Choice model with a latent variable: sequential estimation

Mixture of logit. Measurement equation for the indicators. Sequential estimation.

author:

Michel Bierlaire, EPFL

date:

Thu Apr 13 17:32:34 2023

import sys
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.exceptions import BiogemeError
import biogeme.distributions as dist
import biogeme.results as res
from biogeme.expressions import (
    Beta,
    RandomVariable,
    exp,
    log,
    Integrate,
)

from read_or_estimate import read_or_estimate

from biogeme.data.optima import (
    read_data,
    age_65_more,
    moreThanOneCar,
    moreThanOneBike,
    individualHouse,
    male,
    haveChildren,
    haveGA,
    highEducation,
    WaitingTimePT,
    Choice,
    TimePT_scaled,
    TimeCar_scaled,
    MarginalCostPT_scaled,
    CostCarCHF_scaled,
    distance_km_scaled,
    PurpHWH,
    PurpOther,
    ScaledIncome,
)

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b04latent_choice_seq.py')

Example b04latent_choice_seq.py

Read the estimates from the structural equation estimation.

MODELNAME = 'b02one_latent_ordered'
try:
    struct_results = res.bioResults(pickle_file=f'saved_results/{MODELNAME}.pickle')
except BiogemeError:
    print(
        f'Run first the script {MODELNAME}.py in order to generate the '
        f'file {MODELNAME}.pickle, and move it to the directory saved_results'
    )
    sys.exit()
struct_betas = struct_results.get_beta_values()

Coefficients.

coef_intercept = struct_betas['coef_intercept']
coef_age_65_more = struct_betas['coef_age_65_more']
coef_haveGA = struct_betas['coef_haveGA']
coef_moreThanOneCar = struct_betas['coef_moreThanOneCar']
coef_moreThanOneBike = struct_betas['coef_moreThanOneBike']
coef_individualHouse = struct_betas['coef_individualHouse']
coef_male = struct_betas['coef_male']
coef_haveChildren = struct_betas['coef_haveChildren']
coef_highEducation = struct_betas['coef_highEducation']

Latent variable: structural equation.

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

omega = RandomVariable('omega')
density = dist.normalpdf(omega)
sigma_s = Beta('sigma_s', 1, None, None, 0)

thresholds = [None, 4, 6, 8, 10, None]
formula_income = models.piecewise_formula(
    variable=ScaledIncome,
    thresholds=thresholds,
    betas=[
        struct_betas['beta_ScaledIncome_minus_inf_4'],
        struct_betas['beta_ScaledIncome_4_6'],
        struct_betas['beta_ScaledIncome_6_8'],
        struct_betas['beta_ScaledIncome_8_10'],
        struct_betas['beta_ScaledIncome_10_inf'],
    ],
)


CARLOVERS = (
    coef_intercept
    + coef_age_65_more * age_65_more
    + formula_income
    + coef_moreThanOneCar * moreThanOneCar
    + coef_moreThanOneBike * moreThanOneBike
    + coef_individualHouse * individualHouse
    + coef_male * male
    + coef_haveChildren * haveChildren
    + coef_haveGA * haveGA
    + coef_highEducation * highEducation
    + sigma_s * omega
)

Choice model.

ASC_CAR = Beta('ASC_CAR', 0, None, None, 0)
ASC_PT = Beta('ASC_PT', 0, None, None, 1)
ASC_SM = Beta('ASC_SM', 0, None, None, 0)
BETA_COST_HWH = Beta('BETA_COST_HWH', 0, None, None, 0)
BETA_COST_OTHER = Beta('BETA_COST_OTHER', 0, None, None, 0)
BETA_DIST = Beta('BETA_DIST', 0, None, None, 0)
BETA_TIME_CAR_REF = Beta('BETA_TIME_CAR_REF', 0, None, 0, 0)
BETA_TIME_PT_REF = Beta('BETA_TIME_PT_REF', 0, None, 0, 0)
BETA_WAITING_TIME = Beta('BETA_WAITING_TIME', 0, None, None, 0)

The coefficient of the latent variable should be initialized to something different from zero. If not, the algorithm may be trapped in a local optimum, and never change the value.

BETA_TIME_PT_CL = Beta('BETA_TIME_PT_CL', -0.01, None, None, 0)
BETA_TIME_PT = BETA_TIME_PT_REF * exp(BETA_TIME_PT_CL * CARLOVERS)
BETA_TIME_CAR_CL = Beta('BETA_TIME_CAR_CL', -0.01, None, None, 0)
BETA_TIME_CAR = BETA_TIME_CAR_REF * exp(BETA_TIME_CAR_CL * CARLOVERS)

Definition of utility functions:.

V0 = (
    ASC_PT
    + BETA_TIME_PT * TimePT_scaled
    + BETA_WAITING_TIME * WaitingTimePT
    + BETA_COST_HWH * MarginalCostPT_scaled * PurpHWH
    + BETA_COST_OTHER * MarginalCostPT_scaled * PurpOther
)


V1 = (
    ASC_CAR
    + BETA_TIME_CAR * TimeCar_scaled
    + BETA_COST_HWH * CostCarCHF_scaled * PurpHWH
    + BETA_COST_OTHER * CostCarCHF_scaled * PurpOther
)

V2 = ASC_SM + BETA_DIST * distance_km_scaled

Associate utility functions with the numbering of alternatives.

V = {0: V0, 1: V1, 2: V2}

Conditional on omega, we have a logit model (called the kernel).

condprob = models.logit(V, None, Choice)

We integrate over omega using numerical integration.

loglike = log(Integrate(condprob * density, 'omega'))

Read the data

database = read_data()

Create the Biogeme object.

the_biogeme = bio.BIOGEME(database, loglike)
the_biogeme.modelName = 'b04latent_choice_seq'

Biogeme parameters read from biogeme.toml.

If estimation results are saved on file, we read them to speed up the process. If not, we estimate the parameters.

results = read_or_estimate(the_biogeme=the_biogeme, directory='saved_results')

print(f'Estimated betas: {len(results.data.betaValues)}')
print(f'Final log likelihood: {results.data.logLike:.3f}')
print(f'Output file: {results.data.htmlFileName}')

Estimated betas: 11
Final log likelihood: -1094.811
Output file: b04latent_choice_seq.html

results.get_estimated_parameters()

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ASC_CAR	0.818249	0.123298	6.636358	3.215295e-11
ASC_SM	1.979021	0.267020	7.411508	1.247891e-13
BETA_COST_HWH	-1.715963	0.185522	-9.249369	0.000000e+00
BETA_COST_OTHER	-0.890702	0.110066	-8.092461	6.661338e-16
BETA_DIST	-5.871872	0.736174	-7.976202	1.554312e-15
BETA_TIME_CAR_CL	-0.151834	0.017993	-8.438370	0.000000e+00
BETA_TIME_CAR_REF	-14.372652	1.008616	-14.249870	0.000000e+00
BETA_TIME_PT_CL	-0.141156	0.017664	-7.991069	1.332268e-15
BETA_TIME_PT_REF	-4.373281	0.504807	-8.663266	0.000000e+00
BETA_WAITING_TIME	-0.035181	0.006745	-5.215922	1.829051e-07
sigma_s	9.737609	1.154931	8.431337	0.000000e+00