Question? Leave a message!




Data Mining Association Analysis: Basic Concepts and Algorithms

Data Mining Association Analysis: Basic Concepts and Algorithms
Data Mining Association Analysis: Basic Concepts and Algorithms © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1Association 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 MarketBasket transactions Example of Association Rules TID Items Diaper  Beer, 1 Bread, Milk Milk, Bread  Eggs,Coke, 2 Bread, Diaper, Beer, Eggs Beer, Bread  Milk, 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer Implication means cooccurrence, not causality 5 Bread, Milk, Diaper, Coke © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Definition: Frequent Itemset  Itemset – A collection of one or more items  Example: Milk, Bread, Diaper TID Items – kitemset 1 Bread, Milk  An itemset that contains k items 2 Bread, Diaper, Beer, Eggs  Support count () 3 Milk, Diaper, Beer, Coke – Frequency of occurrence of an itemset 4 Bread, Milk, Diaper, Beer – E.g. (Milk, Bread,Diaper) = 2 5 Bread, Milk, Diaper, Coke  Support – 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 TID Items – An implication expression of the form 1 Bread, Milk X  Y, where X and Y are itemsets 2 Bread, Diaper, Beer, Eggs – Example: 3 Milk, Diaper, Beer, Coke Milk, Diaper  Beer 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke  Rule Evaluation Metrics – Support (s) Example:  Fraction of transactions that contain both X and Y Milk,Diaper Beer – Confidence (c)  (Milk,Diaper,Beer) 2  Measures how often items in Y s 0.4 appear in transactions that T 5 contain X  (Milk, Diaper,Beer) 2 c 0.67  (Milk,Diaper) 3 © 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  Bruteforce 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 Milk,Diaper  Beer (s=0.4, c=0.67) 1 Bread, Milk Milk,Beer  Diaper (s=0.4, c=1.0) 2 Bread, Diaper, Beer, Eggs Diaper,Beer  Milk (s=0.4, c=0.67) 3 Milk, Diaper, Beer, Coke Beer  Milk,Diaper (s=0.4, c=0.67) 4 Bread, Milk, Diaper, Beer Diaper  Milk,Beer (s=0.4, c=0.5) 5 Bread, Milk, Diaper, Coke 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  Twostep 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 BCDE Given d items, there d are 2 possible candidate itemsets ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Frequent Itemset Generation  Bruteforce approach: – Each itemset in the lattice is a candidate frequent itemset – Count the support of each candidate by scanning the database List of Transactions Candidates TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke M N 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke w – Match each transaction against every candidate d – Complexity O(NMw) = Expensive since M = 2 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Computational Complexity  Given d unique items: d – Total number of itemsets = 2 – Total number of possible association rules: d dk  d1 dk  R   k1 j1 k j   d d1  3 2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) d – Complete search: M=2 – Use pruning techniques to reduce M  Reduce the number of transactions (N) – Reduce size of N as the size of itemset increases – Used by DHP and verticalbased mining algorithms  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 antimonotone property of support © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Illustrating Apriori Principle n nu ullll A A B B C C D D E E A AB B A AC C A AD D A AE E B BC C B BD D B BE E C CD D C CE E D DE E Found to be Infrequent A AB BC C A AB BD D A AB BE E A AC CD D A AC CE E A AD DE E B BC CD D B BC CE E B BD DE E C CD DE E A AB BC CD D A AB BC CE E A AB BD DE E A AC CD DE E B BC CD DE E Pruned A AB BC CD DE E supersets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Illustrating Apriori Principle Items (1itemsets) Item Count Bread 4 Coke 2 Milk 4 Pairs (2itemsets) Itemset Count Beer 3 Bread,Milk 3 Diaper 4 Bread,Beer 2 (No need to generate Eggs 1 Bread,Diaper 3 candidates involving Coke Milk,Beer 2 or Eggs) Milk,Diaper 3 Beer,Diaper 3 Minimum Support = 3 Triplets (3itemsets) Itemset Count If every subset is considered, 6 6 6 Bread,Milk,Diaper 3 C + C + C = 41 1 2 3 With supportbased pruning, 6 + 6 + 1 = 13 © Tan,Steinbach, Kumar Introduction to