Methods are provided for determining discriminant functions of minimum risk quadratic classification systems, wherein a discriminant function is represented by a geometric locus of a principal eigenaxis of a quadratic decision boundary. A geometric locus of a principal eigenaxis is determined by solving a system of fundamental locus equations of binary classification, subject to geometric and statistical conditions for a minimum risk quadratic classification system in statistical equilibrium. Feature vectors and machine learning algorithms are used to determine discriminant functions and ensembles of discriminant functions of minimum risk quadratic classification systems, wherein a discriminant function of a minimum risk quadratic classification system exhibits the minimum probability of error for classifying given collections of feature vectors and unknown feature vectors related to the collections.
