Estimation of a logit model

Three alternatives:

• train,

• car and,

• Swissmetro.

Stated preferences data.

author:

Michel Bierlaire, EPFL

date:

Sun Apr 9 17:02:18 2023

import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import Beta

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b01logit_bis.py')

Example b01logit_bis.py

See the data processing script: Data preparation for Swissmetro.

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

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)

Definition of the utility functions.

V1 = ASC_TRAIN + B_TIME * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
V2 = ASC_SM + B_TIME * SM_TT_SCALED + B_COST * SM_COST_SCALED
V3 = ASC_CAR + B_TIME * CAR_TT_SCALED + B_COST * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

V = {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}

Definition of the model. This is the contribution of each observation to the log likelihood function.

logprob = models.loglogit(V, av, CHOICE)

Create the Biogeme object.

the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = 'b01logit'

Default values of the Biogeme parameters are used.
File biogeme.toml has been created

Calculate the null log likelihood for reporting.

the_biogeme.calculate_null_loglikelihood(av)

np.float64(-6964.662979191462)

Estimate the parameters.

results = the_biogeme.estimate()

As the model is not too complex, we activate the calculation of second derivatives. If you want to change it, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds"
*** Initial values of the parameters are obtained from the file __b01logit.iter
Cannot read file __b01logit.iter. Statement is ignored.
As the model is not too complex, we activate the calculation of second derivatives. If you want to change it, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds"
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: Newton with trust region for simple bounds
Iter.         ASC_CAR       ASC_TRAIN          B_COST          B_TIME     Function    Relgrad   Radius      Rho
    0           -0.49           -0.92           -0.88           -0.67      5.4e+03      0.041       10      1.1   ++
    1           -0.18           -0.73              -1            -1.2      5.3e+03     0.0072    1e+02      1.1   ++
    2           -0.16            -0.7            -1.1            -1.3      5.3e+03    0.00018    1e+03        1   ++
    3           -0.16            -0.7            -1.1            -1.3      5.3e+03    1.1e-07    1e+03        1   ++
Results saved in file b01logit.html
Results saved in file b01logit.pickle

print(results.short_summary())

Results for model b01logit
Nbr of parameters:              4
Sample size:                    6768
Excluded data:                  3960
Null log likelihood:            -6964.663
Final log likelihood:           -5331.252
Likelihood ratio test (null):           3266.822
Rho square (null):                      0.235
Rho bar square (null):                  0.234
Akaike Information Criterion:   10670.5
Bayesian Information Criterion: 10697.78

Get the results in a pandas table

pandas_results = results.get_estimated_parameters()
print(pandas_results)

              Value  Rob. Std err  Rob. t-test  Rob. p-value
ASC_CAR   -0.154633      0.058163    -2.658590      0.007847
ASC_TRAIN -0.701187      0.082562    -8.492857      0.000000
B_COST    -1.083790      0.068225   -15.885521      0.000000
B_TIME    -1.277859      0.104254   -12.257120      0.000000