Separating Topological Noise from Features Using Persistent Entropy

Topology is the branch of mathematics that studies shapes and maps among them. From the algebraic definition of topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of k−dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of k−dimensional holes along such sequence is called the persistence barcode. k−dimensional holes with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological feature associated to the given data (i.e. the filtration). In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy.Ministerio de Economía y Competitividad MTM2015-67072-

On the stability of persistent entropy and new summary functions for Topological Data Analysis

Persistent entropy of persistence barcodes, which is based on the Shannon entropy, has been recently defined and successfully applied to different scenarios: characterization of the idiotypic immune network, detection of the transition between the preictal and ictal states in EEG signals, or the classification problem of real long-length noisy signals of DC electrical motors, to name a few. In this paper, we study properties of persistent entropy and prove its stability under small perturbations in the given input data. From this concept, we define three summary functions and show how to use them to detect patterns and topological features

Characterising epithelial tissues using persistent entropy

In this paper, we apply persistent entropy, a novel topological statis- tic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geomet- ric information encoded by -complexes and persistent homology. After using some statistical tests, we can guarantee the existence of signi cant di erences in the studied tissues.Ministerio de Economía y Competitividad MTM2015-67072-

Persistent entropy: a scale-invariant topological statistic for analyzing cell arrangements

In this work, we develop a method for detecting differences in the topological distribution of cells forming epithelial tissues. In particular, we extract topological information from their images using persistent homology and a summary statistic called persistent entropy. This method is scale invariant, robust to noise and sensitive to global topological features of the tissue. We have found significant differences between chick neuroepithelium and epithelium of Drosophila wing discs in both, larva and prepupal stages. Besides, we have tested our method, with good results, with images of mathematical tesselations that model biological tissues

Stable topological summaries for analyzing the organization of cells in a packed tissue

We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.Ministerio de Ciencia e Innovación PID2019-107339GB-I0

K-Factores en nubes bicromáticas

Consideramos una colección de puntos bicromática y nos preguntamos cuántos puntos adicionales son necesarios considerar para asegurar la existencia de un k {factor. Dos tipos de puntos adicionales serán tratados: puntos de Steiner y puntos blancos (con posición prefijada pero no así su color

Cover contact graphs

We study problems that arise in the context of covering certain geometric objects called seeds (e.g., points or disks) by a set of other geometric objects called cover (e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, respectively, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in three types of tasks, both in the general case and in the special case of seeds on a line: (a) deciding whether a given seed set has a connected CCG, (b) deciding whether a given graph has a realization as a CCG on a given seed set, and (c) bounding the sizes of certain classes of CCG’s. Concerning (a) we give efficient algorithms for the case that seeds are points and show that the problem becomes hard if seeds and covers are disks. Concerning (b) we show that this problem is hard even for point seeds and disk covers (given a fixed correspondence between graph vertices and seeds). Concerning (c) we obtain upper and lower bounds on the number of CCG’s for point seeds

Cover Contact Graphs

Es una ponencia presentada al 15th International Symposium on Graph Drawing (2007)We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).German Research Foundation WO 758/4-

Clinical impact of defibrillation testing at the time of implantable cardioverter-defibrillator insertion

Background: Ventricular fibrillation is routinely induced during implantable cardioverter- -defibrillator insertion to assess defibrillator performance, but this strategy is experiencing a progressive decline. We aimed to assess the efficacy of defibrillator therapies and long-term outcome in a cohort of patients that underwent defibrillator implantation with and without defibrillation testing. Methods: Retrospective observational series of consecutive patients undergoing initial defibrillator insertion or generator replacement. We registered spontaneous ventricular arrhythmias incidence and therapy efficacy, and mortality. Results: A total of 545 patients underwent defibrillator implantation (111 with and 434 without defibrillation testing). After 19 (range 9–31) months of follow-up, the death rate per observation year (4% vs. 4%; p = 0.91) and the rate of patients with defibrillator-treated ventricular arrhythmic events per observation year (with test: 10% vs. without test: 12%; p = 0.46) were similar. The generalized estimating equations-adjusted first shock probability of success in patients with test (95%; CI 88–100%) vs. without test (98%; CI 96–100%; p = 0.42) and the proportion of successful antitachycardia therapies (with test: 87% vs. without test: 80%; p = 0.35) were similar between groups. There was no difference in the annualized rate of failed first shock per patient and per shocked patient between groups (5% vs. 4%; p = 0.94). Conclusions: In this observational study, that included an unselected population of patients with a defibrillator, no difference was found in overall mortality, first shock efficacy and rate of failed shocks regardless of whether defibrillation testing was performed or not.Hadid, C.; Atienza, F.; Strasberg, B.; Arenal, Á.; Codner, P.; González-Torrecilla, E.; Datino, T.... (2015). Clinical impact of defibrillation testing at the time of implantable cardioverter-defibrillator insertion. Cardiology Journal. 22(3):253-259. doi:10.5603/Cj.a2014.0062S25325922

