# Exploring the Direct and Indirect Use of ICT Measurements in DODME (Dynamic OD Matrix Estimation) +
## 6 Consideration on the Quality of the DODME Results
\begin{array}{l}l(\mathbf{x},\mathbf{y})=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x} ^{2}+\mu_{y}^{2}+C_{1}}\\ c(\mathbf{x},\mathbf{y})=\frac{2\mu_{x}^{2}+\mu_{y}^{2}+C_{2}}{\mu_{x}^{2}+C_{2}}\\ s(\mathbf{x},\mathbf{y})=\frac{\mu_{y}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}}\end{array} \right. \tag{16}\]  
where \(\mu_{x},\ \mu_{y},\ \sigma_{x},\ \sigma_{y}\), and \(\sigma_{xy}\) are the means, standard deviations, and the covariance of the vectors \(\mathbf{x}\) and \(\mathbf{y}\), respectively. \(C_{1},C_{2}\), and \(C_{3}\) are stability constants to avoid numerical problems; these are typically set to \(C_{1}=C_{2}=2\cdot C_{3}=1\). \(\alpha,\ \beta,\ \gamma\) are weighting coefficients, which are typically set to 1 [32]. During the image comparison, the MSSIM is computed as the mean of the SSIM of all the sub-matrices of dimension N because pixel proximity is crucial in image pattern recognition. A variant of MSSIM is proposed in [15,18] including it in the objective function of the formulation of the mathematical model in Equation (3) after setting specific relationships between the measured and estimated travel times that are measured using ad hoc layouts of Bluetooth antennas, which are not reproducible when travel times are measured by other technologies, such as GPS. Regarding the original problem, ref. [32] obtained the MSSIM by averaging the SSIM using sliding windows, which were submatrices of size \(N_{s}\). When extrapolating the sliding windows to OD matrices, ref. [4] proposed reordering the origin-to-destination (OD) matrix by volume, both by rows and by columns, in order to obtain the MSSIM using the same sliding submatrices because the term S of the SSIM is highly sensitive to the order of the OD pairs. Another open question is the dimension of the submatrices \(N_{s}\), which should be fixed, and it does affect the final measure. Reference [31] also reorders the OD pairs, but in this case, they proposed clustering them in greater regional areas. In the same article, they showed how the dimension of the submatrices affects the MSSIM measure; as such, this new proposal solves the problem of tuning \(N_{s}\), which is fixed automatically by the dimension of these regional areas. Here we proposed a more meaningful variant that is easier to apply in practice and considered the physical meaning of the origin-to-destination (OD) matrices. This variant consisted of calculating the averages of the SSIM according to rows and columns rather than submatrices, that is, by using rectangular sliding rules that correspond to either rows or columns in the OD matrix. One row in an OD matrix for a given period represents the distribution of trips that depart from a single origin zone while, analogously, one column is the distribution of trips arriving at a single destination zone. This, therefore, corresponds to a physical interpretation of mobility patterns in the underlying transport system. Thus, the SSIM will capture the similarity between these described distributions by considering the mean, variance, and structure of the departure and arrival distributions, all of which correspond to the structural property of the trip patterns that are described by the OD. Moreover, this proposal also fixes Ns to the number of origins and destinations in the network. Then, if the MSSIM is averaged over \(N_{s}\) sliding windows, a key question arises regarding whether all windows have the same weight or whether their role in the total demand requires that they have different weights. In the case of OD matrices, it is obvious that not all origins and destinations are equivalent in a transport network. Therefore, a weighted MSSIM, as in [33], prioritizes those origins and destinations with more impact on the network.