arXiv subjects

Manuel Cuerno

Publications and source records attributed to Manuel Cuerno.

7 recordsLinked to original sources

arXiv2025

Quantifying displacement: an urban expansion consequence via persistent homology

Population displacement is a housing-related involuntary residential dislocation. It has become increasingly widespread in many cities, particularly in neighbourhoods undergoing rapid economic and demographic change, and measuring it is essential to assess the social consequences of urban transformation and housing market pressures. Despite its relevance, quantifying displacement presents difficulties due to limited replicability across cities and time periods and the need to analyse long time spans: displacement is a gradual process, impossible to capture in one data snapshot. We introduce a novel tool to overcome these difficulties. Using publicly available address change data, we construct four cubical complexes simultaneously incorporating geographical and temporal information of people moving, and analyse using Topological Data Analysis tools. Finally, we demonstrate this method through a 20-year case study in Madrid, Spain. The results reveal its ability to capture displacement and identify the neighbourhoods and years affected--patterns not observable from raw address change data.

cs.CG↗

arXiv2025

Metric pairs and tuples in theory and applications

We present theoretical properties of the space of metric pairs equipped with the Gromov--Hausdorff distance. First, we establish the classical metric separability and the geometric geodesicity of this space. Second, we prove an Arzel\`a--Ascoli-type theorem for metric pairs. Third, extending a result by Cassorla, we show that the set of pairs consisting of a $2$-dimensional compact Riemannian manifold and a $2$-dimensional submanifold with boundary that can be isometrically embedded in $\mathbb{R}^3$ is dense in the space of compact metric pairs. Finally, to broaden the scope of potential applications, we describe scenarios where the Gromov--Hausdorff distance between metric pairs or tuples naturally arises.

math.MG↗

arXiv2024

Finding patterns of meaning: Reassessing Construal Clustering via Bipolar Class Analysis

Empirical research on \textit{construals}--social affinity groups that share similar patterns of meaning--has advanced significantly in recent years. This progress is largely driven by the development of \textit{Construal Clustering Methods} (CCMs), which group survey respondents into construal clusters based on similarities in their response patterns. We identify key limitations of existing CCMs, which affect their accuracy when applied to the typical structures of available data, and introduce Bipolar Class Analysis (BCA), a CCM designed to address these shortcomings. BCA measures similarity in response shifts between expressions of support and rejection across survey respondents, addressing conceptual and measurement challenges in existing methods. We formally define BCA and demonstrate its advantages through extensive simulation analyses, where it consistently outperforms existing CCMs in accurately identifying construals. Along the way, we develop a novel data-generation process that approximates more closely how individuals map latent opinions onto observable survey responses, as well as a new metric to evaluate the performance of CCMs. Additionally, we find that applying BCA to previously studied real-world datasets reveals substantively different construal patterns compared to those generated by existing CCMs in prior empirical analyses. Finally, we discuss limitations of BCA and outline directions for future research.

cs.SI↗

arXiv2024

The rival coffee shop problem

We show some estimations for the Coffee Shop Problem with a modification respect the original statement: there is a rival competing against us. We present different results based on how fast the rival is able to grow. As main tool, we use a variation of the Wasserstein distance (the Signed Wasserstein Distance presented by Piccoli, Rossi and Tournus in "A Wasserstein norm for signed measures, with application to non local transport equation with source term"), that allow us to work with finite signed measures and fits our problem. We are able to obtain similar inequalities with this distance to the ones produced by the canonical Wasserstein distance.

math.MG↗

arXiv2023

On the reach of isometric embeddings into Wasserstein type spaces

We study the reach (in the sense of Federer) of the natural isometric embedding $X\hookrightarrow W_p(X)$ of $X$ inside its $p$-Wasserstein space, where $(X,\operatorname{dist})$ is a geodesic metric space. We prove that if a point $x\in X$ can be joined to another point $y\in X$ by two minimizing geodesics, then $\operatorname{reach}(x, X\subset W_p(X)) = 0$. This includes the cases where $X$ is a compact manifold or a non-simply connected one. On the other hand, we show that $\operatorname{reach}(X\subset W_p(X)) = \infty$ when $X$ is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding $X\hookrightarrow W_\vartheta(X)$ into an Orlicz--Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of $X$ into $\operatorname{Dgm}_\infty$, the space of persistence diagrams equipped with the bottleneck distance.

math.MG↗

arXiv2023

Topological Data Analysis in Air Traffic Management: the shape of big flight data sets

Analyzing flight trajectory data sets poses challenges due to the intricate interconnections among various factors and the high dimensionality of the data. Topological Data Analysis (TDA) is a way of analyzing big data sets focusing on the topological features this data sets have as point clouds in some metric space. Techniques as the ones that TDA provides are suitable for dealing with high dimensionality and intricate interconnections. This paper introduces TDA and its tools and methods as a way to derive meaningful insights from ATM data. Our focus is on employing TDA to extract valuable information related to airports. Specifically, by utilizing persistence landscapes (a potent TDA tool) we generate footprints for each airport. These footprints, obtained by averaging over a specific time period, are based on the deviation of trajectories and delays. We apply this method to the set of Spanish' airports in the Summer Season of 2018. Remarkably, our results align with the established Spanish airport classification and raise intriguing questions for further exploration. This analysis serves as a proof of concept, showcasing the potential application of TDA in the ATM field. While previous works have outlined the general applicability of TDA in aviation, this paper marks the first comprehensive application of TDA to a substantial volume of ATM data. Finally, we present conclusions and guidelines to address future challenges in the ATM domain.

math.AT↗

arXiv2022

Upper and lower bounds on the filling radius

We give a curvature dependent lower bound for the filling radius of all closed Riemannian manifolds as well as an upper one for manifolds which are the total space of a Riemannian submersion. The latter applies also to the case of submetries. We also see that the reach (in the sense of Federer) of the image of the Kuratowski embedding vanishes, and we finish by giving some inequalities involving the k-intermediate filling radius.

math.DG↗