# Relaxed Rule-Based Learning for Automated Predictive Maintenance: Proof of Concept
## 1 Introduction
The main issue that needs to be addressed is the handling of conflicting predictions. Indeed, suppose that the same instance is covered by two rules, one of them states that the instance belongs to the concept, the other that the instance does not. Another matter to address is the terminating condition for the procedure of forming a rule collection: we add rules one by one to the collection, when are we to stop? Both of the above issues can be relatively straightforwardly addressed by a methodology of _Separate-and-Conquer_ (SeCo) [2; 3]. According to this methodology, when a new rule is formed, the instances covered by this rule are removed. So, the newly formed rules are guaranteed to cover new instances and the process will stop when there are no new instances (of course the terminating condition can be relaxed to avoid overfitting). Unlike in the case of DT, the rules _may overlap_. Indeed, suppose a rule \(R_{1}\) has been formed and the instances covered by it removed. When a new rule \(R_{2}\) is being formed, the procedure growing it does not'see' the removed instances but this does not mean that these instances cannot be covered. However, the rules formed by a SeCo procedure are ordered in a chronological order (according to the time they have been formed). When a prediction is about to be made about some instance \(x\), the prediction is made by the _first_ rule covering this instance. To understand the intuition, assume that the instance \(x\) is covered by the 5 th rule. Then rules 1 to 4 do not cover the instance hence there is no point using them for making the prediction. Rules 6 onward can cover the instance. However, rule 5 has been formed as a result of analysis of a _larger_ training set, so it is rational to assume that it will be more precise on instances covered by it and the subsequent rules. Both DT and SeCo methods suffer from _fragmentation_, though for SeCo algorithm the effect is milder, as demonstrated by Example 2 below. [**Fragmentation**] Consider the same settings of the attributes as in Example 1. Suppose that the concept can be described as the disjunction of the following two rules. 1. \(R_{1}\): \(attr_{1}\in[10,\ldots,20]\wedge attr_{2}\in[15,\ldots 40]\to true\). 2. \(R_{2}\): \(attr_{3}\in[20,\ldots,35]\wedge attr_{4}\in[25,\ldots,50]\to true\). The sets of instances covered by \(R_{1}\) and \(R_{2}\) clearly overlap. Therefore, after a SeCo procedure discovers the first rule, it is likely to be more difficult to discover the second rule. Indeed, suppose such a procedure discovers \(R_{1}\). After that the rule learner will have to look at the part of the dataset that is not covered by \(R_{1}\) and to discover rule \(R_{2}\). A rule learner that learns non-overlapping rules (e.g., a DT algorithm) will have to discover rules that are covered by \(R_{2}\) but not covered by \(R_{1}\). There are several way to present the corresponding set of rules, the most compact of them would look as follows. \(1.\)\(R_{3}\): \(attr_{1}\in[0,\ldots,9]\wedge attr_{3}\in[20,\ldots,35]\wedge attr_{4}\in[25, \ldots,50]\to true\). \(2.\)\(R_{4}\): \(attr_{1}\in[21,\ldots,50]\wedge attr_{3}\in[20,\ldots,35]\wedge attr_{4}\in[25, \ldots,50]\to true\). \(3.\)\(R_{5}\): \(attr_{2}\in[0,\ldots,14]\wedge attr_{3}\in[20,\ldots,35]\wedge attr_{4}\in[25, \ldots,50]\to true\). \(4.\)\(R_{6}\): \(attr_{2}\in[41,\ldots,50]\wedge attr_{3}\in[20,\ldots,35]\wedge attr_{4}\in[25, \ldots,50]\to true\). That is instead of a single rule of length 2, the rule learner will have to learn 4 rules of length 3. In case of \(k\) short rules, the number of rules to be learned grows exponentially with the number of rules already learned. We informally refer to this effect as fragmentation. In case of a more general SeCo method, the situation is not as acute as in the case of non-overlapping rule learning because there is no need to avoid overlapping with \(R_{1}\).