Toward fits to scaling-like data, but with inflection points & generalized Lavalette function

Abstract

Experimental and empirical data are often analyzed on log-log plots in order to find some scaling argument for the observed/examined phenomenon at hands, in particular for rank-size rule research, but also in critical phenomena in thermodynamics, and in fractal geometry. The fit to a straight line on such plots is not always satisfactory. Deviations occur at low, intermediate and high regimes along the log($x$)-axis. Several improvements of the mere power law fit are discussed, in particular through a Mandelbrot trick at low rank and a Lavalette power law cut-off at high rank. In so doing, the number of free parameters increases. Their meaning is discussed, up to the 5 parameter free super-generalized Lavalette law and the 7-parameter free hyper-generalized Lavalette law. It is emphasized that the interest of the basic 2-parameter free Lavalette law and the subsequent generalizations resides in its "noid" (or sigmoid, depending on the sign of the exponents) form on a semi-log plot; something incapable to be found in other empirical law, like the Zipf-Pareto-Mandelbrot law. It remained for completeness to invent a simple law showing an inflection point on a \underline{log-log plot}. Such a law can result from a transformation of the Lavalette law through $x$ $\rightarrow$ log($x$), but this meaning is theoretically unclear. However, a simple linear combination of two basic Lavalette law is shown to provide the requested feature. Generalizations taking into account two super-generalized or hyper-generalized Lavalette laws are suggested, but need to be fully considered at fit time on appropriate data.