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Chapter 6: Mining Frequent Patterns, Association and Correlations Basic concepts Frequent itemset mining methods Constraint-based frequent pattern mining (ch7) Association rules 1 What Is Frequent Pattern Analysis? Frequent pattern: a pattern (a set of items, subsequences, substructures, etc.) that occurs frequently 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. 2 Why Is Freq. Pattern Mining Important? Freq. pattern: intrinsic and important property of data sets Foundation for many essential data mining tasks Association, correlation, and causality analysis Sequential, structural (e.g., sub-graph) patterns Pattern analysis in spatiotemporal, multimedia, timeseries, and stream data Classification: associative classification Cluster analysis: frequent pattern-based clustering Data warehousing: iceberg cube and cube-gradient Semantic data compression: fascicles Broad applications 3 Basic Concepts: Frequent Patterns 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 Customer buys both Customer buys diaper itemset: A set of 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 Customer buys beer 4 Closed Patterns and Max-Patterns A long pattern contains a combinatorial number of sub-patterns, e.g., {a1, …, a100} contains 2100 – 1 = 1.27*1030 sub-patterns! Solution: Mine closed patterns and max-patterns instead An itemset X is a closed pattern if X is frequent and there exist no super-patterns with the same support all super-patterns must have smaller support An itemset X is a max-pattern if X is frequent and there exist no super-patterns that are frequent Relationship between the two? Closed patterns are a lossless compression of freq. patterns, whereas max-patterns are a lossy compression Lossless: can derive all frequent patterns as well as their support Lossy: can derive all frequent patterns 5 Closed Patterns and Max-Patterns DB = {<a1, …, a100>, < a1, …, a50>} min_sup = 1 What is the set of closed patterns? <a1, …, a100>: 1 < a1, …, a50>: 2 How to derive frequent patterns and their support values? What is the set of max-patterns? <a1, …, a100>: 1 How to derive frequent patterns? What is the set of all patterns? {a1}: 2, …, {a1, a2}: 2, …, {a1, a51}: 1, …, {a1, a2, …, a100}: 1 A big number: 2100 – 1 6 Closed Patterns and Max-Patterns For a given dataset with itemset I = {a,b,c,d} and min_sup = 8, the closed patterns are {a,b,c,d} with support of 10, {a,b,c} with support of 12, and {a, b,d} with support of 14. Derive the frequent 2itemsets together with their support values {a,b}: 14 {b,c}: 12 {a,c}: 12 {b,d}: 14 {a,d}: 14 {c,d}: 10 7 Chapter 6: Mining Frequent Patterns, Association and Correlations Basic concepts Frequent itemset mining methods Constraint-based frequent pattern mining (ch7) Association rules 8 Scalable Frequent Itemset Mining Methods Apriori: A Candidate Generation-and-Test Approach Improving the Efficiency of Apriori FPGrowth: A Frequent Pattern-Growth Approach ECLAT: Frequent Pattern Mining with Vertical Data Format 9 Scalable Methods for Mining Frequent Patterns The downward closure (anti-monotonic) property of frequent patterns Any subset of a frequent itemset must be frequent If {beer, diaper, nuts} is frequent, so is {beer, diaper} i.e., every transaction having {beer, diaper, nuts} also contains {beer, diaper} Scalable mining methods: Three major approaches Apriori (Agrawal & Srikant@VLDB’94) Freq. pattern growth (Fpgrowth: Han, Pei & Yin @SIGMOD’00) Vertical data format (Charm—Zaki & Hsiao @SDM’02) 10 Apriori: A Candidate Generation-and-Test Approach Apriori pruning principle: If there is any itemset that is infrequent, its superset should not be generated/tested! (Agrawal & Srikant @VLDB’94, Mannila, et al. @ KDD’ 94) Method: Initially, scan DB once to get frequent 1-itemset Generate length (k+1) candidate itemsets from length k frequent itemsets Test the candidates against DB Terminate when no frequent or candidate set can be generated 11 The Apriori Algorithm—An Example Itemset sup {a} 2 C1 {b} 3 1st scan {c} 3 {d} 1 {e} 3 DB Tid Items 10 a, c, d 20 b, c, e 30 a, b, c, e 40 b, e C2 L2 Itemset {a, c} {b, c} {b, e} {c, e} sup 2 2 3 2 Itemset {a, b} {a, c} {a, e} {b, c} {b, e} {c, e} sup 1 2 1 2 3 2 min_sup= 2 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} {c, e} C3 Itemset {b, c, e} 3rd scan L3 Itemset sup {b, c, e} 2 12 The Apriori Algorithm (Pseudo-code) Ck: Candidate itemset of size k Lk : frequent itemset of size k L1 = {frequent items}; for (k = 1; Lk !