# Exploring the Direct and Indirect Use of ICT Measurements in DODME (Dynamic OD Matrix Estimation) +
## 5 Indirectly Accounting for the ICT Measurements
In [19], we proposed an indirect procedure that was based on the most reliable data supplied by commercial GPS providers, which involves the waypoints that usually take the form \((ID_{k},\,ts_{kl},lat_{kl},long_{kl})\), where \(k\) denotes the trip identity; \(l\) is the ordered \(l\)th waypoint of trip \(k\); \(ts\) is the time stamp; and \(lat\) and \(long\) are the geographic latitude and longitude, respectively. The approach studied in [19], in essence, proposed the following:  
* A graph network model that is supplied either by transport planning software or by a GIS like OSM;
* A standard map matching that associates waypoints with the corresponding points in the links of the network, from which, the link travel times can be heuristically estimated;
* When using appropriate k-shortest paths, a suitable route choice set can be calculated;
* When considering the path overlapping factors, the reliable path proportion \(P_{k(i,j,r)}\), which is the proportion of trips traveling from origin \(i\) to destination \(j\) leaving origin \(i\) at time \(r\) and using path \(k\), can be calculated. The assignment matrix can then be calculated using: \[\vec{\pi}_{ijr}^{lt}=\sum_{k\in K_{ijr}}\delta_{k(i,j,r)}^{lt}P_{k(i,j,r)}\, \forall i,j,r,l,t\] (12) where \(\delta_{k(i,j,r)}^{lt}\) is the link-path incidence matrix: \[\delta_{k(i,j,r)}^{lt}= \begin{cases}1&\text{if path $k(i,j,r)$ uses link $l$ at time $t$}\\ 0&\text{otherwise}\end{cases} \]  
In Equation (11), the estimated assignment matrix \(\vec{\pi}_{ijr}^{lt}\) is combined with the estimated origin-to-destination (OD) matrix \(x_{ijr}\) to obtain the estimated link flows \(y_{lt}\). In the bi-level optimization problem, the OD flows are always the variables that must be adjusted to minimize the objective function. However, from experience with analytical approaches [9], we observed in our experiments that OD flows may be substantially modified to fit traffic counts that imply high volumes to certain OD pairs, which is a drawback that can be overcome with a data-driven approach that aims to preserve the OD pattern that is included in the seed OD matrix. The proposed formulation was inspired by gravity models, where it sets bi-dimensional constraints for rows and columns, as in the double-constrained models that are common for updating gravity distribution models [29]. Therefore, the origin-to-destination (OD) flow appearing in Equation (11) can be decoupled into independent scaling factors that are different for origins and destinations, expanding a seed matrix \(x_{ijr}^{0}\) that, if the collected data are suitable, can be the observed OD matrix:  
\[x_{ijr}=\alpha_{i}\beta_{j}x_{ijr}^{0},\,\,\forall(i,j)\in N,\,\,\forall r\in T \tag{13}\]  
If \(\theta_{lt},\,\,l\in\hat{L}\subseteq L,\,\,t\in T\) are the link flows that are measured at the counting stations and \(\vec{\pi}_{ijr}^{lt}\) is the dynamic assignment matrix that is estimated from GPS data, then data-driven DODME models can be formulated [17; 19]. These modeling approaches do not need to resort to a dynamic traffic assignment to estimate the dynamic assignment matrix since it is estimated from GPS data. Reference [19] formulates the problem as the following nonlinear optimization problem whose variables are the scaling factors \(\alpha_{i},\;i\in I\) and \(\beta_{j},\;j\in J\):  
\[\min_{\alpha_{i},\;\beta_{j}}\left[\sum_{l\in I}\sum_{l\in T}\left(\beta_{lt}- \sum_{(i,j)\in N}\sum_{r=1}^{t}\alpha_{i}\beta_{j}\vec{n}_{ijr}^{tl}x_{ijr}^{0 }\right)^{2}+w_{2}\left(\sum_{(i,j)\in N}\sum_{r=1}^{t}\left(x_{ijr}^{H}-\alpha _{i}\beta_{j}x_{ijr}^{0}\right)^{2}\right)\right] \tag{14}\]  
\(s.\) to \(\alpha_{i}\geq LB_{i},\forall\;i\in I\) and \(\beta_{j}\geq LB_{j},\forall\;j\in J\)  
An advantage of this formulation is the dimensionality reduction from \(|I|\cdot|J|\cdot|T|\) to \(|I|+|J|\). Moreover, using the scaling factors as variables aims to preserve the structure of the seed origin-to-destination (OD) matrix, as gravity models do.