# Numerical Association Rule Mining from a Defined Schema Using the VMO Algorithm
## 1 Introduction  
Technological developments have contributed significantly to solving complex problems; in fact, tasks that, a few years ago, were limited by the number of resources required are feasible in terms of processing today. In addition, the new requirements that they are demanding in terms of speed, veracity, variety, and volume are caused by mechanisms for the capture and generation of massive amounts of data, such as social networks, sensors, and other intelligent sources [1].  
One of the most significant contributions to data science is that of association rule mining (ARM) [2]; from its origin to the present, it has been applied to the discovery of purchasing patterns by starting with transactional data. Several problems have been modeled with this technique, such as customer relationship management (CRM), product recommendation, and promotion and sales associated with products [1, 3]. ARM is not limited only to marketing and sales problems, but has also been adapted in various disciplines. Its application is possible for any area for which data from which rules can be generated are available in order to discover knowledge [4, 5, 6].  
It is known that the datasets are not composed only of qualitative attributes, but also quantitative ones, which have different characteristics, such as the wide range in which the values are defined. They do not occur with a significant frequency in the set, making it difficult to identify patterns with techniques similar to those used with categorical attributes. We refer to quantitative association rule mining (QARM) [7]. This is a subdivision that encompasses numerical attributes in the search process for patterns through rules in the data.  
This approach to the problem is not new. Investigations have already been developed by using the same qualitative principles, but they have reflected limited results in terms ofthe quality and high consumption of computing resources [8]. This gives place to an open field for the definition of promising lines of investigation.  
An important research line inside the problem combines both categorical and numerical attributes, especially those with a class attribute. These types of rules provide interesting alternatives for classification association rules (CARs) [9]. The reduction of classes generated by classical association rule classifiers is a prominent approach, which, among other purposes, seeks to develop significantly smaller rules on big datasets compared to traditional classifiers while maintaining the classification accuracy [10].  
One of the first approaches to QARM studied was through discretization, which consisted of domain partitioning with consideration of the most suitable partitioning method [11; 12; 13]. Several researchers have evaluated this procedure. The disadvantage that they pointed out is its low precision caused by the definition of the width of the intervals due to the partitioning technique applied. Consequently, this leads to a loss of valuable information by raising the sensitivity of the basic quality metrics used in the evaluation, that is, support and confidence.  
Another approach is the use of the statistical measures obtained with the characterization of the numerical attributes. In addition, both supervised and unsupervised clustering techniques have been used to identify dense areas in the distribution. Fuzzy techniques were helpful when there was considerable imprecision in the definition of the interval. Finally, optimization methods have been based on the adaptation of evolutionary algorithms to face requirements that demand speed and large volumes of data.