3.A. Derivation of Logit Models from the GEV Model

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.

3.A Derivation of Logit Models from the GEV Model 153 In the same section, it was also stated that multinomial logit, hierarchical logit and cross-nested logit models can be derived as GEV models. For the multinomial logit and the hierarchical logit this is possible by specifying the functionG(·)as

G(e^V1, . . . , e^Vm)=e^Yo (3.A.2) whereYois the logsum variable relative to the root node of the choice tree for the model under study. The following sections carry out these derivations.

3.A.1 Derivation of the Multinomial Logit Model

In the case of the multinomial logit model, the choice tree has the root nodeo di- rectly connected to all the elementary alternativesj (see Fig.3.2).

In this case the variableY_ocan be expressed as Y_o=ln

m

i=1

e^Vi/θ

and (3.A.2) becomes:

G(e^V1, . . . , e^Vm)=

m

i=1

e^Vi/θ (3.A.3)

It can easily be verified that this function satisfies the four properties mentioned in Sect.3.3.5, given some restrictions on parameterθ.

In fact:

(1) G≥0 for any valueθandV_i (i=1, . . . , m).

(2)

G(αe^V1, . . . , αe^Vm)=

m

i=1

(αe^Vi)^1/θ =α^1/θ

m

i=1

(e^Vi)^1/θ

=α^1/θG(e^V1, . . . , e^Vm);

that is,G(.)is homogeneous of degree 1/θ, which is positive ifθ >0.

(3)

lim

e^Vi→∞

G(e^V1, . . . , e^Vm)= lim

e^Vi→∞

m

i=1

e^Vi/θ= ∞, fori=1,2, . . . , m.

(4) The first derivative ofG(·)with respect to anye^Vj is equal to Gk=∂G(.)/∂e^Vj =e^{Vj[(1/θ )−1]}

θ

which is nonnegative for anyθ≥0. Furthermore, higher-order mixed deriva- tives are all zero, and therefore both nonnegative and nonpositive. Condition (4) is therefore certainly verified if Condition (2) on the positivity of the coefficient θis verified.

Substituting expression (3.A.3) in (3.A.1), it follows that p[j] = e^Vj

1/θ ·1/θ·e^{Vj(1/θ )−1} m

i=1e^Vi/θ = e^Vj/θ m

i=1e^Vi/θ which is the expression of the multinomial logit model of parameterθ.

To complete the demonstration, the joint probability distribution of the random residuals can be derived. In fact, substituting expression (3.A.3) in the joint probabil- ity distribution function (3.3.51), the product ofmGumbel probability distribution functions with parameterθis obtained:

F (ε₁, . . . , ε_m)=exp

−

m

i=1

e^−εi/θ

=

m

i=1

exp[−e^−εi/θ]

Thus expression (3.A.3) for the functionG(·)implies that the random residuals are identically and independently distributed as Gumbel variables with parameter θand therefore with variances and covariances defined by expressions (3.3.2) and (3.3.3). Note that the inclusion of Euler’s constantΦ in the systematic utilitiesV_i entails no loss of generality because, as stated in Sect.3.3.1, MNL choice probabil- ities are invariant with respect to the addition of a constant to all utilities.

3.A.2 Derivation of the Single-Level Hierarchical Logit Model

In the single-level hierarchical logit model with equal covariances, the choice tree has the root nodeoconnected to intermediate nodeskto which elementary alterna- tivesj are connected (see Fig.3.5). The parametersθassociated with all intermedi- ate nodeskare equal.

With this tree structure, the variableY_obecomes:

Y_o=ln

k

exp θ

θ_o·Y_k

=ln

k

i∈Ik

e^Vi/θ θ/θo

with

Yk=ln

i∈Ik

e^Vi/θ Consequently (3.A.2) becomes:

G(e^V1, . . . , e^Vm)=

k

i∈Ik

e^Vi/θ θ/θo

(3.A.4)

3.A Derivation of Logit Models from the GEV Model 155 In this case, it can again be shown thatG(.) satisfies the four properties men- tioned above, given some restrictions on the parametersθandθ_o.

In fact:

(1) G≥0 for any value ofθ, θ_o, andV_k, fork=1, . . . , m.

