Note
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Choice model with a latent variable: maximum likelihood estimation (Monte-Carlo)
Choice model with the latent variable. Mixture of logit with Monte-Carlo integration. Measurement equation for the indicators. Maximum likelihood (full information) estimation.
- author:
Michel Bierlaire, EPFL
- date:
Thu Apr 13 18:11:54 2023
import sys
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
import biogeme.exceptions as excep
from biogeme import models
import biogeme.results as res
from biogeme.expressions import (
Beta,
log,
bioDraws,
MonteCarlo,
Elem,
bioNormalCdf,
exp,
)
from read_or_estimate import read_or_estimate
from optima import (
database,
age_65_more,
moreThanOneCar,
moreThanOneBike,
individualHouse,
male,
haveChildren,
haveGA,
highEducation,
WaitingTimePT,
Envir01,
Envir02,
Envir03,
Mobil11,
Mobil14,
Mobil16,
Mobil17,
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 b05latent_choice_full_mc.py')
Example b05latent_choice_full_mc.py
Read the estimates from the structural equation estimation.
MODELNAME = 'b02one_latent_ordered'
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 = Beta('coef_intercept', struct_betas['coef_intercept'], None, None, 0)
coef_age_65_more = Beta(
'coef_age_65_more', struct_betas['coef_age_65_more'], None, None, 0
)
coef_haveGA = Beta('coef_haveGA', struct_betas['coef_haveGA'], None, None, 0)
coef_moreThanOneCar = Beta(
'coef_moreThanOneCar', struct_betas['coef_moreThanOneCar'], None, None, 0
)
coef_moreThanOneBike = Beta(
'coef_moreThanOneBike', struct_betas['coef_moreThanOneBike'], None, None, 0
)
coef_individualHouse = Beta(
'coef_individualHouse', struct_betas['coef_individualHouse'], None, None, 0
)
coef_male = Beta('coef_male', struct_betas['coef_male'], None, None, 0)
coef_haveChildren = Beta(
'coef_haveChildren', struct_betas['coef_haveChildren'], None, None, 0
)
coef_highEducation = Beta(
'coef_highEducation', struct_betas['coef_highEducation'], None, None, 0
)
Latent variable: structural equation.
Define a random parameter, normally distributed, designed to be used for Monte-Carlo integration
sigma_s = Beta('sigma_s', 1, None, None, 0)
error_component = sigma_s * bioDraws('EC', 'NORMAL_MLHS')
Piecewise linear specification for income.
thresholds = [None, 4, 6, 8, 10, None]
betas_thresholds = [
Beta(
'beta_ScaledIncome_minus_inf_4',
struct_betas['beta_ScaledIncome_minus_inf_4'],
None,
None,
0,
),
Beta(
'beta_ScaledIncome_4_6',
struct_betas['beta_ScaledIncome_4_6'],
None,
None,
0,
),
Beta(
'beta_ScaledIncome_6_8',
struct_betas['beta_ScaledIncome_6_8'],
None,
None,
0,
),
Beta(
'beta_ScaledIncome_8_10',
struct_betas['beta_ScaledIncome_8_10'],
None,
None,
0,
),
Beta(
'beta_ScaledIncome_10_inf',
struct_betas['beta_ScaledIncome_10_inf'],
None,
None,
0,
),
]
formula_income = models.piecewiseFormula(
variable=ScaledIncome,
thresholds=thresholds,
betas=betas_thresholds,
)
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
+ error_component
)
Measurement equations.
Intercepts.
INTER_Envir01 = Beta('INTER_Envir01', 0, None, None, 1)
INTER_Envir02 = Beta('INTER_Envir02', 0, None, None, 0)
INTER_Envir03 = Beta('INTER_Envir03', 0, None, None, 0)
INTER_Mobil11 = Beta('INTER_Mobil11', 0, None, None, 0)
INTER_Mobil14 = Beta('INTER_Mobil14', 0, None, None, 0)
INTER_Mobil16 = Beta('INTER_Mobil16', 0, None, None, 0)
INTER_Mobil17 = Beta('INTER_Mobil17', 0, None, None, 0)
Coefficients.
