1 Introduction
As data mining, decision support systems, and artificial intelligence are quickly entering mainstream technologies, we have to account for the human nature that is innate in people. The recent interest in the field of User Experience (UX) is an important demonstration that software and hardware need to be easily and quickly understood by the usershassenzahl2006user . As the user interface is only a filter that mediates between a very subjective user and a very matteroffact series of algorithms, we also need to identify ways to let the underlying processes understand and account for the very human concept of imprecision.
Keeping the human nature in mind is not just a concept that is applicable to user interfaces, but also to the processes powering decision support systemsbarthelemy2002human . The vague and often imprecise nature of humans is an essential element to take into account also in the backend of analytical software. In particular, we have to account for the arbitrary nature of linguistic variables and quantifiersmartinez2010computing , such as ”People purchase more umbrellas when it is very cloudy,” and temporal associationsale2000approach , such as ”Sales in football jerseys spike in the days following a team’s victory.” The need for a more elastic reasoning model is even more significant when we put the two concepts together, for example when we warn a friend that ”traffic gets very intense on highway XYZ shortly after it starts raining, even if it is just a light rain.”
This paper presents a practical approach to creating an association analysis system that leads to a more meaningful and actionable set of recommendations. The workshoporiented nature of the venue in which this paper is presented lends a great opportunity to explore the innerworkings of this analysis methodology as well as some of the fundamental concepts on which it is based.
2 Background Information
The process of decisionmaking cannot be isolated from imprecision as well as temporality, given that the main task of decision support systems is to effectively summarize situations that will then drive actionsaronson2005decision . Even though time is perhaps the most classic univariate dimension associated with actiondetectionyamato1992recognizing , we can utilize utilize different granularities to study how events are relatedghorbani2017methodology . Combining these concepts with the basic nature of fuzzy set theory applied to association analysischen2009mining , we can build a powerful engine for decision support systems.
2.1 Association Analysis
One of the principal algorithms in data mining scouts for frequent associations in different commercial transactions agrawal1994fast . Its name, MarketBasket Analysis, originates from the goal of identifying which products were frequently purchased together at the grocery store by looking at the pool of items scanned at checkout.
Associations generated through this algorithm are in the form of an implication: . The premise represents the itemset that then triggers the purchase of the itemset represented by the conclusion . Both and may contain only one item, or may contain several. A sample association is the following: .
As different people may purchase different itemsets, and not everyone who purchases peanut butter and bread may also purchase jelly, the associations must have some metric that quantifies their relevance. The two main metrics are support and confidence. A third metric is lift, but we will not address it in this paper.
Support indicates the strength of the association over all of the associations identified in the analysis. The value is calculated in the following manner:
(1) 
where denotes the support count, or frequency, of occurrences of through all associations, divided by the number of all associations identified.
The second metric, confidence, instead quantifies the reliability of the association identified over all the other associations that contain the same trigger itemset. This metric is calculated in the following manner:
(2) 
where the numerator is the count of instances of the association, and the denominator is the total number of associations that included the same premise, for example both Peanut Butter and Bread, whether the customer may have also purchased Jelly or not. Both metrics give a result in the domain .
2.2 Fuzzy Sets
The domain of computing is often associated with precision, and in particular with the true/false dichotomy that is at the core of this field. This concept is easily implemented through hard boundaries and checks, such as whether a number is greater than another or not, and which is at the base of classic, or Cantorian, set theory. The moment we apply computing to real life though, we find the necessity of implementing solutions that account for ambiguity in terms of classification, or at least that account for the possibility of labeling the same event in multiple ways. The concept of Fuzzy Sets extends the classic set theory to account for this uncertainty
zadeh1965fuzzy .Although we can have membership functions of several kinds klir1995fuzzy , in this work we will only utilize trapezoidal sets, also called fuzzy intervals. Each set is associated with a membership function, as shown in Equation 3.
(3) 
where the terms , , , and represent the points that compose a fuzzy interval, shown in Figure 1.
This set will be utilized to assign a linguistic variable zadeh1975concept to continuous values. For example, a value of or would not belong to fuzzy set . Instead values between and would have full membership (). A value of would have a membership value of . In order to facilitate the discussion, we will refer to as any instance of that belongs to stream A, and to the membership weight associated with that linguistic variable as . This will also be done for streams B and C, as well as the temporal relationship .
3 Scouting for Imprecise Temporal Associations
The algorithm discussed in this paper addresses two different aspects related to association analysis. First of all, we include a temporal relationship between the triggering events and the consequence, then we account for imprecision using fuzzy sets. The work is formally described by Sudkamp sudkamp2005examples and the initial implementation has been published previously by the author and others vincenti2005data .
Similarly to association analysis, the associations produced by this method include a premise, which we can call a set of triggers, and a conclusion, which we call a consequence. The triggers itemset currently contains two events, and the consequence itemset contains the temporal relationship as well as an event that is linked to the triggers. The following is an example of an association: .
The events , , and are linguistic variables computed using the process explained in Section 2.2, and the is also a linguistic variable, and it represents the time elapsed between the triggers itemset and the event . A detailed example in Section 3.1
will walk through the process of going from a data stream to a set of imprecise temporal associations and the corresponding decision tree.