(2)

G(αe^V1, . . . , αe^Vm)=

k

i∈Ik

(αe^Vi)^1/θ θ/θ_o

=

k

(α)^1/θ

i∈Ik

(e^Vi)^1/θ θ/θ_o

=

k

(α)^1/θo·

i∈Ik

(e^Vi)^1/θ θ/θo

=(α)^1/θo·

k

i∈Ik

(e^Vi)^1/θ θ/θo

=(α)^1/θo·G(e^V1, . . . , e^Vm);

that is,Gis homogeneous of degree 1/θo, which is positive ifθ_o>0.

(3) lim_eVk→∞G(e^V1, . . . , e^Vm)= ∞, fork=1,2, . . . , m.

(4) The first-order partial derivative ofG(.)with respect to anye^Vh is equal to:

Gh=∂G(.)/∂e^Vh=θ/θo·

i∈Ik

e^Vi/θ

(θ/θo)−1

·1/θ·e^{Vh[(1/θ )−1]} withh∈Ik

which is nonnegative if:

θ_o≥0 (3.A.5)

Inequality (3.A.5) is implied by Condition (2) on the positivity of the homo- geneity coefficient.

Moreover, second-order mixed derivatives are equal to:

∂²G(.)/∂e^Vj∂e^Vh

=

⎧

⎪⎨

⎪⎩

1

θo ·e^{Vj[(1/θ )−1]}·(_θ^θ

o −1)·(

i∈Ike^Vi/θ)^{(θ/θo)−2 1}_θe^{Vh[(1/θ )−1]}

forj, h∈I_k ∀k 0, otherwise

which, given (3.A.5), are nonpositive if:

0≤θ≤θo (3.A.6)

It can be easily shown that if (3.A.6) holds, Condition (4) is always satisfied for higher-order mixed derivatives.

Also in this case, therefore, Conditions (2) and (4) impose restrictions on the two parametersθandθ_o(0< θ≤θ_o)analogous to those described in Sect.3.3.2.

Fig. 3.A.1 Choice tree for a multilevel hierarchical logit model

Choice probabilities can be obtained by substituting function (3.A.4) in (3.A.1):

p[j] =e^Vj

1 θ_o

·

θ θo ·(

i∈Ihe^Vi/θ)^θoθ−1·_θ¹·e^{Vj[(1/θ )−1]}

k(

i∈Ike^Vi/θ)^θoθ

= e^Vj/θ

i∈Ike^Vi/θ · (

i∈Ihe^Vi/θ)^θoθ

k(

i∈Ike^Vi/θ)^θoθ

(3.A.7)

which is the expression of the single-level hierarchical logit model with parameters θoandθ. Introducing the parameterδ=θ/θoand the logsum variableYk:

Yk=ln

i∈Ik

exp(Vi/θ )

(3.A.7) becomes:

p[j] = e^Vj/θ

i∈Ike^Vi/θ · e^δYh

ke^δYk

which is the expression of the single-level hierarchical logit model (see (3.3.19)).

3.A.3 Derivation of the Multilevel Hierarchical Logit Model

The demonstration that the multilevel hierarchical logit (tree-logit) can be derived from function (3.A.2) satisfying the four properties mentioned cannot be easily gen- eralized, because it is difficult to express the choice tree structure in a general form.

To demonstrate the statement that the multilevel hierarchical logit model is a GEV model, reference to an easily generalizable example is made.

Consider the structure of the choice tree in Fig.3.A.1.

There are two intermediate levels and three parameters:θ_o, θ₁, θ₂. Let V_A, V_B, V_C, andV_D be the systematic utilities of the four elementary nodes. According to what was stated in Sect.3.3.3, it follows that

3.A Derivation of Logit Models from the GEV Model 157 δ₁=θ₁/θ_o

δ₂=θ₂/θ₁

Y2=ln(e^VC/θ2+e^VD/θ2) (3.A.8)

Y₁=ln(e^VB/θ1+e^δ2Y2)=ln

e^VB/θ1+(e^VC/θ2+e^VD/θ2)^θ2/θ1 Yo=ln(e^VA/θo+e^δ1Y1)=ln

e^VA/θo+

e^VB/θ1+(e^VC/θ2+e^VD/θ2)^θ2/θ1θ1/θo p[A] =e^VA

e^Yo p[B] =e^VB/θ1

e^Yo ·e^(δ1−1)Y1 (3.A.9)

p[C] =e^VC/θ2

e^Yo ·e^(δ1−1)Y1·e^(δ2−1)Y2

Substituting in (3.A.2) the expression forY_ogiven by (3.A.8) yields:

G(e^VA, . . . , e^VD)=e^VA/θo+

e^VB/θ1+(e^VC/θ2+e^VD/θ2)^θ2/θ1θ₁/θo

(3.A.10) It can be verified that, given some restrictions on the parametersθ, this function satisfies the four properties required ofG(·).

