Stochastic Convergence of Persistence Landscapes and Silhouettes

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory

Interpretable statistics for complex modelling: quantile and topological learning

As the complexity of our data increased exponentially in the last decades, so has our need for interpretable features. This thesis revolves around two paradigms to approach this quest for insights. In the first part we focus on parametric models, where the problem of interpretability can be seen as a “parametrization selection”. We introduce a quantile-centric parametrization and we show the advantages of our proposal in the context of regression, where it allows to bridge the gap between classical generalized linear (mixed) models and increasingly popular quantile methods. The second part of the thesis, concerned with topological learning, tackles the problem from a non-parametric perspective. As topology can be thought of as a way of characterizing data in terms of their connectivity structure, it allows to represent complex and possibly high dimensional through few features, such as the number of connected components, loops and voids. We illustrate how the emerging branch of statistics devoted to recovering topological structures in the data, Topological Data Analysis, can be exploited both for exploratory and inferential purposes with a special emphasis on kernels that preserve the topological information in the data. Finally, we show with an application how these two approaches can borrow strength from one another in the identification and description of brain activity through fMRI data from the ABIDE project

A statistical framework for analyzing shape in a time series of random geometric objects

We introduce a new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds. At the core of our approach is the point of view that the data arises as sampled recordings from a metric space-valued stochastic process, possibly of nonstationary nature, thereby integrating geometric data analysis into the realm of functional time series analysis. We focus on the descriptors coming from topological data analysis. Our framework allows for natural incorporation of spatial-temporal dynamics, heterogeneous sampling, and the study of convergence rates. Further, we derive complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relate these invariants to so-called ball volume processes. Under mild dependence conditions, a weak invariance principle in $D([0,1]\times [0,\mathscr{R}])$ is established for sequential empirical versions of the latter, assuming the probabilistic structure possibly changes over time. Finally, we use this result to introduce novel test statistics for topological change, which are distribution free in the limit under the hypothesis of stationarity.Comment: Submission versio

Topological Microstructure Analysis Using Persistence Landscapes

International audiencePhase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn-Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations

Análisis topológico de datos: aplicación al reconocimiento de emociones

Emotion recognition consists of a series of processes to detect human emotions from facial human expressions. Humans interact with each other primarily through speech, but also through body gestures to emphasize a certain part of the conversation and exhibit emotions. The automatic recognition of a person’s emotional state has become a very active research eld that involves scientists specialized in di erent areas such as arti cial intelligence, computer vision or psychology, among others. Our main objective in thiswork is to develop a novel approach, based on topological data analysis, for the recognition and classi cation of emotions that combines features extracted from videos and audios, in their respective spaces of representation, of people expressing emotions.El área de reconocimiento de emociones consiste en una serie de procesos para detectar emociones humanas a partir de expresiones faciales. Los humanos interactuamos entre nosotros utilizando el habla, pero también a través de los gestos corporales para así enfatizar una cierta parte de la conversación y exhibir emociones. El reconocimiento automático del estado emocional de una persona se ha convertido en un campo activo de investigación que involucra científicos especializados en diferentes áreas tales como la inteligencia artificial, la visión por ordenador o la psicología, entre otras. Nuestro principal objetivo en este trabajo es desarrollar un novedoso enfoque, basado en el análisis topológico de datos, para el reconocimiento y clasificación de emociones que combine características extraídas de videos y audios, en sus respectivos espacios de representación, de personas expresando emociones.Universidad de Sevilla. Máster Universitario en Matemática Avanzad

5th International Conference on Advanced Research Methods and Analytics (CARMA 2023)

Research methods in economics and social sciences are evolving with the increasing availability of Internet and Big Data sources of information. As these sources, methods, and applications become more interdisciplinary, the 5th International Conference on Advanced Research Methods and Analytics (CARMA) is a forum for researchers and practitioners to exchange ideas and advances on how emerging research methods and sources are applied to different fields of social sciences as well as to discuss current and future challenges.Martínez Torres, MDR.; Toral Marín, S. (2023). 5th International Conference on Advanced Research Methods and Analytics (CARMA 2023). Editorial Universitat Politècnica de València. https://doi.org/10.4995/CARMA2023.2023.1700

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Maverick Mathematician

Mathematics; Quantum theory; Aerospace engineers; Biography; Australi

Evaluation of climate change impacts on the hydrologic response of a sparsely-monitored basin in Sardinia, Italy, through distributed hydrologic simulations and hydrometeorological downscaling

The water resources and hydrologic extremes in Mediterranean basins are heavily influenced by climate variability. Modeling these watersheds is difficult due to the complex nature of the hydrologic response as well as the sparseness of hydrometeorological observations. In this work, we first present a strategy to calibrate a distributed hydrologic model, known as TIN-based Real-time Integrated Basin Simulator (tRIBS), in the Rio Mannu basin, a mediumsized watershed (472.5 km2) located in an agricultural area in Sardinia, Italy. In the basin, precipitation, streamflow and meteorological data were collected within different historical periods and at diverse temporal resolutions. We designed two statistical tools for downscaling precipitation and potential evapotranspiration data to create the hourly, high-resolution forcing for the hydrologic model from daily records. Despite the presence of several sources of uncertainty in the observations and model parameterization, the use of the disaggregated forcing led to good calibration and validation performances for the tRIBS model, when daily discharge observations were available. Future climate projections based on global and regional climate models (GCMs and RCMs) indicate that the Mediterranean basins will most likely suffer a decrease in water availability and an intensification of hydrologic extremes. Process-based distributed hydrologic models (DHMs), like tRIBS, have the potential to simulate the complex hydrologic response of Mediterranean watersheds. Thus, when used in combination with RCMs, DHMs can reduce the uncertainty in the quantification of the local impacts of climate change on water resources. In this study, we apply the calibrated tRIBS model in the Rio Mannu basin to evaluate the effects of climate changes reducing related uncertainties. The two downscaling algorithms and the DHM were used to simulate the watershed response to a set of bias-corrected outputs from four RCMs for two simulation extents: a reference (1971 to 2000) and a future (2041 to 2070) period. The time series and spatial maps simulated by the DHM were then post-processed by computing several metrics to quantify the changes on water resource availability and hydrologic extremes in the future climate scenarios as compared to historical conditions. The research was carried out within the CLIMB project, founded by the 7th Framework Programme of the European Commission