An Introduction to Sequential Rule Mining

In this blog post, I will discuss an interesting topic in data mining, which is the topic of sequential rule mining. It consists of discovering rules in sequences. This data mining task has many applications for example for analyzing the behavior of customers in supermarkets or users on a website.


Before, discussing this topic, let me talk a little bit about the context. There has been a lot of work in the field of data mining about pattern mining . The goal of pattern mining is to discover useful, novel and/or unexpected patterns in databases. In this blog post, we will be interested by a specific type of database called sequences databases.  A sequence database contains some sequences. For example, consider the following database:

sequence database

A sequence database containing four sequences

This database contains four sequences named seq1, seq2, seq3 and seq4.  For our example, consider that the symbols “a”, “b”, “c”, d”, “e”, “f”, “g” and “h” respectively represents some items sold in a supermarket. For example, “a” could represent an “apple”, “b” could be some “bread”, etc.

Now, a sequence is an ordered list of sets of items. For our example, we will assume that each sequence represents what a customer has bought in our supermarket over time.  For example, consider the second sequence “seq2”. This sequence indicates that the second customer bought items “a” and “d” together, than bought item “c”, then bought “b”, and then bought “a”, “b”, “e” and “f” together.

Sequences are a very common type of data structures that can be found in many domains such as bioinformatics (DNA sequence), sequences of clicks on websites, the behavior of learners in e-learning,  sequences of what customers buy in retail stores, sentences in a text, etc.

Discovering sequential patterns in sequences

An important data mining problem is to design algorithm for discovering hidden patterns in sequences. There have been a lot of research on this topic in the field of data mining and various algorithms have been proposed.

In the following, Iwill discuss two types of patterns that can be found. Iwill first discuss sequential patterns. Then, I will explain some of their limitations and then discuss sequential rules.

A sequential pattern is  a subsequence that appear in several sequences of a database.  For example, the sequential pattern <{a}{c}{e}> appears in the two first sequences of our database. This pattern is quite interesting. It indicates that customers who bought {a}, often bought {c} after, followed by buying {e}.  Such a pattern is said to have a support of two sequences because it appears in two sequences from the database. Several algorithms have been proposed for finding all sequential patterns in a database such as CM-SPADE, PrefixSpan and GSP. These algorithms takes as input a sequence database and a minimum support  threshold (minsup). Then, they will output all sequential patterns having a support no less than minsup.  Those patterns are said to be the frequent sequential patterns.

For example, for the above example, if we run CM-SPADE with minsup = 3,  we will find the following frequent sequential patterns:

<{a}> with a support of 3 sequences
<{a},{e}> with a support of 3 sequences
<{a},{f}> with a support of 3 sequences
<{b},{e}> with a support of 3 sequences
<{b},{f}> with a support of 4 sequences

Sequential patterns can be quite interesting. In the example, we can learn that buying item “b” is followed by buying item “e” in 3 sequences. However, sequential patterns can be misleading. An important limitation of sequential patterns is that there is no assessment of the probability that a pattern will be followed. Let me explain this in more details. For example, if we consider again the pattern <{b},{e}>. This pattern is said to appear in 3 sequences. It may thus seems likely that if someone buy “b”, he will also buy “e” after. But how likely?  We can observe that item “b” appears in four sequences. Thus, the probability that “e” appears after “b” is actually 3 / 4 = 75 %  (i.e. P(e|b)= 75%).  But sequential patterns only indicate how often the pattern appears. They do not provide any indication about this probability.

Discovering sequential rules in sequences

This now lead us to the main topic of this post which is sequential rule mining. Sequential rule mining has been proposed as an alternative to  sequential pattern mining to take into account the probability that a pattern will be followed. I will provide a few definitions and then we will look at a full example.

A sequential rule is a rule of the form X -> Y  where X and Y are sets of items (itemsets).  A rule X ->Y is interpreted as if  items in X occurs (in any order), then it will be followed by the items in Y (in any order).  For example, consider the rule {a} -> {e,f}. It means that if a customer buy item “a”, then the customer will later buy the items “e” and “f”.  But the order among items in {e,f} is not important. This means that a customer may buy “e” before “f” or “f” before “e”.

To find sequential rules, two measures are generally used:  the support and the confidence. The support of a rule X -> Y  is how many sequences contains the items from X followed by the items from Y.  For example, the support of the rule {a} -> {e,f} is 3 sequences because {a} appears before the items from {e,f} in three sequences (seq1, seq2 and seq3).

