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A practical method of determining weights for crossproduct and power terms in the variable inputs to an adaptive threshold element used for statistical pattern classification is derived. The objective is to make it possible to realize general nonlinear decision surfaces, in contrast with the linear (hyperplanar) decision surfaces that can be realized by a threshold element using only first-order terms as inputs. The method is based on nonparametric estimation of a probability density function for each category to be classified so that the Bayes decision rule can be used for classification. The decision surfaces thus obtained have good extrapolating ability (from training patterns to test patterns) even when the number of training patterns is quite small. Implementation of the method, both in the form of computer programs and in the form of polynomial threshold devices, is discussed, and some experimental results are described.