What is a Logistic Function?
Definition [^1]
The logistic function is derived as $y ' = y(1-y)$, which is a solution to the differential equation. $$ y(t) = {{ 1 } \over { 1 + e^{-t} }} $$
Explanation
In a more general form, it can also be expressed as $\displaystyle f(x) := {{ L } \over { 1 + e^{-k(x-x_{0})} }}$. The logistic function, which is a sigmoid function, is widely mentioned in various fields such as dynamics, statistics, deep learning, biology, due to its many applications.
Logistic?
The question is why it is called the ’logistic function’. Looking up the meaning of Logistic reveals terms like ‘related to logistics’, ‘related to military logistics’, or ‘related to symbolic logic’, which seem to have no connection to the function’s form at a glance. Logistic regression is a classification technique that applies logistic function within regression analysis, and this again has no relation to military logistics or symbolic logic.
The most plausible explanation perhaps lies in considering the meaning behind the logistic differential equation $y’=y(1-y)$, the framework from which the logistic function emerged. The logistic differential equation is often mentioned as a population growth model and is extensively used in many fields. It describes a scenario where the growth of a population explosively increases and then slows down due to factors like food scarcity.
Here, ‘food’ can be interpreted in various contexts: as jobs for employees, as oxygen for aerobic bacteria, etc. Perhaps ‘Logistic’ was chosen as a sophisticated term capable of aptly describing ’that something’. Logistics does not solely refer to food supplies but covers everything needed for war, and likewise, logistics doesn’t only denote specific goods. To align with its meaning outside the realm of science and engineering, this may be the most fitting interpretation.