Relative Curvature Measures of Nonlinearity

Abstract

Simple relative curvature measures of nonlinearity are developed to measure the extent of the nonlinearity in a model-experimental design-parameterization combination. We review the geometric aspects of linear and nonlinear least squares and, using the geometric concept of curvature, compute the maximum relative intrinsic curvature of the solution locus as well as the maximum relative parameter-effects curvature. The relative curvatures are independent of scale changes of the data and of the parameters so they can be used to compare different data sets as well as different parameterizations of the same data set. The methods are applied to 24 published nonlinear data sets, and in all cases it is found that the intrinsic curvature is less than the parameter-effects curvature. The nonlinearity measures of Beale (1960) and the bias expressions developed by Box (1971) are shown to be related to the curvature measures.