• Basic Concepts
  • Frequent Itemset Mining Methods
  • Which Patterns Are Interesting?—Pattern Evaluation Methods
  • Summary

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.

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
    • Sequential, structural (e.g., sub-graph) patterns
    • Pattern analysis in spatiotemporal, multimedia, time-series, and stream data
    • Classification: discriminative, frequent pattern analysis
    • Cluster analysis: frequent pattern-based clustering
    • Data warehousing: iceberg cube and cube-gradient
    • Semantic data compression: fascicles
    • Broad applications

Basic Concepts: Frequent Patterns

  • 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

Basic Concepts: Association Rules

  • Find all the rules X → Y with minimum support and confidence
    • support, s, probability that a transaction contains X U Y
    • confidence, c, conditional probability that a transaction having X also contains Y

      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%)


Closed Patterns and Max-Patterns

  • A long pattern contains a combinatorial number of sub-patterns, e.g.,
    \[ {a_{1}, …, a_{100}} \]
    \[ (100^1)+(100^2)+...+(1^{1}0^{0}0^{0}) = 2^{100}-1 = 1.27 * 10^{30}\]
  • Solution: Mine closed patterns and max-patterns instead
  • An itemset X is closed if X is frequent and there exists no super-pattern Y כ X, 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 כ X (proposed by Bayardo @ SIGMOD’98)
  • Closed pattern is a lossless compression of freq. patterns
    • Reducing the # of patterns and rules

Closed Patterns and Max-Patterns

  • Exercise. DB = {< a1, …, a100>, < a1, …, a50>}
    • Min_sup = 1.
  • What is the set of closed itemset?
    • < a1, …, a100>: 1
    • < a1, …, a50>: 2
  • What is the set of max-pattern?
    • < a1, …, a100>: 1
  • What is the set of all patterns?
    • !!

Computational Complexity of Frequent Itemset Mining

  • How many itemsets are potentially to be generated in the worst case?
    • The number of frequent itemsets to be generated is senstive to the minsup threshold
    • When minsup is low, there exist potentially an exponential number of frequent itemsets
    • The worst case: MN where M: # distinct items, and N: max length of transactions
  • The worst case complexty vs. the expected probability
    • Ex. Suppose Walmart has 10^4 kinds of products
      • The chance to pick up one product 10^(-4)
      • The chance to pick up a particular set of 10 products: ~10^(-40)
      • What is the chance this particular set of 10 products to be frequent 10^3 times in 10^9 transactions?

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

The Downward Closure Property and Scalable Mining Methods

  • The downward closure 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 approach (Charm—Zaki & Hsiao @SDM’02)

Apriori: A Candidate Generation & Test Approach

  • Apriori pruning principle: If there is any itemset which 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

The Apriori Algorithm—An Example

The Apriori Algorithm (Pseudo-Code)

Ck: Candidate itemset of size k

Lk : frequent itemset of size k

L1 = {frequent items};

for (k = 1; Lk !=Ø; 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


return Uk Lk;

Implementation of Apriori

  • 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}

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

Counting Supports of Candidates Using Hash Tree

Candidate Generation: An SQL Implementation

  • SQL Implementation of candidate generation
    • Suppose the items in L(k-1) are listed in an order
    • Step 1: self-joining L(k-1)
      insert into Ck
      select p.item1, p.item2, …, p.item(k-1), q.item(k-1)
      from L(k-1) p, L(k-1) q
      where p.item1=q.item1, …, p.item(k-2)=q.item(k-2), p.item(k-1) < q.item(k-1)
    • Step 2: pruning
      forall itemsets c in Ck do
              forall (k-1)-subsets s of c do
                   if (s is not in L(k-1)) then delete c from Ck
  • Use object-relational extensions like UDFs, BLOBs, and Table functions for efficient implementation [S. Sarawagi, S. Thomas, and R. Agrawal. Integrating association rule mining with relational database systems: Alternatives and implications. SIGMOD’98]

