SPMF documentation > Mining Minimal Non Redundant Association Rules

This example explains how to mine minimal non-redundant association rules using the SPMF open-source data mining library.

How to run this example?

What is this algorithm?

This algorithm discover the set of "minimal non redundant association rules" (Kryszkiewicz, 1998), which is a lossless and compact set of association rules.

In this implementation we use the Zart algorithm for discovering closed itemsets and their associated generators. Then, this information is used to generate the "minimal non redundant association rules".

What is the input?

The input is a transaction database (aka binary context), a threshold named minconf (a value in [0,1] that represents a percentage) and a threshold named minsup (a value in [0,1] that represents a percentage).

A transaction database is a set of transactions. Each transaction is a set of items. For example, consider the following transaction database. It contains 5 transactions (t1, t2, ..., t5) and 5 items (1,2, 3, 4, 5). For example, the first transaction represents the set of items 1, 2, 4 and 5. This database is provided as the file contextZart.txt in the SPMF distribution. It is important to note that an item is not allowed to appear twice in the same transaction and that items are assumed to be sorted by lexicographical order in a transaction.

Transaction id Items
t1 {1, 2, 4, 5}
t2 {1, 3}
t3 {1, 2, 3, 5}
t4 {2, 3, 5}
t5 {1, 2, 3, 5}

What is the output?

This algorithm returns the set of minimal non redundant association rules.

To explain what is the set of minimal non redundant association rules, it is necessary to review some definitions. An itemset is a set of distinct items. The support of an itemset is the number of times that it appears in the database divided by the total number of transactions in the database. For example, the itemset {1 3} has a support of 33 % because it appears in 2 out of 6 transactions from the database.

An association rule X--> Y is an association between two itemsets X and Y that are disjoint. The support of an association rule is the number of transactions that contains X and Y divided by the total number of transactions. The confidence of an association rule is the number of transactions that contains X and Y divided by the number of transactions that contains X.

A closed itemset is an itemset that is strictly included in no itemset having the same support. An itemset Y is the closure of an itemset X if Y is a closed itemset, X is a subset of Y and X and Y have the same support. A generator Y of a closed itemset X is an itemset such that (1) it has the same support as X and (2) it does not have a subset having the same support.

The set of minimal non redundant association rules is defined as the set of association rules of the form P1 ==> P2 / P1, where P1 is a generator of P2, P2 is a closed itemset, and the rule has a support and confidence respectively no less than minsup and minconf.

For example, by applying this algorithm with minsup = 60 %, minconf= 60% on the previous database, we obtains 14 minimal non redundant associations rules:

2 3 ==> 5 support:: 0.6 confidence: 1
3 5 ==> 2 support: 0.6 confidence: 1
1 ==> 3 support: 0.6 confidence: 0,75
1 ==> 2 5 support: 0.6 confidence: 0,75
1 2 ==> 5 support: 0.6 confidence: 1
1 5 ==> 2 support: 0.6 confidence: 1
3 ==> 1 support: 0.6 confidence: 0,75
3 ==> 2 5 support: 0.6 confidence: 0,75
2 ==> 3 5 support: 0.6 confidence: 0,75
2 ==> 1 5 support: 0.6 confidence: 0,75
2 ==> 5 support: 0.8 confidence: 1
5 ==> 2 3 support: 0.6 confidence: 0,75
5 ==> 1 2 support: 0.6 confidence: 0,75
5 ==> 2 support: 0.8 confidence: 1

Input file format

The input file format is a text file containing transactions. Each lines represents a transaction. The items in the transaction are listed on the corresponding line. An item is represented by a positive integer. Each item is separated from the following item by a space. It is assumed that items are sorted according to a total order and that no item can appear twice in the same transaction. For example, for the previous example, the input file is defined as follows:

1 2 4 5
1 3
1 2 3 5
2 3 5
1 2 3 5

This file contains five lines (five transactions). Consider the first line. It means that the first transaction is the itemset {1, 2, 4, 5}. The following lines follow the same format.

