Heteroscedastic specification

Illustrates a heteroscedastic specification. A different scale is associated with different segments of the sample.

author:

Michel Bierlaire, EPFL

date:

Sun Apr 9 17:23:03 2023

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

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,
    GROUP,
)

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)
Scale_group3 = Beta('Scale_group3', 1, 0.001, 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

Scale associated with group 3 is estimated.

scale = (GROUP != 3) + (GROUP == 3) * Scale_group3

Scale the utility functions, and associate them with the numbering of alternatives.

V = {1: scale * V1, 2: scale * V2, 3: scale * 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)

These notes will be included as such in the report file.

USER_NOTES = (
    'Illustrates heteroscedastic specification. A different scale is'
    ' associated with different segments of the sample.'
)

Create the Biogeme object.

the_biogeme = bio.BIOGEME(database, logprob, user_notes=USER_NOTES)
the_biogeme.modelName = 'b03scale'

Estimate the parameters.

results = the_biogeme.estimate()

print(results.short_summary())

Results for model b03scale
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -4976.691
Akaike Information Criterion:   9963.381
Bayesian Information Criterion: 9997.481

Get the results in a pandas table

pandas_results = results.getEstimatedParameters()
pandas_results

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ASC_CAR	-0.015332	0.018508	-0.828415	0.407435
ASC_TRAIN	-0.447096	0.041146	-10.866099	0.000000
B_COST	-0.357349	0.038418	-9.301647	0.000000
B_TIME	-0.374455	0.044514	-8.412151	0.000000
Scale_group3	4.177737	0.370552	11.274371	0.000000