Toward Fits to Scaling-Like Data, but with Inflection Points &Generalized Lavalette Function

Keywords

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-generealized 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 log-log plot. Such a law can result from a transformation of the Lavalette law through x → 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-generealized Lavalette laws are suggested, but need to be fully considered at fit time on appropriate data.

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