The use of a distance measure in regularised discriminant analysis
Let \(X\) be \(p\)-dimensional measurements on each object, where each object belongs to one of \(K\) classes, and let the distribution of measurements from each class be the multivariate normal distribution with population mean \(\mu_{i}\) and covariance function \(\Sigma_{i}\) for class \(i\). For the sample regularized discriminant function (SRDF) an estimation of the covariance matrix is proposed of the form \[ \hat\Sigma_{i}(\lambda,\gamma)=(1-\gamma)\hat\Sigma_{i}(\lambda)+ \gamma p^{-1}tr\left(\hat\Sigma_{i}(\lambda)\right)I, \] where \[ \hat\Sigma_{i}(\lambda)=[(1-\lambda)(n_{i}-1)S_{i}+\lambda S_{p}][(1-\lambda)(n_{i}-1)+\lambda(N-K)]^{-1},\quad 0\leq\lambda\leq 1; \] where \(n_{i}\) is the sample size, \(N=\sum_{i=1}^{K}n_{i}\), \(S_{p}=(N-K)^{-1}\sum_{i=1}^{K}S_{i}(n_{i}-1)\), \(S_{i}\) is the sample covariance matrix for class \(i\), and \(I\) is the identity matrix. The authors investigate the use of the Bhattacharyya distance measure \(B\) as an alternative to the cross-validation procedure of the SRDF in determining the optimal estimated values of \(\hat\lambda\) and \(\hat\gamma\). A classification rule which uses regularization parameters obtained from \(B\) is presented for two and three populations cases. Using simulation, experiment comparisons of commonly used discriminant functions and SRDF are carried out. Various rules are compared in terms of their performances in correctly classifying observations from several real data sets.
scientific article; zbMATH DE number 1865447
scientific article