## 2 Materials and Methods
### A Motivation Example  
In this section, we present a comprehensive motivation example. The role of this example is to (i) introduce new ideas and some terminology underlying the proposed approach, (ii) show the way of dealing with the data and the necessity of supplementary knowledge introduction, and (iii) show why classical approaches to ML/XAI have some intrinsic limitations. The data table below is used as input for ML algorithms. In fact, Table 1 gives only an idea of how the data look and covers only a very small sample of the data in use. A complete data table consists of 100 observations. The structure of the data includes three independent input columns (labeled \(a\), \(b\), and _c_) and one output or decision column (labeled decision). The input columns generally contain real numbers, whereas the output is an integer with possible values of 0, 1, and 2. Consequently, the input triples are deterministically classified into three distinct classes. The ML task is to create a prediction model that can determine the output value (decision) for any given set of input values. Some ML paradigms selected for this experiment are Linear Regression (LR), Decision Trees (DT), and Random Forest (RF). A series of experiments implemented in the R language (using the stats and caret packages) was carried out in order to create models that attempt to discover the decision function from example data. A total of 100 observations were employed for training purposes, and the models were subsequently validated on an additional set of 100 observations, which were distributed similarly to the training dataset. Results of the experiments were assessed using the following metrics: Root Mean Squared Error (RMSE), calculated as \(\text{RMSE}(y,\hat{y})=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}\), and RMSE normalized by the mean (\(NRMSE=\frac{RMSE}{\hat{y}}\)). The results are presented in Table 2 below. The experiment shows that the prediction results are far from being perfect. None of the proposed techniques is capable of creating an accurate ML model. While LR returns the mean decision value of training data and performs similarly on the test dataset, the DT and RF models are overfit. For the learned RF model, we applied the most popular XAI tools, namely LIME and SHAP. The results for representative prediction are presented in Figure 3 below. LIME suggests variable \(b\) has the biggest impact on the prediction, while variable \(c\) contributes the least. In contrast to that, the SHAP explanation puts forward \(c\) as the most impactful variable and \(b\) as the least important. Moreover, in LIME, both \(b\) and \(c\) make positive contributions to the outcome of the model, while in SHAP, the contribution of both is negative. As there are considerable discrepancies between the explanations proposed by LIME and SHAP, it is not straightforward to decide which explanation should be respected as the correct one. Moreover, there is no single explanation technique that is universally superior, and there are no common choice criteria with respect to which technique should be used for explanation generation for the model. This is the reason a clear definition of the purpose of the XAI task and key properties of explanation for a given case is needed. \begin{table}
\begin{tabular}{c c c c} \hline
**a** & **b** & **c** & **Decision** \\ \hline
3 & 9 & 7 & 0 \\
1 & 4 & 4 & 1 \\
5 & 9 & 4 & 2 \\ � & � & � & � \\ \hline \end{tabular}
\end{table} 
Table 1: A schematic presentation of the input data. The table contains a sample of the data used for experiments. \begin{table}
\begin{tabular}{c c c c c} \hline  & **RMSE Train** & **NRMSE Train** & **RMSE Test** & **NRMSE Test** \\ \hline Linear & 0.8267 & 0.8026 & 0.8239 & 0.8368 \\ Regression & 0.5431 & 0.5273 & 0.8142 & 0.8224 \\ Random Forest & 0.2284 & 0.2218 & 0.5883 & 0.5942 \\ \hline \end{tabular}
\end{table} 
Table 2: RMSE for selected ML techniques.