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Data Mining Association Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 6 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1
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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 Example of Association Rules TID Items 1 Bread, Milk {Diaper} → {Beer}, {Milk, Bread} → {Eggs,Coke}, 2 Bread, Diaper, Beer, Eggs {Beer, Bread} → {Milk}, 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer Implication means co-occurrence, 5 Bread, Milk, Diaper, Coke not causality! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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Definition: Frequent Itemset Itemset – A collection of one or more items ◆ Example: {Milk, Bread, Diaper} – k-itemset TID Items An itemset that contains k items 1 Bread, Milk ◆ Support count () 2 Bread, Diaper, Beer, Eggs – Frequency of occurrence of an itemset 3 Milk, Diaper, Beer, Coke 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 ‹#›
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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: {Milk, Diaper} → {Beer} 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke Rule Evaluation Metrics – Support (s) Fraction of transactions that contain Example: ◆ both X and Y {Milk ,Diaper} Beer – Confidence (c) Measures how often items in Y (Milk,Diaper,Beer) 2 ◆ 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 ‹#›
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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 ‹#›
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Mining Association Rules Example of Rules: TID Items 1 Bread, Milk {Milk,Diaper} → {Beer} (s=0.4, c=0.67) 2 Bread, Diaper, Beer, Eggs {Milk,Beer} → {Diaper} (s=0.4, c=1.0) {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 ‹#›
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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 ‹#›
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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 are 2d possible ABCDE candidate itemsets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 List of Candidates TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs N 3 Milk, Diaper, Beer, Coke M 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke w – Match each transaction against every candidate – Complexity ~ O(NMw) => Expensive since M = 2d !!! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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Computational Complexity Given d unique items: – Total number of itemsets = 2d – Total number of possible association rules: d d −k R = d−1 d−k k=1 k j=1 j = 3d −2d+1 +1 If d=6, R = 602 rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 – Used by DHP and vertical-based 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 ‹#›
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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 ‹#›
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Illustrating Apriori Principle nullnull AA BB CC DD EE ABAB ACAC ADAD AEAE BCBC BDBD BEBE CDCD CECE DEDE Found to be Infrequent ABCABC ABDABD ABEABE ACDACD ACEACE ADEADE BCDBCD BCEBCE BDEBDE CDECDE ABCDABCD ABCEABCE ABDEABDE ACDEACDE BCDEBCDE Pruned supersets ABCDEABCDE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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Illustrating Apriori Principle Item Count Items (1-itemsets) Bread 4 Coke 2 Milk 4 Itemset Count Pairs (2-itemsets) 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 (3-itemsets) If every subset is considered, Itemset Count 6C + 6C + 6C = 41 {Bread,Milk,Diaper} 3 1 2 3 With support-based pruning, 6 + 6 + 1 = 13 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 ‹#›
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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 Hash Structure TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs N 3 Milk, Diaper, Beer, Coke k 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke Buckets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 2 3 4 3,6,9 5 6 7 1,4,7 1 4 5 3 4 5 3 5 6 3 6 7 1 3 6 3 6 8 2,5,8 3 5 7 1 2 4 6 8 9 4 5 7 1 2 5 1 5 9 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 1 2 4 1 2 5 1 5 9 6 8 9 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 1 2 4 1 2 5 1 5 9 6 8 9 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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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 1 2 4 1 2 5 1 5 9 6 8 9 4 5 7 4 5 8 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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