Behind the Logistic Function

All you don’t have to know about or you do …

The famous logistic function takes the form of

centered around x₀. A simple case with L=1, x₀=0, k=1 —

is also called the Sigmoid function.

The Sigmoid function has a beautiful form, being centered at (0, 1/2), roughly linear between (-1, 1), and saturating around x=-5.0 and 5.0. Since the early days of Artificial Neural Networks research, people have been using it to model the activation function for neurons.

using Plots

function logit(x::Float64)
    1.0/(1.0+exp(-x))
end


x=collect(-5.0:0.1:5.0)
plot(x, logit.(x), lw=2, label="Sigmoid", xlabel="x", legend=:right)

The logistic function has been also widely used in modeling population growth, diffusion of innovations, and of course, in logistic regression.

From the figure above we can see that the curve is symmetric at x=1/2, i.e., f(x)−1/2 is an odd function. This can be easily verified. We first have

i.e., f(x)−1/2=−(f(−x)−1/2). Done.