# A Comparative Study of Two Rule-Based Explanation Methods for Diabetic Retinopathy Risk Assessment
## 3 Preliminaries
### Dominance-Based Rough Set Approach (DRSA)
\(\mathcal{Y}\leq\) decision rules, providing upper profile descriptions for objects belonging at most to class \(Cl_{k}\) (so they belong to \(Cl_{k}\) or a lower class, \(Cl_{k-1},Cl_{k-2},\ldots\)): If \(f_{1}\leq v_{1}\) AND \(f_{2}\leq v_{2}\) AND \(\ldots f_{n}\leq v_{n}\) THEN \(y\leq y_{k}\). In this notation, we must take into account that all condition attributes in \(\mathcal{F}\) are considered to be maximisation functions (the higher the value, the higher the class label), which are called Gain attributes. In case an attribute has to be minimised, it is called a Cost attribute, and the lower its value, the higher the class label. It is also possible to introduce a criterion as both Cost and Gain. In this case, the attribute may appear twice in the rule and define an interval of values. An important feature of the DRSA method coupled with the VC-DomLEM algorithm is the fact that particular rules are minimal (without redundant conditions) and the whole set of rules is non-redundant (if any rule would be removed, some consistent objects would not be covered by any rule). Algorithm 2 shows the main steps of the DRSA method, assuming that only certain decision rules are considered (as in our case). ```
input :\(\mathcal{D}\) - training set of objects (decision table) output :\(\gamma\) - quality of classification, \(\mathcal{R}\) - set of decision rules generated on \(\mathcal{D}\)
1\(\mathcal{X}^{\geq}\leftarrow CalculateUpwardClassUntions(\mathcal{D})\);
2\(\mathcal{X}^{\leq}\leftarrow CalculateDownwardClassUntions(\mathcal{D})\);
3 foreach\(X\in\mathcal{X}^{\geq}\cup\mathcal{X}^{\leq}\)do
4\(X.LowerApproximation\leftarrow CalculateLowerApproximation(X,\mathcal{D})\);
5
6 end for
7\(\gamma=CalculateQualityOfClassification(\mathcal{X}^{\geq},\mathcal{X}^{\leq},\mathcal{D})\);
8\(\mathcal{R}^{\geq}\leftarrow VC-DomLEM(\mathcal{X}^{\geq})\);
9\(\mathcal{R}^{\leq}\leftarrow VC-DomLEM(\mathcal{X}^{\leq})\);
10\(\mathcal{R}\leftarrow\mathcal{R}^{\geq}\cup\mathcal{R}^{\leq}\);
```  
**Algorithm 2**DRSA method  
In lines 1-2, all upward and downward unions of decision classes are identified, depending on the class labels of \(\mathcal{Y}\). In the loop defined in the following lines 3-5, for each upward/downward union its lower approximation is calculated. These approximations are stored inside objects representing particular unions of classes. In line 6, the quality of classification is calculated. This is a typical rough set descriptor related to consistency of data, defined as a ratio of the number of consistent objects and all objects in \(\mathcal{D}\). During calculation of \(\gamma\), one takes into account the lower approximations calculated previously. In line 7, the VC-DomLEM algorithm is invoked for the upward unions of classes to induce decision rules. It generates rules describing objects from the lower approximations of subsequent unions, iterating from the most specific to the least specific union to control rule minimality. Suppose decision attribute \(\mathcal{Y}\) has labels \(1,2,3,4,5\), and the higher the label, the more preferred the respective decision class. Then, VC-DomLEM will first generate rules for class \(Cl_{5}\), then for upward union of classes \(Cl_{4}^{\geq}=Cl_{4}\cup Cl_{5}\), then for upward union \(Cl_{3}^{\geq}\), and finally for upward union \(Cl_{2}^{\geq}\). Obviously, considering union \(Cl_{1}^{\geq}\) does not make sense (set of all objects). In line 8, the VC-DomLEM algorithm is invoked to induce decision rules for the downward unions of classes. This is realized analogously, with the only difference that this time first class \(Cl_{1}\) will be taken into account, then downward union of classes \(Cl_{5}^{\leq}=Cl_{1}\cup Cl_{2}\), next downward union \(Cl_{5}^{\leq}\), and finally downward union \(Cl_{4}^{\leq}\). Remark that VC-DomLEM algorithm was introduced for the Variable Consistency DRSA (VC-DRSA), being an extension of the classical DRSA.