B_Envir01_F1 = Beta('B_Envir01_F1', -1, None, None, 1)
B_Envir02_F1 = Beta('B_Envir02_F1', -1, None, None, 0)
B_Envir03_F1 = Beta('B_Envir03_F1', 1, None, None, 0)
B_Mobil11_F1 = Beta('B_Mobil11_F1', 1, None, None, 0)
B_Mobil14_F1 = Beta('B_Mobil14_F1', 1, None, None, 0)
B_Mobil16_F1 = Beta('B_Mobil16_F1', 1, None, None, 0)
B_Mobil17_F1 = Beta('B_Mobil17_F1', 1, None, None, 0)
Linear models.
MODEL_Envir01 = INTER_Envir01 + B_Envir01_F1 * CARLOVERS
MODEL_Envir02 = INTER_Envir02 + B_Envir02_F1 * CARLOVERS
MODEL_Envir03 = INTER_Envir03 + B_Envir03_F1 * CARLOVERS
MODEL_Mobil11 = INTER_Mobil11 + B_Mobil11_F1 * CARLOVERS
MODEL_Mobil14 = INTER_Mobil14 + B_Mobil14_F1 * CARLOVERS
MODEL_Mobil16 = INTER_Mobil16 + B_Mobil16_F1 * CARLOVERS
MODEL_Mobil17 = INTER_Mobil17 + B_Mobil17_F1 * CARLOVERS
Scale parameters
SIGMA_STAR_Envir01 = Beta('SIGMA_STAR_Envir01', 1, 1.0e-5, None, 1)
SIGMA_STAR_Envir02 = Beta('SIGMA_STAR_Envir02', 1, 1.0e-5, None, 0)
SIGMA_STAR_Envir03 = Beta('SIGMA_STAR_Envir03', 1, 1.0e-5, None, 0)
SIGMA_STAR_Mobil11 = Beta('SIGMA_STAR_Mobil11', 1, 1.0e-5, None, 0)
SIGMA_STAR_Mobil14 = Beta('SIGMA_STAR_Mobil14', 1, 1.0e-5, None, 0)
SIGMA_STAR_Mobil16 = Beta('SIGMA_STAR_Mobil16', 1, 1.0e-5, None, 0)
SIGMA_STAR_Mobil17 = Beta('SIGMA_STAR_Mobil17', 1, 1.0e-5, None, 0)
Symmetric thresholds.
delta_1 = Beta('delta_1', 0.1, 1.0e-5, None, 0)
delta_2 = Beta('delta_2', 0.2, 1.0e-5, None, 0)
tau_1 = -delta_1 - delta_2
tau_2 = -delta_1
tau_3 = delta_1
tau_4 = delta_1 + delta_2
Ordered probit models.
Envir01_tau_1 = (tau_1 - MODEL_Envir01) / SIGMA_STAR_Envir01
Envir01_tau_2 = (tau_2 - MODEL_Envir01) / SIGMA_STAR_Envir01
Envir01_tau_3 = (tau_3 - MODEL_Envir01) / SIGMA_STAR_Envir01
Envir01_tau_4 = (tau_4 - MODEL_Envir01) / SIGMA_STAR_Envir01
IndEnvir01 = {
1: bioNormalCdf(Envir01_tau_1),
2: bioNormalCdf(Envir01_tau_2) - bioNormalCdf(Envir01_tau_1),
3: bioNormalCdf(Envir01_tau_3) - bioNormalCdf(Envir01_tau_2),
4: bioNormalCdf(Envir01_tau_4) - bioNormalCdf(Envir01_tau_3),
5: 1 - bioNormalCdf(Envir01_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Envir01 = Elem(IndEnvir01, Envir01)
Envir02_tau_1 = (tau_1 - MODEL_Envir02) / SIGMA_STAR_Envir02
Envir02_tau_2 = (tau_2 - MODEL_Envir02) / SIGMA_STAR_Envir02
Envir02_tau_3 = (tau_3 - MODEL_Envir02) / SIGMA_STAR_Envir02
Envir02_tau_4 = (tau_4 - MODEL_Envir02) / SIGMA_STAR_Envir02
IndEnvir02 = {
1: bioNormalCdf(Envir02_tau_1),
2: bioNormalCdf(Envir02_tau_2) - bioNormalCdf(Envir02_tau_1),
3: bioNormalCdf(Envir02_tau_3) - bioNormalCdf(Envir02_tau_2),
4: bioNormalCdf(Envir02_tau_4) - bioNormalCdf(Envir02_tau_3),
5: 1 - bioNormalCdf(Envir02_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Envir02 = Elem(IndEnvir02, Envir02)
Envir03_tau_1 = (tau_1 - MODEL_Envir03) / SIGMA_STAR_Envir03
Envir03_tau_2 = (tau_2 - MODEL_Envir03) / SIGMA_STAR_Envir03
Envir03_tau_3 = (tau_3 - MODEL_Envir03) / SIGMA_STAR_Envir03
Envir03_tau_4 = (tau_4 - MODEL_Envir03) / SIGMA_STAR_Envir03
IndEnvir03 = {
1: bioNormalCdf(Envir03_tau_1),
