hill_saturation

pymc_marketing.mmm.transformers.hill_saturation(x, sigma, beta, lam)

Hill Saturation Function

\[f(x) = \frac{\sigma}{1 + e^{-\beta(x - \lambda)}}\]

where:

• \(\sigma\) is the maximum value (upper asymptote)

• \(\beta\) is the slope parameter

• \(\lambda\) is the transition point on the X-axis

• \(x\) is the independent variable

This function computes the Hill sigmoidal response curve, which is commonly used to describe the saturation effect in biological systems. The curve is characterized by its sigmoidal shape, representing a gradual transition from a low, nearly zero level to a high plateau, the maximum value the function will approach as the independent variable grows large.

Parameters:

xfloat or array_like

The independent variable, typically representing the concentration of a substrate or the intensity of a stimulus.

sigmafloat

The upper asymptote of the curve, representing the maximum value the function will approach as x grows large.

betafloat

The slope parameter, determining the steepness of the curve.

lamfloat

The x-value of the midpoint where the curve transitions from exponential growth to saturation.

Returns:

float or array_like

The value of the Hill function for each input value of x.