Data Mining 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 ‹›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 Hash Structure Transactions TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs k 3 Milk, Diaper, Beer, Coke N 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke 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) 2 3 4 Hash function 5 6 7 3,6,9 1,4,7 3 6 7 1 4 5 3 5 6 3 4 5 1 3 6 3 6 8 3 5 7 2,5,8 6 8 9 1 2 4 1 2 5 1 5 9 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Association Rule Discovery: Hash tree Hash Function Candidate Hash Tree 1,4,7 3,6,9 2,5,8 2 3 4 5 6 7 1 4 5 1 3 6 3 4 5 3 5 6 3 6 7 Hash on 3 5 7 3 6 8 1, 4 or 7 6 8 9 1 2 4 1 5 9 1 2 5 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Association Rule Discovery: Hash tree Hash Function Candidate Hash Tree 1,4,7 3,6,9 2,5,8 2 3 4 5 6 7 1 4 5 1 3 6 3 4 5 3 5 6 3 6 7 Hash on 3 5 7 3 6 8 2, 5 or 8 6 8 9 1 2 4 1 5 9 1 2 5 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Association Rule Discovery: Hash tree Hash Function Candidate Hash Tree 1,4,7 3,6,9 2,5,8 2 3 4 5 6 7 1 4 5 1 3 6 3 4 5 3 5 6 3 6 7 Hash on 3 5 7 3 6 8 3, 6 or 9 6 8 9 1 2 4 1 5 9 1 2 5 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Subset Operation Given a transaction t, what Transaction, t are the possible subsets of size 3 1 2 3 5 6 Level 1 1 2 3 5 6 2 3 5 6 3 5 6 Level 2 1 2 3 5 6 1 3 5 6 1 5 6 2 3 5 6 2 5 6 3 5 6 1 2 3 1 3 5 2 3 5 1 2 5 1 5 6 2 5 6 3 5 6 1 3 6 2 3 6 1 2 6 Level 3 Subsets of 3 items © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Subset Operation Using Hash Tree Hash Function transaction 1 2 3 5 6 1 + 2 3 5 6 1,4,7 3,6,9 2 + 3 5 6 2,5,8 3 + 5 6 2 3 4 5 6 7 1 4 5 1 3 6 3 4 5 3 5 6 3 6 7 3 5 7 3 6 8 6 8 9 1 2 4 1 5 9 1 2 5 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Subset Operation Using Hash Tree Hash Function transaction 1 2 3 5 6 1 + 2 3 5 6 1,4,7 3,6,9 2 + 3 5 6 1 2 + 3 5 6 2,5,8 3 + 5 6 1 3 + 5 6 2 3 4 1 5 + 6 5 6 7 1 4 5 1 3 6 3 4 5 3 5 6 3 6 7 3 5 7 3 6 8 6 8 9 1 2 4 1 5 9 1 2 5 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Subset Operation Using Hash Tree Hash Function transaction 1 2 3 5 6 1 + 2 3 5 6 1,4,7 3,6,9 2 + 3 5 6 1 2 + 3 5 6 2,5,8 3 + 5 6 1 3 + 5 6 2 3 4 1 5 + 6 5 6 7 1 4 5 1 3 6 3 4 5 3 5 6 3 6 7 3 5 7 3 6 8 6 8 9 1 2 4 1 5 9 1 2 5 4 5 7 4 5 8 Match transaction against 11 out of 15 candidates © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Factors Affecting Complexity  Choice of minimum support threshold – 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 – 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 – 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 ‹›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 10   3  Number of frequent itemsets k1 k   Need a compact representation © 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 A B C D E Itemsets AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE BCDE Infrequent Itemsets Border ABCD E © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Closed Itemset  An itemset is closed if none of its immediate supersets has the same support as the itemset Itemset Support A 4 TID Items Itemset Support B 5 1 A,B A,B,C 2 C 3 2 B,C,D A,B,D 3 D 4 3 A,B,C,D A,C,D 2 A,B 4 4 A,B,D B,C,D 3 A,C 2 5 A,B,C,D A,B,C,D 2 A,D 3 B,C 3 B,D 4 C,D 3 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Maximal vs Closed Itemsets Transaction Ids null TID Items 1 ABC 124 123 1234 245 345 A B C D E 2 ABCD 3 BCE 12 124 24 123 4 ACDE 4 2 3 24 34 45 AB AC AD AE BC BD BE CD CE DE 5 DE 12 24 2 2 4 4 3 4 ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE 4 2 ABCD ABCE ABDE ACDE BCDE Not supported by any transactions ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Maximal vs Closed Frequent Itemsets Closed but null Minimum support = 2 not maximal 124 123 1234 245 345 A B C D E Closed and maximal 12 124 24 123 4 2 3 24 34 45 AB AC AD AE BC BD BE CD CE DE 12 24 2 2 4 4 3 4 ABC ABD ABE ACD ACE ADE BCD 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 – Generaltospecific vs Specifictogeneral Frequent Frequent itemset null null itemset null border border .. .. .. .. .. .. Frequent a ,a ,...,a a ,a ,...,a itemset a ,a ,...,a 1 2 n 1 2 n 1 2 n border (a) Generaltospecific (b) Specifictogeneral (c) Bidirectional © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Alternative Methods for Frequent Itemset Generation  Traversal of Itemset Lattice – Equivalent Classes null null A B D A B D C C AB AC AD BC BD AB AC CD BC AD BD CD ABD ACD BCD ABC ABD ACD ABC BCD ABCD ABCD (a) Prefix tree (b) Suffix tree © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Alternative Methods for Frequent Itemset Generation  Traversal of Itemset Lattice – Breadthfirst vs Depthfirst (a) Breadth first (b) Depth 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 Vertical Data Layout Data Layout TID Items A B C D E 1 A,B,E 1 1 2 2 1 2 B,C,D 4 2 3 4 3 3 C,E 5 5 4 5 6 4 A,C,D 6 7 8 9 5 A,B,C,D 7 8 9 6 A,E 8 10 7 A,B 9 8 A,B,C 9 A,C,D 10 B © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›FPgrowth Algorithm  Use a compressed representation of the database using an FPtree  Once an FPtree has been constructed, it uses a recursive divideandconquer approach to mine the frequent itemsets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›FPtree construction null After reading TID=1: A:1 TID Items 1 A,B B:1 2 B,C,D 3 A,C,D,E 4 A,D,E After reading TID=2: null 5 A,B,C 6 A,B,C,D B:1 A:1 7 B,C 8 A,B,C B:1 C:1 9 A,B,D 10 B,C,E D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›FPTree Construction TID Items Transaction 1 A,B Database 2 B,C,D null 3 A,C,D,E 4 A,D,E 5 A,B,C B:3 A:7 6 A,B,C,D 7 B,C 8 A,B,C B:5 C:3 9 A,B,D C:1 D:1 10 B,C,E Header table D:1 C:3 E:1 D:1 E:1 Item Pointer D:1 A B E:1 D:1 C Pointers are used to assist D frequent itemset generation E © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›FPgrowth Conditional Pattern base null for D: P = (A:1,B:1,C:1), (A:1,B:1), A:7 B:1 (A:1,C:1), (A:1), B:5 (B:1,C:1) C:1 C:1 D:1 Recursively apply FP growth on P D:1 C:3 D:1 Frequent Itemsets found D:1 (with sup 1): AD, BD, CD, ACD, BCD D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Tree Projection null Set enumeration tree: A B C D E Possible Extension: E(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 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 Projected Database Original Database: for node A: TID Items TID Items 1 A,B 1 B 2 B,C,D 2 3 A,C,D,E 3 C,D,E 4 A,D,E 4 D,E 5 A,B,C 5 B,C 6 A,B,C,D 6 B,C,D 7 B,C 7 8 A,B,C 8 B,C 9 A,B,D 9 B,D 10 B,C,E 10 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 Vertical Data Layout TID Items A B C D E 1 A,B,E 1 1 2 2 1 2 B,C,D 4 2 3 4 3 3 C,E 5 5 4 5 6 4 A,C,D 6 7 8 9 5 A,B,C,D 7 8 9 6 A,E 8 10 7 A,B 9 8 A,B,C 9 A,C,D TIDlist 10 B © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›ECLAT  Determine support of any kitemset by intersecting tidlists of two of its (k1) subsets. AB B A 1 1 1 2 5 4 5 5 7  7 6 8 8 7 10 8 9  3 traversal approaches: – topdown, bottomup and hybrid  Advantage: very fast support counting  Disadvantage: intermediate tidlists may become too large for memory © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rule Generation  Given a frequent itemset L, find all nonempty 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, ABD C, ACD B, BCD A, A BCD, B ACD, C ABD, D ABC AB CD, AC  BD, AD  BC, BC AD, BD AC, CD AB, k  If L = k, then there are 2 – 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 anti monotone property c(ABC D) can be larger or smaller than c(AB D) – But confidence of rules generated from the same itemset has an antimonotone property – e.g., L = A,B,C,D: c(ABC  D)  c(AB  CD)  c(A  BCD)  Confidence is antimonotone 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 A AB BC CD D= = Low Confidence Rule B BC CD D= = A A A AC CD D= = B B A AB BD D= = C C A AB BC C= = D D C CD D= = A AB B B BD D= = A AC C B BC C= = A AD D A AD D= = B BC C A AC C= = B BD D A AB B= = C CD D D D= = A AB BC C C C= = A AB BD D B B= = A AC CD D A A= = B BC CD D 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 antimonotone  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 AB ABC Item MS(I) Sup(I) AC ABD A A 0.10 0.25 AD ABE AE ACD B B 0.20 0.26 BC ACE C C 0.30 0.29 BD ADE BE BCD D D 0.50 0.05 CD BCE E E 3 4.20 CE BDE DE CDE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Multiple Minimum Support AB ABC Item MS(I) Sup(I) AC ABD A A 0.10 0.25 AD ABE AE