2: bioNormalCdf(Envir03_tau_2) - bioNormalCdf(Envir03_tau_1),
3: bioNormalCdf(Envir03_tau_3) - bioNormalCdf(Envir03_tau_2),
4: bioNormalCdf(Envir03_tau_4) - bioNormalCdf(Envir03_tau_3),
5: 1 - bioNormalCdf(Envir03_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Envir03 = Elem(IndEnvir03, Envir03)
Mobil11_tau_1 = (tau_1 - MODEL_Mobil11) / SIGMA_STAR_Mobil11
Mobil11_tau_2 = (tau_2 - MODEL_Mobil11) / SIGMA_STAR_Mobil11
Mobil11_tau_3 = (tau_3 - MODEL_Mobil11) / SIGMA_STAR_Mobil11
Mobil11_tau_4 = (tau_4 - MODEL_Mobil11) / SIGMA_STAR_Mobil11
IndMobil11 = {
1: bioNormalCdf(Mobil11_tau_1),
2: bioNormalCdf(Mobil11_tau_2) - bioNormalCdf(Mobil11_tau_1),
3: bioNormalCdf(Mobil11_tau_3) - bioNormalCdf(Mobil11_tau_2),
4: bioNormalCdf(Mobil11_tau_4) - bioNormalCdf(Mobil11_tau_3),
5: 1 - bioNormalCdf(Mobil11_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Mobil11 = Elem(IndMobil11, Mobil11)
Mobil14_tau_1 = (tau_1 - MODEL_Mobil14) / SIGMA_STAR_Mobil14
Mobil14_tau_2 = (tau_2 - MODEL_Mobil14) / SIGMA_STAR_Mobil14
Mobil14_tau_3 = (tau_3 - MODEL_Mobil14) / SIGMA_STAR_Mobil14
Mobil14_tau_4 = (tau_4 - MODEL_Mobil14) / SIGMA_STAR_Mobil14
IndMobil14 = {
1: bioNormalCdf(Mobil14_tau_1),
2: bioNormalCdf(Mobil14_tau_2) - bioNormalCdf(Mobil14_tau_1),
3: bioNormalCdf(Mobil14_tau_3) - bioNormalCdf(Mobil14_tau_2),
4: bioNormalCdf(Mobil14_tau_4) - bioNormalCdf(Mobil14_tau_3),
5: 1 - bioNormalCdf(Mobil14_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Mobil14 = Elem(IndMobil14, Mobil14)
Mobil16_tau_1 = (tau_1 - MODEL_Mobil16) / SIGMA_STAR_Mobil16
Mobil16_tau_2 = (tau_2 - MODEL_Mobil16) / SIGMA_STAR_Mobil16
Mobil16_tau_3 = (tau_3 - MODEL_Mobil16) / SIGMA_STAR_Mobil16
Mobil16_tau_4 = (tau_4 - MODEL_Mobil16) / SIGMA_STAR_Mobil16
IndMobil16 = {
1: bioNormalCdf(Mobil16_tau_1),
2: bioNormalCdf(Mobil16_tau_2) - bioNormalCdf(Mobil16_tau_1),
3: bioNormalCdf(Mobil16_tau_3) - bioNormalCdf(Mobil16_tau_2),
4: bioNormalCdf(Mobil16_tau_4) - bioNormalCdf(Mobil16_tau_3),
5: 1 - bioNormalCdf(Mobil16_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Mobil16 = Elem(IndMobil16, Mobil16)
Mobil17_tau_1 = (tau_1 - MODEL_Mobil17) / SIGMA_STAR_Mobil17
Mobil17_tau_2 = (tau_2 - MODEL_Mobil17) / SIGMA_STAR_Mobil17
Mobil17_tau_3 = (tau_3 - MODEL_Mobil17) / SIGMA_STAR_Mobil17
Mobil17_tau_4 = (tau_4 - MODEL_Mobil17) / SIGMA_STAR_Mobil17
IndMobil17 = {
1: bioNormalCdf(Mobil17_tau_1),
2: bioNormalCdf(Mobil17_tau_2) - bioNormalCdf(Mobil17_tau_1),
3: bioNormalCdf(Mobil17_tau_3) - bioNormalCdf(Mobil17_tau_2),
4: bioNormalCdf(Mobil17_tau_4) - bioNormalCdf(Mobil17_tau_3),
5: 1 - bioNormalCdf(Mobil17_tau_4),
6: 1.0,
-1: 1.0,
-2: 1.0,
}
P_Mobil17 = Elem(IndMobil17, Mobil17)
Choice model Read the estimates from the sequential estimation, and use them as starting values
MODELNAME = 'b04latent_choice_seq'
try:
choice_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()
choice_betas = choice_results.getBetaValues()
Parameters to estimate. We use the previously estimated values as starting points.
