## 3 Results
### Time-Dependent Data Cloud from an Urban Multivariate Description  
The time-dependent data cloud comes from the PCA output of a multivariate description of the city of Barcelona in [2]. Since 1987, the city has been divided into 10 administrative districts, which are the largest territorial units of the city and can be compared with neighborhoods in a common metropolitan area: Ciutat Vella, Eixample, Gracia, Les Corts, Sarria, Sant Andreu, Sant Marti, Horta, Sants, and Nou Barris. Barcelona has a population of approximately 1.6 million inhabitants living in 10,216 ha. The inclusion of all 10 districts in the multivariate description was aimed at representing the city at its overall scale and allowing comparisons between them.  
The raw multivariate description--from which the PCA is calculated--comprises the data of 40 environmental, economic, and social indicators for the ten districts in the time span of \(t^{0}=2003\leq t^{n}\leq 2015=t^{N}\), \(n=0,1,...,12\). Hence, the case study data cloud comes from a PCA reduction of the higher-dimensional multivariate dataset \(\mathcal{Y}_{n}\left(t^{n}\right)\in\mathbb{R}^{40}\), into a lower-dimensional one \(\mathcal{X}_{n}\left(t^{n}\right)\in\mathbb{R}^{3}\) that possesses only three independent dimensions: PC1, PC2, and PC3. The dimensionally-reduced dataset from the application of the PCA is presented in Appendix A. Hence, the three-dimensional and time-dependent data cloud is composed of the coordinates \(\mathcal{X}_{n}\left(t^{n}\right)\)of the ten \(p_{i}\) points defined in the sequence of \(N=12\) time-steps from 2003 to 2015, with the time-step size of \(\delta t=1\) year. These points are displayed in Figure 2, where all observations--districts each year--in the time-span are included.  
As the first step of our methodology, we applied Delaunay Triangulation (DT) to the data cloud. Specifically, we calculated the DT to the set of coordinates at each time-step \(\mathcal{X}_{n}\left(t^{n}\right)\). This resulted in a mesh \(\mathcal{T}_{h}(t^{n})\) composed of \(nel=|K|\) non-overlapping tetrahedrons. Table 1 expands the resulting triangulation for year 2003, with the vertices information for the \(nel=17\) tetrahedrons. This triangulation is also depicted in Figure 3, where the following notation for the districts in the vertices is used: C is Ciutat Vella, E is Example, G is Gracia, H is Horta, L is Les Corts, N is Nou Barris, A is Sant Andreu, M is Sant Marti, S is Sants, and R is Sarria. Since the position \(\boldsymbol{x}_{i}\left(t^{n}\right)\) of a given point \(p_{i}\) at a later time-step can surpass the initial tetrahedron's circumscribed sphere, we recalculated the mesh triangulation at each time step \(t^{n}\), \(n=1,...,11\).  
Figure 2: Three-dimensional and time-dependent data cloud from the case study.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Element (id)** & **First Vertex** & **Second Vertex** & **Third Vertex** & **Fourth Vertex** \\ \hline
1 & Example & LesCorts & Gracia & Sarria \\
2 & SantAndreu & Horta & SantMarti & NouBarris \\
3 & Sants & SantAndreu & SantMarti & NouBarris \\
4 & LesCorts & Eixample & Sants & CittatVella \\
5 & Gracia & SantMarti & Horta & Sarria \\
6 & Gracia & SantMarti & Sarria & LesCorts \\
7 & Eixample & Gracia & Sants & CittatVella \\
8 & CittatVella & SantAndreu & Sants & NouBarris \\
9 & Sarria & SantMarti & Horta & LesCorts \\
10 & Gracia & SantAndreu & Sants & CittatVella \\
11 & Gracia & SantAndreu & CiutatVella & NouBarris \\
12 & Sarria & Eixample & LesCorts & CiutatVella \\
13 & Horta & SantAndreu & Gracia & NouBarris \\
14 & SantAndreu & SantMarti & Horta & Gracia \\
15 & LesCorts & Eixample & Gracia & Sants \\
16 & LesCorts & Gracia & SantMarti & Sants \\
17 & Gracia & SantAndreu & SantMarti & Sants \\ \hline \hline \end{tabular}
\end{table}
Table 1: Tetrahedral elements derived from the Delaunay Triangulation of the set of points.  
Figure 3: Delaunay triangulation of the data cloud from the case study.