The confidence of a rule X -> Y is the support of the rule divided by the number of sequences containing the items from X. It can be understood as the conditional probability P(Y|X).  For example, the confidence of the rule {a} -> {e,f} is 1 (or 100 % if written as a precentage), because every time that a customer buy item “a”, he then buy “e” and “f” in the example database.  Another example is the rule {a} -> {b}. This rule has a support of 2 sequences and a confidence of 0.66 (that is 66%).

A sequential rule mining algorithm such as RuleGrowth, ERMiner and CMRules will output all sequential rules having a support and a confidence respectively no less than some thresholds minsup and minconf set by the user.  For example, consider again the example database and suppose that the user set  minsup = 0.5 and minconf = 60%.  The following rules are found by RuleGrowth:

{a,b,c} -> {e} support = 2 sequences confidence = 100 %
{a} -> {c,e,f} support = 2 sequences confidence = 66%
{a,b} -> {e,f} support = 3 sequences confidence = 100%
{b} -> {e,f} support = 3 sequences confidence = 75 %
{a} -> {e,f} support = 3 sequences confidence = 100%
{c} -> {e,f} support = 2 sequences confidence = 100%
{a} -> {b} support = 2 sequences confidence = 66%

These rules can be viewed as more interesting than sequential patterns since they give a measure of confidence that they will be followed.  For example, it is very informative to know that some rules such as {c} -> {e,f} have a confidence of 100 %.

In the past, I have carried a study with my student to compare the prediction accuracy of sequential patterns and sequential rules. In that study, we found sequential rules can provide a much higher prediction accuracy than sequential patterns when the patterns are used for sequence prediction. The reason is that sequential rules consider the probability (confidence), while sequential patterns do not.

Extensions of the task of sequential rule mining

In the previous paragraphs, I have introduced the topic of sequential rule mining. But note there also exists several extensions of the problem of sequential rule mining. These extensions have been proposed to address specific needs. I will provide a brief overview of a few extensions.

  • Discovering the top-k sequential rules.   The idea is to discover the k most frequent rules in a dataset having at least a confidence no less than minconf. For example, a user may specify that he wants to find the top 1000 rules having a confidence of at least 75 %. Some algorithms for this task are TopSeqRules and TNS.
  • Discovering sequential rules with a window size constraint. This algorithm let the user find rules of the form X -> Y where X and Y must be close to each other with respect to time. For example, a user may want to find rules appearing whithin three consecutive itemsets in sequences. This is interesting for example for analyzing sequence of web clicks. An algorithm for this task is TRuleGrowth.
  • Discovering high-utility sequential rules. Another extension is to discover rules where items may be annotated with quantities in sequences and each item may have a unit profit. For example, we may have a sequence where a customer bought three breads, then two apples and two bottle of milk and these items may have some unit profit of 1$, 2$ and 1.50$. The goal of high-utility sequential rule mining is to find rules that generate a high profit and have a high confidence (high-utility rules).  An algorithm for this task is HUSRM.

Open-source implementations and datasets

There exists several algorithms for sequential rule mining and sequential pattern mining that have been proposed. Java implementations of the state-of-the art algorithms are currently offered in my open-source data mining library named SPMF.

It offers several state-of-the-art algorithms for sequential rule mining such as ERMiner (2014),  TNS (2013), RuleGrowth (2011), TopSeqRules (2011), and CMRules (2010). Besides, SPMF offers several algorithms for sequential pattern mining such as CM-SPADE (2014),  VMSP (2014), LAPIN (2005) and PrefixSpan (2004).  To our knowledge, ERMiner is the fastest sequential rule mining algorithm. But RuleGrowth is still quite fast and consumes less memory. You can try the above algorithms by going to the SPMF website. On the website, you will find instructions about how to run algorithms and some datasets on the dataset page.

Applications of sequential rule mining

Some example of applications of sequential rule mining are e-learning,  manufacturing simulation,  quality control, web page prefetching, anti-pattern detection in service based systems, embedded systems,  alarm sequence analysis, restaurant recommendation. For example, here are a few papers describing such applications:


Fournier-Viger, P., Faghihi, U., Nkambou, R., Mephu Nguifo, E.: CMRules: Mining
Sequential Rules Common to Several Sequences. Knowledge-based Systems, Elsevier,
25(1): 63-76 (2012)

Toussaint, Ben-Manson, and Vanda Luengo. “Mining surgery phase-related sequential rules from vertebroplasty simulations traces.” Artificial Intelligence in Medicine. Springer International Publishing, 2015. 35-46.

Faghihi, Usef, Philippe Fournier-Viger, and Roger Nkambou. “CELTS: A Cognitive Tutoring Agent with Human-Like Learning Capabilities and Emotions.” Intelligent and Adaptive Educational-Learning Systems. Springer Berlin Heidelberg, 2013. 339-365.

