Logistic Derivatives

logistic_derivatives(first_constant, second_constant, third_constant, precision=4)

Calculates the first and second derivatives of a logistic function

Parameters

  • first_constant (int or float) – Carrying capacity of the original logistic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • second_constant (int or float) – Growth rate of the original logistic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • third_constant (int or float) – Value of the sigmoid’s midpoint of the original logistic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the resultant roots

Raises

  • TypeError – First three arguments must be integers or floats

  • ValueError – Last argument must be a positive integer

Returns

  • derivatives[‘first’][‘constants’] (list of float) – Coefficients of the resultant first derivative

  • derivatives[‘first’][‘evaluation’] (func) – Function for evaluating the resultant first derivative at any float or integer argument

  • derivatives[‘second’][‘constants’] (list of float) – Coefficients of the resultant second derivative

  • derivatives[‘second’][‘evaluation’] (func) – Function for evaluating the resultant second derivative at any float or integer argument

See also

logistic_equation(), logistic_integral(), logistic_roots(), logistic_model()

Notes

  • Standard form of a logistic function: \(f(x) = \frac{a}{1 + \text{e}^{-b\cdot(x - c)}}\)

  • First derivative of a logistic function: \(f'(x) = \frac{ab\cdot{\text{e}^{-b\cdot(x - c)}}}{(1 + \text{e}^{-b\cdot(x - c)})^2}\)

  • Second derivative of a logistic function: \(f''(x) = \frac{2ab^2\cdot{\text{e}^{-2b\cdot(x - c)}}}{(1 + \text{e}^{-b\cdot(x - c)})^3} - \frac{ab^2\cdot{\text{e}^{-b\cdot(x - c)}}}{(1 + \text{e}^{-b\cdot(x - c)})^2}\)

  • Basic Differentiation Forumulas

  • Chain Rule

  • Derivatives of Exponential Functions

Examples

Import logistic_derivatives function from regressions library
>>> from regressions.analyses.derivatives.logistic import logistic_derivatives
Generate the derivatives of a logistic function with coefficients 2, 3, and 5, then display the coefficients of its first and second derivatives
>>> derivatives_constants = logistic_derivatives(2, 3, 5)
>>> print(derivatives_constants['first']['constants'])
[6.0, 3.0, 5.0]
>>> print(derivatives_constants['second']['constants'])
[18.0, 3.0, 5.0]
Generate the derivatives of a logistic function with coefficients 100, 5, and 11, then evaluate its first and second derivatives at 10
>>> derivatives_evaluation = logistic_derivatives(100, 5, 11)
>>> print(derivatives_evaluation['first']['evaluation'](10))
3.324
>>> print(derivatives_evaluation['second']['evaluation'](10))
16.3977
Generate the derivatives of a logistic function with all inputs set to 0, then display the coefficients of its first and second derivatives
>>> derivatives_zeroes = logistic_derivatives(0, 0, 0)
>>> print(derivatives_zeroes['first']['constants'])
[0.0001, 0.0001, 0.0001]
>>> print(derivatives_zeroes['second']['constants'])
[0.0001, 0.0001, 0.0001]