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

Partition: Scan Database Only Twice

  • Any itemset that is potentially frequent in DB must be frequent in at least one of the partitions of DB
    • Scan 1: partition database and find local frequent patterns
    • Scan 2: consolidate global frequent patterns
  • A. Savasere, E. Omiecinski and S. Navathe, VLDB’95

DHP: Reduce the Number of Candidates

  • A k-itemset whose corresponding hashing bucket count is below the threshold cannot be frequent
    • Candidates: a, b, c, d, e
    • Hash entries
      • {ab, ad, ae}
      • {bd, be, de}
    • Frequent 1-itemset: a, b, d, e
    • ab is not a candidate 2-itemset if the sum of count of {ab, ad, ae} is below support threshold
  • J. Park, M. Chen, and P. Yu. An effective hash-based algorithm for mining association rules. SIGMOD’95

Sampling for Frequent Patterns

  • Select a sample of original database, mine frequent patterns within sample using Apriori
  • Scan database once to verify frequent itemsets found in sample, only borders of closure of frequent patterns are checked
    • Example: check abcd instead of ab, ac, …, etc.
  • Scan database again to find missed frequent patterns
  • H. Toivonen. Sampling large databases for association rules. In VLDB’96

DIC: Reduce Number of Scans

Pattern-Growth Approach: Mining Frequent Patterns Without Candidate Generation

  • Bottlenecks of the Apriori approach
    • Breadth-first (i.e., level-wise) search
    • Candidate generation and test
      • Often generates a huge number of candidates
  • The FPGrowth Approach (J. Han, J. Pei, and Y. Yin, SIGMOD’ 00)
    • Depth-first search
    • Avoid explicit candidate generation
  • Major philosophy: Grow long patterns from short ones using local frequent items only
    • “abc” is a frequent pattern
    • Get all transactions having “abc”, i.e., project DB on abc: DB|abc
    • “d” is a local frequent item in DB|abc  abcd is a frequent pattern

Construct FP-tree from a Transaction Database

Partition Patterns and Databases

  • Frequent patterns can be partitioned into subsets according to f-list
    • F-list = f-c-a-b-m-p
    • Patterns containing p
    • Patterns having m but no p
    • Patterns having c but no a nor b, m, p
    • Pattern f
  • Completeness and non-redundency

Find Patterns Having P From P-conditional Database

  • Starting at the frequent item header table in the FP-tree
  • Traverse the FP-tree by following the link of each frequent item p
  • Accumulate all of transformed prefix paths of item p to form p’s conditional pattern base

From Conditional Pattern-bases to Conditional FP-trees

  • For each pattern-base
    • Accumulate the count for each item in the base
    • Construct the FP-tree for the frequent items of the pattern base

Recursion: Mining Each Conditional FP-tree

A Special Case: Single Prefix Path in FP-tree

  • Suppose a (conditional) FP-tree T has a shared single prefix-path P
  • Mining can be decomposed into two parts
    • Reduction of the single prefix path into one node
    • Concatenation of the mining results of the two parts

Benefits of the FP-tree Structure

  • Completeness
    • Preserve complete information for frequent pattern mining
    • Never break a long pattern of any transaction
  • Compactness
    • Reduce irrelevant info—infrequent items are gone
    • Items in frequency descending order: the more frequently occurring, the more likely to be shared
    • Never be larger than the original database (not count node-links and the count field)

The Frequent Pattern Growth Mining Method

  • Idea: Frequent pattern growth
    • Recursively grow frequent patterns by pattern and database partition
  • Method
    • For each frequent item, construct its conditional pattern-base, and then its conditional FP-tree
    • Repeat the process on each newly created conditional FP-tree
    • Until the resulting FP-tree is empty, or it contains only one path—single path will generate all the combinations of its sub-paths, each of which is a frequent pattern

