Optimal choice of IHS-type of transformations for log-linear modeling

Abstract

The widely used log-linear transformation for fitting scattered positive observations to power models can be generalized to zero and negative values by embedding shape parameters into an inverse hyperbolic sine (IHS) function. The most common IHS-type of functions are analyzed and compared for log-linear modeling. In addition, a novel parametrized log-linear (ParLo) transformation is introduced that generalizes not only the ordinary linear and log-linear regressions but as well the most popular IHS type of transformation, since it allows to reduce the regression’s residual for the case of exclusively positive values. Residual computations for a single regressor are used to benchmark the IHS-type of transformations. An optimization of the shape parameter may lead to residuals lower than those that are achievable by nonlinear least-squares (NLLS) regression for mono-exponential models since reverse IHS transformations are bi-exponential. Thus, the NLLS regression and the covariance-invariant mapping are applied to deterministic bi-exponential models for further benchmarking.