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

TL;DR

This paper explores advanced fitting functions for log-log data with inflection points, extending Lavalette laws with multiple parameters to better model deviations from simple power laws in various phenomena.

Contribution

It introduces generalized Lavalette functions with multiple parameters, capable of modeling inflection points on log-log plots, expanding beyond traditional power law fits.

Findings

Generalized Lavalette laws can fit data with inflection points.

Multi-parameter models improve fit quality over simple power laws.

A simple linear combination of Lavalette laws introduces inflection points.

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.