## 1 Introduction
### Classical Approaches
For a new query sample \(x_{new}\), the leverage values are calculated via a similar formula, defined by Equation (4). From these equations, it is possible to identify that the major problem in the leverage method will be related to the inversion of the matrix \(X^{T}X\). This inversion becomes problematic when the descriptor matrix \(X\) contains redundant or correlated information, or when the number of descriptors exceeds the number of samples [26]. Besides, the calculation of the matrix \(X^{T}X\) is sensitive to outliers, as highlighted in [25; 34]. **Hat matrix \(H\) and leverage values \(h_{ii}\), \(h_{new}\):**  
\[H=X(X^{T}X)^{-1}X^{T} \tag{2}\]  
\[h_{ii}=x_{i}^{T}(X^{T}X)^{-1}x_{i} \tag{3}\]  
\[h_{new}=x_{new}^{T}(X^{T}X)^{-1}x_{new} \tag{4}\]  
where \(X\) is the descriptor matrix based on training data (with n rows/observations and p columns/descriptors), \(x_{i}\) is the column vector of a molecule in the training data and \(x_{new}\) is the column vector of any new molecule. In practice, leverage values (X-outliers) are plotted against standardized residuals (Model-outliers) on a Williams plot, with their corresponding thresholds. Standardized residuals are defined by Equation (5). Generally, a point belongs to the AD if its leverage value is below \(h^{*}=3(p+1)/n\) and if its standardized residual is between \(-3\) and \(3\)[35; 36]. These user-defined rule-of-thumb thresholds can be understood as follows. For the leverage threshold, the idea is that one compares the value of the leverage of a single point with the mean leverage value of all points, \((p+1)/n\). As for standardized residuals, the cutoff value of \(\pm 3\) covers 99% of the assumed normally distributed standardized residuals. Williams plots are more a graphical way to verify whether a developed model is statistically acceptable during model construction [29; 36]. However, it is only possible to check if a new query molecule is an X-outlier, and not if it is a Model-outlier, as the calculation of the standardized residual is impossible (i.e., the observed property value is unknown). **Standardized residuals:**  
\[r_{k}=\frac{y_{k}^{observed}-y_{k}^{model}}{\sqrt{Var(r_{train})}} \tag{5}\]  
where \(y_{k}^{observed}\) and \(y_{k}^{model}\) are, respectively, the observed and the predicted responses for a molecule k and \(Var(r_{train})\) is the variance of residuals of the training data. Other distance-based methods also include techniques that utilize the k-Nearest Neighbors (kNN) algorithm. In this case, it is the distance to the nearest training point or the average distance to the k-nearest training neighbors that can be used as a means to decide whether a point lies inside or outside the AD [16]. Similar to range-based and geometric methods, empty spaces remain undetected with distance-based methods. Most importantly, all these methods suffer from the consequences of the curse of dimensionality. As the number of descriptors increases, points become equidistant whatever the chosen distance metrics. Despite several attempts to develop distance metrics that are more compatible with high-dimensional data [11], it is far more common to use dimensionality reduction methods instead. Finally, another way to improve the AD definition when using distance-based methods is to calculate the distance metrics by considering different weights for the descriptors according to their influence (e.g., coefficients in linear regression) [1]. A different class of AD definition methods is based on the use of probability density distributions. Contrary to all previous methods, this is the only class of methods that can identify empty zones inside a convex hull. Indeed, the probability density function of the training data is first estimated via a parametric or a non-parametric method. Whenever possible, the latter is generally preferred, as it does not make any assumption about the shape of the function and learns it from the data. Then, the smallest region containing a given amount of the total probability is identified as the AD.