sigmoid

Sigmoid refers to an S-shaped curve or anything exhibiting that form. In mathematics and applied disciplines, the term most often denotes a sigmoid function: a smooth, monotone, S-shaped map from the real numbers to a bounded interval, typically [0,1].

The best-known example is the logistic function, f(x) = 1/(1+e^-x). It is differentiable, strictly increasing, and approaches 0 as x tends to negative infinity and 1 as x tends to positive infinity. Its derivative is f(x)(1−f(x)). The inverse function is the logit, defined as logit(y) = ln(y/(1−y)) for y in (0,1).

Variants of the sigmoid include the hyperbolic tangent, tanh(x), which is also S-shaped and maps R to (−1,1). There is a standard relationship: tanh(x) = 2·sigmoid(2x) − 1. Sigmoid functions are often chosen for their properties of smoothness, bounded output, and ease of interpretation as probabilities.

Applications are widespread. In statistics and machine learning, the logistic sigmoid maps a linear predictor to a probability and underpins logistic regression. In neural networks, sigmoids have served as activation functions, helping to introduce nonlinearity. They also model dose–response relationships and growth processes in biology and pharmacology, where the response saturates at high inputs.

Other uses of the term arise from the shape rather than the function. In anatomy, the sigmoid shape appears in structures such as the sigmoid colon, a passage in the large intestine named for its characteristic S form. The word “sigmoid” derives from the Greek word for “the letter sigma,” reflecting the curved outline.