# Exploring the Direct and Indirect Use of ICT Measurements in DODME (Dynamic OD Matrix Estimation) +
## 6 Consideration on the Quality of the DODME Results  
The quality of the results of the origin-to-destination (OD) estimation problem is usually established in terms of the convergence of the optimization algorithm that is used to solve the bi-level problem and the fitting of the simulated to the observed link flows. In addition, when a reliable historical OD matrix \(X^{H}\) is available, a measure comparing the estimated OD \(X\) and the historical \(X^{H}\) is added as a quality indicator. The typical measures that are used are based on classical vector matrices by considering both of the matrices \(X=\left[X_{ij}\right],\;X^{H}=\left[X_{ij}^{H}\right]\in\mathcal{M}_{mxx}( \mathbb{R})\) as vectors of \(\mathbb{R}^{mxn}\). However, refs. [4; 9; 31] proved that the measures that are defined using Euclidean (\(RMSE\big{(}X,X^{H}\big{)}\)), Manhattan (\(RMAE\big{(}X,X^{H}\big{)}\)), or other similar vector distances fail to capture the differences and similarities between the two matrices in many aspects, for example, the structure of the matrices. They present numerical examples where a reference OD matrix can be perturbed to generate two structurally different matrices that are indistinguishable in terms of these measures. This is a critical aspect to account for given that the underdetermination of the problem formulation means that many different OD matrices can generate the same observed link flow measurements; therefore, if they are structurally different, which is the most credible? On the other hand, proving that the optimization algorithm numerically converges only proves that it converges to a solution, but how good is that solution? If the complementary quality indicator is the \(R^{2}\) goodness of fit between the observed and the simulated link flows at the available counting stations in the subset of links \(l\in\hat{L}\subseteq L\), it may happen, as shown in [9], that the optimization algorithm behaves as a kind of meta-regression model that pulls from or pushes to the OD entries \(X_{ij}\) just to improve the fitting to traffic counts and travel times, ignoring the underlying reality of the physical transportation problem, where trips from origin \(i\) to destination \(j\) are a consequence of a socioeconomic reality that cannot increase or decrease the number of trips between this OD pair just for numerical needs and, thus, modify the underlying OD pattern. In practical applications, if a reliable historical origin-to-destination (OD) matrix \(X^{H}\) is available, the number of trips may be outdated for the structure of the matrix, but will still be relevant; therefore, it would be desirable that the structure of the estimated OD matrix be as close as possible to the structure of the historical matrix \(X^{H}\). A measure of similarity borrowed from image quality assessment to compare two different images is proposed in [4]. Reference [32] presents SSIM (structural similarity) for a matrix of pixels as a product of three different comparison components: luminance, contrast, and structure. Luminance corresponds to the intensity of illumination, which is the mean of the different pixels in a sub-matrix. Contrast is the root of the squared average between pixels once the luminance is removed, i.e., it is the standard deviation, and finally, the structure comparison is done by using the covariance between the two matrices. These three factors are first transformed with the aim to adjust them to the interval \([0,1]\), where 1 means a perfect match and 0 means totally different. SSIM, therefore, is a similarity measure that is independent of the magnitude of the values in the matrix. The formula summarizing this explanation is below:  
\[SSIM(\mathbf{x},\mathbf{y})=l(\mathbf{x},\mathbf{y})^{\alpha}c(\mathbf{x},\mathbf{y})^{\beta}s(\mathbf{x},\mathbf{y})^{\gamma} \tag{15}\]  
where luminance, contrast, and structure are respectively defined as:  
\[\left\{