Search results for "topological data analysis"
showing 7 items of 7 documents
Eksperimenti ar topoloģisko datu analīzi
2022
Maģistra darba mērķis ir iepazīstināt ar topoloģisko datu analīzi, kas ir pieeja datu kopu analīzei, izmantojot topoloģijas, kā matemātikas novirziena, metodes. Šī inovatīvā datu analīzes metode pasaulē pēdējos gados strauji attīstās un ar vien plašāk tiek pielietota, lai iegūtu informāciju no sarežģītiem, liela apjoma, daudzdimensionāliem datiem. Pašreiz nekur nav atrodams topoloģiskās datu analīzes apraksts un pielietojamība, latviešu valodā. Darbā tiek apskatīti divi dažādi uz topoloģiskās datu analīzes balstīti algoritmi - Mapper un ToMATo, kuru veiksmīgā izmantošanā noteicošais ir pareizu parametru izvēle. Darbā tiek pētītas un piedāvātas šo algoritmu parametru optimizācijas metodes un…
Ventricular Fibrillation and Tachycardia Detection Using Features Derived from Topological Data Analysis
2022
A rapid and accurate detection of ventricular arrhythmias is essential to take appropriate therapeutic actions when cardiac arrhythmias occur. Furthermore, the accurate discrimination between arrhythmias is also important, provided that the required shocking therapy would not be the same. In this work, the main novelty is the use of the mathematical method known as Topological Data Analysis (TDA) to generate new types of features which can contribute to the improvement of the detection and classification performance of cardiac arrhythmias such as Ventricular Fibrillation (VF) and Ventricular Tachycardia (VT). The electrocardiographic (ECG) signals used for this evaluation were obtained from…
General framework for testing Poisson-Voronoi assumption for real microstructures
2020
Modeling microstructures is an interesting problem not just in Materials Science but also in Mathematics and Statistics. The most basic model for steel microstructure is the Poisson-Voronoi diagram. It has mathematically attractive properties and it has been used in the approximation of single phase steel microstructures. The aim of this paper is to develop methods that can be used to test whether a real steel microstructure can be approximated by such a model. Therefore, a general framework for testing the Poisson-Voronoi assumption based on images of 2D sections of real metals is set out. Following two different approaches, according to the use or not of periodic boundary conditions, thre…
A new measure for the attitude to mobility of Italian students and graduates: a topological data analysis approach
2022
AbstractStudents’ and graduates’ mobility is an interesting topic of discussion especially for the Italian education system and universities. The main reasons for migration and for the so called brain drain, can be found in the socio-economic context and in the famous North–South divide. Measuring mobility and understanding its dynamic over time and space are not trivial tasks. Most of the studies in the related literature focus on the determinants of such phenomenon, in this paper, instead, combining tools coming from graph theory and Topological Data Analysis we propose a new measure for the attitude to mobility. Each mobility trajectory is represented by a graph and the importance of the…
Convergence Rates for Persistence Diagram Estimation in Topological Data Analysis
2014
International audience; Computational topology has recently seen an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and that persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.
Topology-based goodness-of-fit tests for sliced spatial data
2023
In materials science and many other application domains, 3D information can often only be extrapolated by taking 2D slices. In topological data analysis, persistence vineyards have emerged as a powerful tool to take into account topological features stretching over several slices. In the present paper, we illustrate how persistence vineyards can be used to design rigorous statistical hypothesis tests for 3D microstructure models based on data from 2D slices. More precisely, by establishing the asymptotic normality of suitable longitudinal and cross-sectional summary statistics, we devise goodness-of-fit tests that become asymptotically exact in large sampling windows. We illustrate the test…
Optimal rates of convergence for persistence diagrams in Topological Data Analysis
2013
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.