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Sometimes, a discrete choice is made from a very large pool of possible choices. In these circumstances, it may be useful to aggregate choices together, and represent a set of choices as a single meta-choice. This is particularly common in destination choice models, where the individual possible destinations are aggregated together as traffic analysis zones.

The aggregate choice in many ways represents a nested logit model, with the aggregations corresponding to the nests, except we only observe the choice at the nest level, not at the elemental alternative level.

Basic Aggregate Models

To start with, we can make some assumptions:

  1. The individual elemental alternatives within each zone or aggregate are homogeneous. That is, each such alternative has the same systematic utility, Vi=βXi
  2. The particular locations of the zonal or aggregation boundaries are arbitrary, and have no systematic meaning themselves.
  3. The number of individual elemental alternatives within each zone or aggregate is directly observable.

Using these assumptions, we can derive a reasonably simple aggregate/zonal choice model.

The usual form of the nested logit model calculates the probability of an alternative as Pnest Palt|nest.

In the case of aggregate choices, we do not observe the choice, but only the nest, so we only care about Pnest. The nested formula for that term is

Agg choice eq1.png

Using assumption 2, we know that μnest must be 1, as we want the aggregation nesting structure to collapse to a multinomial logit model. Further, our first assumption is that all the Vi are equal, so the terms inside the summation can collapse together, leaving

Agg choice eq2.png

with Nnest as the number of discrete elemental alternatives inside the nest.

Under the assumptions we laid out above, estimating an aggregate model is actually quite simple. We can simply define a variable for each aggregate alternative that has a value of log(Nnest), and including it in a MNL model, with a beta coefficient constrained to be equal to 1.

One thing to be careful of in these models: the log likelihood at “zeros” model should include the parameter on log(Nnest) equal to 1, not 0. This is because this is not a parameter we are estimating in the model, it is a direct function of the structure of aggregation, which we have imposed externally.

In application, however, sometimes we want to relax some of the assumptions we outlined above, which can introduce some complications.

Relax Arbitrary Boundaries Assumption

Relaxing the assumption of arbitrary boundaries puts μnest back into the equation for Vnest:

(equation here)

The logsum parameter thus appears as a coefficient on log(Nnest). This may or may not be a good idea for transportation models. In an intra-urban model, if the boundaries of zones are at the TAZ level, which are small sectors drawn only for modelling purposes, relaxing this assumption probably doesn’t make sense. If the boundaries are aligned with political boundaries (counties, towns) that have differing taxing, administration, or other policies, it might be OK to relax this assumption. In a long distance travel model, if the boundaries are aligned with metropolitan areas, then it is certainly reasonable to relax the arbitrary bounds assumption.

References

Content Charrette: Destination Choice Models

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