Apriori Algorithm
28.2.2 Apriori Algorithm
The first algorithm to use the downward closure and antimontonicity properties was the Apriori algorithm, shown as Algorithm 28.1.
We illustrate Algorithm 28.1 using the transaction data in Figure 28.1 using a mini- mum support of 0.5. The candidate 1-itemsets are {milk, bread, juice, cookies, eggs, coffee} and their respective supports are 0.75, 0.5, 0.5, 0.5, 0.25, and 0.25. The first
four items qualify for L 1 since each support is greater than or equal to 0.5. In the first iteration of the repeat-loop, we extend the frequent 1-itemsets to create the candi- date frequent 2-itemsets, C 2 .C 2 contains {milk, bread}, {milk, juice}, {bread, juice}, {milk, cookies}, {bread, cookies}, and {juice, cookies}. Notice, for example, that {milk, eggs} does not appear in C 2 since {eggs} is small (by the antimonotonicity property) and does not appear in L 1 . The supports for the six sets contained in C 2 are 0.25, 0.5, 0.25, 0.25, 0.5, and 0.25 and are computed by scanning the set of trans- actions. Only the second 2-itemset {milk, juice} and the fifth 2-itemset {bread, cookies} have support greater than or equal to 0.5. These two 2-itemsets form the
frequent 2-itemsets, L 2 . Algorithm 28.1. Apriori Algorithm for Finding Frequent (Large) Itemsets
Input: Database of m transactions, D, and a minimum support, mins, represented as
a fraction of m.
1042 Chapter 28 Data Mining Concepts
Output: Frequent itemsets, L 1 ,L 2 , ..., L k
Begin /* steps or statements are numbered for better readability */
1. Compute support(i j ) = count(i j )/m for each individual item, i 1 ,i 2 , ..., i n by scanning the database once and counting the number of transactions that
item i j appears in (that is, count(i j ));
2. The candidate frequent 1-itemset, C 1 , will be the set of items i 1 ,i 2 , ..., i n ;
3. The subset of items containing i j from C 1 where support(i j ) >= mins
becomes the frequent 1-itemset, L 1 ;
4. k = 1; termination = false;
repeat
1. L k+1 =;
2. Create the candidate frequent (k+1)-itemset, C k+1 , by combining members of L k that have k–1 items in common (this forms candidate frequent (k+1)-
itemsets by selectively extending frequent k-itemsets by one item);
3. In addition, only consider as elements of C k+1 those k+1 items such that every subset of size k appears in L k ;
4. Scan the database once and compute the support for each member of C k+1 ; if the support for a member of C k+1 >= mins then add that member to L k+1 ;
5. If L k+1 is empty then termination = true else k = k + 1;
until termination; End;
In the next iteration of the repeat-loop, we construct candidate frequent 3-itemsets by adding additional items to sets in L 2 . However, for no extension of itemsets in L 2 will all 2-item subsets be contained in L 2 . For example, consider {milk, juice, bread}; the 2-itemset {milk, bread} is not in L 2 , hence {milk, juice, bread} cannot be a fre- quent 3-itemset by the downward closure property. At this point the algorithm ter- minates with L 1 equal to {{milk}, {bread}, {juice}, {cookies}} and L 2 equal to {{milk, juice}, {bread, cookies}}.
Several other algorithms have been proposed to mine association rules. They vary mainly in terms of how the candidate itemsets are generated, and how the supports for the candidate itemsets are counted. Some algorithms use such data structures as bitmaps and hashtrees to keep information about itemsets. Several algorithms have been proposed that use multiple scans of the database because the potential number of itemsets, 2 m , can be too large to set up counters during a single scan. We will examine three improved algorithms (compared to the Apriori algorithm) for asso- ciation rule mining: the Sampling algorithm, the Frequent-Pattern Tree algorithm, and the Partition algorithm.
