## 1 Introduction  
One challenging problem in data analysis is the representation of three-dimensional discrete data [1]. This analysis becomes harder when a time-dependent change of the data takes place, introducing a new temporal dimension. In the present article, we explore formal approaches to quantify the temporal change of discrete three-dimensional data. Specifically, we build a methodology to assess the transformation of a data cloud that is derived from a _Principal Component Analysis_ (PCA): a 13-year multivariate description in [2] that provides a reduced description of an urban system given only by the first three principal components. Since the points represent an abstraction of an urban system, one main goal is to understand the temporal variation of the multivariate description of the districts in order to analyze the behavior of the overall city in the time-span. Our main hypothesis is that these three-dimensional datasets can be thought of as a cloud of points that gradually deforms. However, the challenging issue is that deformation between consecutive times cannot be visualized accurately. There are some methods to overcome this difficulty. One is the vector plot of three-dimensional displacements or velocities, which is typically used to visualize results in computational mechanics applications [3; 4; 5]. For example, these methods are used in the kinematic visualization of motion [6; 7; 8; 9; 10], but they are restricted to display the relative motion among the data, and are not able to identify the most dynamic regions of the dataset. Another is the parallel coordinate technique that successfully exhibits the temporal change of highly-dimensional statistical and information datasets [11]. However, multi-dimensional data are mostly segmented into two-dimensional subsets, which are easier to deal with, Similar to the computer tomographic scans of medical imaging [1; 12; 13]. However, none of these methods are suitable for understanding the patterns of diversification or conformation, which are closely related to the temporal change of the differences between join data values: the maximum and minimum magnitudes of variation and the evaluation of their direction can be significantly helpful to identify differentiation patterns in the data [14]. Conversely, they can be beneficial for locating uniformity in a dataset that was previously differentiated. One common approach to understanding the temporal change of the dataset is to use _confidence ellipses_ in dimensionally-reduced representations (i.e., PCA) [15; 16; 17]. This type of approach has been applied to several time-dependent problems (e.g., [18; 19; 20; 21]). Confidence ellipses quantify the dispersion of the results: the main directions and intervals of variance accounting for the spatiotemporal distribution are calculated first, and then, the ellipses (or ellipsoids in 3D) are drawn using those intervals as the main axes. A temporal analysis can be achieved by evaluating the geometrical characteristics of the ellipses, especially their area, to which the change of the distribution can be associated. Their main drawback lies in the failure to locate the most dynamic areas in the dataset [22]. The field of continuum mechanics provides a measure of the temporal variation of the distance in between points: _the strain-rate tensor_ (see, for instance, [23]). The continuum mechanics theory--which arises from the classical Newtonian mechanics--analyzes the causes and effects of motion for a deformable media composed by an infinite group of particles. When a continuous media is being deformed in various directions at different rates, the strain-rate of a certain position in the medium cannot be expressed by a scalar value solely. It cannot even be expressed by using a single vector. Instead, the rate of deformation must be expressed by the rank-two strain-rate tensor with its components determined by the positional derivatives along each spatial dimension.