## 3 Results
### Explanation Examples: IRIS  
In this section, we consider several white-box, model-driven XAI models for the widely explored IRIS dataset. The IRIS dataset consists of 150 instances, where each instance represents an iris flower sample. Each sample has four features/attributes, namely sepal length, sepal width, petal length, and petal width. Based on these features, each sample in the dataset is labeled with one of three classes, representing different species of iris flowers, namely _Setosa_, _Versicolor_, or _Virginica_. The dataset is balanced, with 50 samples for each of the three species. Moreover, it demonstrates good separability among the different species of iris flowers. The simplicity and small size of the dataset make it a popular choice for algorithm testing in the field of ML. Initially, we applied the GE approach to four features from the dataset (minimizing the number of misclassified labels, allowing 1000 iterations, and with default parameters), utilizing the following grammar:  
expr = grule((expr) & (sub.expr),  (expr) | (sub.expr),  sub.expr),  sub.expr = grule(comparison(var, func.var)),  comparison = grule('>', '<', '==', '>=', '<='),  func.var = grule(num, var, func(var)),  func = grule(mean, max, min, sd),  var = grule(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width),  num = grule(1, 1.5, 2, 2.5, 3, 4, 5)  
Below, we present illustrative examples that demonstrate the separability of the dataset, including the resultant formulas and plots for the _Setosa_ (Figure 6) and _Versicolor_ (Figure 7) classes:  
ifelse(Petal.Length <= Sepal.Width, \"setosa\", \"other\")  
Figure 6: _Setosa_ species classification using GE with sepal width and petal length parameters. While the classification accuracy for the _Setosa_ species was perfect, the accuracy for the other classes suggests potential for improvement (e.g., 94.66% for _Versicolor_). To estimate confidence intervals (CI) for the accuracy of rule-based classification models, which traditionally lack direct uncertainty measures due to their deterministic nature, we applied a bootstrap resampling technique. Utilizing the boot package in R, we performed 1000 resamplings of the IRIS dataset. Each sample was evaluated using a predefined model that classifies instances as a given species or other based on fixed thresholds of the input features. This process of recalculating the model accuracy for each bootstrap sample generated a distribution of accuracies. From this distribution, percentile-based confidence intervals were derived, quantifying the variability and reliability of the model performance across different samples. The estimated CI (with a 95% confidence level) for _Versicolor_ is (0.9067, 0.9800). To improve the classification, we introduced MIVs. In order to maintain simplicity, several models are presented in order but according to the same, simple scheme. * Causal model--data flow;
* Input, output, and MIVs;
* Functional model;
* Accuracy;
* Simplicity. The first model is very simple; only one MIV is introduced, and it is a value intended to be proportional to the petal area. The inputs are petal.length and petal.width  
\[PLmPW=petal.length*petal.width \tag{1}\]  
We allowed for MIVs in the grammar definition. For each species, we enabled 1000 iterations of the evolutionary process. The complete functional model introduces a simple comparison of the MIV value defined by Equation (1) with the threshold values presented below. ifelse(PLmPW <= 2, \"setosa\", \"other\")  
ifelse((PLmPW <= 8) & (PLmPW >= 2), \"versicolor\", \"other\")  
ifelse(PLmPW >= 8, \"virginica\", \"other\")  
For intuition, in human-understandable terms, \(PLmPW\) can be interpreted as the \"rough area\" of a petal. The rules can be put together to create the functional model shown in Figure 8.  
Figure 7: _Versicolor_ species classification using GE with petal width and petal length parameters.