SPMF documentation > Mining Periodic Frequent Patterns using the PFPM algorithm

This example explains how to run the PFPM algorithm using the SPMF open-source data mining library.

How to run this example?

What is PFPM ?

PFPM is an algorithm for discovering periodic frequent itemsets in a sequence of transactions (a transaction database). It was proposed by Fournier-Viger et al. (2016). PFPM can discover patterns that periodically appears in a sequence of transactions. Periodic pattern mining has many applications such as discovering periodic behavior of customers, and finding recurring events.

What is the input of the PFPM algorithm?

The input is a transaction database (a sequence of transactions) and four parameters that are set by the user:

  • a minimum periodicity threshold minper ( a positive integer)
  • a maximum periodicity threshold maxper ( a positive integer)
  • a minimum average periodicity threshold minavgper ( a positive integer)
  • a maximum average periodicity threshold maxavgper ( a positive integer)

Note that two optional parameters are also offered to specify constraints on the minimum and maximum number of items that patterns should contain (positive integers).

A transaction database is a set of transactions. Each transaction is a set of items. For example, consider the following transaction database. It contains 7 transactions (t1, t2, ..., t7) and 5 items (1,2, 3, 4, 5). For example, the first transaction represents the set of items 1 and 3. This database is provided as the file contextPFPM.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, 3}
t2 {5}
t3 {1, 2, 3, 4, 5}
t4 {2, 3, 4, 5}
t5 {1, 3, 4}
t6 {1,3,5}
t7 {2, 3, 5}

What is the output of the algorithm?

PFPM is an algorithm for discovering periodic frequent patterns, which are also called periodic frequent itemsets. An itemset is a group of items. The PFPM algorithm finds itemsets that appears periodically in a sequence of transactions. To measure if an itemset is periodic, the algorithm calculates its periods. To explain this in more details, it is necessary to introduce a few definitions.

The set of transactions containing an itemset X is denoted as g(X). For example, consider the itemset {1, 3}. The set g({1,3}) is equal to {t1, t3, t5, t6}. In other words, the itemset {1, 3} appears in the transactions t1, t3, t5 and t6. It is also said that these are four occurrences of the itemset {1,3}.

Now, to assess the periodicity of an itemset X, its list of periods is calculated. A period is the time in terms of number of transactions between two occurences of an itemset in the database (see the paper for the formal definition). For example, the periods of the itemset {1, 3} are {1,2,2,1,1}. The first period of {1,3} is 1 because {1,3} appears in the first transaction after the creation of the database. The second period of {1,3} is 2 because the itemset appears in transaction t3, which is two transactions after t1. The third period of {1,3} is 2 because the itemset appears in transactions t5, which is two transactions after t3. Then, the fourth period of {1,3} is 1 because the itemset appears in t6, which is one transaction after t5. Finally, the fifth period of {1,3} is 1 because there is one transaction appearing after the last occurrence of {1,3} in the database (in t6).

The PFPM algorithms utilize the list of periods of an itemset X to calculate its average periodicity, minimum periodicity and maximum periodicity. The average periodicity is calculated as the average of the periods of the itemset. The minimum periodicity is the smallest period among the periods of the itemset (note that the first and last periods are excluded from the calculation of the minimum periodicity - see the paper for details). The maximum periodicity is the largest period among the periods of the itemset.

The PFPM algorithm finds all the itemsets that have a minimum periodicity, maximum periodicity that are not less than the minper and maxper thresholds, set by the user, and an average periodicity that is not less than minavgper and not greater than maxavgper.

For example, if PFPM is run on the previous transaction database with a minper = 1, maxper = 3 , minavgper = 1 and maxavgper = 2, the PFPM algorithm finds 11 periodic frequent itemsets.

itemset support (number of transactions where the itemset appear) minimum periodicity maximum periodicity average periodicity
{2} 3 1 3 1.75
{2, 5} 3 1 3 1.75
{2, 3, 5} 3 1 3 1.75
{2, 3} 3 1 3 1.75
{4} 3 1 3 1.75
{3, 4} 3 1 3 1.75
{1} 4 1 2 1.4
{1, 3} 4 1 2 1.4
{5} 5 1 2 1.17
{3, 5} 4 1 3 1.4
{3} 6 1 2 1

How should I interpret the results?

