Numerical integration

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

author:

Michel Bierlaire, EPFL

date:

Thu Apr 13 20:51:32 2023

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

from swissmetro_one 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,
)

Parameters

ASC_CAR = 0.137
ASC_TRAIN = -0.402
ASC_SM = 0
B_TIME = -2.26
B_TIME_S = 1.66
B_COST = -1.29

Define a random parameter, normally distributed, designed to be used for integration

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

integrand = models.logit(util, av, CHOICE)
numerical_integral = Integrate(integrand * density, 'omega')

simulate = {'Numerical': numerical_integral}

biosim = bio.BIOGEME(database, simulate)

results = biosim.simulate(the_beta_values={})
results

	Numerical
0	0.63785

print('Mixture of logit - numerical integration: ', results.iloc[0]['Numerical'])

Mixture of logit - numerical integration:  0.6378498355784457