Aggregation Methods for Random Utility Models

Random Utility Theory

3.7 Aggregation Methods for Random Utility Models

to that of the corresponding multinomial logit model. On the other hand, the cross- elasticity with respect to an attribute of an alternative belonging toB’s nestG(and so correlated withB) is larger for smaller values of parameterθ, that is, for larger covariance between the two alternatives. If two alternatives are perceived as being very similar (i.e., their respective random residuals are highly correlated), the prob- ability of choosing one of them is very sensitive to variations of the attributes of the other. From (3.6.9) it also follows that ifθ=1 the hierarchical logit model becomes a multinomial logit model and the cross-elasticity is analogous to (3.6.8).

Direct and cross-elasticities of the hierarchical logit model, for different values of parameterθ, are shown in Fig.3.16. Forθ=1, the elasticities reported in Fig.3.15 are obtained.

The general conclusion from the above example is that, given equal attributes and coefficients, the more an alternative is perceived as “similar” to other alternatives, the higher are its direct and cross-elasticities. Thus, for any random utility model, variations in the attributes of an alternative will have the greatest effects on the choice probabilities of alternatives that are perceived as close substitutes to it.

3.7 Aggregation Methods for Random Utility Models 149 The average fractionP_jof the population choosing alternativejcan be estimated as

P_j= 1 NT

N_T

i=1

p[j/Xⁱ] =D_j NT

(3.7.3) For populations large enough to replace the sum with an integral, (3.7.3) can be rewritten as

P_j=

X

p[j/X]g(X) dX (3.7.4)

whereg(X) represents the joint probability density function of the vector of at- tributes over the whole population, a measure of the frequency with which the dif- ferent values ofXoccur in the population. In practice, the distributiong(X)is not known and, to calculate the percentageP_j, aggregation techniques that estimatePˆ_j using information on a limited number of individuals must be used.

In the literature, various aggregation methods have been proposed; these can be seen as approximate techniques for integrating (3.7.4).

The methods most frequently applied are:

(1) Average individual (2) Classification (3) Sample enumeration (4) Classification/enumeration

(1) In the first method, an “average individual” is considered, whose attributesX¯ are the average population values calculated from the densityg(X). The aggregated choice percentage is determined as a function of these attributes:

Pˆ_j=p[j/X]¯ (3.7.5)

This method is acceptable only if the relationship between the vector of attributes and the choice probabilitiesp[j/X]is linear or almost linear. Should the probability function be convex or concave, the method would, respectively, underestimate or overestimate the actual value of the fraction of the population choosing alternative j (see Fig.3.17). It can also be shown that the deviation of linear estimatePˆj from its true value is larger for greater dispersion of the values ofX in the population, that is, for larger variances in the marginal distributions ofg(X).

(2) Theclassificationmethod can be seen as an extension of the average individ- ual method described above. In order to reduce the variance ofg(X), the popula- tion is divided into homogeneous and mutually exclusive classes. Letirepresent a generic class withN_imembers. The average individual technique is then applied to each such class, and the estimated fraction of the population choosing alternativej becomes:

Pˆ_j=

I

i=1

Ni

NT

P[j/X¯ⁱ] (3.7.6)

Fig. 3.17 Bias of average individual estimates of population fractions

whereX¯ⁱ is the vector of attributes for the average individual of theith class.

In applications, classes are defined on the basis of a few criteria that are ex- pected to have the greatest effect on systematic utilities. Variables influencing the distribution of the attributes are often adopted as classification criteria, for exam- ple, professional status or income. The numberNi of individuals belonging to each class should be available from statistical sources. The classification technique gives satisfactory results when the number of classes is limited and the individual classes are relatively homogeneous with respect to the attributes included in the model.

(3) With thesample enumerationmethod, it is assumed that the whole population can be represented by a random sample of individuals (decision-makers) extracted from it. The average fraction of individuals choosing alternativej in the overall population is estimated from the probability thatj is chosen by the individuals be- longing to the random sample. IfN_s is the number of individuals in the sample, then:

Pˆj= 1 N_s

Ns

h=1

p[j/X^h] (3.7.7)

whereX^his the vector of the attributes relative to thehth individual in the sample.

Expression (3.7.7) applies to the estimation of the mean population choice fraction when the individuals are chosen using simple random sampling.²⁶

(4) Sample enumeration and classification methods can be combined; this is equivalent to assuming a stratified random sample of decision-makers. A random sample of individuals is extracted from each of theI strata (the homogeneous and

26Further elements of sample theory are discussed in Chap. 8 on demand estimation and its bibli- ography.

