import numpy
from matplotlib import pyplot
import ipywidgets

import calibr8

WARNING (theano.tensor.blas): Using NumPy C-API based implementation for BLAS functions.

The Generalized Logistic Function

Real-world calibration curves rarely follow linear dose/response relationships, but often exhibit lower & upper saturations. These relationships can be described with S-shaped functions. The logistic function (the sigmoid curve is a special case of it) is often well suited for real-world calibration curves.

Several different flavors of S-shaped curves are available: + sigmoid (1 parameter: variable slope) + logistic (3 parameters: variable upper limit, variable x-value of the inflection point, variable slope) + generalized logistic (5 parameters: variable limits, variable inflection point, variable slope, variable symmetry)

Here, we’re going to use the generalized logistic, because it is most useful. However, we use a different parametrization that is derived in the notebook Background_AsymmetricLogistic.

\[f(x)= L_L \cdot (L_U - L_L) \cdot (e^{(\frac{S}{L_U - L_L} (I_{x} - x) + c (e^{c} + 1)^{-(e^{c} + 1) e^{- c}}) (e^{c} + 1)^{(e^{c} + 1) e^{- c}}} + 1)^{- e^{- c}}\]

The following table shows the bounds & interpretation of its 5 paramters:

parameter	interpretation
\(L_L \in \mathcal{R}\)	lower asymptote
\(L_U \in \mathcal{R}\)	upper asymptote
\(I_x \in \mathcal{R}\)	x-value of the inflection point
\(S \in \mathcal{R}\)	slope at the inflection point I_x
\(c \in \mathcal{R}\)	moves the inflection point closer to L_L (c < 0) or closer to L_U (0 < c).

Example

Let’s say we have an assay with noticeable lower- and upper satuation limits, as well as some asymmetry.

To visualize the meaning of the parameters, we can draw an interactive plot of the calibr8.asymmetric_logistic function:

help(calibr8.asymmetric_logistic)

Help on function asymmetric_logistic in module calibr8.core:

asymmetric_logistic(x, theta)
    5-parameter asymmetric logistic model.

    Parameters
    ----------
    x : array-like
        independent variable
    theta : array-like
        parameters of the logistic model
            L_L: lower asymptote
            L_U: upper asymptote
            I_x: x-value at inflection point
            S: slope at the inflection point
            c: symmetry parameter (0 is symmetric)

    Returns
    -------
    y : array-like
        dependent variable

def tangent(I_x, I_y, S):
    """Get x,y to plot a line with slope S around the coordinate <I_x,I_y>."""
    x = numpy.linspace(I_x-1,I_x+1,2)
    y = -S*I_x + I_y + S*x
    return x, y

def plot_logistic(L_L=0, L_U=1, I_x=0.0, S=0.5, c=1.0):
    theta = (L_L, L_U, I_x, S, c)
    X_MIN, X_MAX = -5, 5
    X = numpy.linspace(-5, 5, 100)

    # get key properties to visualize
    I_y = calibr8.asymmetric_logistic(I_x, theta)

    fig, ax = pyplot.subplots(figsize=(16,6))
    ax.plot(X, calibr8.asymmetric_logistic(X, theta))
    ax.plot(*tangent(I_x, I_y, S), label='$tangent$')

    ax.axvline(I_x, linestyle=':', color='red', label='$I_x$')
    ax.axhline(I_y, linestyle=':', color='red', label='$I_y$')
    ax.axhline(L_U, linestyle=':', label='$L_U$')
    ax.axhline(L_L, linestyle='--', label='$L_L$')

    ax.set_xlim(X_MIN, X_MAX)
    ax.set_ylim(L_L-0.5, L_U+0.5)
    ax.legend(loc='center left')
    pyplot.show()

plot_logistic()
ipywidgets.interact(
    plot_logistic,
    L_L=(-5.0,0),
    L_U=(0.0,5),
    I_x=(-5.0, 5),
    S=(-2.0, 3),
    c=(-2.0, 2)
);

Summary

• The generalized logistic function can describe relationships with lower & upper satuation

• Even in cases of “good linear relationships”, the GLF is applicable

• The original parameterization is not very intuitive, but was reparameterized such that 4 of 5 parameters are interpretable

%load_ext watermark
%watermark -n -u -v -iv -w

Last updated: Wed Feb 23 2022

Python implementation: CPython
Python version       : 3.8.12
IPython version      : 7.31.0

matplotlib: 3.5.1
sys       : 3.8.12 | packaged by conda-forge | (default, Oct 12 2021, 21:22:46) [MSC v.1916 64 bit (AMD64)]
ipywidgets: 7.6.5
calibr8   : 6.4.0
numpy     : 1.20.3

Watermark: 2.3.0