## 1 Introduction
Hence, the mathematical framework of tensors can determine exactly the deformation that is accumulated in a location inside the medium--which is typically subject to the imposition of displacements or loads. This tensor is commonly used to detail the amount of elastic energy in the physical descriptions of multiple materials, such as solids or fluids (see [24] for a complete mathematical exposition). Most of those models are formulated as the product of a constitutive tensor and the strain-rate tensor, giving the stress condition of the material that is balanced in the kinetic equations. In the present study, the calculation of the strain-rate tensor is not related to the kinetics of any material, and, thus, it can only be a mathematical tool that supports the examination of the deformation rates in the discrete statistical data. However, the strain-rate tensor arises from the continuum assumption, and discrete displacements of points rather than continuous distributions take place in the variation of the data cloud. Typically, the issue of applying derivatives to discrete displacements of points is solved by several different approaches. Few statistical techniques use co-variance functions to represent directly the strain-rate field (see, e.g., [25]). The common approach is to compute a continuous version of the displacement--or velocity--field, so that the derivatives can be applied to the continuous displacements. Some methods in this line have interpolated the discrete displacements by minimizing the residual--or distance--between the continuous interpolation and the discrete version [26]. Other interpolation techniques weight the distance between an interpolated piece-wise continuous field and the discrete displacement field (see, e.g., [27]). This method results in a minimization technique where a continuous strain-rate field can be derived. For example, the piece-wise continuous field can be defined as being splines or the widely used linear polynomials in variational formulations [28]. These techniques have been applied in Earth science and medical imaging works [29; 30; 31], as well as in the strain-rate calculation of geodetic observations [32; 33]. Another fundamental issue is the representation of the strain-rate state. One of the possible techniques that can help to visualize the deformation rate of the dataset is to plot the main componentsof the tensor using _strain-rate diagrams_, where concentrations of strain-rate patterns can be displayed as vector fields (see, for example, the ones in geodetical observations of the Earth's mantle [34; 35; 36]). The main drawback of strain-rate diagrams is that the strain-rate components are visualized as the projection of three-dimensional vector fields into the two-dimensional framework, and, therefore, the third-dimension component has to be necessarily neglected. Another method, more suitable to two-dimensional plots, is the contour graph of principal stresses, where the stress patterns in structural elements [37; 38] and tectonics [39] are visualized using continuous lines that depend on the stress magnitude. That method overcomes the three-dimensional issue, but it does not give insights about the orientation of the principal stresses. A dual form of the contour graph is to calculate the family of curves that are instantaneously tangent to the _extension_ and _contraction_ components of the strain-rate tensor: the so-called _trajectory_ curves in the continuum mechanics field [23]. The trajectories of each principal component of the stress can be depicted in a separated plot with the stress magnitude colored along the trajectory line such that stress patterns are completely shown in a two-dimensional framework. Since a robust methodology that describes the temporal change of the urban system--represented by a multivariate dataset--has not been carried out before, we perform a quantitative analysis by including the strain-rate tensor as the fundamental metric.