Information conservation in feature selection

The percentage of separability information (energy) contained in the m-D space after feature selection can be found as

$$r=\frac{\sum_{i=1}^m {\bf a}_i^T{\bf S}_{B/W}{\bf a}_i}{\sum... ...^m {\bf a}_i^T{\bf S}_{B/W}{\bf a}_i}{\sum_{i=1}^n \lambda_i}$$

where $\lambda_i$'s are the eigenvalues of ${\bf S}_{B/W}$. When KLT is used, the above can be further written as

$$r= \frac{\sum_{i=1}^m \phi_i^T{\bf S}_{B/W}\phi_i}{\sum_{i=1... ...bda_i} =\frac{\sum_{i=1}^m \lambda_i}{\sum_{i=1}^n \lambda_i}$$

as here ${\bf a}_i={\bf\phi}_i\;\;\;(i=1,\cdots,m)$ are the eigenvectors of ${\bf S}_{B/W}$ (corresponding to the m largest eigenvalues $\lambda_i\;\;\;(i=1,\cdots,m)$).