Understanding the logistic function

The logistic function generates an S-shaped curve; the equation takes the following form:

Figure 6.2 – The logistic function

Here, L is the maximum value of the curve, k is the logistic growth rate, or steepness, of the curve, and x0 is the x-value of the curve's midpoint.

Taking , , and , the logistic function produces the standard logistic function, seen in the following plot:

Figure 6.3 – The standard logistic function, y = 1 / (1 + e-x)

If you have studied logistic regression or neural networks, you may recognize this as the sigmoid function. Any input value for x, from -∞ to ∞, will be squished into an output value, y, between 0 and 1. This equation is what allows a logistic regression model to accept any input value and output a probability between 0 and 1.

The equation was developed by Pierre François Verhulst, a Belgian mathematician, in a series...