Dissertations / Theses on the topic 'Geometric statistics'
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Saive, Yannick. "DirCNN: Rotation Invariant Geometric Deep Learning." Thesis, KTH, Matematisk statistik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-252573.
Nyligen har ämnet geometrisk deep learning presenterat ett nytt sätt för maskininlärningsalgoritmer att arbeta med punktmolnsdata i dess råa form.Banbrytande arkitekturer som PointNet och många andra som byggt på dennes framgång framhåller vikten av invarians under inledande datatransformationer. Sådana transformationer inkluderar skiftning, skalning och rotation av punktmoln i ett tredimensionellt rum. Precis som vi önskar att klassifierande maskininlärningsalgoritmer lyckas identifiera en uppochnedvänd hund som en hund vill vi att våra geometriska deep learning-modeller framgångsrikt ska kunna hantera transformerade punktmoln. Därför använder många modeller en inledande datatransformation som tränas som en del av ett neuralt nätverk för att transformera punktmoln till ett globalt kanoniskt rum. Jag ser tillkortakommanden i detta tillgångavägssätt eftersom invariansen är inte fullständigt garanterad, den är snarare approximativ. För att motverka detta föreslår jag en lokal deterministisk transformation som inte måste läras från datan. Det nya lagret i det här projektet bygger på Edge Convolutions och döps därför till DirEdgeConv, namnet tar den riktningsmässiga invariansen i åtanke. Lagret ändras en aning för att introducera ett nytt lager vid namn DirSplineConv. Dessa lager sätts ihop i olika modeller som sedan jämförs med sina efterföljare på samma uppgifter för att ge en rättvis grund för att jämföra dem. Resultaten är inte lika bra som toppmoderna resultat men de är ändå tillfredsställande. Jag tror även resultaten kan förbättas genom att förbättra inlärningshastigheten och dess schemaläggning. I ett experiment där ablation genomförs på de nya lagren ser vi att lagrens huvudkoncept förbättrar resultaten överlag.
Ho, Pak-kei. "Parametric and non-parametric inference for Geometric Process." Click to view the E-thesis via HKUTO, 2005. http://sunzi.lib.hku.hk/hkuto/record/B31483859.
Ho, Pak-kei, and 何柏基. "Parametric and non-parametric inference for Geometric Process." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2005. http://hub.hku.hk/bib/B31483859.
Keil, Mitchel J. "Automatic generation of interference-free geometric models of spatial mechanisms." Diss., This resource online, 1990. http://scholar.lib.vt.edu/theses/available/etd-08252008-162631/.
Suttmiller, Alexander Gage. "Streamline Feature Detection: Geometric and Statistical Evaluation of Streamline Properties." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1315967677.
Saha, Abhijoy. "A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562679374833421.
Chu, Chi-Yang. "Applied Nonparametric Density and Regression Estimation with Discrete Data| Plug-In Bandwidth Selection and Non-Geometric Kernel Functions." Thesis, The University of Alabama, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10262364.
Bandwidth selection plays an important role in kernel density estimation. Least-squares cross-validation and plug-in methods are commonly used as bandwidth selectors for the continuous data setting. The former is a data-driven approach and the latter requires a priori assumptions about the unknown distribution of the data. A benefit from the plug-in method is its relatively quick computation and hence it is often used for preliminary analysis. However, we find that much less is known about the plug-in method in the discrete data setting and this motivates us to propose a plug-in bandwidth selector. A related issue is undersmoothing in kernel density estimation. Least-squares cross-validation is a popular bandwidth selector, but in many applied situations, it tends to select a relatively small bandwidth, or undersmooths. The literature suggests several methods to solve this problem, but most of them are the modifications of extant error criterions for continuous variables. Here we discuss this problem in the discrete data setting and propose non-geometric discrete kernel functions as a possible solution. This issue also occurs in kernel regression estimation. Our proposed bandwidth selector and kernel functions perform well in simulated and real data.
Carriere, Mathieu. "On Metric and Statistical Properties of Topological Descriptors for geometric Data." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS433/document.
