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/*
 * Copyright (c) 2010-2021 Haifeng Li. All rights reserved.
 *
 * Smile is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * Smile is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with Smile.  If not, see .
 */

/**
 * Frequent item set mining and association rule mining.
 * Association rule learning is a popular and well researched method for
 * discovering interesting relations between variables in large databases.
 * Let I = {i₁, i₂,..., i_n} be a set of n
 * binary attributes called items. Let D = {t₁, t₂,..., t_m}
 * be a set of transactions called the database. Each transaction in D has a
 * unique transaction ID and contains a subset of the items in I.
 * An association rule is defined as an implication of the form X ⇒ Y
 * where X, Y ⊆ I and X ∩ Y = Ø. The item sets X and Y are called
 * antecedent (left-hand-side or LHS) and consequent (right-hand-side or RHS)
 * of the rule, respectively. The support supp(X) of an item set X is defined as
 * the proportion of transactions in the database which contain the item set.
 * Note that the support of an association rule X ⇒ Y is supp(X ∪ Y).
 * The confidence of a rule is defined conf(X ⇒ Y) = supp(X ∪ Y) / supp(X).
 * Confidence can be interpreted as an estimate of the probability P(Y | X),
 * the probability of finding the RHS of the rule in transactions under the
 * condition that these transactions also contain the LHS.
 * 
 * For example, the rule {onions, potatoes} ⇒ {burger} found in the sales
 * data of a supermarket would indicate that if a customer buys onions and
 * potatoes together, he or she is likely to also buy burger. Such information
 * can be used as the basis for decisions about marketing activities such as
 * promotional pricing or product placements.
 * 

 * Association rules are usually required to satisfy a user-specified minimum
 * support and a user-specified minimum confidence at the same time. Association
 * rule generation is usually split up into two separate steps:
 * 

 *  First, minimum support is applied to find all frequent item sets
 * in a database (i.e. frequent item set mining).
 * 
 Second, these frequent item sets and the minimum confidence constraint
 * are used to form rules.
 * 
 * 
 * Finding all frequent item sets in a database is difficult since it involves
 * searching all possible item sets (item combinations). The set of possible
 * item sets is the power set over I (the set of items) and has size 2ⁿ - 1
 * (excluding the empty set which is not a valid item set). Although the size
 * of the power set grows exponentially in the number of items n in I, efficient
 * search is possible using the downward-closure property of support
 * (also called anti-monotonicity) which guarantees that for a frequent item set
 * also all its subsets are frequent and thus for an infrequent item set, all
 * its supersets must be infrequent.
 * 

 * In practice, we may only consider the frequent item set that has the maximum
 * number of items bypassing all the sub item sets. An item set is maximal
 * frequent if none of its immediate supersets is frequent.
 * 

 * For a maximal frequent item set, even though we know that all the sub item
 * sets are frequent, we don't know the actual support of those sub item sets,
 * which are very important to find the association rules within the item sets.
 * If the final goal is association rule mining, we would like to discover
 * closed frequent item sets. An item set is closed if none of its immediate
 * supersets has the same support as the item set.
 * 

 * Some well known algorithms of frequent item set mining are Apriori,
 * Eclat and FP-Growth. Apriori is the best-known algorithm to mine association
 * rules. It uses a breadth-first search strategy to counting the support of
 * item sets and uses a candidate generation function which exploits the downward
 * closure property of support. Eclat is a depth-first search algorithm using
 * set intersection.
 * 

 * FP-growth (frequent pattern growth) uses an extended prefix-tree (FP-tree)
 * structure to store the database in a compressed form. FP-growth adopts a
 * divide-and-conquer approach to decompose both the mining tasks and the
 * databases. It uses a pattern fragment growth method to avoid the costly
 * process of candidate generation and testing used by Apriori.
 * 
 * 
References
 * 
 *  R. Agrawal, T. Imielinski and A. Swami. Mining Association Rules Between Sets of Items in Large Databases, SIGMOD, 207-216, 1993.
 *  Rakesh Agrawal and Ramakrishnan Srikant. Fast algorithms for mining association rules in large databases. VLDB, 487-499, 1994.
 *  Mohammed J. Zaki. Scalable algorithms for association mining. IEEE Transactions on Knowledge and Data Engineering, 12(3):372-390, 2000.
 *  Jiawei Han, Jian Pei, Yiwen Yin, and Runying Mao. Mining frequent patterns without candidate generation. Data Mining and Knowledge Discovery 8:53-87, 2004.
 * 
 * 
 * @author Haifeng Li
 */
package smile.association;