Latent class model

Example of a discrete mixture of logit (or latent_old class model).

Michel Bierlaire, EPFL Sat Jun 21 2025, 15:11:24

from IPython.core.display_functions import display

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data import (
    CAR_AV_SP,
    CAR_CO_SCALED,
    CAR_TT_SCALED,
    CHOICE,
    SM_AV,
    SM_COST_SCALED,
    SM_TT_SCALED,
    TRAIN_AV_SP,
    TRAIN_COST_SCALED,
    TRAIN_TT_SCALED,
    database,
)

from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, log
from biogeme.models import logit
from biogeme.results_processing import get_pandas_estimated_parameters

Parameters to be estimated.

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_cost = Beta('b_cost', 0, None, None, 0)

Class membership probability.

prob_class1 = Beta('prob_class1', 0.5, 0, 1, 0)
prob_class2 = 1 - prob_class1

Definition of the utility functions for latent_old class 1, where the time coefficient is zero.

v_train_class_1 = asc_train + b_cost * TRAIN_COST_SCALED
v_swissmetro_class_1 = asc_sm + b_cost * SM_COST_SCALED
v_car_class_1 = asc_car + b_cost * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

v_class_1 = {1: v_train_class_1, 2: v_swissmetro_class_1, 3: v_car_class_1}

Definition of the utility functions for latent_old class 2, whete the time coefficient is estimated.

v_train_class_2 = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro_class_2 = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car_class_2 = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

v_class_2 = {1: v_train_class_2, 2: v_swissmetro_class_2, 3: v_car_class_2}

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

The choice model is a discrete mixture of logit, with availability conditions

choice_probability_class_1 = logit(v_class_1, av, CHOICE)
choice_probability_class_2 = logit(v_class_2, av, CHOICE)
prob = (
    prob_class1 * choice_probability_class_1 + prob_class2 * choice_probability_class_2
)
log_probability = log(prob)

Create the Biogeme object

the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b07discrete_mixture'

Estimate the parameters

results = the_biogeme.estimate()

print(results.short_summary())

Results for model b07discrete_mixture
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5208.498
Akaike Information Criterion:   10427
Bayesian Information Criterion: 10461.1

pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)

          Name     Value  Robust std err.  Robust t-stat.  Robust p-value
0  prob_class1  0.250792         0.021741       11.535649    0.000000e+00
1    asc_train -0.397586         0.062033       -6.409281    1.462079e-10
2       b_cost -1.264065         0.085606      -14.766051    0.000000e+00
3      asc_car  0.124605         0.050735        2.455992    1.404965e-02
4       b_time -2.797932         0.171663      -16.298949    0.000000e+00