Ridge regression techniques for the optimization of piecewise linear discriminants: application to Fourier transform infrared remote sensing measurements

Abstract

An optimization procedure for piecewise linear discriminant analysis is described based on the application of ridge regression techniques. The method is based on the use of a matrix transformation that stabilizes data that exhibit a high degree of collinearity. Linear discriminants computed from collinear data exhibit imprecision in the estimation of the discriminant coefficients. The transform described here produces a perturbed data matrix in which the degree of collinearity has been decreased and which can be used in place of the original for subsequent discriminant calculations. The degree of stabilization provided by the transform is specified by the user in the form of a perturbation constant. The discriminant optimization procedure is demonstrated with an application in which an automated algorithm is implemented for the detection of SF 6 by Fourier transform infrared remote sensing measurements. A detailed study is described in which 96 data sets are investigated consisting of four different numbers of observations, four different dimensionalities, and six different levels of the perturbation constant. The results of this study indicate that the optimization procedure significantly improves the precision of discriminants computed with collinear data. The level of improvement is greatest in the case of greatest collinearity.