Sigmoid Functions: Gudermannian and Gompertz Curves

The Gudermannian is named after Christoph Gudermann (1798–1852). The Gompertz function is named after Benjamin Gompertz (1779–1865). These are two amongst several sigmoid functions. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions.


Sigmoid Functions

A sigmoid function is an S-shaped function, usually increasing monotonically on {\mathbb{R}} and having finite limits as {x\rightarrow\pm\infty}. It is normally required to have a positive derivative at every real point. Thus, it is bounded and has bounded variation.

Sigmoid functions often arise as the integrals of bell-shaped functions having a single maximum. Here are several examples:

\displaystyle \frac{1}{1+x^2} \qquad \exp(-x^2) \qquad \frac{1}{(1+x^2)^{3/2}} \qquad \mathrm{sech}^2\,x \qquad \mathrm{sech\,}x

They are shown in the Figure below.


The integrals of the five bell shaped functions given above are

\displaystyle \arctan x \qquad \frac{\sqrt{\pi}}{2} \mathrm{erf\,}x \qquad \frac{x}{\sqrt{1+x^2}} \qquad \tanh x \qquad \mathrm{gd\,}x

They are shown in the following Figure.


Sigmoid functions arise as cumulative distributions in probability theory. If a probability density function (pdf) is strictly positive on {\mathbb{R}} then the cumulative distribution function — the integral of the pdf — is strictly monotone increasing.

The Gudermannian

The Gudermannian is defined as the integral of the hyperbolic secant:

\displaystyle \mathrm{gd\,} x = \int_0^x \mathrm{sech\,} x \, \mathrm{d}x = \int_0^x \frac{\mathrm{d}x}{\cosh x} \qquad -\infty < x < +\infty

Since {\mathrm{sech\,} x} is bell-shaped, with a single peak, {\mathrm{gd\,} x} is monotone increasing on {\mathbb{R}}. This function connects the circular and hyperbolic functions in a number of ways, For example,

\displaystyle \mathrm{gd\,} x = \arcsin(\tanh x) = \arctan(\sinh x) = 2\arctan\left(\tanh\frac{x}{2}\right)

It is asymptotic to {\pm\pi/2} as {x\rightarrow\pm\infty}. The gudermannian is valuable in the theory of Mercator’s projection and in some problems in mechanics including the inverted pendulum.


Gompertz Function

So far, all the sigmoid functions considered have symmetry about some central value {x_0}. An interesting sigmoid function arising in acturial studies and population dynamics is the Gompertz function. It has the form

\displaystyle G(x) = a \exp[-b \exp( -c x )] \qquad \mbox{with}\ a, b, c\ \mbox{all positive.}

The limiting values are {G(-\infty)=0} and {G(+\infty)=a} and the function increases monotonically between these limits. Parameter {b} determines where the function increases most steeply and {c} determines the degree of steepness. For example, with {a=b=1}

\displaystyle \frac{\mathrm{d}G}{\mathrm{d}x} = c e^{-cx} G(x) \qquad\mbox{so}\qquad {\frac{\mathrm{d}G}{\mathrm{d}x}}\bigg|_0 = \frac{c}{e}

The Gompertz function is a solution of the differential equation

\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = k(x)\,y \qquad\mbox{with}\qquad k(x) = c \exp(-cx)

where the growth rate, or decay rate, {k(x)} is an exponential function of {x}.

The function {G(t)} is a simple function for mortality with {(a-G(t))} representing the surviving population as a function of time {t}.

In the Figures below, we plot {G(t)} for a range of values of the parameters {a}, {b} and {c}.





Sigmoid functions have a wide range of applications. Their role in artificial neural networks (ANNs) makes the invaluable, as such networks can simulate complex systems and have been proven to be universal approximators, representing the behaviour of an arbitrary function to any desired degree.


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