# Association rule learning

In data mining, association rule learning is a popular and well researched method for discovering interesting relations between variables in large databases. It is intended to identify strong rules discovered in databases using different measures of interestingness.[1] Based on the concept of strong rules, Rakesh Agrawal et al.[2] introduced association rules for discovering regularities between products in large-scale transaction data recorded by point-of-sale (POS) systems in supermarkets. For example, the rule $\{\mathrm{onions, potatoes}\} \Rightarrow \{\mathrm{burger}\}$ 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 hamburger meat. Such information can be used as the basis for decisions about marketing activities such as, e.g., promotional pricing or product placements. In addition to the above example from market basket analysis association rules are employed today in many application areas including Web usage mining, intrusion detection, Continuous production and bioinformatics. As opposed to sequence mining, association rule learning typically does not consider the order of items either within a transaction or across transactions.

## Definition

Example database with 4 items and 5 transactions
transaction ID milk bread butter beer
1 1 1 0 0
2 0 0 1 0
3 0 0 0 1
4 1 1 1 0
5 0 1 0 0

Following the original definition by Agrawal et al.[2] the problem of association rule mining is defined as: Let $I=\{i_1, i_2,\ldots,i_n\}$ be a set of $n$ binary attributes called items. Let $D = \{t_1, t_2, \ldots, t_m\}$ 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 $I$. A rule is defined as an implication of the form $X \Rightarrow Y$ where $X, Y \subseteq I$ and $X \cap Y = \emptyset$. The sets of items (for short itemsets) $X$ and $Y$ are called antecedent (left-hand-side or LHS) and consequent (right-hand-side or RHS) of the rule respectively.

To illustrate the concepts, we use a small example from the supermarket domain. The set of items is $I= \{\mathrm{milk, bread, butter, beer}\}$ and a small database containing the items (1 codes presence and 0 absence of an item in a transaction) is shown in the table to the right. An example rule for the supermarket could be $\{\mathrm{butter, bread}\} \Rightarrow \{\mathrm{milk}\}$ meaning that if butter and bread are bought, customers also buy milk.

Note: this example is extremely small. In practical applications, a rule needs a support of several hundred transactions before it can be considered statistically significant, and datasets often contain thousands or millions of transactions.

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## Useful Concepts

To select interesting rules from the set of all possible rules, constraints on various measures of significance and interest can be used. The best-known constraints are minimum thresholds on support and confidence.

• The support $\mathrm{supp}(X)$ of an itemset $X$ is defined as the proportion of transactions in the data set which contain the itemset. In the example database, the itemset $\{\mathrm{milk, bread, butter}\}$ has a support of $1/5=0.2$ since it occurs in 20% of all transactions (1 out of 5 transactions).
• The confidence of a rule is defined $\mathrm{conf}(X\Rightarrow Y) = \mathrm{supp}(X \cup Y) / \mathrm{supp}(X)$. For example, the rule $\{\mathrm{milk, bread}\} \Rightarrow \{\mathrm{butter}\}$ has a confidence of $0.2/0.4=0.5$ in the database, which means that for 50% of the transactions containing milk and bread the rule is correct (50% of the times a customer buys milk and bread, butter is bought as well). Be careful when reading the expression: here supp(X∪Y) means "support for occurrences of transactions where X and Y both appear", not "support for occurrences of transactions where either X or Y appears", the latter interpretation arising because set union is equivalent to logical disjunction. The argument of $\mathrm{supp}()$ is a set of preconditions, and thus becomes more restrictive as it grows (instead of more inclusive).
• 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.[3]
• The lift of a rule is defined as $\mathrm{lift}(X\Rightarrow Y) = \frac{ \mathrm{supp}(X \cup Y)}{ \mathrm{supp}(X) \times \mathrm{supp}(Y) }$ or the ratio of the observed support to that expected if X and Y were independent. The rule $\{\mathrm{milk, bread}\} \Rightarrow \{\mathrm{butter}\}$ has a lift of $\frac{0.2}{0.4 \times 0.4} = 1.25$.
• The conviction of a rule is defined as $\mathrm{conv}(X\Rightarrow Y) =\frac{ 1 - \mathrm{supp}(Y) }{ 1 - \mathrm{conf}(X\Rightarrow Y)}$. The rule $\{\mathrm{milk, bread}\} \Rightarrow \{\mathrm{butter}\}$ has a conviction of $\frac{1 - 0.4}{1 - .5} = 1.2$, and can be interpreted as the ratio of the expected frequency that X occurs without Y (that is to say, the frequency that the rule makes an incorrect prediction) if X and Y were independent divided by the observed frequency of incorrect predictions. In this example, the conviction value of 1.2 shows that the rule $\{\mathrm{milk, bread}\} \Rightarrow \{\mathrm{butter}\}$ would be incorrect 20% more often (1.2 times as often) if the association between X and Y was purely random chance.
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## Process

