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

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}

The choice model is a logit, with availability conditions

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

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

results = the_biogeme.estimate()

print(results.short_summary())

Results for model 06estimationIntegral
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5214.879
Akaike Information Criterion:   10439.76
Bayesian Information Criterion: 10473.86

pandas_results = results.getEstimatedParameters()
pandas_results

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ASC_CAR	0.137075	0.051715	2.650599	8.034925e-03
ASC_TRAIN	-0.401234	0.065607	-6.115756	9.610024e-10
B_COST	-1.285301	0.086258	-14.900638	0.000000e+00
B_TIME	-2.259140	0.116548	-19.383794	0.000000e+00
B_TIME_S	1.654096	0.124704	13.264163	0.000000e+00