Scaling FP-growth by Database Projection

  • What about if FP-tree cannot fit in memory?
    • DB projection
  • First partition a database into a set of projected DBs
  • Then construct and mine FP-tree for each projected DB
  • Parallel projection vs. partition projection techniques
    • Parallel projection
      • Project the DB in parallel for each frequent item
      • Parallel projection is space costly
      • All the partitions can be processed in parallel
    • Partition projection
      • Partition the DB based on the ordered frequent items
      • Passing the unprocessed parts to the subsequent partitions

Partition-Based Projection

FP-Growth vs. Apriori: Scalability With the Support Threshold

FP-Growth vs. Tree-Projection: Scalability with the Support Threshold

Advantages of the Pattern Growth Approach

  • Divide-and-conquer:
    • Decompose both the mining task and DB according to the frequent patterns obtained so far
    • Lead to focused search of smaller databases
  • Other factors
    • No candidate generation, no candidate test
    • Compressed database: FP-tree structure
    • No repeated scan of entire database
    • Basic ops: counting local freq items and building sub FP-tree, no pattern search and matching
  • A good open-source implementation and refinement of FPGrowth
    • FPGrowth+ (Grahne and J. Zhu, FIMI'03)

Further Improvements of Mining Methods

  • AFOPT (Liu, et al. @ KDD’03)
    • A “push-right” method for mining condensed frequent pattern (CFP) tree
  • Carpenter (Pan, et al. @ KDD’03)
    • Mine data sets with small rows but numerous columns
    • Construct a row-enumeration tree for efficient mining
  • FPgrowth+ (Grahne and Zhu, FIMI’03)
    • Efficiently Using Prefix-Trees in Mining Frequent Itemsets, Proc. ICDM'03 Int. Workshop on Frequent Itemset Mining Implementations (FIMI'03), Melbourne, FL, Nov. 2003
  • TD-Close (Liu, et al, SDM’06)

Extension of Pattern Growth Mining Methodology

  • Mining closed frequent itemsets and max-patterns
    • CLOSET (DMKD’00), FPclose, and FPMax (Grahne & Zhu, Fimi’03)
  • Mining sequential patterns
    • PrefixSpan (ICDE’01), CloSpan (SDM’03), BIDE (ICDE’04)
  • Mining graph patterns
    • gSpan (ICDM’02), CloseGraph (KDD’03)
  • Constraint-based mining of frequent patterns
    • Convertible constraints (ICDE’01), gPrune (PAKDD’03)
  • Computing iceberg data cubes with complex measures
    • H-tree, H-cubing, and Star-cubing (SIGMOD’01, VLDB’03)
  • Pattern-growth-based Clustering
    • MaPle (Pei, et al., ICDM’03)
  • Pattern-Growth-Based Classification
    • Mining frequent and discriminative patterns (Cheng, et al, ICDE’07)

ECLAT: Mining by Exploring Vertical Data Format

  • Vertical format: t(AB) = {T11, T25, …}
    • tid-list: list of trans.-ids containing an itemset
  • Deriving frequent patterns based on vertical intersections
    • t(X) = t(Y): X and Y always happen together
    • t(X) ⊂ t(Y): transaction having X always has Y
  • Using diffset to accelerate mining
    • Only keep track of differences of tids
    • t(X) = {T1, T2, T3}, t(XY) = {T1, T3}
    • Diffset (XY, X) = {T2}
  • Eclat (Zaki et al. @KDD’97)
  • Mining Closed patterns using vertical format: CHARM (Zaki & Hsiao@SDM’02)

Mining Frequent Closed Patterns: CLOSET

  • Flist: list of all frequent items in support ascending order
    • Flist: d-a-f-e-c
  • Divide search space
    • Patterns having d
    • Patterns having d but no a, etc.
  • Find frequent closed pattern recursively
    • Every transaction having d also has cfa → cfad is a frequent closed pattern
  • J. Pei, J. Han & R. Mao. “CLOSET: An Efficient Algorithm for Mining Frequent Closed Itemsets", DMKD'00.