=; k++) do begin Ck+1 = candidates generated from Lk; for each transaction t in database do increment the count of all candidates in Ck+1 that are contained in t Lk+1 = candidates in Ck+1 with min_support end return k Lk; 13 Implementation of Apriori Generate candidates, then count support for the generated candidates How to generate candidates? Step 1: self-joining Lk Step 2: pruning Example: L3={abc, abd, acd, ace, bcd} Self-joining: L3*L3 Pruning: abcd from abc and abd acde from acd and ace acde is removed because ade is not in L3 C4={abcd} The above procedures do not miss any legitimate candidates. Thus Apriori mines a complete set of frequent patterns. 14 How to Count Supports of Candidates? Why counting supports of candidates a problem? The total number of candidates can be very huge One transaction may contain many candidates Method: Candidate itemsets are stored in a hash-tree Leaf node of hash-tree contains a list of itemsets and counts Interior node contains a hash table Subset function: finds all the candidates contained in a transaction 15 Example: Counting Supports of Candidates Subset function 3,6,9 1,4,7 Transaction: 1 2 3 5 6 2,5,8 1+2356 234 567 13+56 145 136 12+356 124 457 125 458 345 356 357 689 367 368 159 16 Further Improvement of the Apriori Method Major computational 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 17 Apriori applications beyond freq. pattern mining Given a set S of students, we want to find each subset of S such that the age range of the subset is less than 5. Apriori algorithm, level-wise search using the downward closure property for pruning to gain efficiency Can be used to search for any subsets with the downward closure property (i.e., anti-monotone constraint) CLIQUE for subspace clustering used the same Apriori principle, where the dimensions are the itemset 18 Chapter 6: Mining Frequent Patterns, Association and Correlations Basic concepts Frequent itemset mining methods Constraint-based frequent pattern mining (ch7) Association rules 19 Constraint-based (Query-Directed) Mining Finding all the patterns in a database autonomously? — unrealistic! Data mining should be an interactive process The patterns could be too many but not focused! User directs what to be mined using a data mining query language (or a graphical user interface) Constraint-based mining User flexibility: provides constraints on what to be mined Optimization: explores such constraints for efficient mining — constraint-based mining: constraint-pushing, similar to push selection first in DB query processing Note: still find all the answers satisfying constraints, not finding some answers in “heuristic search” 20 Constrained Mining vs. Constraint-Based Search Constrained mining vs. constraint-based search/reasoning Both are aimed at reducing search space Finding all patterns satisfying constraints vs. finding some (or one) answer in constraint-based search in AI Constraint-pushing vs. heuristic search It is an interesting research problem on how to integrate them Constrained mining vs. query processing in DBMS Database query processing requires to find all Constrained pattern mining shares a similar philosophy as pushing selections deeply in query processing Constraint-Based Frequent Pattern Mining Pattern space pruning constraints Monotonic: If c is satisfied, no need to check c again Succinct: c must be satisfied, so one can start with the data sets satisfying c Anti-monotonic: If constraint c is violated, its further mining can be terminated Convertible: c is not monotonic nor anti-monotonic, but it can be converted into it if items in the transaction can be properly ordered Data space pruning constraint Data succinct: Data space