The Use of an Adaptive Threshold Element to Design a Linear Optimal Pattern Classifier

Published in: IEEE Transactions on Information Theory, v. 12, no. 1, Jan. 1966, p. 42-50

Posted on on January 01, 1966

by J. S. Koford, Gabriel F. Groner

This paper develops a relationship between two traditional statistical methods of pattern classifier design, and an adaption technique involving minimization of the mean-square error in the output of a linear threshold device. It is shown that the two-category classifier derived by least-mean-square-error adaption using an equal number of sample patterns from each category is equivalent to the optimal statistical classifier if the patterns are multivariate Gaussian random variables having the same covariance matrix for both pattern categories. It is also shown that the classifier is always equivalent to the classifier derived by R. A. Fisher. A simple modification of the least-mean-square-error adaption procedure enables the adaptive structure to converge to a nearly-optimal classifier, even though the numbers of sample patterns are not equal for the two categories. The use of minimization of mean-square error as a technique for designing classifiers has the added advantage that it leads to the optimal classifier for patterns even when the covariance matrix is singular.

This report is part of the RAND Corporation External publication series. Many RAND studies are published in peer-reviewed scholarly journals, as chapters in commercial books, or as documents published by other organizations.

Our mission to help improve policy and decisionmaking through research and analysis is enabled through our core values of quality and objectivity and our unwavering commitment to the highest level of integrity and ethical behavior. To help ensure our research and analysis are rigorous, objective, and nonpartisan, we subject our research publications to a robust and exacting quality-assurance process; avoid both the appearance and reality of financial and other conflicts of interest through staff training, project screening, and a policy of mandatory disclosure; and pursue transparency in our research engagements through our commitment to the open publication of our research findings and recommendations, disclosure of the source of funding of published research, and policies to ensure intellectual independence. For more information, visit

The RAND Corporation is a nonprofit institution that helps improve policy and decisionmaking through research and analysis. RAND's publications do not necessarily reflect the opinions of its research clients and sponsors.