Logistic Integral

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

Generates the integral 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

• integral[‘constants’] (list of float) – Coefficients of the resultant integral

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

See also

logistic_equation(), logistic_derivatives(), logistic_roots(), logistic_model()

Notes

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

• Integral of a logistic function: \(F(x) = \frac{a}{b}\cdot{\ln|\text{e}^{b\cdot(x - c)} + 1|}\)

• Basic Properties of Indefinite Integrals

• Basic Integration Forumulas

• Substitution Rule

Examples

Import logistic_integral function from regressions library

>>> from regressions.analyses.integrals.logistic import logistic_integral

Generate the integral of a logistic function with coefficients 2, 3, and 5, then display its coefficients

>>> integral_constants = logistic_integral(2, 3, 5)
>>> print(integral_constants['constants'])
[0.6667, 3.0, 5.0]

Generate the integral of a logistic function with coefficients 100, 5, and 11, then evaluate its integral at 10

>>> integral_evaluation = logistic_integral(100, 5, 11)
>>> print(integral_evaluation['evaluation'](10))
0.1343

Generate the integral of a logistic function with all inputs set to 0, then display its coefficients

>>> integral_zeroes = logistic_integral(0, 0, 0)
>>> print(integral_zeroes['constants'])
[1.0, 0.0001, 0.0001]