ASC_CAR = Beta('ASC_CAR', choice_betas['ASC_CAR'], None, None, 0)
ASC_PT = Beta('ASC_PT', 0, None, None, 1)
ASC_SM = Beta('ASC_SM', choice_betas['ASC_SM'], None, None, 0)
BETA_COST_HWH = Beta('BETA_COST_HWH', choice_betas['BETA_COST_HWH'], None, None, 0)
BETA_COST_OTHER = Beta(
'BETA_COST_OTHER', choice_betas['BETA_COST_OTHER'], None, None, 0
)
BETA_DIST = Beta('BETA_DIST', choice_betas['BETA_DIST'], None, None, 0)
BETA_TIME_CAR_REF = Beta(
'BETA_TIME_CAR_REF', choice_betas['BETA_TIME_CAR_REF'], None, 0, 0
)
BETA_TIME_CAR_CL = Beta(
'BETA_TIME_CAR_CL', choice_betas['BETA_TIME_CAR_CL'], None, None, 0
)
BETA_TIME_PT_REF = Beta(
'BETA_TIME_PT_REF', choice_betas['BETA_TIME_PT_REF'], None, 0, 0
)
BETA_TIME_PT_CL = Beta(
'BETA_TIME_PT_CL', choice_betas['BETA_TIME_PT_CL'], None, None, 0
)
BETA_WAITING_TIME = Beta(
'BETA_WAITING_TIME', choice_betas['BETA_WAITING_TIME'], None, None, 0
)
Definition of the utility functions.
BETA_TIME_PT = BETA_TIME_PT_REF * exp(BETA_TIME_PT_CL * CARLOVERS)
V0 = (
ASC_PT
+ BETA_TIME_PT * TimePT_scaled
+ BETA_WAITING_TIME * WaitingTimePT
+ BETA_COST_HWH * MarginalCostPT_scaled * PurpHWH
+ BETA_COST_OTHER * MarginalCostPT_scaled * PurpOther
)
BETA_TIME_CAR = BETA_TIME_CAR_REF * exp(BETA_TIME_CAR_CL * CARLOVERS)
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) for the choice.
condprob = models.logit(V, None, Choice)
Conditional on omega, we have the product of ordered probit for the indicators.
condlike = (
P_Envir01
* P_Envir02
* P_Envir03
* P_Mobil11
* P_Mobil14
* P_Mobil16
* P_Mobil17
* condprob
)
We integrate over omega using numerical integration
loglike = log(MonteCarlo(condlike))
Create the Biogeme object. As the objective is to illustrate the syntax, we calculate the Monte-Carlo approximation with a small number of draws. To achieve that, we provide a parameter file different from the default one.
the_biogeme = bio.BIOGEME(database, loglike, parameter_file='few_draws.toml')
the_biogeme.modelName = 'b05latent_choice_full_mc'
File few_draws.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(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: 45
Final log likelihood: -18465.213
Output file: b05latent_choice_full_mc.html
results.getEstimatedParameters()
Total running time of the script: (0 minutes 0.706 seconds)