Data Mining Association Analysis Basic Concepts and Algorithms
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 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 Market-Basket transactions Example of Association Rules {Diaper} {Beer}, {Milk, Bread} {Eggs, Coke}, {Beer, Bread} {Milk}, Implication means co-occurrence, not causality! © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 2
Definition: Frequent Itemset l Itemset – A collection of one or more items u Example: {Milk, Bread, Diaper} – k-itemset u l An itemset that contains k items Support count ( ) – Frequency of occurrence of an itemset – E. g. ({Milk, Bread, Diaper}) = 2 l Support – Fraction of transactions that contain an itemset – E. g. s({Milk, Bread, Diaper}) = 2/5 l 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 l Association Rule – An implication expression of the form X Y, where X and Y are itemsets – Example: {Milk, Diaper} {Beer} l Rule Evaluation Metrics – Support (s) u Fraction of transactions that contain both X and Y Example: – Confidence (c) u Measures how often items in Y appear in transactions that contain X © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 4
Association Rule Mining Task l Given a set of transactions T, the goal of association rule mining is to find all rules having – support ≥ minsup threshold – confidence ≥ minconf threshold l 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 5
Mining Association Rules Example of Rules: {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 6
Mining Association Rules l Two-step approach: 1. Frequent Itemset Generation – Generate all itemsets whose support minsup 2. Rule Generation – l 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 7
Frequent Itemset Generation Given d items, there are 2 d possible candidate itemsets © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 8
Frequent Itemset Generation l Brute-force approach: – Each itemset in the lattice is a candidate frequent itemset – Count the support of each candidate by scanning the database – Match each transaction against every candidate – Complexity ~ O(NMw) => Expensive since M = 2 d !!! © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 9
Computational Complexity l Given d unique items: – Total number of itemsets = 2 d – Total number of possible association rules: If d=6, R = 602 rules © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 10
Reducing Number of Candidates l Apriori principle: – If an itemset is frequent, then all of its subsets must also be frequent l Apriori principle holds due to the following property of the support measure: – 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 11
Illustrating Apriori Principle Found to be Infrequent Pruned supersets © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 12
Illustrating Apriori Principle Items (1 -itemsets) Pairs (2 -itemsets) (No need to generate candidates involving Coke or Eggs) Minimum Support = 3 Triplets (3 -itemsets) If every subset is considered, 6 C + 6 C = 41 1 2 3 With support-based pruning, 6 + 1 = 13 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 13
Apriori Algorithm l Method: – Let k=1 – Generate frequent itemsets of length 1 – Repeat until no new frequent itemsets are identified 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 Introduction to Data Mining 4/18/2004 14
The Apriori Algorithm — Example Database D L 1 C 1 Scan D C 2 Scan D L 2 C 3 © Tan, Steinbach, Kumar Scan D L 3 Introduction to Data Mining 4/18/2004 15
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 16
FP-tree construction null After reading TID=1: A: 1 B: 1 After reading TID=2: A: 1 null B: 1 C: 1 D: 1 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 17
FP-Tree Construction Transaction Database null B: 3 A: 7 B: 5 Header table C: 1 C: 3 D: 1 D: 1 E: 1 Pointers are used to assist frequent itemset generation © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 18
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: 3 D: 1 Frequent Itemsets found (with sup > 1): AD, BD, CD, ACD, BCD D: 1 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 19
Projected Database Original Database: Projected Database for node A: For each transaction T, projected transaction at node A is T E(A) © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 20
ECLAT l For each item, store a list of transaction ids (tids) TID-list © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 21
ECLAT l Determine support of any k-itemset by intersecting tid-lists of two of its (k-1) subsets. l 3 traversal approaches: – top-down, bottom-up and hybrid l l 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 22
Compact Representation of Frequent Itemsets l Some itemsets are redundant because they have identical support as their supersets l Number of frequent itemsets l Need a compact representation © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 23
Maximal Frequent Itemset An itemset is maximal frequent if none of its immediate supersets is frequent Maximal Itemsets Infrequent Itemsets © Tan, Steinbach, Kumar Border Introduction to Data Mining 4/18/2004 24
Closed Itemset l An itemset is closed if none of its immediate supersets has the same support as the itemset © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 25
Maximal vs Closed Frequent Itemsets Closed but not maximal Minimum support = 2 Closed and maximal # Closed = 9 # Maximal = 4 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 26
Structural Similarity Size-4 graph Structural similarity Significance similarity Sibling Size-5 graph Size-6 graph © Tan, Steinbach, Kumar 2021/9/4 Introduction to Data Mining 4/18/2004 27 27
Structural Leap Search Leap on g’ subtree if : leap length, tolerance of structure/frequency dissimilarity g : a discovered graph Mining Part Leap Part g’: a sibling of g © Tan, Steinbach, Kumar 2021/9/4 Introduction to Data Mining 4/18/2004 28 28
Branch-and-Bound vs. LEAP Pruning base Feature Optimality Efficiency © Tan, Steinbach, Kumar 2021/9/4 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 29
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