CLOSET+: Mining Closed Itemsets by Pattern-Growth

  • Itemset merging: if Y appears in every occurrence of X, then Y is merged with X
  • Sub-itemset pruning: if Y כ X, and sup(X) = sup(Y), X and all of X’s descendants in the set enumeration tree can be pruned
  • Hybrid tree projection
    • Bottom-up physical tree-projection
    • Top-down pseudo tree-projection
  • Item skipping: if a local frequent item has the same support in several header tables at different levels, one can prune it from the header table at higher levels
  • Efficient subset checking

MaxMiner: Mining Max-Patterns

CHARM: Mining by Exploring Vertical Data Format

  • Vertical format: t(AB) = {T11, T25, …}
    • tid-list: list of trans.-ids containing an itemset
  • Deriving closed patterns based on vertical intersections
    • t(X) = t(Y): X and Y always happen together
    • t(X) ⊂ t(Y): transaction having X always has Y
  • Using diffset to accelerate mining
    • Only keep track of differences of tids
    • t(X) = {T1, T2, T3}, t(XY) = {T1, T3}
    • Diffset (XY, X) = {T2}
  • Eclat/MaxEclat (Zaki et al. @KDD’97), VIPER(P. Shenoy et al.@SIGMOD’00), CHARM (Zaki & Hsiao@SDM’02)

Visualization of Association Rules: Plane Graph

Visualization of Association Rules: Rule Graph

Visualization of Association Rules (SGI/MineSet 3.0)

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
  • Measure of dependent/correlated events: lift

Are lift and X^2 Good Measures of Correlation?

  • “Buy walnuts buy milk [1%, 80%]” is misleading if 85% of customers buy milk
  • Support and confidence are not good to indicate correlations
  • Over 20 interestingness measures have been proposed (see Tan, Kumar, Sritastava @KDD’02)
  • Which are good ones?

Null-Invariant Measures

Comparison of Interestingness Measures

Analysis of DBLP Coauthor Relationships

Which Null-Invariant Measure Is Better?

  • IR (Imbalance Ratio): measure the imbalance of two itemsets A and B in rule implications

\[ IR(A,B) = \frac {|sup(A)-sup(B)|} {sup(A)+sup(B)-sup(AUB)}\]

  • Kulczynski and Imbalance Ratio (IR) together present a clear picture for all the three datasets D4 through D6
    • D4 is balanced & neutral
    • D5 is imbalanced & neutral
    • D6 is very imbalanced & neutral


  • Basic concepts: association rules, support-confident framework, closed and max-patterns
  • Scalable frequent pattern mining methods
    • Apriori (Candidate generation & test)
    • Projection-based (FPgrowth, CLOSET+, ...)
    • Vertical format approach (ECLAT, CHARM, ...)
  • Which patterns are interesting?
    • Pattern evaluation methods


  • Basic Concepts of Frequent Pattern Mining (Association Rules)
    • R. Agrawal, T. Imielinski, and A. Swami. Mining association rules between sets of items in large databases. SIGMOD'93.  
    • (Max-pattern) R. J. Bayardo. Efficiently mining long patterns from databases. SIGMOD'98.  
    • (Closed-pattern) N. Pasquier, Y. Bastide, R. Taouil, and L. Lakhal. Discovering frequent closed itemsets for association rules. ICDT'99.  
    • (Sequential pattern) R. Agrawal and R. Srikant. Mining sequential patterns. ICDE'95
  • Apriori and Its Improvements
    • R. Agrawal and R. Srikant. Fast algorithms for mining association rules. VLDB'94. 
    • H. Mannila, H. Toivonen, and A. I. Verkamo. Efficient algorithms for discovering association rules. KDD'94. 
    • A. Savasere, E. Omiecinski, and S. Navathe. An efficient algorithm for mining association rules in large databases. VLDB'95.
    • J. S. Park, M. S. Chen, and P. S. Yu. An effective hash-based algorithm for mining association rules. SIGMOD'95.
    • H. Toivonen. Sampling large databases for association rules. VLDB'96.
    • S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket analysis. SIGMOD'97.
    • S. Sarawagi, S. Thomas, and R. Agrawal. Integrating association rule mining with relational database systems: Alternatives and implications. SIGMOD'98.

