

Toward Fits to ScalingLike Data, but with Inflection Points &Generalized Lavalette FunctionKeywordsAbstractExperimental and empirical data are often analyzed on loglog plots in order to find some scaling argument for the observed/examined phenomenon at hands, in particular for ranksize 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 cutoff at high rank. In so doing, the number of free parameters increases. Their meaning is discussed, up to the 5 parameter free supergeneralized Lavalette law and the 7parameter free hypergenerealized Lavalette law. It is emphasized that the interest of the basic 2parameter free Lavalette law and the subsequent generalizations resides in its "noid" (or sigmoid, depending on the sign of the exponents) form on a semilog plot; something incapable to be found in other empirical law, like the ZipfParetoMandelbrot law. It remained for completeness to invent a simple law showing an inflection point on a loglog 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 supergeneralized or hypergenerealized Lavalette laws are suggested, but need to be fully considered at fit time on appropriate data. (top)
