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Data Mining: Concepts and Techniques (2nd ed.) — Chapter 5 — Frequent Pattern Mining 1 Mining Frequent Patterns, Association and Correlations: Basic Concepts and Methods Basic Concepts Frequent Itemset Mining: Apriori Algorithm Improving the efficiency of Apriori algorithm Summary 2 What Is Frequent Pattern Analysis? Frequent pattern: a pattern (a set of items, subsequences, substructures, etc.) that occurs frequently together (or strongly correlated) in a data set First proposed by Agrawal, Imielinski, and Swami [AIS93] in the context of frequent itemsets and association rule mining Motivation: Finding inherent regularities in data What products were often purchased together?— Beer and diapers?! What are the subsequent purchases ….after buying a PC? What kinds of DNA are sensitive to this new drug? Can we automatically classify web documents? Applications Basket data analysis, cross-marketing, catalog design, sale campaign analysis, Web log (click stream) analysis, and DNA sequence analysis. 3 Why Is Freq. Pattern Mining Important? Freq. pattern: An intrinsic and important property of datasets. Foundation for many essential data mining tasks Association, correlation, and causality analysis Mining sequential, structural (e.g., sub-graph) patterns Pattern analysis in spatiotemporal, multimedia, timeseries, and stream data Classification: discriminative based frequent pattern analysis Cluster analysis: frequent pattern-based sub-space clustering Data warehousing: iceberg cube and cube-gradient Semantic data compression: fascicles 4 Basic Concepts: Frequent Patterns and Association rules Tid Items bought 10 Beer, Nuts, Diaper 20 Beer, Coffee, Diaper 30 Beer, Diaper, Eggs 40 Nuts, Eggs, Milk 50 Nuts, Coffee, Diaper, Eggs, Milk • Let minsup=50% • Freq. 1-itemsets: • Beer:3(60%); Nuts:3(60%); Diaper:4(80%); Eggs:3(60%) • Freq. 2-itemsets: • {Beer, Diaper}:3(60%) itemset: A set of one or more items k-itemset X = {x1, …, xk} (absolute) support, or, support count of X: Frequency or occurrence of an itemset X (relative) support, s, is the fraction of transactions that contains X (i.e., the probability that a transaction contains X) An itemset X is frequent if X’s support is no less than a minsup threshold 5 Basic Concepts: Association Rules Tid Items bought 10 Beer, Nuts, Diaper 20 Beer, Coffee, Diaper 30 Beer, Diaper, Eggs 40 50 Nuts, Eggs, Milk Nuts, Coffee, Diaper, Eggs, Milk Customer buys both Customer buys diaper Let minsup = 50%, minconf = 50% Freq. Pat.: Beer:3, Nuts:3, Diaper:4, Eggs:3, {Beer, Diaper}:3 Customer buys beer Note: Itemset: notation! Find all the rules X Y with minimum support and confidence support, s, probability that a transaction contains X Y X Yconfidence, c, conditional probability that a transaction having X also contains Y X Y a subtle Association rules: (any more…!) Beer Diaper (60%, 100%) Diaper Beer (60%, 75%) 6 Closed Patterns and Max-Patterns A long pattern contains a combinatorial number of subpatterns, e.g., {a1, …, a100} contains (1001) + (1002) + … + (110000) = 2100 – 1 = 1.27*1030 sub-patterns! Solution: Mine closed patterns and max-patterns instead An itemset X is closed if X is frequent and there exists no super-pattern Y כX, with the same support as X (proposed by Pasquier, et al. @ ICDT’99) An itemset X is a max-pattern if X is frequent and there exists no frequent super-pattern Y כX (proposed by Bayardo @ SIGMOD’98) Closed pattern is a lossless compression of freq. patterns Reducing the # of patterns and rules 7 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} Itemset {A} {B} {C} {D} {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} 