An improved power law for nonlinear least-squares fitting?

Abstract

Models based on a power law are prevalent in many areas of study. When regression analysis is performed on data sets modeled by a power law, the traditional model uses a lead coefficient. However, the proposed model replaces the lead coefficient with a scaling parameter and reduces uncertainties in best-fit parameters for data sets with exponents close to 3. This study extends previous work by testing each model for a range of parameters. Data sets with known values of scaling parameter and exponent were generated by adding normally distributed random errors with controlled mean and standard deviations to underlying power laws. These data sets were then analyzed for both forms of the power law. For the scaling parameter, the proposed model provided smaller errors in 96/180 cases and smaller uncertainties in 88/180 cases. In most remaining cases, the traditional model provided smaller errors or uncertainties. Examination of conditions indicates that the proposed law has potential in select cases, but due to ambiguity in the conditions which favor one model over the other, an approach similar to the one in this study is encouraged for determining which model will offer reduced errors and uncertainties in data sets where additional accuracy is desired.