Minimum Distance Classifier

• Training

  The kth class $\omega_k$ is represented by its mean vector ${\bf m}_k$ and covariance matrix which can be estimated from the training samples:
  

  $${\bf m}_k=\frac{1}{K_k}\sum_{i=1}^{K_k} {\bf x}_i^{(k)}\;\;\;\;(k=1,\cdots,C)$$
  
  
  and
  
  $${\bf\Sigma}_k=\frac{1}{K_k}\sum_{i=1}^{K_k} ({\bf x}_i^{(k)}-{\bf m}_k)({\bf x}_i^{(k)}-{\bf m}_k)^T$$
  
  

• Classification

  A given pattern ${\bf x}$ of unknown class is classified to $\omega_k$ if its Mahalanobis distance to $\omega_k$ is smaller than those to all other classes:
  

  $${\bf x}\in \omega_k\;\;\;\mbox{iff}\;\;\;d_M({\bf x},\omega_k) =min \{d_M({\bf x},\omega_i)\;\;i=1,\cdots,C \}$$
  
  

For simplicity, the distance $d_p({\bf x},{\bf m}_i)$ can be used to replace $d_M({\bf x},\omega_i)$ above. As now only the mean vector of each class is used, the classification does not take into account how the classes are distributed in the feature space.