A Generalization of the k-NN Rule

A modification of the k-nearest neighbors (k-NN) rule is presented in which classification is made not according to the ``majority vote'' but rather an integer threshold k1 (k1-NN rule). It is shown that while k-NN approximates the minimum expected error rule, k1-NN approximates the minimum expected risk rule with a threshold t. The relationship between t and values of k and k1 is derived. Several practical methods of using k1-NN for minimum expected risk classification and for classification with a reject option are described and illustrated with examples.