Since we are working with fuzzy sets, each association cannot be counted as a single instance. The partial membership of any value to more than one set would lead to results with incorrect weights. For this reason, each association will carry a weight calculated in the following manner:
(4) 
Consequently, also the metrics of support and confidence must be adjusted to account for partial memberships of the different events. Support is calculated in the following manner:
(5) 
where the numerator is the total weight of all associations divided by the combined weight of all associations, . Comparing this way to calculate the support to the traditional method reported in Section 2.2, we need to note that may be equal to
, but it may also be different. This depends on the membership functions that are utilized to classify the stream events, as well as any minimum thresholds that the investigators may set for both support and confidence levels.
Confidence is the second metric, which is calculated using the following equation:
(6) 
where the numerator is the weight of all associations and the denominator is the weight of all associations that have as the trigger itemset.
3.1 Example
In order to better illustrate the process, we will step through an example that will utilize each element that we discussed so far. We start from a generic set of three streams, as reported in Table 1.
Timestamp  Stream 1  Stream 2  Stream 3 
0  2     
3    8   
7      10.5 
13      15 
1000  7     
1003    2   
1013      7 
Each stream event is associated with a timestamp, so that we can establish a temporal relationship between any two events. This will be useful when we are determining whether a pair of triggering events has occurred, and whether a consequence may be linked to a set of triggering events. The streams contain continuous values that are associated with a generic measure of volume, and the timestamp represents a generic unit of time.
In this example, the window between trigger events will be at most of 10 time units, which means that if an event were to occur at Stream 1 at time , we would have a set of triggering events only if an event occurs on Stream 2 between time and . Once the set of triggers is identified, we will evaluate Stream 3 for a consequence. In this case the window between the triggers and the consequence is also of 10 time units.
Association ID  Trigger 1  Trigger 2  Consequence  
2  8  4  10.5  
2  8  10  15  
7  2  10  7  
The data streams above, in conjunction with the time window constraints, will lead to the identification of three numerical associations, reported in Table 2.
Type  Set Name  a  b  c  d 

Data streams  Small Volume  0  0  3  6 
Medium Volume  3  6  9  12  
Large Volume  9  12  15  15  
Temporal associations  Immediately After  0  0  1  3 
Short Time After  1  3  5  7  
Long Time After  5  7  10  10  
The next step involves classifying the events represented by the continuous values into linguistic variables. In order to do so, we will have to utilize fuzzy sets. The boundaries of the fuzzy intervals utilized to assign linguistic variables to the stream data as well as the temporal relationships are reported in Table 3. We are reporting only the values associated with the fuzzy interval, which can be utilized in conjunction with Equation 3 to identify the linguistic variable (set name) and to calculate the related membership value.
Association ID  Trigger 1  Trigger 2  Consequence  Weight  Support  Confidence  
Small  Medium  Shortly After  Medium  0.5  
Small  Medium  Shortly After  Large  0.5  
Small  Medium  Long Time After  Large  1.0  
Medium  Small  Long Time After  Medium  1.0  
The numeric associations reported in Table 2 will then be turned into the fuzzy associations reported in Table 4. One of the first things we can notice is that the number of associations is now greater. This is because not all numeric associations contain values that belong uniquely to one fuzzy set. In the case of association in Table 2, we can see that the value associated with Stream 3 is of 10.5, which reports and . This classification into two different sets for the consequence event leads to fuzzy associations and , in Table 4.
4 Application to Decision Support Systems
The process of mining for imprecise temporal associations is similar to most other data mining processes kurgan2006survey . The process utilized by this method is reported in Figure 2. The solid lines in the figure represent data flows between processes, while the dashed lines represent influences between processes. The iterative nature of identifying and refining membership functions is the core process that regulates the imprecision associated with any associations discovered. Generating membership functions can be as simple as making educated guesses, or it can be as complex as utilizing elaborate artificially intelligent methodsmedasani1998overview .
The knowledge base can assume many forms, but the most useful representation that we have found in the context of decision support systems is inspired by decision trees. The associations reported in Table 4 can easily be turned into a decision tree, as reported in Figure 3.
This representation leverages a very visual layout that will help decisionmakers in the process. The leaf nodes also report Support (Equation 5) and Confidence (Equation 6) to facilitate the consumption of the findings of this algorithm. The resulting tree can be compared to FrequentPattern (FP) Trees han2000mining , already in use in data mining lin2010linguistic .
5 Conclusions and Future Work
The material discussed in this paper guides the reader through the foundations and the basic concepts that are involved with mining for imprecise temporal associations in data streams. The integration of such analysis in the decisionmaking process is essential, especially given the volume and nature of today’s data, as well as its importance in terms of operationalizing the findings quickly and effectively. The work described in this paper has demonstrated its relevance in the fields of medicine and energy utilization, and the findings will be published at a later date.
The next steps for this research focus on algorithmic improvements. In particular, the system is currently limited in the number of streams that it processes to 2 for the triggering itemset and one for the consequence. In order to extend classic association analysis algorithms, this system will also have to accommodate itemsets with one or more streams for either the trigger or the consequence.
Given the nature of continuous stream analysis, the number of potential associations is significant and the runtime for the system can be quite tedious. Incrementing the number of streams to analyze must be accompanied by a different implementation, which should include distributed and parallel processing. Currently the system operates on a single node and does not distributed the work among multiple processors. The distribution of processing during the associations mining phase will significantly increase the capabilities of this system.
Lastly, the process of identifying membership functions is currently driven by domain knowledge and empirical methods. Even though this approach may be appropriate, we will implement other methods of identifying the most meaningful membership functions through statistical methods and the application of artificial intelligence.
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