111: Swissmetro Cross-Nested Logit Mode Choice

This example is a mode choice model built using the Swissmetro example dataset. First we create the DB and Model objects. When we create the DB object, we will redefine the weight value:

d = larch.examples.SWISSMETRO()
m = larch.Model(dataservice=d)

We can attach a title to the model. The title does not affect the calculations as all; it is merely used in various output report styles.

m.title = "swissmetro example 11 (cross nested logit)"

We need to identify the availability and choice variables. These have been conveniently set up in the data.

m.availability_var = 'avail'
m.choice_ca_var = 'choice'

The swissmetro dataset, as with all Biogeme data, is only in co format.

from larch.roles import P,X
m.utility_co[1] = ( P.ASC_TRAIN
                  + P.B_TIME * X.TRAIN_TT
                  + P.B_COST * X("TRAIN_CO*(GA==0)") )
m.utility_co[2] = ( P.B_TIME * X.SM_TT
                  + P.B_COST * X("SM_CO*(GA==0)") )
m.utility_co[3] = ( P.ASC_CAR
                  + P.B_TIME * X.CAR_TT
                  + P.B_COST * X("CAR_CO") )

For this example, we want to nest together the Train and Car modes into a “existing” modes nest, and we want to nest Train and SM together into a “public” modes nest. This creates a structure different from a traditional nested logit model, because the Train mode is “cross-nested”: it appears in more than one nest. The desired nesting structure then looks like this:

digraph Desired_Nesting_Structure {
bgcolor="transparent"
root [label="Root", shape="oval"]
train [label="Train", shape=box, style="rounded", penwidth=2]
sm [label="SM", shape=box, style="rounded", penwidth=2]
car [label="Car", shape=box, style="rounded", penwidth=2]
public [label="Public", shape=oval]
exist [label="Existing", shape=oval]
exist -> train
public -> train
public -> sm
exist -> car
root -> exist
root -> public
}

To create nests, we can use the new_node command, although we’ll need to know what the alternative codes are for the alternatives in our dataset. To find out, we can do:

>>> m.dataservice.alternatives
[(1, 'Train'), (2, 'SM'), (3, 'Car')]

For this example, we want to nest together the Train and Car modes into a “existing” modes nest, and we want to nest together the Train and SM modes into a “public” modes nest. We can use the new_nest command like this:

m.graph.new_node(
        parameter="existing",
        parent=m.graph.root_id,
        children=[1,3],
        name='Existing',
        phi_parameters={1:'PHI'},
)
m.graph.new_node(
        parameter="public",
        parent=m.graph.root_id,
        children=[1,2],
        name='Public',
)

For a cross-nested model, we need to assign an allocation level to each graph link for all entering links of any node that has more than one predecessor. In this case, that applies only to the “Train” node.

Larch employs a logit-type function to manage this allocation, instead of specifying the allocation directly as a parameter. So, the allocation on the link Public->Train (PT) is given by

\[\alpha_{PT} = \frac{\exp ( \phi_{PT} )}{\exp ( \phi_{PT} ) + \exp ( \phi_{ET} )},\]

where \(\phi_{PT}\) is a parameter associated with the link PT, \(\phi_{ET}\) is a similar parameter for the link Public->Existing (ET).

We can attach parameters to each link using the phi_parameters argument to new_node.

Larch will find all the parameters in the model, but we’d like to output them in a rational order. We can use the ordering method to do this:

m.ordering = [
        ("ASCs", 'ASC.*',),
        ("LOS", 'B_.*',),
        ("LogSums", 'Existing', 'Public', ),
]

The swissmetro example models exclude some observations. We will use the selector to identify the observations we would like to keep. There are two selector criteria, and only cases that evaluate True for both are selected.

m.dataservice.selector = ["PURPOSE in (1,3)", "CHOICE != 0"]

We can estimate the models and check the results match up with those given by Biogeme:

>>> m.load_data()
>>> m.maximize_loglike(method='slsqp')
┣ ...Optimization terminated successfully...
>>> m.loglike()
-5214.063...
>>> m.calculate_parameter_covariance()

>>> print(m.pfo()[['value','std err','t stat','robust std err','robust t stat']])  # parameter frame, ordered
                    value    std err  t stat  robust std err  robust t stat
Category Parameter
ASCs     ASC_CAR   -0.238  3.606e-02  -6.606       4.883e-02         -4.878
         ASC_TRAIN  0.092  4.513e-02   2.047       6.503e-02          1.421
LOS      B_COST    -0.008  4.247e-04 -19.335       5.493e-04        -14.949
         B_TIME    -0.008  5.376e-04 -14.497       9.715e-04         -8.022
Other    existing   0.399  2.711e-02 -22.186       3.937e-02        -15.277
         public     0.246  3.035e-02 -24.853       2.068e-02        -36.463

Tip

If you want access to the model in this example without worrying about assembling all the code blocks together on your own, you can load a read-to-estimate copy like this:

m = larch.example(111)