28.2 Association Rules 1043
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» Fundamentals_of_Database_Systems,_6th_Edition
» Characteristics of the Database Approach
» Advantages of Using the DBMS Approach
» A Brief History of Database Applications
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» The Three-Schema Architecture
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» ER-to-Relational Mapping Algorithm
» Discussion and Summary of Mapping for ER Model Constructs
» Mapping EER Model Constructs
» The Role of Information Systems
» The Database Design and Implementation Process
» Use of UML Diagrams as an Aid to Database Design Specification 6
» Rational Rose: A UML-Based Design Tool
» Automated Database Design Tools
» Introduction to Object-Oriented Concepts and Features
» Object Identity, and Objects versus Literals
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» Retrieving Single Tuples with Embedded SQL
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» Specifying Queries at Runtime Using Dynamic SQL
» SQLJ: Embedding SQL Commands in Java
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» Database Programming with SQL/CLI Using C
» JDBC: SQL Function Calls for Java Programming
» Database Stored Procedures and SQL/PSM
» PHP Variables, Data Types, and Programming Constructs
» Overview of PHP Database Programming
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» Normal Forms Based on Primary Keys
» General Definitions of Second and Third Normal Forms
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» Join Dependencies and Fifth Normal Form
» Inference Rules for Functional Dependencies
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» Properties of Relational Decompositions
» Dependency-Preserving Decomposition
» Dependency-Preserving and Nonadditive (Lossless) Join Decomposition into 3NF Schemas
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» Types of Single-Level Ordered Indexes
» Some General Issues Concerning Indexing
» Algorithms for External Sorting
» Implementing the SELECT Operation
» Implementing the JOIN Operation
» Algorithms for PROJECT and Set
» Notation for Query Trees and Query Graphs
» Heuristic Optimization of Query Trees
» Catalog Information Used in Cost Functions
» Examples of Cost Functions for SELECT
» Examples of Cost Functions for JOIN
» Example to Illustrate Cost-Based Query Optimization
» Factors That Influence Physical Database Design
» Physical Database Design Decisions
» An Overview of Database Tuning in Relational Systems
» Transactions, Database Items, Read and Write Operations, and DBMS Buffers
» Why Concurrency Control Is Needed
» Transaction and System Concepts
» Desirable Properties of Transactions
» Serial, Nonserial, and Conflict-Serializable Schedules
» Testing for Conflict Serializability of a Schedule
» How Serializability Is Used for Concurrency Control
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» Dealing with Deadlock and Starvation
» Concurrency Control Based on Timestamp Ordering
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» Validation (Optimistic) Concurrency
» Granularity of Data Items and Multiple Granularity Locking
» Using Locks for Concurrency Control in Indexes
» Other Concurrency Control Issues
» Recovery Outline and Categorization of Recovery Algorithms
» Caching (Buffering) of Disk Blocks
» Write-Ahead Logging, Steal/No-Steal, and Force/No-Force
» Transaction Rollback and Cascading Rollback
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» The ARIES Recovery Algorithm
» Recovery in Multidatabase Systems
» Introduction to Database Security Issues 1
» Discretionary Access Control Based on Granting and Revoking Privileges
» Mandatory Access Control and Role-Based Access Control for Multilevel Security
» Introduction to Statistical Database Security
» Introduction to Flow Control
» Encryption and Public Key Infrastructures
» Challenges of Database Security
» Distributed Database Concepts 1
» Types of Distributed Database Systems
» Distributed Database Architectures
» Data Replication and Allocation
» Example of Fragmentation, Allocation, and Replication
» Query Processing and Optimization in Distributed Databases
» Overview of Transaction Management in Distributed Databases
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» Current Trends in Distributed Databases
» Distributed Databases in Oracle 13
» Generalized Model for Active Databases and Oracle Triggers
» Design and Implementation Issues for Active Databases
» Examples of Statement-Level Active Rules
» Time Representation, Calendars, and Time Dimensions
» Incorporating Time in Relational Databases Using Tuple Versioning
» Incorporating Time in Object-Oriented Databases Using Attribute Versioning
» Temporal Querying Constructs and the TSQL2 Language
» Spatial Database Concepts 24
» Multimedia Database Concepts
» Clausal Form and Horn Clauses
» Datalog Programs and Their Safety
» Evaluation of Nonrecursive Datalog Queries
» Introduction to Information Retrieval
» Types of Queries in IR Systems
» Evaluation Measures of Search Relevance
» Web Analysis and Its Relationship to Information Retrieval
» Analyzing the Link Structure of Web Pages
» Approaches to Web Content Analysis
» Trends in Information Retrieval
» Data Mining as a Part of the Knowledge
» Goals of Data Mining and Knowledge Discovery
» Types of Knowledge Discovered during Data Mining
» Market-Basket Model, Support, and Confidence
» Frequent-Pattern (FP) Tree and FP-Growth Algorithm
» Other Types of Association Rules
» Approaches to Other Data Mining Problems
» Commercial Data Mining Tools
» Data Modeling for Data Warehouses
» Difficulties of Implementing Data Warehouses
» Grouping, Aggregation, and Database Modification in QBE
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