Choice model with latent variable: sequential estimation

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

author:

Michel Bierlaire, EPFL

date:

Fri Apr 14 09:47:53 2023

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

from read_or_estimate import read_or_estimate

from optima import (
    database,
    male,
    age,
    haveChildren,
    highEducation,
    childCenter,
    childSuburb,
    SocioProfCat,
    WaitingTimePT,
    Choice,
    TimePT_scaled,
    TimeCar_scaled,
    MarginalCostPT_scaled,
    CostCarCHF_scaled,
    distance_km_scaled,
    PurpHWH,
    PurpOther,
)

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example m02_sequential_estimation.py')

Example m02_sequential_estimation.py

Read the estimates from the structural equation estimation.

MODELNAME = 'm01_latent_variable'
try:
    struct_results = res.bioResults(pickleFile=f'saved_results/{MODELNAME}.pickle')
except excep.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.getBetaValues()

Coefficients

coef_intercept = struct_betas['coef_intercept']
coef_age_30_less = struct_betas['coef_age_30_less']
coef_male = struct_betas['coef_male']
coef_haveChildren = struct_betas['coef_haveChildren']
coef_highEducation = struct_betas['coef_highEducation']
coef_artisans = struct_betas['coef_artisans']
coef_employees = struct_betas['coef_employees']
coef_child_center = struct_betas['coef_child_center']
coef_child_suburb = struct_betas['coef_child_suburb']

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)

ACTIVELIFE = (
    coef_intercept
    + coef_child_center * childCenter
    + coef_child_suburb * childSuburb
    + coef_highEducation * highEducation
    + coef_artisans * (SocioProfCat == 5)
    + coef_employees * (SocioProfCat == 6)
    + coef_age_30_less * (age <= 30)
    + coef_male * male
    + coef_haveChildren * haveChildren
    + sigma_s * omega
)

Choice model

ASC_CAR = Beta('ASC_CAR', 0.94, None, None, 0)
ASC_PT = Beta('ASC_PT', 0, None, None, 1)
ASC_SM = Beta('ASC_SM', 0.35, None, None, 0)
BETA_COST_HWH = Beta('BETA_COST_HWH', -2.3, None, None, 0)
BETA_COST_OTHER = Beta('BETA_COST_OTHER', -1.9, None, None, 0)
BETA_DIST = Beta('BETA_DIST', -1.3, None, None, 0)
BETA_TIME_CAR_REF = Beta('BETA_TIME_CAR_REF', -6.1, None, 0, 0)
BETA_TIME_PT_REF = Beta('BETA_TIME_PT_REF', 0, None, 0, 0)
BETA_WAITING_TIME = Beta('BETA_WAITING_TIME', -0.075, 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_AL = Beta('BETA_TIME_PT_AL', 1.5, None, None, 0)
BETA_TIME_PT = BETA_TIME_PT_REF * exp(BETA_TIME_PT_AL * ACTIVELIFE)
BETA_TIME_CAR_AL = Beta('BETA_TIME_CAR_AL', -0.11, None, None, 0)
BETA_TIME_CAR = BETA_TIME_CAR_REF * exp(BETA_TIME_CAR_AL * ACTIVELIFE)

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'))

Create the Biogeme object

the_biogeme = bio.BIOGEME(database, loglike)
the_biogeme.modelName = 'm02_sequential_estimation'
the_biogeme.maxiter = 1000

File biogeme.toml has been parsed.

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(results.short_summary())

Results for model m02_sequential_estimation
Nbr of parameters:              11
Sample size:                    1906
Excluded data:                  359
Final log likelihood:           -1121.273
Akaike Information Criterion:   2264.546
Bayesian Information Criterion: 2325.626

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

Final log likelihood: -1121.273
Output file: m02_sequential_estimation.html

results.getEstimatedParameters()

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ASC_CAR	0.622583	0.116322	5.352241	8.687180e-08
ASC_SM	1.654831	0.225762	7.329973	2.302603e-13
BETA_COST_HWH	-1.822974	0.386925	-4.711438	2.459747e-06
BETA_COST_OTHER	-0.929877	0.213730	-4.350703	1.357015e-05
BETA_DIST	-4.920546	0.532287	-9.244152	0.000000e+00
BETA_TIME_CAR_AL	-0.038697	0.022999	-1.682579	9.245669e-02
BETA_TIME_CAR_REF	-9.057901	1.304489	-6.943639	3.821166e-12
BETA_TIME_PT_AL	-0.036915	0.021627	-1.706888	8.784284e-02
BETA_TIME_PT_REF	-3.054210	0.517935	-5.896901	3.703930e-09
BETA_WAITING_TIME	-0.027764	0.011142	-2.491826	1.270885e-02
sigma_s	33.539526	20.078869	1.670389	9.484239e-02