Fig. 2

Ellipses obtained in the case of a two-dimensional space for two classes of observations. Each class k is generated using a different multivariate normal distribution \(\mathcal {N}({\varvec{\mu }}_k, {\varvec{\Sigma }}_k)\). The first graph (a) is obtained considering \({\varvec{\mu }}_1 = t\left( \begin{array}{c} 0 \\ 0 \end{array}\right)\), \({\varvec{\mu }}_2 = \left( \begin{array}{c} 1.5 \\ -1.5 \end{array}\right)\), \({\varvec{\Sigma }}_1 = \left( \begin{array}{cc} 0.3 &{} 0.1 \\ 0.1 &{} 0.9 \end{array}\right)\) and \({\varvec{\Sigma }}_2 = \left( \begin{array}{cc} 0.8 &{} 0 \\ 0 &{} 0.8 \end{array}\right)\), resulting in a positive value of the \(\gamma\)-metric. The second graph (b) is obtained considering \({\varvec{\mu }}_1 =\left( \begin{array}{c} 0.1 \\ 0.1 \end{array}\right)\), \({\varvec{\mu }}_2 = \left( \begin{array}{c} 1.1 \\ 0 \end{array}\right)\), \({\varvec{\Sigma }}_1 = \left( \begin{array}{cc} 0.4 &{} -0.1 \\ -0.1 &{} 0.5 \end{array}\right)\) and \({\varvec{\Sigma }}_2 = \left( \begin{array}{cc} 0.9 &{} 0 \\ 0 &{} 1.2 \end{array}\right)\), resulting in a negative value of the \(\gamma\)-metric