Each frequent periodic itemset is annotated with its support (number of transactions where it appears) as well as its minimum/maximum periodicity and average periodicity. For example, the itemset {1, 3} has a support of 4 because it appears in four transactions (t1, t3, t5 and t6. The average periodicity of {1,3} is 1.4 because on average it appears every 1.4 transactions in terms of time. The smallest period of {1,3} (minimum periodicity) is 1 and the largest period of {1,3} is 2 transactions.This indicates that {3} appears quite periodically.

Input file format

The input file format used by PFPM is defined as follows. It is a text file. An item is represented by a positive integer. A transaction is a line in the text file. In each line (transaction), items are separated by a single space. It is assumed that all items within a same transaction (line) are sorted according to a total order (e.g. ascending order) and that no item can appear twice within the same line.

For example, for the previous example, the input file is defined as follows:

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

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 a frequent periodic itemset. On each line, the items of the itemset are first listed. Each item is represented by an integer and it is followed by a single space. After, all the items, the keyword "#SUP:" appears, which is followed by an integer indicating the support of the itemset, expressed as a number of transactions. Then, the keyword #MINPER: appears and is followed by a space, an integer indicating the minimum periodicity of the itemset, and another space.Then, the keyword #MAXPER: appears and is followed by a space, an integer indicating the maximal periodicity of the itemset, and another space. Then, the keyword #AVGPER: appears and is followed by a space, a double value indicating the average periodicity of the itemset.

For example, here is the output file for this example. The first line indicates that the itemset {2} is a frequent periodic itemset, having a support of 3 transactions, a minimum periodicity of 1 transactions, a maximum periodicity of 3 transactions and an average periodicity f 1.75 transactions..

2 #SUP: 3 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.75
2 5 #SUP: 3 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.75
2 5 3 #SUP: 3 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.75
2 3 #SUP: 3 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.75
4 #SUP: 3 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.75
4 3 #SUP: 3 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.75
1 #SUP: 4 #MINPER: 1 #MAXPER: 2 #AVGPER: 1.4
1 3 #SUP: 4 #MINPER: 1 #MAXPER: 2 #AVGPER: 1.4
5 #SUP: 5 #MINPER: 1 #MAXPER: 2 #AVGPER: 1.1666666666666667
5 3 #SUP: 4 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.4
3 #SUP: 6 #MINPER: 1 #MAXPER: 2 #AVGPER: 1.0

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 "contextPFPM.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
3 1
5
3 5 1 2 4
3 5 2 4
3 1 4
3 5 1
3 5 2

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:

apple tomato #SUP: 4 #MINPER: 1 #MAXPER: 2 #AVGPER: 1.4
bread #SUP: 5 #MINPER: 1 #MAXPER: 2 #AVGPER: 1.1666666666666667
bread tomato #SUP: 4 #MINPER: 1 #MAXPER: 3 #AVGPER: 1.4
tomato #SUP: 6 #MINPER: 1 #MAXPER: 2 #AVGPER: 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.

Performance

PFPM is currently the only algorithm designed to mine periodic frequent itemset in a sequence of transaction (a transaction database), offered in SPMF. Another algorithm called PHM is offered for mining periodic patterns in sequence of transactions containing profit informaiton (items have weights/unit profit, and transactions indicate purchase quantities for items)

Where can I get more information about the PFPM algorithm?

This is the article proposing the PFPM algorithm:

Fournier-Viger, P., Lin, C.-W., Duong, Q.-H., Dam, T.-L., Sevcic, L., Uhrin, D., Voznak, M. (2016). PFPM: Discovering Periodic Frequent Patterns with Novel Periodicity Measures. Proc. 2nd Czech-China Scientific Conference 2016, Elsevier, 10 pages.

You can also view a video presentation of the PFPM algorithm

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