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Data Mining Association Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 6 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Association Rule Mining Given a set of transactions, find rules that will predict the occurrence of an item based on the occurrences of other items in the transaction Market-Basket transactions TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke © Tan,Steinbach, Kumar Introduction to Data Mining Example of Association Rules {Diaper} {Beer}, {Milk, Bread} {Eggs,Coke}, {Beer, Bread} {Milk}, Implication means co-occurrence, not causality! 4/18/2004 ‹#› Definition: Frequent Itemset Itemset – A collection of one or more items Example: {Milk, Bread, Diaper} – k-itemset An itemset that contains k items Support count () – Frequency of occurrence of an itemset – E.g. ({Milk, Bread,Diaper}) = 2 Support TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke – Fraction of transactions that contain an itemset – E.g. s({Milk, Bread, Diaper}) = 2/5 Frequent Itemset – An itemset whose support is greater than or equal to a minsup threshold © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Definition: Association Rule Association Rule – An implication expression of the form X Y, where X and Y are itemsets – Example: {Milk, Diaper} {Beer} Rule Evaluation Metrics TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke – Support (s) Example: Fraction of transactions that contain both X and Y {Milk , Diaper } Beer – Confidence (c) Measures how often items in Y appear in transactions that contain X © Tan,Steinbach, Kumar s (Milk, Diaper, Beer ) |T| 2 0.4 5 (Milk, Diaper, Beer ) 2 c 0.67 (Milk, Diaper ) 3 Introduction to Data Mining 4/18/2004 ‹#› Association Rule Mining Task Given a set of transactions T, the goal of association rule mining is to find all rules having – support ≥ minsup threshold – confidence ≥ minconf threshold Brute-force approach: – List all possible association rules – Compute the support and confidence for each rule – Prune rules that fail the minsup and minconf thresholds Computationally prohibitive! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mining Association Rules Example of Rules: TID Items 1 Bread, Milk 2 3 4 5 Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke {Milk,Diaper} {Beer} (s=0.4, c=0.67) {Milk,Beer} {Diaper} (s=0.4, c=1.0) {Diaper,Beer} {Milk} (s=0.4, c=0.67) {Beer} {Milk,Diaper} (s=0.4, c=0.67) {Diaper} {Milk,Beer} (s=0.4, c=0.5) {Milk} {Diaper,Beer} (s=0.4, c=0.5) Observations: • All the above rules are binary partitions of the same itemset: {Milk, Diaper, Beer} • Rules originating from the same itemset have identical support but can have different confidence • Thus, we may decouple the support and confidence requirements © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mining Association Rules Two-step approach: 1. Frequent Itemset Generation – Generate all itemsets whose support minsup 2. Rule Generation – Generate high confidence rules from each frequent itemset, where each rule is a binary partitioning of a frequent itemset Frequent itemset generation is still computationally expensive © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Frequent Itemset Generation null A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining BCDE Given d items, there are 2d possible candidate itemsets 4/18/2004 ‹#› Frequent Itemset Generation Brute-force approach: – Each itemset in the lattice is a candidate frequent itemset – Count the support of each candidate by scanning the database Transactions N TID 1 2 3 4 5 Items Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke List of Candidates M w – Match each transaction against every candidate – Complexity ~ O(NMw) => Expensive since M = 2d !!! © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Computational Complexity Given d unique items: – Total number of itemsets = 2d – Total number of possible association rules: d d k R k j 3 2 1 d 1 d k k 1 j 1 d d 1 If d=6, R = 602 rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Reducing Number of Candidates Apriori principle: – If an itemset is frequent, then all of its subsets must also be frequent Apriori principle holds due to the following property of the support measure: X , Y : ( X Y ) s( X ) s(Y ) – Support of an itemset never exceeds the support of its subsets – This is known as the anti-monotone property of support © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Illustrating Apriori Principle null A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE Found to be Infrequent ABCD ABCE Pruned supersets © Tan,Steinbach, Kumar Introduction to Data Mining ABDE ACDE BCDE ABCDE 4/18/2004 ‹#› Illustrating Apriori Principle Item Bread Coke Milk Beer Diaper Eggs Count 4 2 4 3 4 1 Items (1-itemsets) Itemset {Bread,Milk} {Bread,Beer} {Bread,Diaper} {Milk,Beer} {Milk,Diaper} {Beer,Diaper} Minimum Support = 3 Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Triplets (3-itemsets) If every subset is considered, 6C + 6C + 6C = 41 1 2 3 With support-based pruning, 6 + 6 + 1 = 13 © Tan,Steinbach, Kumar Count 3 2 3 2 3 3 Introduction to Data Mining Itemset {Bread,Milk,Diaper} Count 3 4/18/2004 ‹#› Apriori