ACD B B 0.20 0.26 BC ACE C BD ADE C 0.30 0.29 BE BCD D D 0.50 0.05 CD BCE E CE BDE E 3 4.20 DE CDE © 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: – L : set of frequent items 1 – F : set of items whose support is  MS(1) 1 where MS(1) is min ( MS(i) ) i – C : candidate itemsets of size 2 is generated from F 2 1 instead of L 1 © 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 f : support of X and Y 11 X f f f f : support of X and Y 11 10 1+ 10 f : support of X and Y X f f f 01 01 00 o+ f : support of X and Y 00 f f T +1 +0 Used to define various measures  support, confidence, lift, Gini, Jmeasure, etc. © Tan,Steinbach, Kumar 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(CoffeeTea) = 0.75 but P(Coffee) = 0.9  Although confidence is high, rule is misleading  P(CoffeeTea) = 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(SB) = 420/1000 = 0.42 – P(S)  P(B) = 0.6  0.7 = 0.42 – 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Statisticalbased Measures  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 ) P(X ,Y )P(X )P(Y ) coefficien t P(X )1P(X )P(Y )1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(CoffeeTea) = 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 ‹›Drawback of Lift Interest Y Y Y Y X 10 0 10 X 90 0 90 X 0 90 90 X 0 10 10 10 90 100 90 10 100 0.1 0.9 Lift10 Lift1.11 (0.1)(0.1) (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 Apriori style support based pruning How does it affect these measuresProperties of A Good Measure  PiatetskyShapiro: 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 f f f f Example 11 10 01 00 10 examples of E1 8123 83 424 1370 E2 8330 2 622 1046 contingency tables: E3 9481 94 127 298 E4 3954 3080 5 2961 E5 2886 1363 1320 4431 E6 1500 2000 500 6000 E7 4000 2000 1000 3000 E8 4000 2000 2000 2000 Rankings of contingency tables E9 1720 7121 5 1154 E10 61 2483 4 7452 using various measures: © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Property under Variable Permutation B A B A A p q B p r r s q s A B Does M(A,B) = M(B,A) Symmetric measures:  support, lift, collective strength, cosine, Jaccard, etc Asymmetric measures:  confidence, conviction, Laplace, Jmeasure, etc © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Property under Row/Column Scaling GradeGender Example (Mosteller, 1968): 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 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 A B C D E F Transaction 1 1 0 0 1 0 0 0 0 1 1 1 0 . 0 0 1 1 1 0 . 0 0 1 1 1 0 0 1 1 0 1 1 . 0 0 1 1 1 0 . 0 0 1 1 1 0 0 0 1 1 1 0 . 0 0 1 1 1 0 Transaction N 1 0 0 1 0 0 (a) (b) (c) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example: Coefficient coefficient is analogous to correlation coefficient for continuous variables Y Y Y Y X 60 10 70 X 20 10 30 X 10 20 30 X 10 60 70 70 30 100 30 70 100 0.6 0.70.7 0.2 0.30.3  0.70.30.70.3 0.70.30.70.3  0.5238 0.5238  Coefficient is the same for both tables © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Property under Null Addition B B B B A p q p q A r s r s + k A A Invariant measures:  support, cosine, Jaccard, etc Noninvariant measures:  correlation, Gini, mutual information, odds ratio, etc © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Different Measures have Different Properties Symbol Measure Range P1 P2 P3 O1 O2 O3 O3' O4  Correlation 1 … 0 … 1 Yes Yes Yes Yes No Yes Yes No  Lambda 0 … 1 Yes No No Yes No No Yes No  Odds ratio 0 … 1 …  Yes Yes Yes Yes Yes Yes Yes No Q Yule's Q 1 … 0 … 1 Yes Yes Yes Yes Yes Yes Yes No Y Yule's Y 1 … 0 … 1 Yes Yes Yes Yes Yes Yes Yes No  Cohen's 1 … 0 … 1 Yes Yes Yes Yes No No Yes No M Mutual Information 0 … 1 Yes Yes Yes Yes No No Yes No J JMeasure 0 … 1 Yes No No No No No No No G Gini Index 0 … 1 Yes No No No No No Yes No s Support 0 … 1 No Yes No Yes No No No No c Confidence 0 … 1 No Yes No Yes No No No Yes L Laplace 0 … 1 No Yes No Yes No No No No V Conviction 0.5 … 1 …  No Yes No Yes No No Yes No I Interest 0 … 1 …  Yes Yes Yes Yes No No No No IS IS (cosine) 0 .. 1 No Yes Yes Yes No No No Yes PS PiatetskyShapiro's 0.25 … 0 … 0.25 Yes Yes Yes Yes No Yes Yes No F Certainty factor 1 … 0 … 1 Yes Yes Yes No No No Yes No AV Added value 0.5 … 1 … 1 Yes Yes Yes No No No No No S Collective strength 0 … 1 …  No Yes Yes Yes No Yes Yes No  Jaccard 0 .. 