In fact:

(1) G≥0 for any value ofθj,(j=o,1,2),Vi (i=A, B, C, D).

(2)

G(αe^VA, . . . , αe^VD)=(αe^VA)^1/θo+

(αe^VB)^1/θ1 +

(αe^VC)^1/θ2+(αe^VD)^1/θ2θ₂/θ₁θ₁/θ_o

=(α)^1/θo·(e^VA)^1/θo+

(α)^1/θ1·(e^VB)^1/θ1 +

(α)^1/θ2·(e^VC)^1/θ2+(α)^1/θ2·(e^VD)^1/θ2θ₂/θ₁θ₁/θ_o

=(α)^1/θo·(e^VA)^1/θo+

(α)^1/θ1·(e^VB)^1/θ1 +(α)^1/θ1·

(e^VC)^1/θ2+(e^VD)^1/θ2θ2/θ1

}^θ1/θo

=(α)^1/θo·(e^VA)^1/θo+(α)^1/θo·

(e^VB)^1/θ1 +

(e^VC)^1/θ2+(e^VD)^1/θ2θ2/θ1θ1/θo

=(α)^1/θo·G(e^VA, . . . , e^VD);

that is,G(.)is homogeneous of degree 1/θo, which is positive ifθo>0.

(3) lim_eVi→∞G(e^VA, . . . , e^VD)= ∞, fori=A, B, C, D;α.

(4) First-order partial derivatives can be expressed as

∂G/∂e^VA=1/θo·e^{VA(1/θo−1)}

∂G/∂e^VB=θ₁/θ_o·(e^VB/θ1+e^δ2Y2)^δ1−1·1/θ1·e^{VB(1/θ1−1)}

∂G/∂e^VC=θ1/θo·(e^VB/θ1+e^δ2Y2)^δ1−1·θ2/θ1·(e^VC/θ2+e^VD/θ2)^δ2−1

·1/θ2·e^{VC(1/θ2−1)}

Note that in this case there is no structural symmetry, and the different deriva- tives differ from each other. First-order derivatives are nonnegative if:

θ_o≥0 (3.A.11)

Other restrictions on the parametersθ can be deduced from the second-order mixed derivatives. In particular, it is sufficient to use only the following two mixed derivatives.

∂²G/∂e^VB∂e^VC = 1 θ_o ·e^VB(

1 θ1−1)

·θ₁−θ_o

θ_o ·(e^VB/θ1+e^δ2Y2)^δ1−2

· 1

θ₁·(e^VC/θ2+e^VD/θ2)^δ2−1·e^VC(

1 θ2−1)

∂²G/∂e^VC∂e^VD = 1 θ_o ·e^VC(

1 θ2−1)

·θ₁−θ_o

θ_o ·(e^VB/θ1+e^δ2Y2)^δ1−2

·θ₂ θ₁·

(e^VC/θ2+e^VD/θ2)^δ2−12

· 1 θ₂·e^VD(

1 θ2−1)

+ 1 θ_o·e^VC(

1 θ2−1)

·θ₂−θ₁

θ₁ ·(e^VC/θ2+e^VD/θ2)^δ2−2

· 1 θ₂·e^VD(

1 θ2−1)

·(e^VB/θ1+e^δ2Y2)^δ1−1 (3.A.12) Invoking inequality (3.A.11), it can be seen that the first one is nonpositive if:

0≤θ₁≤θ_o (3.A.13)

Invoking (3.A.13) in the second one, it follows that the first term is always nonpositive and the second term is nonpositive if:

0≤θ₂≤θ₁ (3.A.14)

Combining expressions (3.A.13) and (3.A.14), it follows that

0≤θ2≤θ1≤θo (3.A.15)

It can be shown that if inequality (3.A.15) holds, Condition (4) is always verified for the other second-order mixed derivatives not included in (3.A.12), as well as for higher-order mixed derivatives.