3.7 Aggregation Methods for Random Utility Models 151 mutually exclusive classes) into which the population is divided. IfN_iis the number of individuals belonging to stratumi andN_si is the number of sample individuals extracted from stratumi, the fractionPˆj can be estimated as

Pˆj=

I

i=1

Wi

1 N_si

Nsi

h=1

p[j/X^h] (3.7.8)

where the ratioW_i=N_i/N_T is the weight of stratumiin the population.

The total number of decision-makers choosing each alternative j (the aggre- gate demand for alternativej) can be calculated by multiplying expressions (3.7.6), (3.7.7) and (3.7.8) byN_T. The ratio between the number of individuals in the pop- ulation (or a class) and the number of individuals in the sample,NT/Ns orNi/Nsi, is called the “expansion factor” of individuals from the sample to the population.

A number of extensions to these basic methods have been proposed to overcome difficulties sometimes encountered in their application.

The sample enumeration method allows significant flexibility in the use of ran- dom utility models, because the attributes considered in vectorXmight include vari- ables relating to the individual for which it is difficult, if not impossible, to obtain mean values over the whole population or subpopulations (classes). This flexibility is achieved at the cost of greater computational complexity. However, this draw- back is becoming less important with the steady increase in available computing power. Another problem associated with the sample enumeration method relates to the availability of samples of decision-makers for each classiand each choice con- text (e.g., each traffic zone in the study area). The samples should be large enough to guarantee adequate coverage of the distribution of attributesX. This would require large samples of decision-makers for each zone. Theprototypical samplemethod overcomes this problem by using the same sample ofNsi decision-makers of class i for different traffic zones, but applying different weightsW_i^z to each class i in each zonez (W_i^z=N_i^z/N_T). This method requires knowledge of the number,N_i^z, of individuals of classiin each zone, which can be obtained from statistical sources (present scenario), or from sociodemographic forecasts (future scenarios).

In methods based on sample enumeration, estimation of the average number of individuals choosing alternativej in zonez,D_j^z, requires the expansion factorsg^z_i of each class in each zone:

D^z_j=

I

i=1

g^z_i

Nsi

h=1

p[j/X^h] (3.7.9)

where these expansion factors can be formally expressed as g^z_i = N_i^z

N_si

Sometimes the numberN_i^z of individuals of class iin zone zis unknown, es- pecially when several classes have been defined. In this case, it is not possible to estimate either the weights of the individual classes(W_i^z=N_i^z/NT)and the aver-

age choice percentages by (3.7.8), or the expansion factorsg_i^zand the total number of individuals choosing alternativej, D_j, by (3.7.9). To overcome this problem, the target variable methodcan be adopted. This method is described here in reference to the calculation of expansion factors; once these are known, the weightsW_i^zcan easily be calculated. The expansion factors are calculated so that, when the proto- typical sample is rescaled to its universe, it reproduces the zonal values of selected aggregated variables, known as target variablesT_t^z. Typical target variables are the number of residents by professional status, age, sex, income group, and so on. For- mally, the expansion factorsg^z_i must satisfy the following equations.

i

g_i^z

N_si

h=1

K(t, h)=T_t^z (3.7.10)

whereK(t, h)is the contribution to thetth target variable of thehth component of the prototypical sample belonging to categoryi. For example, if the tth target variable is the number of workers in the zone, individualhof classiwill contribute one if employed, zero otherwise. In general, the number of unknown expansion factors (i.e., of classes in each zone) is larger than the numberN_tof target variables, so the system of equations (3.7.10) does not have a unique solution. In this case, the vectorg^zof expansion factors for the classes in each zone can be obtained by solving a least squares problem that minimizes the weighted distance from a vector of reference expansion factorsgˆ while, at the same time, satisfying as closely as possible the system of equations (3.7.10):

g^z=argmin

g^z≥0

i

g_i^z− ˆg_i2

+α

N_t

t=1

i

g_i^z

N_si

h=1

K(t, h)−T_t^z 2

(3.7.11) Reference expansion factors can be obtained as sample estimates of the fraction of users belonging to each class. The parameterαis the relative weight of the two parts of the objective function in (3.7.11), that is, the relative weight that the analyst associates with the target variables (3.7.10) and to the initial estimates gˆ in the solution of problem (3.7.11).

Note that this least squares problem imposes nonnegativity constraints on the variables (3.7.11). It is similar in structure to the problem of estimating O-D demand flows from traffic count data that is formulated and discussed in Chap. 8, and can be solved by using the projected gradient algorithm described in Appendix A.