## 3 Methods
### Approximator Learning  
The paper [27] describes the detailed workflow associated with model approximation and diagnostic attributes; in this section we just briefly call the most important assumptions. The main idea underlying this approach is to create a surrogate model [28; 29] based on predictions of the origin model \(M(x)\). We expect that a surrogate model will mimic the behavior of the original model \(M\), which we do not have access to. This assumption is due to the business environment, in which access to the original model is difficult, while our method requires only access to the training data. In addition, an approach based solely on predictions is model-agnostic and universal. The quality of a surrogate model is monitored by the Cohen kappa score calculated between the original predictions and the approximated predictions, and it should be above 0.9 for good properties. To approximate these predictions, we use the rough sets theory [30] and create a surrogate model using an ensemble of approximate reducts \(AR\)[31]. An approximate reduct is an irreducible subset of attributes that is sufficient to express almost the same information about the decisions as the whole attribute set. In this context, we distinguish two data sets: diagnosed (new) \(D=(X_{D},d_{D})\), which is used to create a surrogate model, and training \(R=(X_{R},d_{R})\), which is used to train the original model. In our case, the decision values for which we construct the approximate reducts correspond to predictions of the diagnosed model, i.e., \(M(X_{D})\), not the actual ground truth target values. The result of the above-mentioned steps is the trained approximator model \(\overline{M}\), composed of a set of approximate reducts \(\mathcal{R}=\{AR_{1},\ldots,AR_{i},\ldots,AR_{k}\}\) and approximated prediction for instances of data \(\overline{M}(x)\). The construction of the approximator depends on the selection of two hyper-parameters: an \(\epsilon\) representing the approximation threshold for reducts, and a number of reducts in the ensemble. Since the aim of the procedure is to find the appropriate approximation of the model's predictions, we use the grid search to tune the hyper-parameter settings. We also assume that there is access to the decision attribute (target) in the data set, which will be denoted as \(d\). For \(d(x)\), we will denote its value for an instance \(x\). Moreover, \(q^{M}(x)\) is the estimation of decision class probabilities made by the diagnosed model \(M\) and \(\overline{q^{M}}(x)\) is the approximation of decision class probability predictions. The next step is defining the neighborhood \(N(x)\) for each observation in the diagnosed data set. We want to identify instances that are similar with regard to predictions made by \(M\), and we define a notion of neighborhood based on the prediction process of the model approximator \(\overline{M}\). In particular, the neighborhood \(N^{AR}(x)\) for a diagnosed instance \(x\) with regard to a single reduct \(AR\in\mathcal{R}\) is defined as a subset of instances from \(X_{R}\) that belong to the same indiscernibility class, i.e., \([x]_{AR}\). The final neighborhood is the sum of neighborhoods computed for all reducts in the ensemble. Thus, for each instance in the diagnosed data set, we have information about similar instances in the training data set. Then, it is possible to compare the distribution of targets, predictions, and approximate predictions between given observations in the diagnosed data set and similar observations from the training data set, which were chosen for the neighborhood. Such a defined neighborhood for a given observation from a diagnosed data set in extreme cases can contain all observations from the training data set or none of them. The neighborhood is the basis for calculating diagnostic attributes, which are listed below:  
* Target consistency with approximations in the neighborhood--measures the consistency of the target of the diagnosed instance with the approximations from the neighborhood of this instance.