A Hierarchical Gravity Model with Spatial Correlation: Mathematical Formulation and Parameter Estimation

Abstract

This study presents a hierarchical trip distribution gravity model that can accommodate various spatial correlation structures. It is formulated on the basis of the solution to an equivalent optimization problem, and its parameters are estimated using a sequential maximum likelihood procedure. We conclude that accounting for spatial correlation through a hierarchical structure incorporated into gravity-type trip distribution models significantly increases their explanatory and predictive powers and improves the results they generate for use in transportation system planning processes.

Appendices

Appendix A: Alternative gravity models with spatial correlation

The hierarchical gravity model that first determines the trip generation for each zone and then the trip distribution (see Fig. 8) is equivalent to a model in which the trips generated within a single zone are correlated.

Fig. 8

Hierarchical structure of trip distribution with correlation of trips generated in a single zone

The equivalent optimization problem from which this model is derived is as follows:

$$\mathop {\min }\limits_{\left\{ {T_i ,T_{ij} } \right\}} {\text{ }}Z = \sum\limits_i {\sum\limits_j {T_{ij} C_{ij} } } + \frac{1}{\beta }\sum\limits_i {T_i \left( {\ln T_i - 1} \right)} + \frac{1}{\lambda }\left[ {\sum\limits_i {\sum\limits_j {T_{ij} \left( {\ln T_{ij} - 1} \right)} } - \sum\limits_i {T_i \left( {\ln T_i - 1} \right)} } \right]$$

(45)

s.t.:

$$\sum\limits_j {T_{ij} } = T_i \quad \forall i\quad \left( {\mu _i } \right)$$

(46)

$$\sum\limits_i {T_i } = T\quad \quad \left( \rho \right)$$

(47)

where T _ij is the number of modeled trips between the origin-destination pair (i, j), C _ij is the cost of traveling between an O-D pair, Ti is the number of modeled trips generated by zone i, and T is the total trips in the system (note that T is known and exogenous, but T _i is endogenous).

The corresponding Lagrangian function of problem (45)–(47) is

$$L = Z + \sum\limits_i {\mu _i \left( {\sum\limits_j {T_{ij} } - T_i } \right)} + \rho \left( {\sum\limits_i {T_i } - T} \right)$$

(48)

The optimality conditions for (48) are

$$\frac{{\partial L}}{{\partial T_{ij} }} = C_{ij} + \frac{1}{\lambda }\ln T_{ij} + \mu _i = 0 \to T_{ij} = e^{ - \lambda \left( {C_{ij} + \mu _i } \right)} $$

(49)

$$\frac{{\partial L}}{{\partial T_i }} = \frac{1}{\beta }\ln T_i - \frac{1}{\lambda }\ln T_i - \mu _i + \rho = 0$$

(50)

Given that $\sum\limits_j {T_{ij} = T_i }$, by (49) we obtain the following:

$$\sum\limits_j {T_{ij} } = \sum\limits_j {e^{ - \lambda \left( {C_{ij} + \mu _i } \right)} } = e^{ - \lambda \mu _i } \sum\limits_j {e^{ - \lambda C_{ij} } = T_i } $$

(51)

Dividing T _ij in (49) by T _i in (51) we then have

$$\frac{{T_{ij} }}{{T_i }} = \frac{{e^{ - \lambda C_{ij} } }}{{\sum\limits_j {e^{ - \lambda C_{ij} } } }} = P_{ij/i} $$

(52)

Equation (52) gives the proportion traveling between pair (i, j) generated by zone i.

Taking the natural logarithm of (51), we obtain

$$\ln T_i = - \lambda \mu _i + \ln \sum\limits_j {e^{ - \lambda C_{ij} } } \to \frac{1}{\lambda }\ln T_i + \mu _i = \frac{1}{\lambda }\ln \sum\limits_j {e^{ - \lambda C_{ij} } } $$

(53)

Substituting (53) into (50), we get

$$\frac{1}{\beta }\ln T_i - \frac{1}{\lambda }\ln \sum\limits_j {e^{ - \lambda C_{ij} } } + \rho = 0$$

(54)

$$T_i = e^{ - \rho + \frac{\beta }{\lambda }\ln \sum\limits_j {e^{ - \lambda C_{ij} } } } $$

(55)

Dividing (55) by $\sum\limits_i {T_i } = T$ gives

$$\frac{{T_i }}{T} = \frac{{e^{ - \beta L_i } }}{{\sum\limits_i {e^{ - \beta L_i } } }} = P_i $$

