## 1 Introduction
### Classical Approaches  
A typical approach that is commonly employed in QSPR/QSAR studies involving different molecules is to consider that the AD corresponds to the chemical families on which the model was trained [20; 21; 22; 23]. This approach, otherwise known as \"chemical-based\", relies on the premise that the predicted property will explicitly depend on the molecular characteristics that are used to define the chemical families (e.g., type of bonds, presence of rings or functional groups, etc. ), a dependence that is not always guaranteed a priori. \"Range-based\" methods use the range of each descriptor within the training data to define the AD as a hyper-rectangle of dimension equal to the number of descriptors [1]. Despite their simplicity, the major problem of range-based methods lies in the presence of empty spaces inside the hyper-rectangle making the AD description quite rough. This is even more pronounced for high-dimensional data, where points tend to become equidistant as the number of dimensions increases [9]. Empty spaces do not contain any training data and can be particularly wide in the presence of outliers. This could therefore call into question the reliability of future predictions in these regions, even though they are part of the AD. To limit the extent of the empty spaces caused by the high dimensionality, range-based methods can also be applied to descriptor-derived representations of lower dimension, such as the principal components (PC), issued by the prior implementation of a Principal Component Analysis (PCA). This combination also addresses eventual collinearities within the descriptor space. In \"geometric methods\", the AD corresponds to a convex hull, as this is the smallest convex area that contains the entire training set. In a 2D space, the convex hull can be easily understood via the rubber band analogy: the rubber band tautly surrounding the training set descriptors is the convex hull. Similarly as for range-based methods, empty spaces are not detected. What is more, the calculation of the convex hull becomes computationally complex in high dimensions. In fact, as higher dimensions lead to a higher extrapolation probability [24], both geometric and range-based methods will very likely consider any new point as being outside the AD (i.e., the new point will most probably be extrapolating in at least one dimension). \"Distance-based methods\" are among the most commonly used. They rely on the calculation of a distance-based metric between a given point and the training set. If the distance is below a defined threshold, the point is considered to be part of the AD and vice versa. Among popular distance metrics, one finds the Euclidean and Mahalanobis distances. The latter is popular for the detection of outliers in multivariate data, as is the case with the descriptor space [25]. Contrary to the Euclidean distance, it can identify correlations between features by means of the covariance matrix, similarly as in PCA. Graphically, the AD shape in the case of Mahalanobis distance will be more extended in the directions containing the highest variance (ellipse in 2D), and this will avoid the deletion of points that are not real outliers, as in the case of Euclidean distance (circle in 2D) [26]. Another approach that is popular in recent works is the so-called \"leverage method\" [1; 16; 27; 28; 29; 30; 31; 32]. In these works, the number of features is variable, ranging from less than 10 (principally) to several hundreds or thousands (less frequently). In analogy to the Mahalanobis distance [33], the leverage quantifies the distance between a given point and the centroid of the training data \(X\), in terms of its feature values, and provides an estimation of the potential influence that the observed response will have on its predicted value. Mathematically, the leverage values of the training samples \(X\) can be obtained via the diagonal elements \(h_{ii}\) of the hat matrix \(H\), as defined by Equations (2) and (3).