Mixturas de distribuciones: Modelización de experiencias con asimetría en los datos

Una de las estrategias de la Estadística para describir y explicar una realidad compleja es la representación de la misma mediante modelos probabilísticos. En esta línea se enmarca la presente memoria, teniendo como principal objetivo la modelización de fenómenos aleatorios a través de mixturas finitas de distribuciones. En concreto, aquellas experiencias aleatorias con asimetría en los datos. Inicialmente, esta memoria surge tratando de abordar un problema real al que se tiene que enfrentar la gestión de hospitales y servicios de salud para programar y financiar adecuadamente los mismos. El estudio del tiempo de hospitalización de los pacientes, variable conocida como “estancia hospitalaria”. La estancia hospitalaria es un índice del consumo de recursos ampliamente utilizado en la gestión hospitalaria. En particular, se utiliza en la dirección, planificación y control de calidad del servicio. Usualmente los datos observados de esta variable presentan asimetría positiva, por ello, tradicionalmente se han utilizado modelos Lognormal, Gamma y Weibull. En esta perspectiva, Marazzi y otros, presentan criterios para decidir sobre cual de los tres modelos anteriores representar mejor la variable citada. En esta memoria, se pretende ampliar el estudio a través de un modelo más general basado en mixturas finitas de distribuciones, en particular, de las distribuciones antes citadas. El modelo consiste en una mixtura de tres componentes pertenecientes a cada una de las tres familias. Este modelo de “mixturas mixtas” proporciona una flexibilidad extra necesaria cuando una sola de las familias no conduce a una explicación satisfactoria de la realidad. Conviene resaltar que, además de la gestión hospitalaria, existen otras muchas situaciones reales descritas por variables que se debaten entre la modelización por alguna de las tres distribuciones: lognormal, Gamma o Weibull. Para ilustras esta aplicación, a continuación se recogen algunas de ellas: Microbiología: Estudio de las posibles consecuencias de la preparación de comidas en ciertos patógenos alimenticios. Neurología: Estudio de la actividad de las neuronas. Veterinaria: Estudio de la duración de viremias en ganado vacuno. Mecánica: Estudios de resistencia de materiales. Meteorología: Concentración de vapor de agua en las nubes y en el estudio de pluviometría. Geología: Estudio de los tamaños de las partículas de sedimentos fluviales. Geofísica: Estudio de las frecuencias de terremotos de cierta magnitud. Epidemiología: Estimación del periodo de incubación del SIDA. La memoria se ha estructurado de la siguiente forma: En el primer capítulo se recogen definiciones, propiedades y conceptos necesarios para el desarrollo de la misma. Se presentan las distribuciones objeto de estudio con algunas de sus características. También, se introduce la modelización mediante mixturas de distribuciones. Se revisa el método de estimación de máxima verosimilitud, poniendo de manifiesto sus propiedades y recogiendo condiciones suficientes para que las mismas se verifiquen. Se finaliza este capítulo con el estudio de un procedimiento iterativo, ampliamente utilizado en el área de las mixturas, para el cálculo de los estimadores de máxima verosimilitud de los diferentes parámetros: el algoritmo EM. En el segundo capítulo se estudian condiciones suficientes que permitan comprobar la identificabilidad de experimentos modelizados a través de mixturas finitas de distribuciones. Dado que los resultados existentes sobre identificabilidad de mixturas no son aplicables al modelo bajo estudio en esta memoria, se propone un nuevo resultado que relaja las condiciones de resultados propuestos en la literatura sobre el tema. El capítulo finaliza con la comprobación de la identificabilidad de distintas familias de mixturas, mediante este nuevo resultado. Para comprobar que en el conjunto de mixturas mixtas de las familias Lognormal, Gamma y Weibull, las soluciones de las ecuaciones de verosimilitud cumplen las condiciones de consistencia (recogidas en el Capítulo 1), en el tercer capítulo se proponen algunos resultados que facilitan la verificación de dichas condiciones para las mixturas de uniones de ciertas familias paramétricas de funciones de densidad como son las familias exponenciales y otro tipo de familias más generales que incluyen a las distribuciones Weibull. Por último, el Capítulo 4 aborda el problema de la estimación de máxima verosimilitud de los parámetros del modelo presentado, a través de la aplicación del algoritmo EM. Asimismo, se estudian los elementos necesarios para la aplicación de dicho algoritmo y un método de aceleración del mismo. A continuación, se presenta un conjunto de simulaciones del modelo con el objetivo de ilustrar la validez de las técnicas propuestas en esta memoria. Este capítulo se concluye con la aplicación a conjuntos de datos reales de la variable estancia hospitalaria