Examining the effect of second-order terms in mathematical programming approaches to the classification problem

Research on mathematical programming approaches to the classification problem has focused almost exclusively on linear discriminant functions with only first-order terms. While many of these first-order models have displayed excellent classificatory performance when compared to Fisher's linear discriminant method, they cannot compete with Smith's quadratic discriminant method on certain data sets. In this paper, we investigate the appropriateness of including second-order terms in mathematical programming models. Various issues are addressed, such as performance of models with small to moderate sample size, need for crossproduct terms, and loss of power by the mathematical programming models under conditions ideal for the parametric procedures. A simulation study is conducted to assess the relative performance of first-order and second-order mathematical programming models to the parametric procedures. The simulation study indicates that mathematical programming models using polynomial functions may be prone to overfitting on the training samples which in turn may cause rather poor fits on the validation samples. The simulation study also indicates that inclusion of cross-product terms may hurt a polynomial model's accuracy on the validation samples, although omission of them means that the model is not invariant to nonsingular transformations of the data.