(56)

where $L_i = - \frac{1}{\lambda }\ln \left( {\sum\limits_j {e^{ - \lambda C_{ij} } } } \right)$. Finally, by (52) and (56) we can define

$$\frac{{T_{ij} }}{T} = \underbrace {\frac{{e^{ - \beta L_i } }}{{\sum\limits_i {e^{ - \beta L_i } } }}}_{P_i }\underbrace {\frac{{e^{ - \lambda C_{ij} } }}{{\sum\limits_j {e^{ - \lambda C_{ij} } } }}}_{P_{ij/i} } = P_{ij} $$

(57)

If we now consider a model in which all trips with the same destination zone are correlated, the equivalent optimization problem is the following:

$$\mathop {\min }\limits_{\left\{ {T_j ,T_{ij} } \right\}} {\text{ }}Z = \sum\limits_i {\sum\limits_j {T_{ij} C_{ij} } } + \frac{1}{\beta }\sum\limits_j {T_j \left( {\ln T_j - 1} \right)} + \frac{1}{\lambda }\left[ {\sum\limits_i {\sum\limits_j {T_{ij} \left( {\ln T_{ij} - 1} \right)} } - \sum\limits_j {T_j \left( {\ln T_j - 1} \right)} } \right]$$

(58)

s.t.:

$$\sum\limits_i {T_{ij} } = T_j \quad \forall j\quad \left( {\gamma _j } \right)$$

(59)

$$\sum\limits_j {T_j } = T\quad \quad \left( \rho \right)$$

(60)

where T _j is the number of modeled trips attracted by zone j. The optimality conditions for (58)–(60) are

$$\frac{{T_{ij} }}{{T_j }} = \frac{{e^{ - \lambda C_{ij} } }}{{\sum\limits_i {e^{ - \lambda C_{ij} } } }} = P_{ij/j} $$

(61)

$$\frac{{T_j }}{T} = \frac{{e^{ - \beta L_j } }}{{\sum\limits_j {e^{ - \beta L_j } } }} = P_j $$

(62)

$$\frac{{T_{ij} }}{T} = \underbrace {\frac{{e^{ - \beta L_j } }}{{\sum\limits_j {e^{ - \beta L_j } } }}}_{P_j }\underbrace {\frac{{e^{ - \lambda C_{ij} } }}{{\sum\limits_i {e^{ - \lambda C_{ij} } } }}}_{P_{ij/j} } = P_{ij} $$

(63)

where $L_j = - \frac{1}{\lambda }\ln \left( {\sum\limits_i {e^{ - \lambda C_{ij} } } } \right)$.

Appendix B: Sensitivity analysis

In this sensitivity analysis, the 37 trip zones, defined to coincide with Santiago’s administrative districts, were grouped into 6 areas as shown in Fig. 9. In this sensitivity case, the hierarchical model (called HGM-S) involves two decisions: the first is the choice of area in which a trip is to be made, and the second is the choice of a specific origin-destination pair whose districts lie within the chosen area. It follows that all (i, j) pairs whose origin and destination districts are in the same area are intercorrelated and define Group 1, while the pairs with origin and destination districts located in different areas are intercorrelated and constitute Group 2. In this example, k = 1, 2 (two ρ _k parameters).

Fig. 9

New spatial aggregation of Santiago trip zones

In Table 7 we see the HGM-S results compared with the SGM and the HGM (see Section 3.2).

Table 7 Results of the estimation of HGM-S, HGM and SGM models

We can test the hypothesis that the two models (HGM-S and SGM) are not equivalent using the likelihood ratio test:

$$LR = - 2\left( { - 2,410.13 + 2,407.15} \right) = 5.96 >\chi _{\left[ {0.95;1} \right]}^2 = 3,84$$

(64)

Using Welch’s t test, it was also found that the difference between the values of λ for HGM-S (−0.35) and SGM (−0.32) was significantly different from zero (at 95% confidence):

$$\left. \begin{aligned}& H_0 :\lambda _{HGM - S} - \lambda _{SGM} = 0 \\& H_1 :\lambda _{HGM - S} - \lambda _{SGM} \ne 0 \\ \end{aligned} \right\} \to \left| {\frac{{0.35 - 0.32}}{{0.007}}} \right| = 4.28 >1.96$$

(65)

We therefore again conclude that λ _SGM is biased because of the absence of spatial correlation in SGM.