What is the simplest formula for activation / smooth step function?

Question

I want smooth step function, which is changing from a to b while x changes from 0 to 1, and I want to control both slope and step location.

Which is the simplest formula for this?

Answer 1

I think I have exactly what you want (see the manipulator perrycioc below!). I took the logistic function, $$P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)}.$$ To this I introduced a constant $a$, and then adjusted $K=(b-a)$, took $P_0$, and adjusted that to be $P_0=\frac{b-a}{2}$. Lastly, to control the step location, we can let $t=x-c$. The $r$ term already controls the slope, and so the result is,

$$f(x)=\frac{\frac{ (b-a)^2}{2}\cdot e^{r\cdot (x-c)}}{(b-a)+\frac{ (b-a)}{2} \cdot \left(e^{r (x-c)}-1\right)}+a.$$ Surprisingly enough, this cleans up to be the nice compact function, $$f(x)=\frac{a \cdot e^{c r}+b \cdot e^{r x}}{e^{c r}+e^{r x}}.$$ Parameter descriptions:

1. $a$ and $b$ correspond to your description, with $a<b$.

2. $r \geq 0$ changes the "slope".

3. $c$ acts as the step location shifter corresponding to the $x_1,x_2$ in your graph.

While the function is still asymptotic, and still has sigmoid symmetry (not sure if you wanted to destroy that), the result is that you can pretty much put this anywhere on the plane, and do exactly what you want.

I made this manipulator for you, which I will affectionately call the perrycioc function. You should be able to vary the parameters in browser and see how it goes. Note that I have left both the initial form and the simplified form here, so you can manipulate either and see that they are exactly equivalent.

Enjoy.

Answer 2

Take a look at Sigmoid functions. In particular, try logistic curves