# Solved – Re-parameterization of an asymmetric s-shaped function

Is anyone aware of a re-parameterization of any asymmetric s-shaped function (like, but not necessarily the 5 parameter logistic curve), where one of the parameters is the first inflection point of the first derivative (i.e. the maximum of the second derivative).

I mean point 1 in the upper figure: (The picture shows the point mentioned above for the of case of a symmetric logistic function.)

So far, I had a look at various references (among others Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models. New York, Dekker). However, I have only found parameterizations, where one of the parameters is the inflection point of the function (point 2 in the upper figure).

Unfortunately, calculating this point after the estimation is not a possible solution in my case as this parameter should be integrated in another equation that is estimated simultaneously.

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Pace Procrastinator, this sort of thing can be done.

Consider the five-parameter logistic model. It has many parameterizations, but it's simple and natural to reduce them to something like

\$\$y = nu +tau left(1+e^{frac{x-mu }{sigma }}right)^{-rho }\$\$

with \$rho gt 0\$. We can interpret \$mu\$ and \$nu\$ as \$x\$ and \$y\$ locations and \$sigma\$ and \$tau\$ as \$x\$ and \$y\$ scales; \$rho\$ is the shape or asymmetry parameter.

Let \$x_{+}\$ be the location of one extremum of the second derivative and \$x_{-}\$ be the location of the other extremum. Then, solving for the zeros of the third derivative, I find them at

\$\$x_{pm} = mu + sigma log left(frac{(3 rho +1) pmsqrt{(rho +1) (5 rho +1) }}{2 rho ^2}right).\$\$

Whence, setting

\$\$kappa = exp frac{x_{+}-x_{-}}{sigma}\$\$

we find

\$\$rho = frac{3 kappa pmsqrt{kappa ^3+2 kappa ^2+kappa }}{kappa ^2-7 kappa +1},\$\$

taking the positive sign when \$kappa\$ exceeds the larger root of \$1-7x+x^2=0\$ (about \$6.8541\$) and the negative sign when \$kappa\$ is less than the smaller root (about \$0.145898\$)–other values of \$kappa\$ will not give a sigmoidal curve–and

\$\$mu = frac{x_{+} + x_{-}}{2} + sigma log(rho).\$\$

This allows a parameterization in terms of \$(sigma, nu, tau, x_{-}, x_{+})\$ (with some significant restrictions on \$sigma\$ needed to make the results valid).

Here is a plot of \$y\$ (in blue) and its third derivative (in red) based on these formulas with the parameters set to \$(-1/4, 1/2, 1, 1, 0)\$: Indeed, the ascending zero of the third derivative occurs at \$1\$ and the descending zero at \$0\$, as specified.

This parameterization is not necessarily so messy. If your data are within a narrow range, for instance, the asymptotic values might not matter and you could just use a suitable polynomial. Without knowing more about the particular problem, it's hard to provide a specific recommendation.

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