Persistent homology : computation and applications of a modern data analysis tool

Este trabajo es un esfuerzo por resumir los principios de la homologia persistente y explicar sus intuiciones matemáticas y porque esto lo hace una poderosa herramienta. Para poner esto más en enfasis se revisara su aplicación en el campo de redes sociales, en particular, se mostrara cómo la homologia persistente puede ayudar a desenmascarar importantes relaciones globales antes no exploradas.The present work is an effort to review the principles of persistent homology, explain its mathemathical intuitions and why this makes it a powerful tool for data analysis. In order to underline this importance even more we will also review an application to the field of social networks; moreover, it will be shown how the topological insights from persistent homology can help unravel important global relationships of the networks that were not available before.Magíster en MatemáticasMaestrí

Reconocimiento de patrones para la estimación de características por medio de técnicas no lineales

Las aplicaciones de procesamiento de datos tradicionales son inadecuadas para el manejo de una cantidad elevada de datos. Para lograr una eficiente manipulación y extracción de características o muestras que representa la información, es necesario conocer aspectos como la captación y tratamiento de datos. En este documento se depuró una base de datos correspondiente al comportamiento del consumo de energía eléctrica en una carga residencial. La depuración y análisis estadístico de las muestras se realizó por medio del análisis de componentes principales. El entrenamiento del conjunto de datos de menor dimensión a la base de datos original se hizo por medio de las técnicas de máquina de soporte vectorial y redes neuronales artificiales. Finalmente, se presenta una propuesta de análisis de muestras que se encuentren o no dentro de los límites de operación por medio de la actualización de patrones dinámicos para la validación no supervisada de nuevas muestras.Traditional data processing applications are unsuitable for handling large amounts of data. To achieve an efficient manipulation and extraction of characteristics or samples that the information represents, it is necessary to know aspects such as data collection and treatment. In this document, a database corresponding to the behavior of electrical energy consumption in a residential load was refined. The debugging and statistical analysis of the samples were carried out using the principal component analysis. The training of the smallest data set to the original database was made using vector support machine techniques and artificial neural networks. Finally, a proposal is presented for the analysis of samples that are within the operating limits or not using updating dynamic patterns for the unsupervised validation of new samples

Network Analysis of the Gender Gap in International Remittances by Migrants

Financial inclusion is considered a key enabler of international development goals. Despite the expansion of financial access overall, the gender inequalities in basic access have remained consistent. This research investigates the predictive power of global remittance and migration flows on the gender gap in financial inclusion. First, singular value decomposition is applied to the World Bank’s 2017 Global Findex data to identify the financial inclusion variables that most contribute to the gender gap in financial inclusion. We find that indicators pertaining to account ownership, emergency funding, and receiving payments are especially significant. Based on the identified variables, a novel Financial Inclusion Gender Gap Score is calculated for 143 economies. The score is then incorporated into a complex network analysis of global remittance and migration networks. We analyze how network features such as node attributes, community membership, and bow-tie structure can be used to make inferences about the magnitude of a financial inclusion gender gap. Our findings suggest that weaker linkages in the network, characterized by lower node strength and peripheral positions in the bow-tie structure, are determinants of a notable financial inclusion gender gap. We also highlight communities in the remittance and migration networks with a more substantial gender imbalance, and discuss the the social- and cultural-leaning factors driving community formation in the migration network that seem to predicate a greater gap

Topological tools for understanding complex systems

The behavior of complex systems is often influenced by their structure. In mathematics, the field of algebraic topology has been especially useful for characterizing mathematical structures. Topological data analysis (TDA) is a growing field in which methods from algebraic topology are applied to studying the structure of data. TDA has been used in a variety of applications, including biological data, granular materials, and demography. Social interactions are heavily informed by space and have complex structure due to patterns in the way humans arrange themselves geographically. Consequently, social applications can benefit from the application of TDA.In this dissertation, I develop topological methods for studying spatial networks and apply them to a wide variety of data sets. In particular, I study methods for building topological spaces (specifically, simplicial complexes) based on data. I present two novel simplicial-complex constructions, the adjacency complex and the level-set complex, for spatial data. I apply both constructions to random networks, cities, voting, and scientific images, gaining insights into the structure of these systems. I also propose a novel simplicial complex construction for studying patterns of neighborhood formation based on combining demographic and spatial data. I present case studies in neighborhood segregation for two U.S. cities. In addition to my topological research, I discuss two projects in the study of social systems using methods from network analysis. I present an extension to multilayer networks of the Hegselmann--Krause model for opinion dynamics and discuss preliminary findings on its convergence properties. I also present a framework for estimating homelessness underreporting in California Local Education agencies (LEAs)

Spatial Formats under the Global Condition

Contributions to this volume summarize and discuss the theoretical foundations of the Collaborative Research Centre at Leipzig University which address the relationship between processes of (re-)spatialization on the one hand and the establishment and characteristics of spatial formats on the other hand