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Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent ...
There are two luminous efficiency functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies. The photopic curve is the CIE standard curve used in the CIE 1931 color space.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
I stumbled over the second of the two algebraic functions in the example section: x^n/(x^n + (1-x)^n). as far as I see this is not a sigmoid function. It violates the definition of a sigmoid given in the section Definition. Notably, the derivative is not non-negative and it has multiple inflection points.
The above development has the advantage of basing the new X F Y F Z F color matching functions on the physiologically-based LMS cone response functions. In addition, it offers a one-to-one relationship between the LMS chromaticity coordinates and the new X F Y F Z F chromaticity coordinates, which was not the case for the CIE 1931 color ...
A comparison between a typical normalized M cone's spectral sensitivity and the CIE 1931 luminosity function for a standard observer in photopic vision. In the CIE 1931 model, Y is the luminance, Z is quasi-equal to blue (of CIE RGB), and X is a mix of the three CIE RGB curves chosen to be nonnegative (see § Definition of the CIE XYZ color space).
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.
S-curve (math), a characteristic S-shaped curve of a sigmoid function; S-curve corset, an Edwardian corset style; S-Curve Records, a record company label; Reverse curve, or "S" curve, in civil engineering