1 No Yes Yes Yes No No No Yes  2 1 2  12 30 Klosgen's Yes Yes Yes No No No No No K  © Tan,Steinbach, Kumar 3 Introduct3 ion to Da 3ta 3 Mining 4/18/2004 ‹›  Supportbased 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 supportbased pruning and examine the tables that are removed © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Effect of Supportbased Pruning All Itempairs 1000 900 800 700 600 500 400 300 200 100 0 Correlation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Effect of Supportbased Pruning Support 0.01 Support 0.03 300 300 250 250 200 200 150 150 100 100 50 50 0 0 Correlation Correlation Support 0.05 300 250 200 Supportbased pruning 150 eliminates mostly 100 negatively correlated 50 itemsets 0 Correlation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Effect of Supportbased Pruning  Investigate how supportbased pruning affects other measures  Steps: – Generate 10000 contingency tables – Rank each table according to the different measures – Compute the pairwise correlation between the measures © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Effect of Supportbased Pruning  Without Support Pruning (All Pairs) All Pairs (40.14) 1 Conviction Odds ratio 0.9 Col Strength Correlation 0.8 Interest 0.7 PS CF 0.6 Yule Y Reliability 0.5 Kappa 0.4 Klosgen Yule Q 0.3 Confidence Laplace 0.2 IS 0.1 Support Jaccard 0 Lambda 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Correlation Gini Jmeasure Mutual Info 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  Red cells indicate correlation between the pair of measures 0.85  40.14 pairs have correlation 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› JaccardEffect of Supportbased Pruning  0.5  support  50 0.005 = support = 0.500 (61.45) 1 Interest Conviction 0.9 Odds ratio Col Strength 0.8 Laplace 0.7 Confidence Correlation 0.6 Klosgen Reliability 0.5 PS 0.4 Yule Q CF 0.3 Yule Y Kappa 0.2 IS Jaccard 0.1 Support 0 Lambda 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Gini Correlation Jmeasure Mutual Info 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:  61.45 pairs have correlation 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› JaccardEffect of Supportbased Pruning  0.5  support  30 0.005 = support = 0.300 (76.42) Support 1 Interest 0.9 Reliability Conviction 0.8 Yule Q Odds ratio 0.7 Confidence 0.6 CF Yule Y 0.5 Kappa Correlation 0.4 Col Strength 0.3 IS Jaccard 0.2 Laplace PS 0.1 Klosgen 0 Lambda 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Mutual Info Correlation Gini Jmeasure 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  76.42 pairs have correlation 0.85 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› JaccardSubjective Interestingness Measure  Objective measure: – Rank patterns based on statistics computed from data – e.g., 21 measures of association (support, confidence, Laplace, Gini, mutual information, Jaccard, etc).  Subjective measure: – Rank patterns according to user’s interpretation  A pattern is subjectively interesting if it contradicts the expectation of a user (Silberschatz Tuzhilin)  A pattern is subjectively interesting if it is actionable (Silberschatz Tuzhilin) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Interestingness via Unexpectedness  Need to model expectation of users (domain knowledge) Pattern expected to be frequent + Pattern expected to be infrequent Pattern found to be frequent Pattern found to be infrequent Expected Patterns + Unexpected Patterns +  Need to combine expectation of users with evidence from data (i.e., extracted patterns) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Interestingness via Unexpectedness  Web Data (Cooley et al 2001) – Domain knowledge in the form of site structure – Given an itemset F = X , X , …, X (X : Web pages) 1 2 k i  L: number of links connecting the pages  lfactor = L / (k  k1)  cfactor = 1 (if graph is connected), 0 (disconnected graph) – Structure evidence = cfactor  lfactor P(X X... X ) 1 2 k  – Usage evidence P(XX...X ) 1 2 k – Use DempsterShafer theory to combine domain knowledge and evidence from data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›
Website URL
Comment