Logistic function

A bounded S-shaped function that maps real inputs to values between lower and upper asymptotes. Its standard form is 1/(1+e⁻ˣ), with midpoint 1/2 and steepness set by input scaling.

Logistic equation: A differential equation in which a population grows proportionally to its size while being limited by a carrying capacity. Its solution is a logistic function, giving the curve a model of growth toward a finite limit.
Logistic regression: A statistical model that predicts categorical outcomes by applying a logistic transformation to a linear predictor. It uses the function to turn a linear score into an estimated probability.
Asymptote: A line that a curve approaches arbitrarily closely as its input tends toward a limit or infinity. The logistic function approaches horizontal asymptotes that set its lower and upper bounds.
Pierre François Verhulst: A Belgian mathematician who introduced the logistic population model in the nineteenth century. Verhulst formulated the population-growth equation whose solution is the logistic curve.
Exponential function: A function whose value changes by a constant multiplicative factor for each unit change in its input. The logistic function is built from an exponential term and approaches it at one tail.
Generalized linear model: A statistical framework that relates a response distribution to predictors through a link function. Logistic regression is one member of this model family, using the logit link.
Inflection point: A point on a curve where its concavity changes sign. The standard logistic curve changes concavity at its midpoint, where its slope is greatest.
Thomas Malthus: An English cleric and scholar known for arguing that population can grow geometrically while resources grow arithmetically. Malthusian exponential growth provided the contrast that motivated Verhulst’s bounded model.
Sigmoid function: A function with an S-shaped curve that transitions smoothly between limiting values. The logistic function is a specific, widely used member of this broader family.
Artificial neuron: A computational unit that combines inputs and applies a nonlinear activation function to produce an output. The logistic function can serve as its activation, producing a bounded response.

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