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 Closed pattern is a lossless compression of frequent patterns. It reduces the # of patterns but does not lose the support information. Max-patterns Min_sup=2 Difference from close patterns? Do not care for the real support of the subpatterns of a max-pattern Max-pattern: frequent patterns without proper frequent super pattern BCDE, ACD are max-patterns BCD is not a max-pattern Tid Items 10 20 30 A,B,C,D,E B,C,D,E, A,C,D,F Maximal vs Closed Frequent Itemsets Transaction Ids minsup=2 124 123 A 12 124 AB 12 ABC TID Items 1 ABC 2 ABCD 3 BCE 4 ACDE 5 DE Closed but not maximal null 24 AC B AE 24 ABD ABE 345 D 2 3 BC BD 4 ACD 245 C 123 4 AD 2 1234 BE 2 4 ACE ADE E 24 CD Closed and maximal frequent 34 CE 3 BCD 45 DE 4 BCE BDE CDE # Closed = 9 2 4 ABCD ABCE ABDE ACDE BCDE # Maximal = 4 ABCDE 10 Maximal vs Closed Itemsets Closed Frequent Itemsets are Lossless: the support for any frequent itemset can be deduced from the closed frequent itemsets Max-pattern is a lossy compression. We only know all its subsets are frequent but not the real support. Thus in many applications, mining close-patterns is more desirable than mining max-patterns. Frequent Itemsets Closed Frequent Itemsets Maximal Frequent Itemsets Mining Frequent Patterns, Association and Correlations: Basic Concepts and Methods Basic Concepts Frequent Itemset Mining: Apriori Algorithm Improving the efficiency of Apriori algorithm Summary 12 Key Observation (monotonicity) Any subset of a frequent itemset must also be frequent: Downward clouser property (also called Apriori propery) If {beer, diaper, nuts} is frequent, so is {beer, diaper} Efficient mining methodology: Apriori pruning principle Any superset of an infrequent itemset must also be infrequent. If any subset of an itemset S is infrequent, then there is no chance for S to be frequent - why do we even have to consider S..! Prune….! 13 The Downward Closure Property and Scalable Mining Methods Scalable mining methods: Three major approaches Level-wise, join-based approach:Apriori (Agrawal & Srikant@VLDB’94) Freq. pattern projection and growth (FPgrowth—Han, Pei & Yin @SIGMOD’00) Vertical data format approach (Eclat—Zaki , Parthasarathy Ogihara, Li @KDD’97) 14 Apriori: A Candidate Generation & Test Approach Outline of Apriori (level-wise, candidate generation and testing) Method: Initially, scan DB once to get frequent 1-itemset Repeat Generate length (k+1) candidate itemsets from length k frequent itemsets Test the candidates against DB to find frequent (k+1) itemsets Set k:=k+1 Terminate when no frequent or candidate set can be generated Return all the frequent itemsets derived. 15 The Apriori Algorithm (Pseudo-Code) Ck: Candidate itemset of size k Lk : frequent itemset of size k k:=1 L1 = {frequent items}; //Frequent 1-itemset While ( Lk !=; do { //When Lk is not empty Ck+1 = candidates generated from Lk; // candidates generation. Derive Lk+1 by counting for all candidates in Ck+1 wrt TDB and satisfying minsup; // Lk+1 = candidates in Ck+1 with minsup. k:=k+1 } return k Lk; 16 The Apriori Algorithm—An Example Database TDB Tid Items 10 A, C, D 20 B, C, E 30 A, B, C, E 40 B, E Supmin = 2 Itemset {A, C} {B, C} {B, E} {C, E} C3 sup {A} 2 {B} 3 {C} 3 {D} 1 {E} 3 C1 1st scan C2 L2 Itemset sup 2 2 3 2 Itemset {B, C, E} Itemset {A, B} {A, C} {A, E} {B, C} {B, E} {C, E} 3rd scan sup 1 2 1 2 3 2 L3 Itemset sup {A} 2 {B} 3 {C} 3 {E} 3 L1 C2 2nd scan Itemset {A, B} {A, C} {A, E} {B, C} {B, E} Itemset sup {B, C, E} 2 {C, E} Self-join: members of Lk-1 are joinable if their first (k-2) items are in common Apriori Implementation of Trick How to generate