Combination of many specifications

We combine the following specifications:

• 3 models

  • logit

  • nested logit with two nests: public and private transportation

  • nested logit with two nests existing and future modes

• 3 functional forms for the travel time variables

  • linear specification,

  • Box-Cox transform,

  • power series,

• 2 specifications for the cost coefficients:

  • generic

  • alternative specific

• 2 specifications for the travel time coefficients:

  • generic

  • alternative specific

• 4 segmentations for the constants:

  • not segmented

  • segmented by GA (yearly subscription to public transport)

  • segmented by luggage

  • segmented both by GA and luggage

• 3 segmentations for the time coefficients:

  • not segmented

  • segmented with first class

  • segmented with trip purpose

This leads to a total of 432 specifications. See Bierlaire and Ortelli (2023).

author:

Michel Bierlaire, EPFL

date:

Sat Jul 15 15:40:33 2023

import numpy as np
from biogeme import models
from biogeme.expressions import Expression, Beta
from biogeme.nests import OneNestForNestedLogit, NestsForNestedLogit
from biogeme.catalog import (
    Catalog,
    segmentation_catalogs,
    generic_alt_specific_catalogs,
)

from biogeme.data.swissmetro import (
    read_data,
    CHOICE,
    SM_AV,
    CAR_AV_SP,
    TRAIN_AV_SP,
    TRAIN_TT_SCALED,
    TRAIN_COST_SCALED,
    SM_TT_SCALED,
    SM_COST_SCALED,
    CAR_TT_SCALED,
    CAR_CO_SCALED,
)

Read the data

database = read_data()

Definition of the segmentations.

segmentation_ga = database.generate_segmentation(
    variable='GA', mapping={0: 'noGA', 1: 'GA'}
)

segmentation_luggage = database.generate_segmentation(
    variable='LUGGAGE', mapping={0: 'no_lugg', 1: 'one_lugg', 3: 'several_lugg'}
)

segmentation_first = database.generate_segmentation(
    variable='FIRST', mapping={0: '2nd_class', 1: '1st_class'}
)

We consider two trip purposes: ‘commuters’ and anything else. We need to define a binary variable first.

database.data['COMMUTERS'] = np.where(database.data['PURPOSE'] == 1, 1, 0)

segmentation_purpose = database.generate_segmentation(
    variable='COMMUTERS', mapping={0: 'non_commuters', 1: 'commuters'}
)

Parameters to be estimated.

ASC_CAR = Beta('ASC_CAR', 0, None, None, 0)
ASC_TRAIN = Beta('ASC_TRAIN', 0, None, None, 0)
B_TIME = Beta('B_TIME', 0, None, None, 0)
B_COST = Beta('B_COST', 0, None, None, 0)

Parameter of the Box-Cox transform.

ell_travel_time = Beta('lambda_travel_time', 1, -10, 10, 0)

Coefficients of the power series.

square_tt_coef = Beta('square_tt_coef', 0, None, None, 0)
cube_tt_coef = Beta('cube_tt_coef', 0, None, None, 0)

Function calculation the power series.

def power_series(the_variable: Expression) -> Expression:
    """Generate the expression of a polynomial of degree 3

    :param the_variable: variable of the polynomial
    """
    return (
        the_variable
        + square_tt_coef * the_variable**2
        + cube_tt_coef * the_variable * the_variable**3
    )

Train travel time

Linear specification.

linear_train_tt = TRAIN_TT_SCALED

Box-Cox transform.

boxcox_train_tt = models.boxcox(TRAIN_TT_SCALED, ell_travel_time)

Power series.

power_train_tt = power_series(TRAIN_TT_SCALED)

Definition of the catalog.

train_tt_catalog = Catalog.from_dict(
    catalog_name='train_tt_catalog',
    dict_of_expressions={
        'linear': linear_train_tt,
        'boxcox': boxcox_train_tt,
        'power': power_train_tt,
    },
)

Swissmetro travel time

Linear specification.

linear_sm_tt = SM_TT_SCALED

Box-Cox transform.

