## 4 Conclusions  
In this study, we quantified the temporal change of a time-dependent and three-dimensional dataset. Contrary to other approaches [1; 7; 11; 14], we calculated the three-dimensional strain-rate state of the dataset based on the interpolation of discrete point-wise displacements--or data variations. We applied a technique in continuum mechanics using a FEM interpolation of non-overlapping linear tetrahedral elements that covers the three-dimensional dataset. The methodology was demonstrated to exhibit regions of major deformation-rate. Departing from the calculation of the numerical strain-rate values, we introduced some data-visualization techniques that help to locate the magnitudes and orientations of the strain-rate state in the three-dimensional framework. This was the case of the principal strain-rate trajectories, which where demonstrated to be more detailed than other possible visualization techniques, e.g., strain-rate diagrams [32; 36] or confidence ellipsoids [18; 19; 20; 21]. The main difference with strain-rate diagrams is the ability of the present approach to display a continuum version of the strain-rate inside the dataset, and to separate the analysis into each principal component of the strain-rate state, while being mesh independent. Compared to the method of confidence ellipsoids, the kinematic basis of the proposed methodology makes an accurate description of the deformation patterns in the complete three-dimensional domain, rather than in a compact and partial region that contains the data distribution. It also allows a detailed \begin{table}
\begin{tabular}{r c c c c c c c} \hline \hline
**Element(id)** & \(L^{\infty}(\lambda_{1})\) & **Year** & \(n_{1}^{\top}\implies L^{\infty}(\lambda_{1})\) & \(L^{-\infty}(\lambda_{3})\) & **Year** & \(n_{3}^{\top}\implies L^{-\infty}(\lambda_{3})\) \\ \hline  
0.0935\,0.9952\,0.0288]\) & \(-1.1715\) & 2010 & \([0.1262\,0.9888\,0.0798]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum and minimum eigenspace components in the time span. temporal description of the change between one instant and another. This is specifically not possible using the method of the ellipsoids since that method accounts for the complete temporal distribution in order to evaluate change. The calculation of the strain-rate state shows that the methodology is suitable for quantifying the temporal change of a reduced three-dimensional dataset describing the social, economic and environmental state of the city of Barcelona. In particular, high strain-rates are associated with the localized deformation of regions that represent the districts of the city in the time span of 13 years. The method not only portrays the similarities and differences--distances--between the districts of the city, similar to _cluster analysis_[46; 47; 48] or _distance measures_[49; 50], which explore similarities and groups in data. The distinctive attribute of the present work is the feasibility to quantify the districts' differentiation with time: the strain-rate tensor provides quantitative information about local regions of extension and contraction, where extension-rate patterns indicate differentiation and contraction-rate patterns indicate clustering--or homogenization. In this regard, the strain-rate methodology can be potentially used as an extension of clustering analysis, since it reveals contraction and extension patterns of clusters. Conclusions about the divergence or clustering of districts in time can therefore be stated. For example, it reveals the time, location, and orientation of pressures affecting the inhabitants of certain districts of the city, essentially those which are rapidly diverging from the rest (e.g., the case of Ciutat Vella for the case study). This methodology locates regions where detailed action is necessary, as well as the foretelling of possible ruptures in the system.