3.A Derivation of Logit Models from the GEV Model 159 Choice probabilities for the multilevel hierarchical logit model described here can be obtained by substituting expression (3.A.10) in (3.A.1), yielding:

p[A] = e^VA 1/θo

·1/θo·e^{VA(1/θo−1)}

e^Yo =e^VA/θo e^Yo p[B] = e^VB

1/θo

·θ₁/θ_o·(e^VB/θ1+e^δ2Y2)^δ1−1·1/θ1·e^{VB(1/θ1−1)}

e^Yo =e^VB/θ1

e^Yo ·e^(δ1−1)Y1 p[C] = e^VC

1/θo

·^θ1/θo·(e

VB/θ1+e^δ2Y2)^δ1−1·θ2/θ1·(e^VC/θ2+e^VD/θ2)^δ2−1·1/θ2·e^{VC(1/θ2−1)} e^Yo

=e^VC/θ2

e^Yo ·e^(δ1−1)Y1·e^(δ2−1)Y2 equal to the expressions (3.A.9)

The conditions on parametersθ obtained for the three models described so far are both necessary and sufficient; if they are not satisfied the functionG(·)does not have the properties (1) through (4) and the models are not compatible with random utility theory.

3.A.4 Derivation of the Cross-nested Logit Model

The cross-nested logit model has a choice graph shown in Fig. 3.11and can be obtained as a GEV model by specifying the functionG(.)as

G(.)=

k

i∈Ik

α^1/δ_ik ^ke^Vi/θk δk

(3.A.16) withδ_k=θ_k/θ_o and the membership parametersα_ik in the interval[0,1]. In this case as well, it can be verified thatG(.) satisfies the four properties, given some restrictions on parametersθ_k.

In fact:

(1) G≥0 for any value ofθ_k, V_i (i=1, . . . , m), aim[0,1].

(2)

G(βe^V1, . . . , βe^Vm)=

k

i∈I_k

α_ik^1/δk(βe^Vi)^1/θk δ_k

=

k

β^1/θk

i∈Ik

α_ik^1/δk(e^Vi)^1/θk δk

=β^1/θo

k

i∈Ik

α^1/δ_ik ^k(e^Vi)^1/θk δ_k

=β^1/θo·G(e^V1, . . . , e^Vm);

that is,G(.)is homogeneous of degree 1/θo, which is positive ifθ_o≥0.

(3) lim_eVk→∞G(e^V1, . . . , e^Vm)= ∞, fork=1,2, . . . , m.

(4) The first-order partial derivative ofG(·)with respect to anye^Vj is equal to:

G_j=∂G(.)/∂e^Vj=

k

δ_k·

i∈Ik

α_ik^1/δke^Vi/θk δ_k−1

·α_{j k}^1/δk θ_k ·(e^Vj)

1 θk−1

and is nonnegative if

θ_o≥0 (3.A.17)

Inequality (3.A.17) is implied by Condition (2) on the positivity of the homo- geneity coefficient.

Moreover, second-order mixed derivatives are equal to:

∂²G(.)/∂e^Vj∂e^Vh=

k

α^1/δ_hk^k· 1 θk

(e^Vh)

1 θk−1

·(δ_k−1)·

i∈Ik

α_ik^1/δke^Vi/θk δ_k−2

·α_{j k}^1/δk θ_o ·(e^Vj)

1 θk−1

If inequality (3.A.17) is satisfied, all terms of the summation are nonpositive if:

0≤θ_k≤θ_o ∀k (3.A.18)

Thus the condition of nonpositivity is always satisfied (for any value ofVi, aik) if (3.A.18) is true.

It can be easily shown that Condition (4) for higher-order mixed derivatives is always verified if (3.A.18) holds.