can be pruned at the initial pattern mining process Data anti-monotonic: If a transaction t does not satisfy c, t can be pruned from its further mining 22 Anti-Monotonicity in Constraint Pushing Anti-monotonicity When an itemset S violates the constraint, so does any of its superset sum(S.Price) v is anti-monotonic sum(S.Price) v is not anti-monotonic C: range(S.profit) 15 is anti-monotonic TDB (min_sup=2) TID Transaction 10 a, b, c, d, f 20 b, c, d, f, g, h 30 a, c, d, e, f 40 c, e, f, g Item Profit a 40 Itemset ab violates C b 0 So does every superset of ab c -20 d 10 e -30 f 30 g 20 h -10 support count >= min_sup is antimonotonic core property used in Apriori Monotonicity for Constraint Pushing Monotonicity When an itemset S satisfies the constraint, so does any of its superset sum(S.Price) v is monotonic min(S.Price) v is monotonic C: range(S.profit) 15 Itemset ab satisfies C So does every superset of ab Item Profit a 40 b 0 c -20 d 10 e -30 f 30 g 20 h -10 24 Succinctness Given A1, the set of items satisfying a succinctness constraint C, then any set S satisfying C is based on A1 , i.e., S contains a subset belonging to A1 Idea: Without looking at the transaction database, whether an itemset S satisfies constraint C can be determined based on the selection of items If a constraint is succinct, we can directly generate precisely the sets that satisfy it, even before support counting begins. Avoids substantial overhead of generate-and-test, i.e., such constraint is pre-counting pushable min(S.Price) v is succinct sum(S.Price) v is not succinct Constraint-Based Mining—A General Picture Constraint Antimonotone Monotone Succinct vS no yes yes SV no yes yes SV yes no yes min(S) v no yes yes min(S) v yes no yes max(S) v yes no yes max(S) v no yes yes count(S) v yes no weakly count(S) v no yes weakly sum(S) v ( a S, a 0 ) yes no no sum(S) v ( a S, a 0 ) no yes no range(S) v yes no no range(S) v no yes no avg(S) v, { , , } convertible convertible no support(S) yes no no support(S) no yes no 26 Chapter 6: Mining Frequent Patterns, Association and Correlations Basic concepts Frequent itemset mining methods Constraint-based frequent pattern mining (ch7) Association rules 27 Basic Concepts: Association Rules An association rule is of the form X Y with minimum support and confidence, where X,Y I, X Y = support(X->Y): probability that a transaction contains X Y, i.e., support(X->Y) = P(X U Y) confidence(X->Y): conditional probability that a transaction having X also contains Y, i.e. confidence(X->Y) = P(Y|X) Can be estimated by the percentage of transactions in DB that contain X Y. Not to be confused with P(X or Y) confidence(X->Y) = P(Y|X) = support(X Y) / support (X) = support_count(X Y) / support_count(X) confidence(X->Y) can be easily derived from the support count of X and the support count of X Y. Thus association rule mining can be reduced to frequent pattern mining 28 Basic Concepts: 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 Customer buys both Customer buys diaper Let minsup = 50%, minconf = 50% Freq. Pat.: Beer:3, Nuts:3, Diaper:4, Eggs:3, {Beer, Diaper}:3 Association rules: (many more!) Beer Diaper (60%, 100%) Diaper Beer (60%, 75%) If {a} => {b} is an association rule, then {b} => {a} is also an association rule? Same support, different confidence If {a,b} => {c} is an association rule, then {b} => {c} is also an association rule? Customer buys beer If {b} => {c} is an association rule then {a,b} => {c} is also an association rule? 29 Interestingness Measure: Correlations (Lift) play basketball eat cereal [40%, 66.7%] is misleading The overall % of students eating cereal is 75% > 66.7%. play basketball not eat cereal [20%, 33.3%] is more accurate, although with lower support and confidence Support and confidence are not good to indicate correlations Measure of dependent/correlated events: lift P( A B) lift P( A) P( B) Basketball Not basketball Sum (row) Cereal 2000 1750 3750 Not cereal 1000 250 1250 Sum(col.) 3000 2000 5000 2000 / 5000 lift ( B, C ) 0.89 3000 / 5000 * 3750 / 5000 lift ( B, C ) 1000 / 5000 1.33 3000 / 5000 *1250 / 5000 30