Estimation of mixtures of logit

Estimation of a mixtures of logit models where the integral is calculated using numerical integration.

author:

Michel Bierlaire, EPFL

date:

Thu Apr 13 21:03:03 2023

import biogeme.biogeme as bio
import biogeme.distributions as dist
from biogeme import models
from biogeme.expressions import Beta, RandomVariable, Integrate, log

from swissmetro import (
    database,
    TRAIN_TT_SCALED,
    TRAIN_COST_SCALED,
    SM_TT_SCALED,
    SM_COST_SCALED,
    CAR_TT_SCALED,
    CAR_CO_SCALED,
    TRAIN_AV_SP,
    SM_AV,
    CAR_AV_SP,
    CHOICE,
)

ASC_CAR = Beta('ASC_CAR', 0, None, None, 0)
ASC_TRAIN = Beta('ASC_TRAIN', 0, None, None, 0)
ASC_SM = Beta('ASC_SM', 0, None, None, 1)
B_TIME = Beta('B_TIME', 0, None, None, 0)
B_TIME_S = Beta('B_TIME_S', 1, None, None, 0)
B_COST = Beta('B_COST', 0, None, None, 0)

Define a random parameter, normally distirbuted, designed to be used for Monte-Carlo simulation

omega = RandomVariable('omega')
density = dist.normalpdf(omega)
b_time_rnd = B_TIME + B_TIME_S * omega

Definition of the utility functions

v1 = ASC_TRAIN + b_time_rnd * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
v2 = ASC_SM + b_time_rnd * SM_TT_SCALED + B_COST * SM_COST_SCALED
v3 = ASC_CAR + b_time_rnd * CAR_TT_SCALED + B_COST * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives

util = {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}

The choice model is a logit, with availability conditions

condprob = models.logit(util, av, CHOICE)
prob = Integrate(condprob * density, 'omega')
logprob = log(prob)

the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = '06estimation_integral'

results = the_biogeme.estimate()

print(results.short_summary())

Results for model 06estimation_integral
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5214.88
Akaike Information Criterion:   10439.76
Bayesian Information Criterion: 10473.86

pandas_results = results.get_estimated_parameters()
pandas_results

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ASC_CAR	0.134900	0.047631	2.832218	4.622627e-03
ASC_TRAIN	-0.403873	0.058895	-6.857458	7.009726e-12
B_COST	-1.284092	0.046169	-27.812680	0.000000e+00
B_TIME	-2.254195	0.118544	-19.015637	0.000000e+00
B_TIME_S	1.649665	0.136564	12.079820	0.000000e+00