sigmoidx

Sigmoidx is a term used in some mathematical texts and discussions to denote a parametric family of sigmoidal (S-shaped) functions whose shape is defined with explicit dependence on the input axis x. It is not a single standardized function, but rather a label for models that generalize the classic sigmoid by allowing adjustable location, slope, and sometimes asymmetry.

A common instantiation is the logistic form f(x) = 1 / (1 + exp(-a (x - b))). Here a controls the steepness and b the midpoint (inflection point) at which f(x) = 0.5. For a > 0, f is increasing; for a < 0, decreasing. The derivative is f'(x) = a f(x) (1 - f(x)).

Equivalent representations: f(x) = 0.5 [1 + tanh( (a/2) (x - b) )].

Relationship to generalized logistic or Richards curve. The Richards form introduces extra shape parameters to model asymmetry, and is sometimes described as a sigmoidx variant in applied contexts.

Applications: Used as an activation function in neural networks, for modeling dose–response curves in pharmacology, and as a link function in generalized linear models and logistic regression.

Notes: The exact form and parameter interpretation vary by author; sigmoidx is not an official standard term. In practice, it denotes a sigmoid-like transfer with explicit x-dependence.

See also: logistic function, sigmoid, activation function, generalized logistic function.

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