Frequent itemset lattice, where the color of the box indicates how many transactions contain the combination of items. Note that lower levels of the lattice can contain at most the minimum number of their parents' items; e.g. {ac} can have only at most $min(a,c)$ items. This is called the downward-closure property.[2]

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:

1. First, minimum support is applied to find all frequent itemsets in a database.
2. Second, these frequent itemsets and the minimum confidence constraint are used to form rules.

While the second step is straightforward, the first step needs more attention.

Finding all frequent itemsets in a database is difficult since it involves searching all possible itemsets (item combinations). The set of possible itemsets is the power set over $I$ and has size $2^n-1$ (excluding the empty set which is not a valid itemset). Although the size of the powerset grows exponentially in the number of items $n$ in $I$, efficient search is possible using the downward-closure property of support[2][4] (also called anti-monotonicity[5]) which guarantees that for a frequent itemset, all its subsets are also frequent and thus for an infrequent itemset, all its supersets must also be infrequent. Exploiting this property, efficient algorithms (e.g., Apriori[6] and Eclat[7]) can find all frequent itemsets.

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## History

The concept of association rules was popularised particularly due to the 1993 article of Agrawal et. al.,[2] which has acquired more than 6000 citations according to Google Scholar, as of March 2008, and is thus one of the most cited papers in the Data Mining field. However, it is possible that what is now called "association rules" is similar to what appears in the 1966 paper[8] on GUHA, a general data mining method developed by Petr Hájek et al.[9]

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## Alternative measures of interestingness

Next to confidence also other measures of interestingness for rules were proposed. Some popular measures are:

• Collective strength[11]
• Lift (originally called interest)[14]

A definition of these measures can be found here. Several more measures are presented and compared by Tan et al.[15] Looking for techniques that can model what the user has known (and using this models as interestingness measures) is currently an active research trend under the name of "Subjective Interestingness"

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## Statistically sound associations

One limitation of the standard approach to discovering associations is that by searching massive numbers of possible associations to look for collections of items that appear to be associated, there is a large risk of finding many spurious associations. These are collections of items that co-occur with unexpected frequency in the data, but only do so by chance. For example, suppose we are considering a collection of 10,000 items and looking for rules containing two items in the left-hand-side and 1 item in the right-hand-side. There are approximately 1,000,000,000,000 such rules. If we apply a statistical test for independence with a significance level of 0.05 it means there is only a 5% chance of accepting a rule if there is no association. If we assume there are no associations, we should nonetheless expect to find 50,000,000,000 rules. Statistically sound association discovery[16][17] controls this risk, in most cases reducing the risk of finding any spurious associations to a user-specified significance level.

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## Algorithms

Many algorithms for generating association rules were presented over time.

Some well known algorithms are Apriori, Eclat and FP-Growth, but they only do half the job, since they are algorithms for mining frequent itemsets. Another step needs to be done after to generate rules from frequent itemsets found in a database.

### Apriori algorithm

Apriori[6] is the best-known algorithm to mine association rules. It uses a breadth-first search strategy to count the support of itemsets and uses a candidate generation function which exploits the downward closure property of support.

### Eclat algorithm

Eclat[7] is a depth-first search algorithm using set intersection.

### FP-growth algorithm

FP stands for frequent pattern.

In the first pass, the algorithm counts occurrence of items (attribute-value pairs) in the dataset, and stores them to 'header table'. In the second pass, it builds the FP-tree structure by inserting instances. Items in each instance have to be sorted by descending order of their frequency in the dataset, so that the tree can be processed quickly. Items in each instance that do not meet minimum coverage threshold are discarded. If many instances share most frequent items, FP-tree provides high compression close to tree root.

