As before in the traditional modelling framework, we have the link counts x and the route flows,y, which, although ultimately of interest, are usually not directly observed. However, we now imagine that a fraction of vehicles deliver routing information directly,
ygps = (ygps,1, . . . , ygps,N)T, where N is the number of routes in the network. We
also assume that each vehicle on route r has probability pr of being tracked, with a
corresponding vector of probabilitiesp= (p1, . . . , pN)T.
Conditional on the overall number of route flows, the tracked route flows are bino- mially distributed withygps|y∼Binom(y,p).
Due to the fact that we consider the total route flows, including both tracked and non-tracked vehicles, as Poisson distributed with mean rate vectorλ, we consider the unconditional distribution of the tracked route flows as a thinned Poisson process. That is, each time we observe a vehicle on route r it can either be classified as being tracked with probabilitypr or categorised as non-tracked with probability 1−pr. We
define a diagonal matrixP =diag(p) with the tracking probabilities as elements and a diagonal matrix with the probabilities of not being tracked asQ=I−P withI being the appropriately dimensioned identity matrix. The matrices P and Q are diagonal matrices because the routes are independent of each other. Under these settings, the tracked vehicle flows are randomlyPois(P λ) distributed and the non-tracked vehicles follow aPois(Qλ) distribution in accordance to standard Poisson thinning arguments. If the route flows of all vehicles are large enough we can approximate their distri- bution by the normal distribution as follows:
y∼Norm(λ,Σ), (4.1)
in which case the unconditional distribution of the tracked route flows is
ygps∼Norm(P λ,PΛ). (4.2)
According to Equation4.1we can apply a normal approximation to the link counts as well, that is
x∼Norm(Aλ,AΣAT), (4.3)
The rationale for assuming we are dealing with high demand, where the normal approximation holds, is that otherwise in the low demand case, where Poisson and sim- ilar models are necessary, there will be too small a chance of observing GPS equipped vehicles for the kind of analysis we examine here to be possible. In addition, using a normal approximation offers greater computational efficiency, which improves the abil- ity to compute estimates for all kinds of networks, including large and more complicated ones.
We can develop a likelihood individually for each of the two sources of data, the first representing the link count information, while the second relates how likely certain average flows are given the routing information.
These two likelihoods can then be multiplied with each other, where we can consider the information from the tracked vehicles as an update of the link count data likeli- hood. This factorisation of the likelihood is only valid, however, if the two elements are statistically independent of each other.
This may not be the case if in the data collection process the link counts include both the tracked and the non-tracked vehicles. Since we assume here that they are observed contemporaneously, we cannot consider the link counts xand monitored route counts
ygpsto be independent.
We recommend separating the two sources of information by decomposing the link counts intox=xgps+xnot where xgps=Aygpsis the contribution to the link counts
from tracked vehicles, and xnot is the contribution from those vehicles that are not
tracked. The vectorsygpsandxnot are independent under standard assumptions.
We know the distribution of the tracked route flows from Equation 4.2, while the corresponding result forxnot is
xnot∼Norm(AQλ,AQΛAT). (4.4)
Thus, the updated likelihood has the form
L(λ,p) =f(xnot|λ,p)·f(ygps|λ,p) (4.5)
where f generically denotes probability mass functions. Equation 4.5 emphasises the fact that the vectorpof tracking probabilities is a nuisance parameter that will generally need estimating in tandem with λ. We assume for now that the probability of being tracked is known and will return to its inference at a later point.
Both the normal models defined in Equations 4.2and 4.4 have a functional rela- tionship between the mean and the variance, that is, the variance is a function of the mean. While the dependence of the covariance matrices onλ can provide important information (Hazelton, 2003) it also leads to a more complex log-likelihood. A sim- plification of the likelihood is to assume that the covariance matrix of y is fixed; i.e.
Var(y) = Σ =diag(σ) not dependent onλ.
We then get the corresponding simplified models defined by
ygps∼Norm(P λ,PΣ) (4.6)
and
xnot∼Norm(AQλ,AQΣAT). (4.7)
We examine the likelihood theory for both the original4.2/4.4and simplified4.6/4.7 models in the next two sections. One reason for looking at both scenarios is that the computational needs for obtaining estimates for the simple model are considerably less than in the case of the original model. Thus, it may be more practical, especially with very large networks, to use the simplified version.
A particular line of interest in our likelihood analysis is obtaining large sample approximations to the properties of estimators of the mean route flows. The asymptotic process that we have in mind is an increase in the length of the observation period,, so that the elements ofλ, and hence the likely vehicle counts, become larger.
In other words, we setλ≡λ() =λ0 whereλ0is the vector of mean route flows per unit time (e.g. an hour), and then consider the asymptotic process → ∞. We assume throughout thatis known so that we can switch between estimation ofλand
λ0 by application of this scaling factor.
The distinction between these parameters is important when we discuss the asymp- totic likelihood theory, since then we will require a fixed (non-asymptotic) parameter
λ0 rather than the ‘moving target’ λ. For this reason Λ0 is defined to be diag(λ0), as well as Σ =Σ0for the simplified model, so that the scale of these dispersion matrices match that of the data.