biogeme.distributions

Example of usage of the distributions module. This is for programmers who need examples of use of the functions of the class. The examples are designed to illustrate the syntax.

author:

Michel Bierlaire

date:

Fri Nov 17 08:27:24 2023

import biogeme.version as ver
import biogeme.distributions as dist
from biogeme.expressions import Beta

print(ver.getText())

biogeme 3.2.13 [2023-12-23]
Home page: http://biogeme.epfl.ch
Submit questions to https://groups.google.com/d/forum/biogeme
Michel Bierlaire, Transport and Mobility Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL)

pdf of the normal distributio: returns the biogeme expression of the probability density function of the normal distribution:

\[f(x;\mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp{-\frac{(x-\mu)^2}{2\sigma^2}}.\]

Calculated for a numeric value.

resulting_expression = dist.normalpdf(0)
resulting_expression

(exp((((-(`0.0` - `0.0`)) * (`0.0` - `0.0`)) / ((`2.0` * `1.0`) * `1.0`))) / (`1.0` * `2.506628275`))

resulting_expression.getValue()

0.39894228034270457

Calculated for an expression.

a_parameter = Beta('a_parameter', 0, None, None, 1)
mu = Beta('mu', 0, None, None, 1)
sigma = Beta('sigma', 1, None, None, 1)

resulting_expression = dist.normalpdf(a_parameter, mu=mu, s=sigma)
resulting_expression

(exp((((-(Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('mu', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) * (Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('mu', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) / ((`2.0` * Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)))) / (Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) * `2.506628275`))

resulting_expression.getValue()

0.39894228034270457

pdf of the lognormal distribution: returns the biogeme expression of the probability density function of the lognormal distribution

\[f(x;\mu, \sigma) = \frac{1}{x\sigma \sqrt{2\pi}} \exp{-\frac{(\ln x-\mu)^2}{2\sigma^2}}\]

Calculated for a numeric value.

resulting_expression = dist.lognormalpdf(1)
resulting_expression

(((`1.0` > `0.0`) * exp((((-(log(`1.0`) - `0.0`)) * (log(`1.0`) - `0.0`)) / ((`2.0` * `1.0`) * `1.0`)))) / ((`1.0` * `1.0`) * `2.506628275`))

resulting_expression.getValue()

0.39894228034270457

Calculated for an expression.

a_parameter = Beta('a_parameter', 1, None, None, 1)
mu = Beta('mu', 0, None, None, 1)
sigma = Beta('sigma', 1, None, None, 1)

resulting_expression = dist.lognormalpdf(a_parameter, mu=mu, s=sigma)
resulting_expression

(((Beta('a_parameter', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) > `0.0`) * exp((((-(log(Beta('a_parameter', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) - Beta('mu', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) * (log(Beta('a_parameter', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) - Beta('mu', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) / ((`2.0` * Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))))) / ((Beta('a_parameter', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) * Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * `2.506628275`))

resulting_expression.getValue()

0.39894228034270457

pdf of the uniform distribution: returns the biogeme expression of the probability density function of the uniform distribution

\[\begin{split}f(x; a, b) = \left\{ \begin{array}{ll} \frac{1}{b-a} & \mbox{for } x \in [a, b] \\ 0 & \mbox{otherwise}\end{array} \right.\end{split}\]

Calculated for a numeric value

resulting_expression = dist.uniformpdf(0)
resulting_expression

((((`0.0` < `-1.0`) * `0.0`) + ((`0.0` > `1.0`) * `0.0`)) + (((`0.0` >= `-1.0`) * (`0.0` <= `1.0`)) / (`1.0` - `-1.0`)))

resulting_expression.getValue()

0.5

Calculated for an expression

a_parameter = Beta('a_parameter', 0, None, None, 1)
a = Beta('a', -1, None, None, 1)
b = Beta('b', 1, None, None, 1)

resulting_expression = dist.uniformpdf(a_parameter, a=a, b=b)
resulting_expression

((((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) < Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * `0.0`) + ((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) > Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * `0.0`)) + (((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) >= Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * (Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) <= Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) / (Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))))

resulting_expression.getValue()

0.5

pdf of the triangular distribution: returns the biogeme expression of the probability density function of the triangular distribution

\[\begin{split}f(x;a, b, c) = \left\{ \begin{array}{ll} 0 & \text{if } x < a \\\frac{2(x-a)}{(b-a)(c-a)} & \text{if } a \leq x < c \\\frac{2(b-x)}{(b-a)(b-c)} & \text{if } c \leq x < b \\0 & \text{if } x \geq b. \end{array} \right.\end{split}\]

It is assumed that \(a < c < b\). It is not verified.

Calculated for a numeric value.

resulting_expression = dist.triangularpdf(0)
resulting_expression

bioMultSum(((`0.0` < `-1.0`) * `0.0`), ((((`0.0` >= `-1.0`) * (`0.0` < `0.0`)) * `2.0`) * ((`0.0` - `-1.0`) / ((`1.0` - `-1.0`) * (`0.0` - `-1.0`)))), (((`0.0` == `0.0`) * `2.0`) / (`1.0` - `-1.0`)), (((((`0.0` > `0.0`) * (`0.0` <= `1.0`)) * `2.0`) * (`1.0` - `0.0`)) / ((`1.0` - `-1.0`) * (`1.0` - `0.0`))), ((`0.0` > `1.0`) * `0.0`))

resulting_expression.getValue()

1.0

Calculated for an expression.

a_parameter = Beta('a_parameter', 0, None, None, 1)
a = Beta('a', -1, None, None, 1)
b = Beta('b', 1, None, None, 1)
c = Beta('c', 0, None, None, 1)

resulting_expression = dist.triangularpdf(a_parameter, a=a, b=b, c=c)
resulting_expression

bioMultSum(((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) < Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * `0.0`), ((((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) >= Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * (Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) < Beta('c', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) * `2.0`) * ((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) / ((Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * (Beta('c', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))))), (((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) == Beta('c', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * `2.0`) / (Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))), (((((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) > Beta('c', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * (Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) <= Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) * `2.0`) * (Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) / ((Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('a', -1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * (Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('c', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)))), ((Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) > Beta('b', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)) * `0.0`))

resulting_expression.getValue()

1.0

CDF of the logistic distribution: returns the biogeme expression of the cumulative distribution function of the logistic distribution

\[f(x;\mu, \sigma) = \frac{1}{1+\exp\left(-\frac{x-\mu}{\sigma} \right)}\]

Calculated for a numeric value

resulting_expression = dist.logisticcdf(0)
resulting_expression

(`1.0` / (`1.0` + exp(((-(`0.0` - `0.0`)) / `1.0`))))

resulting_expression.getValue()

0.5

Calculated for an expression

a_parameter = Beta('a_parameter', 0, None, None, 1)
mu = Beta('mu', 0, None, None, 1)
sigma = Beta('sigma', 1, None, None, 1)

resulting_expression = dist.logisticcdf(a_parameter, mu=mu, s=sigma)
resulting_expression

(`1.0` / (`1.0` + exp(((-(Beta('a_parameter', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1) - Beta('mu', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 1))) / Beta('sigma', 1, -1.3407807929942596e+154, 1.3407807929942596e+154, 1)))))

resulting_expression.getValue()

0.5