Recursive processing of this compressed version of main dataset grows large item sets directly, instead of generating candidate items and testing them against the entire database. Growth starts from the bottom of the header table (having longest branches), by finding all instances matching given condition. New tree is created, with counts projected from the original tree corresponding to the set of instances that are conditional on the attribute, with each node getting sum of it's children counts. Recursive growth ends when no individual items conditional on the attribute meet minimum support threshold, and processing continues on the remaining header items of the original FP-tree.

Once the recursive process has completed, all large item sets with minimum coverage have been found, and association rule creation begins.

[18]

### GUHA procedure ASSOC

GUHA is a general method for exploratory data analysis that has theoretical foundations in observational calculi.[19]

The ASSOC procedure[20] is a GUHA method which mines for generalized association rules using fast bitstrings operations. The association rules mined by this method are more general than those output by apriori, for example "items" can be connected both with conjunction and disjunctions and the relation between antecedent and consequent of the rule is not restricted to setting minimum support and confidence as in apriori: an arbitrary combination of supported interest measures can be used.

### OPUS search

OPUS is an efficient algorithm for rule discovery that, in contrast to most alternatives, does not require either monotone or anti-monotone constraints such as minimum support.[21] Initially used to find rules for a fixed consequent[21][22] it has subsequently been extended to find rules with any item as a consequent.[23] OPUS search is the core technology in the popular Magnum Opus association discovery system.

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## Lore

A famous story about association rule mining is the "beer and diaper" story. A purported survey of behavior of supermarket shoppers discovered that customers (presumably young men) who buy diapers tend also to buy beer. This anecdote became popular as an example of how unexpected association rules might be found from everyday data. There are varying opinions as to how much of the story is true.[24] Daniel Powers says:[24]

In 1992, Thomas Blischok, manager of a retail consulting group at Teradata, and his staff prepared an analysis of 1.2 million market baskets from about 25 Osco Drug stores. Database queries were developed to identify affinities. The analysis "did discover that between 5:00 and 7:00 p.m. that consumers bought beer and diapers". Osco managers did NOT exploit the beer and diapers relationship by moving the products closer together on the shelves.

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## Other types of association mining

Contrast set learning is a form of associative learning. Contrast set learners use rules that differ meaningfully in their distribution across subsets.[25]

Weighted class learning is another form of associative learning in which weight may be assigned to classes to give focus to a particular issue of concern for the consumer of the data mining results.

High-order pattern discovery techniques facilitate the capture of high-order (polythetic) patterns or event associations that are intrinsic to complex real-world data. [26]

K-optimal pattern discovery provides an alternative to the standard approach to association rule learning that requires that each pattern appear frequently in the data.

Generalized Association Rules hierarchical taxonomy (concept hierarchy)

Quantitative Association Rules categorical and quantitative data [27]

Interval Data Association Rules e.g. partition the age into 5-year-increment ranged

Maximal Association Rules

Sequential pattern mining discovers subsequences that are common to more than minsup sequences in a sequence database, where minsup is set by the user. A sequence is an ordered list of transactions. [28]

Sequential Rules discovering relationships between items while considering the time ordering. It is generally applied on a sequence database. For example, a sequential rule found in database of sequences of customer transactions can be that customers who bought a computer and CD-Roms, later bought a webcam, with a given confidence and support.