Algorithm Method: – Let k=1 – Generate frequent itemsets of length 1 – Repeat until no new frequent itemsets are identified Generate length (k+1) candidate itemsets from length k frequent itemsets Prune candidate itemsets containing subsets of length k that are infrequent Count the support of each candidate by scanning the DB Eliminate candidates that are infrequent, leaving only those that are frequent © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› The Apriori Algorithm — Example Database D TID 100 200 300 400 itemset sup. C1 {1} 2 {2} 3 Scan D {3} 3 {4} 1 {5} 3 Items 134 235 1235 25 C2 itemset sup L2 itemset sup 2 2 3 2 {1 {1 {1 {2 {2 {3 C3 itemset {2 3 5} Scan D {1 3} {2 3} {2 5} {3 5} © Tan,Steinbach, Kumar 2} 3} 5} 3} 5} 5} 1 2 1 2 3 2 L1 itemset sup. {1} {2} {3} {5} 2 3 3 3 C2 itemset {1 2} Scan D {1 {1 {2 {2 {3 3} 5} 3} 5} 5} L3 itemset sup {2 3 5} 2 Introduction to Data Mining 4/18/2004 ‹#› FP-growth Algorithm Use a compressed representation of the database using an FP-tree Once an FP-tree has been constructed, it uses a recursive divide-and-conquer approach to mine the frequent itemsets © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› FP-tree construction null After reading TID=1: TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E} A:1 B:1 After reading TID=2: null A:1 B:1 B:1 C:1 D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› FP-Tree Construction TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E} Header table Item Pointer A B C D E © Tan,Steinbach, Kumar Transaction Database null B:3 A:7 B:5 C:1 C:3 D:1 D:1 C:3 D:1 D:1 D:1 E:1 E:1 E:1 Pointers are used to assist frequent itemset generation Introduction to Data Mining 4/18/2004 ‹#› FP-growth C:1 Conditional Pattern base for D: P = {(A:1,B:1,C:1), (A:1,B:1), (A:1,C:1), (A:1), (B:1,C:1)} D:1 Recursively apply FPgrowth on P null A:7 B:5 B:1 C:1 C:3 D:1 D:1 Frequent Itemsets found (with sup > 1): AD, BD, CD, ACD, BCD D:1 D:1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Projected Database Original Database: TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E} Projected Database for node A: TID 1 2 3 4 5 6 7 8 9 10 Items {B} {} {C,D,E} {D,E} {B,C} {B,C,D} {} {B,C} {B,D} {} For each transaction T, projected transaction at node A is T E(A) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› ECLAT For each item, store a list of transaction ids (tids) Horizontal Data Layout TID 1 2 3 4 5 6 7 8 9 10 © Tan,Steinbach, Kumar Items A,B,E B,C,D C,E A,C,D A,B,C,D A,E A,B A,B,C A,C,D B Vertical Data Layout A 1 4 5 6 7 8 9 B 1 2 5 7 8 10 C 2 3 4 8 9 D 2 4 5 9 E 1 3 6 TID-list Introduction to Data Mining 4/18/2004 ‹#› ECLAT Determine support of any k-itemset by intersecting tid-lists of two of its (k-1) subsets. A 1 4 5 6 7 8 9 B 1 2 5 7 8 10 AB 1 5 7 8 3 traversal approaches: – top-down, bottom-up and hybrid Advantage: very fast support counting Disadvantage: intermediate tid-lists may become too large for memory © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Compact Representation of Frequent Itemsets Some itemsets are redundant because they have identical support as their supersets TID A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 10 Number of frequent itemsets 3 k Need a compact representation 10 k 1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Maximal Frequent Itemset An itemset is maximal frequent if none of its immediate supersets is frequent null Maximal Itemsets A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE Infrequent Itemsets ABCD E © Tan,Steinbach, Kumar Introduction to Data Mining ACDE BCDE Border 4/18/2004 ‹#› Closed Itemset An itemset is closed if none of its immediate supersets has the same support as the itemset TID 1 2 3 4 5 Items {A,B} {B,C,D} {A,B,C,D} {A,B,D} {A,B,C,D} © Tan,Steinbach, Kumar Itemset {A} {B} {C} {D} {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} Introduction to Data Mining Support 4 5 3 4 4 2 3 3 4 3 Itemset Support {A,B,C} 2 {A,B,D} 3 {A,C,D} 2 {B,C,D} 2 {A,B,C,D} 2 4/18/2004 ‹#› Maximal vs Closed Frequent Itemsets Minimum support = 2 124 123 A 12 124 AB 12 ABC 24 AC AD ABD ABE 1234 B AE 345 D 2 3 BC BD 4 ACD 245 C 123 4 24 2 Closed but not maximal null 24 BE 2 4 ACE E ADE CD Closed and maximal 34 CE 3 BCD 45 DE 4 BCE BDE CDE 4 2 ABCD ABCE ABDE ACDE BCDE # Closed = 9 # Maximal = 4 ABCDE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Structural Similarity Size-4 graph Structural similarity Significance similarity g ~ g' F (g) ~ F (g') Sibling Size-5 graph Size-6 graph © Tan,Steinbach, Kumar 2017/5/22 Introduction to Data Mining 4/18/2004 ‹#› 27 Structural Leap Search Leap on g’ subtree if 2 ( g , g ' ) sup ( g ) sup ( g ' ) 2 ( g , g ' ) sup ( g ) sup ( g ' ) : leap length, tolerance of structure/frequency dissimilarity g : a discovered graph Mining Part Leap Part g’: a sibling of g © Tan,Steinbach, Kumar 2017/5/22 Introduction to Data Mining 4/18/2004 ‹#› 28 Branch-and-Bound vs. LEAP Pruning base Feature Optimality Efficiency © Tan,Steinbach, Kumar 2017/5/22 Branch-and-Bound LEAP Parent-child bound Sibling similarity (“vertical”) (“horizontal”) strict pruning approximate pruning Guaranteed Near optimal Good Better Introduction to Data Mining 4/18/2004 ‹#› 29