Choice probabilities can be obtained by substituting the functionG(.)expressed by (3.A.16) in (3.A.1):

p[j] = e^Vj 1/θo

·

k[^α

1/δk j k

θo ·(

i∈Ikα_ik^1/δke^Vi/θk)^δk−1·(e^Vj)

1 θk−1

]

k(

i∈Ikα^1/δ_ik ^ke^Vi/θk)^δk

=

k[α^1/δ_{j k}^ke^Vj/θk·(

i∈Ikα^1/δ_ik ^ke^Vi/θk)^δk−1]

k(

i∈Ikα^1/δ_ik ^ke^Vi/θk)^δk (3.A.19) which is the expression for the cross-nested logit model (3.3.49).

3.B Random Variables Relevant for Random Utility Models 161

3.B. Random Variables Relevant for Random Utility Models 3.B.1 The Gumbel Random Variable

The Gumbel random variable is a continuous variable that plays a very important role in building logit-form random utility models. Below we describe the probability functions of this variable and illustrate some of its important properties. To facili- tate the immediate application of the results to random utility models, the Gumbel variable is indicated byU (instead ofX_G)and its expected value byV (instead of E[X_G]).

The probability density function of a Gumbel r.v. U with mean V and scale parameterθis given by:

f_U(u)=1/θ·exp

−(u−V )/θ−Φ exp

−exp

−(u−V )/θ−Φ

(3.B.1) and its distribution function is:

F_U(u)=exp

−exp

−(u−V )/θ−Φ

(3.B.2) whereΦis Euler’s constant, approximately equal to 0.577.

The mean and the variance of the Gumbel variable are:

E[U] =V

Var[U] =σ_U² =π²θ² 6

(3.B.3)

From expressions (3.B.3) it can be deduced that the standard deviation of the Gumbel r.v. is directly proportional to the parameterθ. Figure3.B.1 shows some probability density functions of the zero mean Gumbel r.v. for different values of parameterθ.

It can easily be demonstrated, by substitution in expression (3.B.2), that if U is a Gumbel variable with parameters(V , θ ), any r.v. obtained from it by a linear transformation

Y =aU+b is also a Gumbel r.v. with mean

E[Y] =aV +b

and the same parameterθ(same variance). From this result, it follows immediately that the residual of a random utility modelε=U−V (a=1, b= −V )is a Gumbel r.v. with zero mean and parameterθ.

The Gumbel r.v. has the important property of stability with respect to maximiza- tion. In other words, ifUj, j=1, . . . , N, are independent Gumbel r.v. with different meansV_j but the same parameterθ, the maximum of these variables:

U_M= max

j=1,...,N[U_j] (3.B.4)

Fig. 3.B.1 Probability density functions of a Gumbel r.v.

is also a Gumbel r.v. with parameterθ.

In fact, the probability distribution function ofUM can be obtained as F_UM(u)=Pr(U_M< u)=Pr

j=1,...,Nmax {U_j} ≤u and from the independence of theU_j, it follows that:

Pr

j=1,...,Nmax {U_j} ≤u

=

j=1,...,N

Pr[U_j< u] =

j=1,...,N

F_Uj(u)

Substituting expression (3.B.2) for the Gumbel probability distribution function into the previous expression, it follows that

F_UM(u)=

j=1,...,N

exp

−exp

−(u−V_j)/θ−Φ

which yields:

F_UM(u)=exp

−exp(−Φ)·exp(−u/θ )·

j

exp(Vj/θ )

(3.B.5)

If the EMPU variable described in Chap.3is denoted byV_M then:

V_M=θln

j

exp(Vj/θ ) (3.B.6)

3.B Random Variables Relevant for Random Utility Models 163 and, when this is substituted in expression (3.B.5), the result is

F_UM(u)=exp

−exp

−(u−V_M)/θ−Φ

which is still the probability distribution function of a Gumbel random variable with meanV_Mand parameterθ, as can be immediately seen by comparison with (3.B.2).

The multinomial logit model can be obtained by using the definition of a random utility model (3.2.1) and the property of stability with respect to maximization of the Gumbel r.v. described above.