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## References

1. ^ Piatetsky-Shapiro, Gregory (1991), Discovery, analysis, and presentation of strong rules, in Piatetsky-Shapiro, Gregory; and Frawley, William J.; eds., Knowledge Discovery in Databases, AAAI/MIT Press, Cambridge, MA.
2. Agrawal, R.; Imieliński, T.; Swami, A. (1993). "Mining association rules between sets of items in large databases". Proceedings of the 1993 ACM SIGMOD international conference on Management of data - SIGMOD '93. p. 207. doi:10.1145/170035.170072. ISBN 0897915925. edit
3. ^ Hipp, J.; Güntzer, U.; Nakhaeizadeh, G. (2000). "Algorithms for association rule mining --- a general survey and comparison". ACM SIGKDD Explorations Newsletter 2: 58. doi:10.1145/360402.360421. edit
4. ^ Tan, Pang-Ning; Michael, Steinbach; Kumar, Vipin (2005). "Chapter 6. Association Analysis: Basic Concepts and Algorithms". Introduction to Data Mining. Addison-Wesley. ISBN 0-321-32136-7.
5. ^ Pei, Jian; Han, Jiawei; and Lakshmanan, Laks V. S.; Mining frequent itemsets with convertible constraints, in Proceedings of the 17th International Conference on Data Engineering, April 2–6, 2001, Heidelberg, Germany, 2001, pages 433-442
6. ^ a b Agrawal, Rakesh; and Srikant, Ramakrishnan; Fast algorithms for mining association rules in large databases, in Bocca, Jorge B.; Jarke, Matthias; and Zaniolo, Carlo; editors, Proceedings of the 20th International Conference on Very Large Data Bases (VLDB), Santiago, Chile, September 1994, pages 487-499
7. ^ a b Zaki, M. J. (2000). "Scalable algorithms for association mining". IEEE Transactions on Knowledge and Data Engineering 12 (3): 372–390. doi:10.1109/69.846291. edit
8. ^ Hájek, Petr; Havel, Ivan; Chytil, Metoděj; The GUHA method of automatic hypotheses determination, Computing 1 (1966) 293-308
9. ^ Hájek, Petr; Feglar, Tomas; Rauch, Jan; and Coufal, David; The GUHA method, data preprocessing and mining, Database Support for Data Mining Applications, Springer, 2004, ISBN 978-3-540-22479-2
10. ^ Omiecinski, Edward R.; Alternative interest measures for mining associations in databases, IEEE Transactions on Knowledge and Data Engineering, 15(1):57-69, Jan/Feb 2003
11. ^ Aggarwal, Charu C.; and Yu, Philip S.; A new framework for itemset generation, in PODS 98, Symposium on Principles of Database Systems, Seattle, WA, USA, 1998, pages 18-24
12. ^ Brin, Sergey; Motwani, Rajeev; Ullman, Jeffrey D.; and Tsur, Shalom; Dynamic itemset counting and implication rules for market basket data, in SIGMOD 1997, Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD 1997), Tucson, Arizona, USA, May 1997, pp. 255-264
13. ^ Piatetsky-Shapiro, Gregory; Discovery, analysis, and presentation of strong rules, Knowledge Discovery in Databases, 1991, pp. 229-248
14. ^ Brin, Sergey; Motwani, Rajeev; Ullman, Jeffrey D.; and Tsur, Shalom; Dynamic itemset counting and implication rules for market basket data, in SIGMOD 1997, Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD 1997), Tucson, Arizona, USA, May 1997, pp. 265-276
15. ^ Tan, Pang-Ning; Kumar, Vipin; and Srivastava, Jaideep; Selecting the right objective measure for association analysis, Information Systems, 29(4):293-313, 2004
16. ^ Webb, Geoffrey I. (2007); Discovering Significant Patterns, Machine Learning 68(1), Netherlands: Springer, pp. 1-33 online access
17. ^ Gionis, Aristides; Mannila, Heikki; Mielikäinen, Taneli; and Tsaparas, Panayiotis; Assessing Data Mining Results via Swap Randomization, ACM Transactions on Knowledge Discovery from Data (TKDD), Volume 1, Issue 3 (December 2007), Article No. 14
18. ^ Witten, Frank, Hall: Data mining practical machine learning tools and techniques, 3rd edition
19. ^ Rauch, Jan; Logical calculi for knowledge discovery in databases, in Proceedings of the First European Symposium on Principles of Data Mining and Knowledge Discovery, Springer, 1997, pp. 47-57
20. ^ Hájek, Petr; and Havránek, Tomáš (1978). Mechanizing Hypothesis Formation: Mathematical Foundations for a General Theory. Springer-Verlag. ISBN 3-540-08738-9.
21. ^ a b Webb, Geoffrey I. (1995); OPUS: An Efficient Admissible Algorithm for Unordered Search, Journal of Artificial Intelligence Research 3, Menlo Park, CA: AAAI Press, pp. 431-465 online access
22. ^ Bayardo, Roberto J., Jr.; Agrawal, Rakesh; Gunopulos, Dimitrios (2000). "Constraint-based rule mining in large, dense databases". Data Mining and Knowledge Discovery 4 (2): 217–240. doi:10.1023/A:1009895914772.
23. ^ Webb, Geoffrey I. (2000); Efficient Search for Association Rules, in Ramakrishnan, Raghu; and Stolfo, Sal; eds.; Proceedings of the Sixth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2000), Boston, MA, New York, NY: The Association for Computing Machinery, pp. 99-107 online access