Logistic function#

Several forms of the logistic function and their first and second derivatives

The current functions are the following. Each function has a second form with _p appended to the name, where parameters are given as an iterable, e.g. logistic and logistic_p = logistic(x, *p):

Function	Description
logistic	Logistic function L/(1+exp(-k(x-x0)))
dlogistic	First derivative of logistic function
d2logistic	Second derivative of logistic function
logistic_offset	logistic function with offset L/(1+exp(-k(x-x0))) + a
dlogistic_offset	First derivative of logistic function with offset
d2logistic_offset	Second derivative of logistic function with offset
logistic2_offset	Double logistic function with offset L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a
dlogistic2_offset	First derivative of double logistic function with offset
d2logistic2_offset	Second derivative of double logistic function with offset

This module was written by Matthias Cuntz while at Department of Computational Hydrosystems, Helmholtz Centre for Environmental Research - UFZ, Leipzig, Germany, and continued while at Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Nancy, France.

copyright:: Copyright 2015-2022 Matthias Cuntz, see AUTHORS.rst for details.
license:: MIT License, see LICENSE for details.

Functions:

`logistic`(x, L, k, x0)	Logistic function
`logistic_p`(x, p)	Wrapper function for `logistic()`: logistic(x, *p)
`dlogistic`(x, L, k, x0)	First derivative of `logistic()` function
`dlogistic_p`(x, p)	Wrapper function for `dlogistic()`: dlogistic(x, *p)
`d2logistic`(x, L, k, x0)	Second derivative of `logistic()` function
`d2logistic_p`(x, p)	Wrapper function for `d2logistic()`: d2logistic(x, *p)
`logistic_offset`(x, L, k, x0, a)	Logistic function with offset
`logistic_offset_p`(x, p)	Wrapper function for `logistic_offset()`: logistic_offset(x, *p)
`dlogistic_offset`(x, L, k, x0, a)	First derivative of `logistic_offset()` function
`dlogistic_offset_p`(x, p)	Wrapper function for `dlogistic_offset()`: dlogistic_offset(x, *p)
`d2logistic_offset`(x, L, k, x0, a)	Second derivative of `logistic_offset()` function
`d2logistic_offset_p`(x, p)	Wrapper function for `d2logistic_offset()`: d2logistic_offset(x, *p)
`logistic2_offset`(x, L1, k1, x01, L2, k2, x02, a)	Double logistic function with offset
`logistic2_offset_p`(x, p)	Wrapper function for `logistic2_offset()`: logistic2_offset(x, *p)
`dlogistic2_offset`(x, L1, k1, x01, L2, k2, x02, a)	First derivative of `logistic2_offset()` function
`dlogistic2_offset_p`(x, p)	Wrapper function for `dlogistic2_offset()`: dlogistic2_offset(x, *p)
`d2logistic2_offset`(x, L1, k1, x01, L2, k2, ...)	Second derivative of `logistic_offset()` function
`d2logistic2_offset_p`(x, p)	Wrapper function for `d2logistic2_offset()`: d2logistic2_offset(x, *p)

History

Written Mar 2015 by Matthias Cuntz (mc (at) macu (dot) de)
Added functions logistic_p and logistic_offset_p, Dec 2017, Matthias Cuntz
Changed to Sphinx docstring and numpydoc, Dec 2019, Matthias Cuntz
Distinguish iterable and array_like parameter types, Jan 2020, Matthias Cuntz
Make systematically function_p versions of all logistic functions and its derivatives, Feb 2020, Matthias Cuntz
Split logistic and curvature into separate files, May 2020, Matthias Cuntz
More consistent docstrings, Jan 2022, Matthias Cuntz

d2logistic(x, L, k, x0)[source]#

Second derivative of logistic() function

\[L/(1+exp(-k(x-x0)))\]

which is

\[-k^2.L.sinh(k(x-x0))/(2(cosh(k(x-x0))+1)^2)\]

Parameters:

x (array_like) – Independent variable to evalute derivative of logistic function
L (float) – Maximum of logistic function
k (float) – Steepness of logistic function
x0 (float) – Inflection point of logistic function

Returns:

Second derivative of logistic function at x with maximum L, steepness k and inflection point x0

Return type:

float or ndarray

d2logistic2_offset(x, L1, k1, x01, L2, k2, x02, a)[source]#

Second derivative of logistic_offset() function

\[L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\]

which is

\[- k1^2.L1.sinh(k1(x-x01)) / (2(cosh(k1(x-x01))+1)^2) + k2^2.L2.sinh(k2(x-x02)) / (2(cosh(k2(x-x02))+1)^2)\]

Parameters:

x (array_like) – Independent variable to evalute logistic function
L1 (float) – Maximum of first logistic function
k1 (float) – Steepness of first logistic function
x01 (float) – Inflection point of first logistic function
L2 (float) – Maximum of second logistic function
k2 (float) – Steepness of second logistic function
x02 (float) – Inflection point of second logistic function
a (float) – Offset of double logistic function

Returns:

