nlsq.core.functions.sigmoid¶

nlsq.core.functions.sigmoid(x, L, x0, k, b)[source]¶

Sigmoid (logistic) function: y = L / (1 + exp(-k*(x-x0))) + b

Common for dose-response curves, growth saturation, S-curves.

Parameters:

x (array_like) – Independent variable
L (float) – Maximum value (saturation level)
x0 (float) – Midpoint (inflection point, x value at half-maximum)
k (float) – Steepness (growth rate, positive)
b (float) – Baseline offset (minimum asymptote)

Returns:

y – Dependent variable

Return type:

array_like

Notes

At x=x0: y = L/2 + b (midpoint)
As x→-∞: y → b (lower asymptote)
As x→+∞: y → L + b (upper asymptote)
Steeper curve: larger k

Examples

>>> from nlsq import curve_fit
>>> from nlsq.core.functions import sigmoid
>>> import numpy as np
>>>
>>> # Dose-response curve
>>> x = np.linspace(0, 10, 100)
>>> y = 5 / (1 + np.exp(-2*(x-5))) + 1 + np.random.normal(0, 0.1, 100)
>>> popt, pcov = curve_fit(sigmoid, x, y, p0='auto')
>>> print(f"EC50 (midpoint): {popt[1]:.2f}")