## 4 Classifier-Independent Level of Separability Measures
### Discriminant Ratio  
Similar to Equation (1), by \(X_{\Omega}^{i}\), we denote the subset of \(X_{\Omega}\) that consists solely of the samples from class \(\omega_{i}\), i.e.,  
\[X_{\Omega}^{i}=\{x\in X|(x,y)\in X_{\Omega}\wedge y=\omega_{i}\}. \tag{9}\]  
Moreover, by \(\bar{x}^{(i)}\), we define the _centroid_ of the set \(X_{\Omega}^{i}\). More precisely, with \(N_{i}=|X_{\Omega}^{i}|\) denoting the number of samples in \(X_{\Omega}^{i}\),  
\[\bar{x}^{(i)}=\frac{1}{N_{i}}\sum_{x\in X_{\Omega}^{i}}x. \tag{10}\]  
Furthermore, by \(\sigma_{i}^{2}\in\mathbb{R}^{d}\), we denote the \(d\)-dimensional (\(X\subset\mathbb{R}^{d}\), \(d\in\mathbb{N}\)) variance in \(X_{\Omega}^{i}\), which we define as follows:\[\sigma_{i}^{2}=\frac{1}{N_{i}-1}\sum_{x\in X_{\Omega}^{i}}(x-\bar{x}^{(i)})\circ(x- \bar{x}^{(i)}). \tag{11}\]  
Note that the \(\circ\)-symbol denotes the Hadamard product, also known as the Schur product, which is an element-wise product. More precisely, for \(u,v\in\mathbb{R}^{d}\) with \(w=u\circ v\), it holds, \(w\in\mathbb{R}^{d}\) with \(w_{i}=u_{i}v_{i}\), \(\forall i=1,\ldots,d\).  
Inspired by [PERSON]'s discriminant analysis [24], in this work, we define the _discriminant ratio_ (DR) between the classes \(\omega_{i}\) and \(\omega_{j}\) as follows,  
\[DR_{i,j}:=\frac{\|\bar{x}^{(i)}-\bar{x}^{(j)}\|^{2}}{\|\sigma_{i}^{2}\|+\| \sigma_{j}^{2}\|}. \tag{12}\]  
Obviously, it holds, \(DR_{i,j}\in\mathbb{R}_{\geq 0}\), and \(DR_{i,j}\) is undefined if and only if \(\sigma_{i}^{2}=\sigma_{j}^{2}=0\). Therefore, \(DR_{i,j}\) is undefined if and only if each of the sets \(X_{\Omega}^{i}\) and \(X_{\Omega}^{j}\) consists of solely one data point or of arbitrarily many identical data points, respectively. However, this is an unrealistic classification task scenario. Therefore, in this work, we will always assume that it holds, \(\|\sigma_{i}^{2}\|+\|\sigma_{j}^{2}\|>0\).  
Note that the DR measure constitutes an LSM mapping by Definition 1. More precisely, changing the class labels for the samples from class \(\omega_{i}\) to \(\omega_{j}\), and changing the class labels for the samples from class \(\omega_{j}\) to \(\omega_{i}\), obviously leads to the same \(DR_{i,j}\) value as for the initial class labels. Therefore, the DR mapping is label invariant, and therefore fulfils property (P2) of Equation (2). Thus, we only have to check property (P1) of Equation (2), since the validity of property (P0), i.e., the non-negativity, directly follows from Equation (12).  
To this end, let \(Y=\{0,1\}\) be a binary label set, and \(X\subset\mathbb{R}^{d},d\in\mathbb{N}\), be a \(d\)-dimensional data set, with at least two unequal elements, i.e., \(|X|>1\). For \(X_{Y}^{*}=X\times Y\), it holds, \(\bar{x}^{(0)}=\bar{x}^{(1)}\), and thus \(\|\bar{x}^{(0)}-\bar{x}^{(1)}\|^{2}=0\). Therefore, for \(X_{Y}^{*}=X\times Y\), it holds that \(DR_{0,1}=0\), which is the lower bound of the defined measure. Thus, the DR mapping fulfils property (P1) of Equation (2), and therefore constitutes an LSM mapping by Definition 1.  
An exemplary illustration of the discriminant ratio-specific components is provided in Figure 5, for a 2-dimensional class pair, based on a toy data set.