Note that it is also possible to use the ARFF format as an alternative to the default input format. The specification of the ARFF format can be found here. Most features of the ARFF format are supported except that (1) the character "=" is forbidden and (2) escape characters are not considered. Note that when the ARFF format is used, the performance of the data mining algorithms will be slightly less than if the native SPMF file format is used because a conversion of the input file will be automatically performed before launching the algorithm and the result will also have to be converted. This cost however should be small.

Output file format

The output file format is defined as follows. It is a text file, where each line represents an association rule. On each line, the items of the rule antecedent are first listed. Each item is represented by an integer, followed by a single space. After, that the keyword "==>" appears followed by a space. Then, the items of the rule consequent are listed. Each item is represented by an integer, followed by a single space. Then, the keyword " #SUP: " appears followed by the support of the rule represented by an double value indicating a p. Then, the keyword " #CONF: " appears followed by the confidence of the rule represented by a double value (a value between 0 and 1, inclusively). For example, here is the output file for this example:

2 ==> 5 #SUP: 0,8 #CONF: 1
2 ==> 3 5 #SUP: 0,6 #CONF: 0,75
2 ==> 1 5 #SUP: 0,6 #CONF: 0,75
5 ==> 2 #SUP: 0,8 #CONF: 1
5 ==> 2 3 #SUP: 0,6 #CONF: 0,75
5 ==> 1 2 #SUP: 0,6 #CONF: 0,75
3 ==> 2 5 #SUP: 0,6 #CONF: 0,75
3 ==> 1 #SUP: 0,6 #CONF: 0,75
2 3 ==> 5 #SUP: 0,6 #CONF: 1
3 5 ==> 2 #SUP: 0,6 #CONF: 1
1 2 ==> 5 #SUP: 0,6 #CONF: 1
1 5 ==> 2 #SUP: 0,6 #CONF: 1
1 ==> 3 #SUP: 0,6 #CONF: 0,75
1 ==> 2 5 #SUP: 0,6 #CONF: 0,75

For example, the last line indicates that the association rule {1} --> {2, 5} has a support of 60 % and a confidence of 75%. The other lines follow the same format.

Note that if the ARFF format is used as input instead of the default input format, the output format will be the same except that items will be represented by strings instead of integers.

Optional feature: giving names to items

Some users have requested the feature of given names to items instead of using numbers. This feature is offered in the user interface of SPMF and in the command line of SPMF. To use this feature, your file must include @CONVERTED_FROM_TEXT as first line and then several lines to define the names of items in your file. For example, consider the example database "contextZart.txt". Here we have modified the file to give names to the items: 

@CONVERTED_FROM_TEXT
@ITEM=1=apple
@ITEM=2=orange
@ITEM=3=tomato
@ITEM=4=milk
@ITEM=5=bread
1 2 4 5
1 3
1 2 3 5
2 3 5
1 2 3 5

In this file, the first line indicates, that it is a file where names are given to items. Then, the second line indicates that the item 1 is called "apple". The third line indicates that the item 2 is called "orange". Then the following lines define four sequences in the SPMF format.

Then, if we apply a sequential pattern mining algorithm using this file using the user interface of SPMF or the command line, the output file contains several patterns, including the following ones:

orange ==> bread #SUP: 4 #CONF: 1.0
bread ==> apple orange #SUP: 3 #CONF: 0.75
bread ==> orange tomato #SUP: 3 #CONF: 0.75
bread ==> orange #SUP: 4 #CONF: 1.0

Note that this feature could be also used from the source code of SPMF using the ResultConverter class. However, there is currently no example provided for using it from the source code.

Where can I get more information about closed association rules?

The following article provides detailed information about Minimal Non Redundant Association Rules:

M. Kryszkiewicz (1998). Representative Association Rules and Minimum Condition Maximum Consequence Association Rules. Proc. of PKDD '98, Nantes, France, September 23-26.

For a good overview of itemset mining and association rule mining, you may read this survey paper.

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