In fact, from (3.2.1) it follows that

p[j] =Pr(U_j> U_M^′) with

U_M^′=max

k=j{Uk}

This probability can therefore be expressed as the product of the probability that the perceived utilityUj has a value within an infinitesimal neighborhood ofx and the probability thatU_M^′ has a value less thanx. The resulting probability element must obviously be integrated with respect to all possible values ofx:

p[j] =Pr(U_j> U_M^′)= +∞

−∞

F_U

M′(x)·f_Uj(x) dx (3.B.7) whereF_U

M′ andf_Uj are the probability distribution function and the probability density function of the random variablesU_M^′ andU_j, respectively. If the U_k are i.i.d. Gumbel variables with parameterθand meanV_k, thenU_M^′, as shown above, is also a Gumbel variable with the same parameterθand mean equal to:

V_M^′=θln

k=j

exp(Vk/θ ) (3.B.8)

Expression (3.B.7) then becomes:

p[j] = +∞

−∞

exp

−exp

−(x−V_M^′)/θ−Φ

·exp

−exp

−(x−V_j)/θ−Φ

×exp

−(x−Vj)/θ−Φ

·(1/θ ) dx

= +∞

−∞

exp

−exp

−(x−V_j)/θ−Φ

−exp

−(x−V_M^′)/θ−Φ

×exp

−(x−Vj)/θ−Φ

·(1/θ ) dx

=exp(Vj/θ−Φ)· +∞

−∞

exp

−exp(−x/θ )·

exp(Vj/θ−Φ) +exp[VM^′/θ−Φ]

exp(−x/θ )·(1/θ ) dx

=exp(Vj/θ−Φ)

× +∞

−∞

exp

−exp(−x/θ )[exp(V_j/θ−Φ)+exp[V_M′/θ−Φ]

exp(−x/θ )·(1/θ ) dx

= exp(Vj/θ−Φ)

exp(Vj/θ−Φ)+exp(VM^′/θ−Φ)

× exp

−exp(−x/θ )[exp(Vj/θ−Φ)+exp[V_M′/θ−Φ]

+∞

−∞

= exp(Vj/θ ) exp(Vj/θ )+exp(VM^′/θ )

and, substituting expression (3.B.8) forV_M^′, it follows that p[j] = exp(Vj/θ )

exp(Vj/θ )+

k=jexp(Vk/θ )= exp(Vj/θ )

kexp(Vk/θ ) which is the multinomial logit model described in Sect.3.3.1.

3.B.2 The Multivariate Normal Random Variable

The multivariate normal r.v.,XMVN, is the generalization of the normal r.v. to n dimensions. Its probability density function is given by

f_XMVN(x)=

(2π )ⁿdet(ΣX)−1/2

exp

−1/2(x−µ_X)^TΣ⁻¹_X (x−µ_X)

(3.B.9) where det(Σ)denotes the determinant of the matrixΣ.

The parameters of a multivariate normal r.v. are the vectorµ_Xof the means, with componentsμX_i, and the positive semidefinite variance–covariance (or dispersion) matrixΣX. In other words:

E[X_MVN] =µ_X, ΣX_MVN=ΣX

The equiprobability surfaces of the multivariate normal variable, or the loci of points in then-dimensional Euclidean space for which the density function is con- stant, have the equation:

(x−µ_X)^TΣ⁻¹_X (x−µ_X)=C² (3.B.10) whereCis a constant. Expression (3.B.10) is the equation of an ellipsoid withµ_X as its center (see Fig.3.B.2).

Recall that the sum of two univariate normal random variables is again a nor- mal random variable, a property known as invariance with respect to summation.

Specifically, ifX is distributed asN (μX, σ_X²)andY is distributed asN (μY, σ_Y²),

3.B Random Variables Relevant for Random Utility Models 165

Fig. 3.B.2 Equiprobable surfaces of the multivariate normal r.v.

thenX+Y is distributed asN (μX+μY, σ_X²+σ_Y²+2 cov(X, Y )); similarly,X−Y is distributed asN (μ_X−μ_Y, σ_X²+σ_Y²−2 cov(X, Y )).

The multivariate normal r.v. has the property of invariance with respect to linear transformations, which can be considered an extension of the property of invariance with respect to summation of the univariate normal r.v. In other words, ifXis a random vector with probability multivariate normal density function (3.B.9) andA is a matrix of dimensions(m×n), the vectorY=AXis also multivariate normal with mean vector and dispersion matrix given by

E[Y] =AE[X] =AµX, ΣY=E

A(X−µX)(X−µX)^TA^T

=AΣXA^T

Furthermore, from (3.B.9) it can be easily deduced that if thencomponents of XMVN are noncorrelated (i.e., the matrixΣ is diagonal), then they are also inde- pendent; that is, the probability density function (3.B.9) is the product ofndensity functions of univariate normal random variables with meansμ_Xiand variancesσ_X²

i. It is worth recalling that two independent random variables are noncorrelated in any case.

