Cross-nested logit

Example of a cross-nested logit model with two nests:

• one with existing alternatives (car and train),

• one with public transportation alternatives (train and Swissmetro)

This illustrates the possibility to ignore all membership parameters that are 0.

author:

Michel Bierlaire, EPFL

date:

Sun Apr 9 18:06:44 2023

import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import Beta
from biogeme.nests import OneNestForCrossNestedLogit, NestsForCrossNestedLogit

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,
)

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b11cnl.py')

Example b11cnl.py

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)

MU_EXISTING = Beta('MU_EXISTING', 1, 1, 10, 0)
MU_PUBLIC = Beta('MU_PUBLIC', 1, 1, 10, 0)

Nest membership parameters.

ALPHA_EXISTING = Beta('ALPHA_EXISTING', 0.5, 0, 1, 0)
ALPHA_PUBLIC = 1 - ALPHA_EXISTING

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 nests.

nest_existing = OneNestForCrossNestedLogit(
    nest_param=MU_EXISTING,
    dict_of_alpha={1: ALPHA_EXISTING, 3: 1.0},
    name='existing',
)

nest_public = OneNestForCrossNestedLogit(
    nest_param=MU_PUBLIC, dict_of_alpha={1: ALPHA_PUBLIC, 2: 1.0}, name='public'
)

nests = NestsForCrossNestedLogit(
    choice_set=[1, 2, 3], tuple_of_nests=(nest_existing, nest_public)
)

The choice model is a cross-nested logit, with availability conditions.

logprob = models.logcnl(V, av, nests, CHOICE)

Create the Biogeme object

the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = 'b11cnl'

Biogeme parameters read from biogeme.toml.

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 __b11cnl.iter
Parameter values restored from __b11cnl.iter
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
Results saved in file b11cnl~00.html
Results saved in file b11cnl~00.pickle

print(results.short_summary())

Results for model b11cnl
Nbr of parameters:              7
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5214.049
Akaike Information Criterion:   10442.1
Bayesian Information Criterion: 10489.84

pandas_results = results.get_estimated_parameters()
pandas_results

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ALPHA_EXISTING	0.495086	0.034723	14.258046	0.000000e+00
ASC_CAR	-0.240519	0.053433	-4.501346	6.752446e-06
ASC_TRAIN	0.098094	0.069916	1.403026	1.606089e-01
B_COST	-0.818963	0.058942	-13.894456	0.000000e+00
B_TIME	-0.776852	0.102331	-7.591581	3.153033e-14
MU_EXISTING	2.514925	0.248163	10.134152	0.000000e+00
MU_PUBLIC	4.108825	0.494954	8.301433	0.000000e+00