambda Gini -0.8 -0.6 -0.4 -0.2 0 0.2 Correlation 0.4 0.6 0.8 J-measure J-measure Mutual Info Mutual Info 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 u Red cells indicate correlation between the pair of measures > 0.85 u 40.14% pairs have correlation > 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining Scatter Plot between Correlation & Jaccard Measure 1 u 4/18/2004 77 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Scatter Plot between Correlation & Jaccard Measure: 61.45% pairs have correlation > 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 78 1 Subjective Interestingness Measure Effect of Support-based Pruning 0.5% ≤ support ≤ 30% u l 0.005 <= support <= 0.300 (76.42%) Interest 0.9 Reliability Conviction 0.8 Yule Q 0.7 Odds ratio Confidence 0.6 Jaccard CF Yule Y Kappa Objective measure: – Rank patterns based on statistics computed from data – e.g., 21 measures of association (support, confidence, Laplace, Gini, mutual information, Jaccard, etc). 1 Support 0.5 0.4 Correlation Col Strength l 0.3 IS Jaccard 0.2 Subjective measure: Laplace 0.1 PS – Rank patterns according to user’s interpretation Klosgen 0 -0.4 Lambda Mutual Info -0.2 0 0.2 0.4 Correlation 0.6 0.8 1 Gini u A pattern is subjectively interesting if it contradicts the expectation of a user (Silberschatz & Tuzhilin) u A pattern is subjectively interesting if it is actionable (Silberschatz & Tuzhilin) J-measure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Scatter Plot between Correlation & Jaccard Measure u 76.42% pairs have correlation > 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 79 Interestingness via Unexpectedness l Introduction to Data Mining l Web Data (Cooley et al 2001) Pattern expected to be infrequent u L: number of links connecting the pages Pattern found to be frequent u lfactor = L / (k × k-1) Pattern found to be infrequent u cfactor = 1 (if graph is connected), 0 (disconnected graph) – Structure evidence = cfactor × lfactor Expected Patterns – Usage evidence = Unexpected Patterns P ( X Ι X Ι ... Ι X ) P( X ∪ X ∪ ... ∪ X ) 1 1 Need to combine expectation of users with evidence from data (i.e., extracted patterns) © Tan,Steinbach, Kumar Introduction to Data Mining 80 – Domain knowledge in the form of site structure – Given an itemset F = {X1, X2, …, Xk} (Xi : Web pages) Pattern expected to be frequent + - + l 4/18/2004 Interestingness via Unexpectedness Need to model expectation of users (domain knowledge) + - © Tan,Steinbach, Kumar 4/18/2004 81 2 2 k k – Use Dempster-Shafer theory to combine domain knowledge and evidence from data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 82