Manufacturing simulation

Kamsu-Foguem, B., Rigal, F., Mauget, F.: Mining association rules for the quality
improvement of the production process. Expert Systems and Applications 40(4),
1034-1045 (2012)

Quality control

Bogon, T., Timm, I. J., Lattner, A. D., Paraskevopoulos, D., Jessen, U., Schmitz,
M., Wenzel, S., Spieckermann, S.: Towards Assisted Input and Output Data Analysis
in Manufacturing Simulation: The EDASIM Approach. In: Proc. 2012 Winter
Simulation Conference, pp. 257–269 (2012)

Web page prefetching

Fournier-Viger, P. Gueniche, T., Tseng, V.S.: Using Partially-Ordered Sequential
Rules to Generate More Accurate Sequence Prediction. Proc. 8th International Conference
on Advanced Data Mining and Applications, pp. 431-442, Springer (2012)

Anti-pattern detection in service based

Nayrolles, M., Moha, N., Valtchev, P.: Improving SOA antipatterns detection in
Service Based Systems by mining execution traces. In: Proc. 20th IEEE Working
Conference on Reverse Engineering, pp. 321-330 (2013)

Embedded systems

Leneve, O., Berges, M., Noh, H. Y.: Exploring Sequential and Association Rule
Mining for Pattern-based Energy Demand Characterization. In: Proc. 5th ACM
Workshop on Embedded Systems For Energy-Efficient Buildings. ACM, pp. 1–2

Alarm sequence analysis

Celebi, O.F., Zeydan, E., Ari, I., Ileri, O., Ergut, S.: Alarm Sequence Rule Mining
Extended With A Time Confidence Parameter. In: Proc. 14th Industrial Conference
on Data Mining (2014)

Ileri, Omer, and Salih Ergüt. “Alarm Sequence Rule Mining Extended With A Time Confidence Parameter.” (2014).


Jannach, Dietmar, and Simon Fischer. “Recommendation-based modeling support for data mining processes.” Proceedings of the 8th ACM Conference on Recommender systems. ACM, 2014.

Interestingly, the above work found that sequential rules found by CMRules provided better results than other compared patterns found using FPGrowth and other algorithms.

Jannach, D., Jugovac, M., & Lerche, L. (2015, March). Adaptive Recommendation-based Modeling Support for Data Analysis Workflows. In Proceedings of the 20th International Conference on Intelligent User Interfaces (pp. 252-262). ACM.

Restaurant recommendation

Han, M., Wang, Z., Yuan, J.: Mining Constraint Based Sequential Patterns and
Rules on Restaurant Recommendation System. Journal of Computational Information
Systems 9(10), 3901-3908 (2013)

Customer behavior analysis

Noughabi, Elham Akhond Zadeh, Amir Albadvi, and Behrouz Homayoun Far. “How Can We Explore Patterns of Customer Segments’ Structural Changes? A Sequential Rule Mining Approach.” Information Reuse and Integration (IRI), 2015 IEEE International Conference on. IEEE, 2015.


In this blog post, I have given an overview of the tasks of sequential rule mining and sequential pattern mining, which aims at discovering patterns in sequences. Hope that you enjoyed reading it 😉 For researchers, there are many possibility of research on this topic.


Philippe Fournier-Viger is a professor of Computer Science and also the founder of the open-source data mining software SPMF, offering more than 80 data mining algorithms.


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5 Responses to An Introduction to Sequential Rule Mining

  1. Pingback: An introduction to frequent pattern mining - The Data Mining & Research Blog

  2. Another awesome article, thank you again!

  3. Pingback: An introduction to the discovery of periodic patterns in data - The Data Mining & Research Blog

  4. Tho says:

    Thanks for the post. However, I have a question: what are differences between sequence mining and rule mining ? They look similar, aren’t they?

    • Hello, Glad you like the post. Sometimes the terms may have different meaning. But for me “Sequence mining” is more general. It means to discover patterns in sequences. But it does not say what kind of patterns. In a sequence, we could find many kind of patterns such as “sequential patterns”, “sequential rules”, “periodic patterns”, etc. So, sequential rule mining is one task that you could call sequence mining, but there are also other tasks. Besides, some people will also discover patterns in 1 sequence, while other will find patterns that are common to multiple sequences. There are in fact many variations of these tasks. For example, if you do sequential rule mining in a single sequence, it is usually called episode rule mining instead of sequential rule mining, but you could still call it sequence mining.

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