# Exploring the Direct and Indirect Use of ICT Measurements in DODME (Dynamic OD Matrix Estimation) +
## 8 Computational Results  
The minimization problem that was used for the experiments is the one shown in Equation (21):  
\[ \begin{array}{c}\min Z(\mathbf{X})=w_{1}F_{1}\left(\mathbf{Y},\hat{\mathbf{Y}}\right)+w_{ 2}F_{2}\left(\mathbf{X},\mathbf{X^{H}}\right)+w_{3}F_{3}\left(\mathbf{tt},\hat{\mathbf{tt}} \right)\\ \left(\mathbf{Y},\ \mathbf{tt}\right)=Assignmt(\mathbf{X})\\ \mathbf{X}\in\Omega\subset\mathbb{R}_{+}^{1\times T}\end{array}  \tag{21}\]  
The three methods that were used in the computational experiments were:  
* An SPSA (stochastic perturbation stochastic approximation) with travel times and the objective function defined in Equation (4);
* A hybrid SPSA with travel times and the objective function defined in Equation (5);
* A data-driven approach with the objective function defined in Equation (14). Following the computational experience in [1], when the SPSA (stochastic perturbation stochastic approximation) is used, a constrained SPSA variant is applied in which  
Figure 4: Hillsboro network that was covered by the 450 subpaths. Figure 3: OD path flows that were intercepted by link \(L_{1}\). the feasible set \(\Omega\) is determined by the bounding constraints, as shown in Equation (22), whose objective is to help to preserve the structural similarity:  
\[\Omega=\left\{(1-\beta)x_{ijr}^{H}\leq x_{ijr}\leq(1+\beta)x_{ijr}^{H}\,\ \forall x_{ijr}\in\mathbf{X}\right\} \tag{22}\]  
The term \(\beta\) of Equation (22) represents an acceptable relative change of the solution relative to the historical OD matrix, which serves as a reference, and is expected to be reliable based on the empirical background provided by the demand analysis practice. In these experiments, 0.2 was used based on common practices in traffic analysis applications. A total of eight experiments were performed in order to see the effects of each of the proposed innovations. Three main factors were combined to generate the set of experiments:  
* The term of the reference historical OD matrix on the objective function (\(w_{2}=0\) or \(w_{2}=1\));
* The term of travel times on the objective function (\(w_{3}=0\) or \(w_{3}=1\));
* The hybridization, or not, of the SPSA (stochastic perturbation stochastic approximation) gradient. All the experiments used the matrix that was proposed for benchmarking purposes in MULTITUDE Cost action [1] as the historical OD matrix, normalizing the variables in the objective function. The weight \(w_{3}\) was set to \(w_{3}=F_{2}\left(\mathbf{X}^{0}\right)/F_{3}\left(\mathbf{X}^{0}\right)\) in order to equalize the importance of traffic counts and travel times. Figure 5 summarizes the computational results in terms of the structural similarity index MSSIM and the total number of trips. On the one hand, the addition of the term comparing the historical origin-to-destination (OD) matrix (\(w_{2}=1\)) and the addition of travel times to the problem allowed for an increase in the total number of trips to get closer to the ground truth \(NT\left(\mathbf{X}^{GT}\right)=9973\), and the estimated OD matrices were more reliable from the structural point of view, namely, in the hybrid approach, which showed a more consistent behavior. It can be appreciated that in the case of the hybrid SPSA without travel times, the effect of adding the historical OD matrix (\(w_{2}=1\)) was negligible because the curves overlapped. This was because the optimization process was not stochastic anymore and the contribution of the \(F_{2}\) term was not significant. On the other hand, it was experimentally shown that the hybrid SPSA (stochastic perturbation stochastic approximation) gradient in Equation (6) outperformed the SPSA gradient. As expected, the analytical part of the gradient of the linearized objective function was a better approximation of the maximum descent direction. This effect was notably appreciated in the cases of \(w_{3}=0\). However, when \(w_{3}\
eq 0\), the effects of the hybrid SPSA vanished because of the SPSA-gradient-based part.