Reference Notes

Random utility theory has stimulated, both in theory and in applications, the un- derstanding and modeling of the mechanisms underlying travel demand. One of the first systematic accounts of its foundation can be found in the book by Domencich and McFadden (1975). The book formalizes the theoretical work carried out in the early 1970s on random utility models and on multinomial logit models in particular.

Theoretical analyses of random utility models can be found in Williams (1977), Manski (1977), and the book by Manski and McFadden (1981). The book by Ben- Akiva and Lerman (1985) gives a very comprehensive account of random utility theory, of logit family models, and of many applied issues dealt with in this chapter and in Chap. 8. A recent contribution covering advanced topics in random utility theory is represented by Train (2003).

Williams and Ortùzar (1982) analyze the limitations of random utility (or “com- pensatory”) models and compare them with other behavioral discrete choice models.

The paper also contains a comprehensive, albeit dated, bibliography on noncompen- satory models. Detailed analysis of the state of the art in the mid-1980s on the use of random utility models in modeling travel demand can be found in the note by Horowitz (1985). More recent systematic reviews of random utility models can be found in Bath (1997) and in Ben-Akiva and Bierlaire (1999).

As for specific random utility models, references to the single-level hierarchical logit model can be found in Williams (1977) and Daly and Zachary (1978), and Da- ganzo and Kusnic (1993) discuss the multilevel hierarchical logit model in its most general form. The cross-nested logit model is implicitly encompassed in McFad- den (1978); the first explicit formulation called “ordered GEV” can be traced back to Small (1987). Vovsha (1997), Vovsha and Bekhor (1998), Wen and Koppelman (2001), Papola (2004), and Abbe et al. (2007) provide further theoretical formula- tions and developments. The paired combinatorial logit model was first proposed by Chu (1989), and was subsequently elaborated by Koppelman and Wen (2000). The formulation reported in Sect.3.3.4is from Papola (2004).

Theoretical analysis of the covariances underlying the cross-nested logit model is provided by Marzano and Papola (2008). The GEV model was proposed by McFad- den (1978) and subsequently generalized by Ben-Akiva and Francois (1983). The demonstration that GEV models are random utility models and the derivation of hi- erarchical logit models as GEV models is from Papola (1996) and the derivation of the cross-nested logit model as a GEV model is from Papola (2004).

Detailed analysis of the probit model is contained in the book by Daganzo (1979);

for the calculation of probit choice probabilities reference can be made to Horowitz et al. (1982) and Langdon (1984). Reference to the factor analytic probit can be found in Ben-Akiva and Bierlaire (1999) and reference to the random coefficients (tastes) approach can be found in Ben-Akiva and Lerman (1985) and in Ortuzar and Willumsen (2001). The GHK method derives the name from its authors: Geweke (1991), Hajivassiliou and McFadden (1998), and Keane (1994); a different formu- lation can be found in Bolduc (1999).

The mixed logit model is also a rather recent development of random utility mod- els. One of the first papers dealing with its theoretical and computational aspects was

Reference Notes 167 by Ben-Akiva and Bolduc (1996). Other references to this model may be found in Bolduc et al. (1996) and in Ben-Akiva and Bierlaire (1999); more recent develop- ments and detailed analysis of model properties and applications can be found in Train (2003) and in the doctoral dissertation by Walker (2001).

The general approach to modeling choice set alternatives is contained in Manski (1977). A state-of-the-art review of explicit models of choice set generation and a number of specifications may be found in Ben-Akiva and Boccara (1995). The implicit availability perception approach is described in Cascetta and Papola (2001).

The expected maximum perceived utility function and its mathematical proper- ties are dealt with in Daganzo’s volume (1979). Reference can also be made to the work of Cantarella (1997), which draws on and generalizes Daganzo’s results.

The definition of elasticity associated with random utility models and the expres- sions for the multinomial logit model are given in various texts; particular reference can be made to Domencich and McFadden (1975) and to Ben-Akiva and Lerman (1985). The results on elasticities of the single-level hierarchical logit model are from Koppelman (1989).