candidates? Step 1: self-joining Lk Step 2: pruning Example of Candidate-generation L3={abc, abd, acd, ace, bcd} Self-joining: L3*L3 abcd from abc and abd acde from acd and ace Pruning: acde is removed because ade is not in L3 C4 = {abcd} Any (k-1)-itemset that is not frequent cannot be a subset of a frequent k-itemset 18 Challenges of Frequent Pattern Mining Challenges Multiple scans of transaction database Huge number of candidates Tedious workload of support counting for candidates Improving Apriori: general ideas Reduce passes of transaction database scans Shrink number of candidates Facilitate support counting of candidates 19 Apriori: Improvements and Alternatives Reduce passes of transaction database scans Partitioning (e.g. Savasere, et al., 1995) Dynamic itemset counting (Brin, et al.,1997) Shrink the number of candidates Hash-based technique (e.g., DHP: Park, et al., 1995) Transaction reduction (e.g., Bayardo 1998) Sampling (e.g., Toivonen, 1996) 20 Partitioning : Scan Database Only Twice Theorem: Any itemset that is potentially frequent in TDB must be frequent in at least one of the partitions of TDB Method: Scan 1: Partition database (how?) and find local frequent patterns. Scan 2: Consolidate global frequent patterns (how to ?) 21 Direct Hashing & Pruning (DHP) When generating L1, the algorithm also generates all the 2itemsets for each transaction, hashes them to a hash table and keeps a count. 22 Hash Function Used For each pair, a numeric value is obtained by first representing B by 1, C by 2, E 3, J 4, M 5 and Y 6. Now each pair can be represented by a two digit number, for example (B, E) by 13 and (C, M) by 26. The two digits are then coded as modulo 8 number (dividing by 8 and using the remainder). This is the bucket address. A count of the number of pairs hashed is kept. Those addresses that have a count above the support value have the bit vector set to 1 otherwise 0. All pairs in rows that have zero bit are removed. 23 Transaction Reduction As discussed earlier, any transaction that does not contain any frequent k-itemsets cannot contain any frequent (k+1)-itemsets and such a transaction may be marked or removed. TID Items bought Frequent items (L1) are A, B, D, M, T. We are not able to use these to eliminate any transactions since all transactions have at least one of the items in L1. The frequent pairs (C2) are {A,B} and {B,M}. How can we reduce transactions using these? 001 B, M, T, Y 002 B, M 003 T, S, P 004 A, B, C, D 005 A, B 006 T, Y, E 007 A, B, M 008 B, C, D, T, P 009 D, T, S 010 A, B, M 24 Sampling [Toivonen, 1995] A random sample (usually large enough to fit in the main memory) may be obtained from the overall set of transactions and the sample is searched for frequent itemsets. These frequent itemsets are called sample frequent itemsets. Not guaranteed to be accurate but we sacrifice accuracy for efficiency. A lower support threshold may be used for the sample to ensure not missing any frequent datasets. Sample size is such that the search for frequent itemsets for the sample can be done in main memory. 25 Dynamic Itemset Counting Interrupt algorithm after every M transactions while scanning. Itemsets which are already frequent are combined in pairs to generate higher order itemsets. The technique is dynamic in that, it starts estimating support for all the itemsets if all of their subsets are already found frequent. The resulting algorithm requires fewer database scans than Apriori. 26 DIC: Reduce Number of Scans 27 Summary Frequent patterns Closed patterns and Max-patterns Apriori algorithm for mining frequent patterns Improving the efficiency of apriori: Partitioning, DHP, DIC 28