# A Comparative Study of Two Rule-Based Explanation Methods for Diabetic Retinopathy Risk Assessment
## 3 Preliminaries
### Dominance-Based Rough Set Approach (DRSA)  
The DRSA method [15] can be used to construct a set of decision rules in line 6 of Algorithm 1. The main difference of these rules with respect to those obtained from a decision tree is that they are classification rules with a set of conditions that take into account the preference directions of the input variables. In this subsection, we explain how the rules are constructed from the set of examples \(\mathcal{D}\). Rough set theory (RST) [28] is a formal theory derived from fundamental research on the logical properties of information systems. The main goal of the rough set analysis is the approximation of concepts. In addition, it offers mathematical tools to discover patterns hidden in data. As a result, it has a wide range of applications, including feature and pattern extraction, data reduction and decision rules generation (our goal in this work). It can also identify partial or total dependencies in data, among other things. Rough set analysis concerns data stored in a table known as an Information Table. Each row represents an object \(x_{i}\), evaluated with respect to multiple attributes representing different points of view; the information table is defined as a pair \((X,\mathcal{F})\), where \(X\) is a non-empty finite set of objects and \(\mathcal{F}\) is a non-empty finite set of attributes. A special kind of information table is a Decision Table \((X,\mathcal{F}\cup Dec)\), where the attributes are divided into condition attributes \(\mathcal{F}\) and decision attributes \(Dec\). The former are related to features of objects, while the latter relate to decisions about objects. Often there is just a single decision attribute \(\mathcal{Y}\). Distinct values \(y_{k}\) of this attribute, called class labels, induce a partition of the set of objects into so-called decision classes \(Cl_{k}\). DRSA is an extension of RST, suitable for analysis of decision tables where both condition attributes from \(\mathcal{F}\) and the output decision variable (decision attribute) \(\mathcal{Y}\) are ordinal, and there exist monotonic relationships between attributes from \(\mathcal{F}\) and \(\mathcal{Y}\). A positive relationship means that the greater the value of the condition attribute, the higher the class label. A negative relationship means that the greater the value of the condition attribute, the lower the class label. Both types of relationships are captured by induced decision rules. In general, in DRSA the number of decision classes can be more than two. Then, one has to consider upward and downward unions of decision classes. However, in the case of REITPROGRAM, we only have two classes: 0 for the absence of DR risk and 1 for the presence of DR risk. Thus, using DRSA, we calculate rough approximations and induce decision rules for exactly these two classes. Rules are constructed using elementary building blocks, known as dominance cones, with origins in each object in the attribute space. Based on the rough set concept, rules for a lower or/and an upper approximation of each decision class are obtained from a training set (\(\mathcal{D}\) in our case) [29]. The choice of DRSA for explainability in the Diabetic Retinopathy disease is motivated by the fact that the values of the attributes are mainly ordinal, and a change from one value to another may be an indicator of the risk of developing DR. Moreover, using the VC-DomLEM algorithm [30], one can induce a set of rules being a minimal cover of consistent objects from both classes. This enables to efficiently distinguish between the two possible decision outputs [30], which is one of the aims of a surrogate model. Two types of rules may be distinguished:  
1. \(\mathcal{Y}\geq\) decision rules, providing lower profile descriptions for objects belonging at least to class \(Cl_{k}\) (so they belong to \(Cl_{k}\) or a better class, \(Cl_{k+1},Cl_{k+2},\ldots\)): If \(f_{1}\geq v_{1}\) AND \(f_{2}\geq v_{2}\) AND \(\ldots f_{n}\geq v_{n}\) THEN \(y\geq y_{k}\)
2.