Second derivative of double logistic function with offset at x

Return type:

float or ndarray

d2logistic2_offset_p(x, p)[source]#: Wrapper function for d2logistic2_offset(): d2logistic2_offset(x, *p)

d2logistic_offset(x, L, k, x0, a)[source]#

Second derivative of logistic_offset() function

\[L/(1+exp(-k(x-x0))) + a\]

which is

\[-k^2.L.sinh(k(x-x0))/(2(cosh(k(x-x0))+1)^2)\]

Parameters:

x (array_like) – Independent variable to evalute derivative of logistic function
L (float) – Maximum of logistic function
k (float) – Steepness of logistic function
x0 (float) – Inflection point of logistic function
a (float) – Offset of logistic function

Returns:

Second derivative of logistic function at x with maximum L, steepness k, inflection point x0, and offset a

Return type:

float or ndarray

d2logistic_offset_p(x, p)[source]#: Wrapper function for d2logistic_offset(): d2logistic_offset(x, *p)

d2logistic_p(x, p)[source]#: Wrapper function for d2logistic(): d2logistic(x, *p)

dlogistic(x, L, k, x0)[source]#

First derivative of logistic() function

\[L/(1+exp(-k(x-x0)))\]

which is

\[k.L/(2(cosh(k(x-x0))+1))\]

Parameters:

x (array_like) – Independent variable to evalute derivative of logistic function
L (float) – Maximum of logistic function
k (float) – Steepness of logistic function
x0 (float) – Inflection point of logistic function

Returns:

First derivative of logistic function at x with maximum L, steepness k and inflection point x0

Return type:

float or ndarray

dlogistic2_offset(x, L1, k1, x01, L2, k2, x02, a)[source]#

First derivative of logistic2_offset() function

\[L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\]

which is

\[k1.L1/(2(cosh(k1(x-x01))+1)) - k2.L2/(2(cosh(k2(x-x02))+1))\]

Parameters:

x (array_like) – Independent variable to evalute logistic function
L1 (float) – Maximum of first logistic function
k1 (float) – Steepness of first logistic function
x01 (float) – Inflection point of first logistic function
L2 (float) – Maximum of second logistic function
k2 (float) – Steepness of second logistic function
x02 (float) – Inflection point of second logistic function
a (float) – Offset of double logistic function

Returns:

First derivative of double logistic function with offset at x

Return type:

float or ndarray

dlogistic2_offset_p(x, p)[source]#: Wrapper function for dlogistic2_offset(): dlogistic2_offset(x, *p)

dlogistic_offset(x, L, k, x0, a)[source]#

First derivative of logistic_offset() function

\[L/(1+exp(-k(x-x0))) + a\]

which is

\[k.L/(2(cosh(k(x-x0))+1))\]

Parameters:

x (array_like) – Independent variable to evalute derivative of logistic function
L (float) – Maximum of logistic function
k (float) – Steepness of logistic function
x0 (float) – Inflection point of logistic function
a (float) – Offset of logistic function

Returns:

First derivative of logistic function with offset at x with maximum L, steepness k, inflection point x0, and offset a

Return type:

float or ndarray

dlogistic_offset_p(x, p)[source]#: Wrapper function for dlogistic_offset(): dlogistic_offset(x, *p)

dlogistic_p(x, p)[source]#: Wrapper function for dlogistic(): dlogistic(x, *p)

logistic(x, L, k, x0)[source]#

Logistic function

\[L/(1+exp(-k(x-x0)))\]

Parameters:

x (array_like) – Independent variable to evalute logistic function
L (float) – Maximum of logistic function
k (float) – Steepness of logistic function
x0 (float) – Inflection point of logistic function

Returns:

Logistic function at x with maximum L, steepness k and inflection point x0

Return type:

float or ndarray

logistic2_offset(x, L1, k1, x01, L2, k2, x02, a)[source]#

Double logistic function with offset

\[L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\]

Parameters:

x (array_like) – Independent variable to evalute logistic function
L1 (float) – Maximum of first logistic function
k1 (float) – Steepness of first logistic function
x01 (float) – Inflection point of first logistic function
L2 (float) – Maximum of second logistic function
k2 (float) – Steepness of second logistic function
x02 (float) – Inflection point of second logistic function
a (float) – Offset of double logistic function

Returns:

Double Logistic function at x

Return type:

float or ndarray

logistic2_offset_p(x, p)[source]#: Wrapper function for logistic2_offset(): logistic2_offset(x, *p)

logistic_offset(x, L, k, x0, a)[source]#

Logistic function with offset

\[L/(1+exp(-k(x-x0))) + a\]

Parameters:

x (array_like) – Independent variable to evalute logistic function
L (float) – Maximum of logistic function
k (float) – Steepness of logistic function
x0 (float) – Inflection point of logistic function
a (float) – Offset of logistic function

Returns:

Logistic function at x with maximum L, steepness k, inflection point x0 and offset a

Return type:

float or ndarray

logistic_offset_p(x, p)[source]#: Wrapper function for logistic_offset(): logistic_offset(x, *p)

logistic_p(x, p)[source]#: Wrapper function for logistic(): logistic(x, *p)

Logistic function

Contents

Logistic function#