boxcox_sm_tt = models.boxcox(SM_TT_SCALED, ell_travel_time)

Power series.

power_sm_tt = power_series(SM_TT_SCALED)

Definition of the catalog. Note that the controller is the same as for train.

sm_tt_catalog = Catalog.from_dict(
    catalog_name='sm_tt_catalog',
    dict_of_expressions={
        'linear': linear_sm_tt,
        'boxcox': boxcox_sm_tt,
        'power': power_sm_tt,
    },
    controlled_by=train_tt_catalog.controlled_by,
)

Car travel time

Linear specification.

linear_car_tt = CAR_TT_SCALED

Box-Cox transform.

boxcox_car_tt = models.boxcox(CAR_TT_SCALED, ell_travel_time)

Power series.

power_car_tt = power_series(CAR_TT_SCALED)

Definition of the catalog. Note that the controller is the same as for train.

car_tt_catalog = Catalog.from_dict(
    catalog_name='car_tt_catalog',
    dict_of_expressions={
        'linear': linear_car_tt,
        'boxcox': boxcox_car_tt,
        'power': power_car_tt,
    },
    controlled_by=train_tt_catalog.controlled_by,
)

Catalogs for the alternative specific constants.

ASC_TRAIN_catalog, ASC_CAR_catalog = segmentation_catalogs(
    generic_name='ASC',
    beta_parameters=[ASC_TRAIN, ASC_CAR],
    potential_segmentations=(
        segmentation_ga,
        segmentation_luggage,
    ),
    maximum_number=2,
)

Catalog for the travel time coefficient.

(B_TIME_catalog_dict,) = generic_alt_specific_catalogs(
    generic_name='B_TIME',
    beta_parameters=[B_TIME],
    alternatives=('TRAIN', 'SM', 'CAR'),
    potential_segmentations=(
        segmentation_first,
        segmentation_purpose,
    ),
    maximum_number=1,
)

Catalog for the travel cost coefficient.

(B_COST_catalog_dict,) = generic_alt_specific_catalogs(
    generic_name='B_COST', beta_parameters=[B_COST], alternatives=('TRAIN', 'SM', 'CAR')
)

Definition of the utility functions.

V1 = (
    ASC_TRAIN_catalog
    + B_TIME_catalog_dict['TRAIN'] * train_tt_catalog
    + B_COST_catalog_dict['TRAIN'] * TRAIN_COST_SCALED
)
V2 = (
    B_TIME_catalog_dict['SM'] * sm_tt_catalog
    + B_COST_catalog_dict['SM'] * SM_COST_SCALED
)
V3 = (
    ASC_CAR_catalog
    + B_TIME_catalog_dict['CAR'] * car_tt_catalog
    + B_COST_catalog_dict['CAR'] * CAR_CO_SCALED
)

Associate utility functions with the numbering of alternatives.

V = {1: V1, 2: V2, 3: V3}

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

Definition of the logit model. This is the contribution of each observation to the log likelihood function.

logprob_logit = models.loglogit(V, av, CHOICE)

Nested logit model: nest with existing alternatives.

mu_existing = Beta('mu_existing', 1, 1, 10, 0)
existing = OneNestForNestedLogit(
    nest_param=mu_existing, list_of_alternatives=[1, 3], name='Existing'
)

nests_existing = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(existing,))
logprob_nested_existing = models.lognested(V, av, nests_existing, CHOICE)

Nested logit model: nest with public transportation alternatives.

mu_public = Beta('mu_public', 1, 1, 10, 0)
public = OneNestForNestedLogit(
    nest_param=mu_public, list_of_alternatives=[1, 2], name='Public'
)

nests_public = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(public,))
logprob_nested_public = models.lognested(V, av, nests_public, CHOICE)

Catalo for models.

model_catalog = Catalog.from_dict(
    catalog_name='model_catalog',
    dict_of_expressions={
        'logit': logprob_logit,
        'nested existing': logprob_nested_existing,
        'nested public': logprob_nested_public,
    },
)