A distribution-free geometric upper bound for the probability of error of a minimum distance classifier

An upperbound to the probability of error per class in a multivariate pattern classification is derived. The bound, given by

$\text{P(E|class w}{}_{\mathrm{i}}\text{)≤}\text{N}\text{R}{}^2{}_{\mathrm{i}}$

is derived with minimal assumptions; specifically the mean vectors exist and are distinct and the covariance matrices exist and are non-singular. No other assumptions are made about the nature of the distributions of the classes. In equation (i) N is the number of features in the feature (vector) space and R_i is a measure of the “radial neighbourhood” of a class. An expression for R_i is developed. A comparison to the multivariate Gaussian hypothesis is presented.