biogeme.models.piecewise module
Implements piecewise linear formulation
- author:
Michel Bierlaire
- date:
Wed Oct 25 09:43:41 2023
- biogeme.models.piecewise.piecewiseFormula(variable, thresholds, betas=None)[source]
Warning
This function is deprecated. Use
piecewise_formula()instead.- Return type:
- Parameters:
- biogeme.models.piecewise.piecewiseFunction(x, thresholds, betas)[source]
Warning
This function is deprecated. Use
piecewise_function()instead.- Return type:
float- Parameters:
x (float)
thresholds (list[float])
betas (list[float])
- biogeme.models.piecewise.piecewiseVariables(variable, thresholds)[source]
Warning
This function is deprecated. Use
piecewise_variables()instead.- Return type:
list[Expression]- Parameters:
variable (Expression | str)
thresholds (list[float])
- biogeme.models.piecewise.piecewise_as_variable(variable, thresholds, betas=None)[source]
Generate the formula for a piecewise linear specification, seen as a transformed variable.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to \(-\infty\) and \(+\infty\), respectively. If \(t\) is the variable of interest, for each interval \([a:a+b[\), we define a variable defined as:
\[\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}\]The specification this is returned is
\[x_{T1} + \sum_{i=2}^{K-1} eta_i x_{Ti}\]- Parameters:
variable (string) – name of the variable for which we need the piecewise linear transform.
thresholds (list(float)) – list of thresholds
betas (list(biogeme.expresssions.Beta)) – list of Beta parameters to be used in the specification. The number of entries should be the number of thresholds, minus two. If None, for each interval, the parameter Beta(‘beta_VAR_interval’,0, None, None, 0) is used, where var is the name of the variable. Default: none.
- Returns:
expression of the piecewise linear specification.
- Return type:
biogeme.expressions.expr.Expression
- Raises:
BiogemeError – if the thresholds are not defined properly, which means that only the first and the last threshold can be set to None.
BiogemeError – if the length of list
initialexpr.Betasis not equal to the length ofthresholdsminus one.
- biogeme.models.piecewise.piecewise_formula(variable, thresholds, betas=None)[source]
Generate the formula for a piecewise linear specification.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to \(-\infty\) and \(+\infty\), respectively. If \(t\) is the variable of interest, for each interval \([a:a+b[\), we define a variable defined as:
\[\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}\]New variables and new parameters are automatically created to obtain the specification
\[\sum_{i=1}^{K-1} \beta_i x_{Ti}\]- Parameters:
variable (
str|Variable) – name of the variable for which we need the piecewise linear transform.thresholds (
list[float]) – list of thresholdsbetas (
list[Expression|float|int|bool] |None) – list of Beta parameters to be used in the specification. The number of entries should be the number of thresholds, minus one. If None, for each interval, the parameter Beta(‘beta_VAR_interval’,0, None, None, 0) is used, where var is the name of the variable. Default: none.
- Return type:
- Returns:
expression of the piecewise linear specification.
- Raises:
BiogemeError – if the thresholds are not defined properly, which means that only the first and the last threshold can be set to None.
BiogemeError – if the length of list
initialexpr.Betasis not equal to the length ofthresholdsminus one.
See also
- biogeme.models.piecewise.piecewise_function(x, thresholds, betas)[source]
Plot a piecewise linear specification.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to \(-\infty\) and \(+\infty\), respectively. If \(t\) is the variable of interest, for each interval \([a:a+b[\), we define a variable defined as:
\[\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}\]- Parameters:
x (float) – value at which the piecewise specification must be avaluated
thresholds (list(float)) – list of thresholds
betas (list(float)) – list of the Beta parameters. The number of entries should be the number of thresholds, plus one.
- Returns:
value of the numpy function
- Return type:
float
- Raises:
BiogemeError – if the thresholds are not defined properly, which means that only the first and the last threshold can be set to None.
- biogeme.models.piecewise.piecewise_variables(variable, thresholds)[source]
Generate the variables to include in a piecewise linear specification.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to \(-\infty\) and \(+\infty\),respectively. If \(t\) is the variable of interest, for each interval \([a:a+b[\), we define a variable defined as:
\[\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}\]- Parameters:
variable (
Expression|str) – variable for which we need the piecewise linear transform. The expression itself or the name of the variable can be given.thresholds (
list[float]) – list of thresholds
- Return type:
list[Expression]- Returns:
list of variables to for the piecewise linear specification.
- Raises:
BiogemeError – if the thresholds are not defined properly, as only the first and the last thresholds can be set to None.
See also