References (cont')

  • Depth-First, Projection-Based FP Mining
    • R. Agarwal, C. Aggarwal, and V. V. V. Prasad. A tree projection algorithm for generation of frequent itemsets. J. Parallel and Distributed Computing:02.
    • J. Han, J. Pei, and Y. Yin. Mining frequent patterns without candidate generation . SIGMOD’ 00.
    • J. Liu, Y. Pan, K. Wang, and J. Han. Mining Frequent Item Sets by Opportunistic Projection. KDD'02.
    • J. Han, J. Wang, Y. Lu, and P. Tzvetkov. Mining Top-K Frequent Closed Patterns without Minimum Support. ICDM'02.
    • J. Wang, J. Han, and J. Pei. CLOSET+: Searching for the Best Strategies for Mining Frequent Closed Itemsets. KDD'03.
    • G. Liu, H. Lu, W. Lou, J. X. Yu. On Computing, Storing and Querying Frequent Patterns. KDD'03.
    • G. Grahne and J. Zhu, Efficiently Using Prefix-Trees in Mining Frequent Itemsets, Proc. ICDM'03 Int. Workshop on Frequent Itemset Mining Implementations (FIMI'03), Melbourne, FL, Nov. 2003

References (cont')

  • Vertical Format and Row Enumeration Methods
    • M. J. Zaki, S. Parthasarathy, M. Ogihara, and W. Li. Parallel algorithm for discovery of association rules. DAMI:97.
    • Zaki and Hsiao. CHARM: An Efficient Algorithm for Closed Itemset Mining, SDM'02.
    • C. Bucila, J. Gehrke, D. Kifer, and W. White. DualMiner: A Dual-Pruning Algorithm for Itemsets with Constraints. KDD’02.
    • F. Pan, G. Cong, A. K. H. Tung, J. Yang, and M. Zaki , CARPENTER: Finding Closed Patterns in Long Biological Datasets. KDD'03.
    • H. Liu, J. Han, D. Xin, and Z. Shao, Mining Interesting Patterns from Very High Dimensional Data: A Top-Down Row Enumeration Approach, SDM'06.

References (cont')

  • Mining Correlations and Interesting Rules
    • M. Klemettinen, H. Mannila, P. Ronkainen, H. Toivonen, and A. I. Verkamo. Finding interesting rules from large sets of discovered association rules. CIKM'94.
    • S. Brin, R. Motwani, and C. Silverstein. Beyond market basket: Generalizing association rules to correlations. SIGMOD'97.
    • C. Silverstein, S. Brin, R. Motwani, and J. Ullman. Scalable techniques for mining causal structures. VLDB'98.
    • P.-N. Tan, V. Kumar, and J. Srivastava. Selecting the Right Interestingness Measure for Association Patterns. KDD'02.
    • E. Omiecinski. Alternative Interest Measures for Mining Associations. TKDE’03.
    • T. Wu, Y. Chen and J. Han, “Association Mining in Large Databases: A Re-Examination of Its Measures”, PKDD'07
  •  Freq. pattern Mining Applications
    • Y. Huhtala, J. Kärkkäinen, P. Porkka, H. Toivonen. Efficient Discovery of Functional and Approximate Dependencies Using Partitions. ICDE’98.
    • H. V. Jagadish, J. Madar, and R. Ng. Semantic Compression and Pattern Extraction with Fascicles. VLDB'99.
    •  T. Dasu, T. Johnson, S. Muthukrishnan, and V. Shkapenyuk. Mining Database Structure; or How to Build a Data Quality Browser. SIGMOD'02.
    • K. Wang, S. Zhou, J. Han. Profit Mining: From Patterns to Actions. EDBT’02. 

  • March 13, 2013

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ali1k (VU Amsterdam)

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