In the context of supervised Machine Learning, finding alternate representations, or descriptors, for data is of primary interest since it can greatly enhance the performance of algorithms. Among them, topological descriptors focus on and encode the topological information contained in geometric data. One advantage of using these descriptors is that they enjoy many good and desireable properties, due to their topological nature. For instance, they are invariant to continuous deformations of data. However, the main drawback of these descriptors is that they often lack the structure and operations required by most Machine Learning algorithms, such as a means or scalar products. In this thesis, we study the metric and statistical properties of the most common topological descriptors, the persistence diagrams and the Mappers. In particular, we show that the Mapper, which is empirically instable, can be stabilized with an appropriate metric, that we use later on to conpute confidence regions and automatic tuning of its parameters. Concerning persistence diagrams, we show that scalar products can be defined with kernel methods by defining two kernels, or embeddings, into finite and infinite dimensional Hilbert spaces
Pedersen, Morten Akhøj. "Méthodes riemanniennes et sous-riemanniennes pour la réduction de dimension." Electronic Thesis or Diss., Université Côte d'Azur, 2023. http://www.theses.fr/2023COAZ4087.
In this thesis, we propose new methods for dimension reduction based on differential geometry, that is, finding a representation of a set of observations in a space of lower dimension than the original data space. Methods for dimension reduction form a cornerstone of statistics, and thus have a very wide range of applications. For instance, a lower dimensional representation of a data set allows visualization and is often necessary for subsequent statistical analyses. In ordinary Euclidean statistics, the data belong to a vector space and the lower dimensional space might be a linear subspace or a non-linear submanifold approximating the observations. The study of such smooth manifolds, differential geometry, naturally plays an important role in this last case, or when the data space is itself a known manifold. Methods for analysing this type of data form the field of geometric statistics. In this setting, the approximating space found by dimension reduction is naturally a submanifold of the given manifold. The starting point of this thesis is geometric statistics for observations belonging to a known Riemannian manifold, but parts of our work form a contribution even in the case of data belonging to Euclidean space, mathbb{R}^d.An important example of manifold valued data is shapes, in our case discrete or continuous curves or surfaces. In evolutionary biology, researchers are interested in studying reasons for and implications of morphological differences between species. Shape is one way to formalize morphology. This application motivates the first main contribution of the thesis. We generalize a dimension reduction method used in evolutionary biology, phylogenetic principal component analysis (P-PCA), to work for data on a Riemannian manifold - so that it can be applied to shape data. P-PCA is a version of PCA for observations that are assumed to be leaf nodes of a phylogenetic tree. From a statistical point of view, the important property of such data is that the observations (leaf node values) are not necessarily independent. We define and estimate intrinsic weighted means and covariances on a manifold which takes the dependency of the observations into account. We then define phylogenetic PCA on a manifold to be the eigendecomposition of the weighted covariance in the tangent space of the weighted mean. We show that the mean estimator that is currently used in evolutionary biology for studying morphology corresponds to taking only a single step of our Riemannian gradient descent algorithm for the intrinsic mean, when the observations are represented in Kendall's shape space. Our second main contribution is a non-parametric method for dimension reduction that can be used for approximating a set of observations based on a very flexible class of submanifolds. This method is novel even in the case of Euclidean data. The method works by constructing a subbundle of the tangent bundle on the data manifold via local PCA. We call this subbundle the principal subbundle. We then observe that this subbundle induces a sub-Riemannian structure and we show that the resulting sub-Riemannian geodesics with respect to this structure stay close to the set of observations. Moreover, we show that sub-Riemannian geodesics starting from a given point locally generate a submanifold which is radially aligned with the estimated subbundle, even for non-integrable subbundles. Non-integrability is likely to occur when the subbundle is estimated from noisy data, and our method demonstrates that sub-Riemannian geometry is a natural framework for dealing which such problems. Numerical experiments illustrate the power of our framework by showing that we can achieve impressively large range reconstructions even in the presence of quite high levels of noise
I denne afhandling præsenteres nye metoder til dimensionsreduktion, baseret p˚adifferential geometri. Det vil sige metoder til at finde en repræsentation af et datasæti et rum af lavere dimension end det opringelige rum. S˚adanne metoder spiller enhelt central rolle i statistik, og har et meget bredt anvendelsesomr˚ade. En laveredimensionalrepræsentation af et datasæt tillader visualisering og er ofte nødvendigtfor efterfølgende statistisk analyse. I traditionel, Euklidisk statistik ligger observationernei et vektor rum, og det lavere-dimensionale rum kan være et lineært underrumeller en ikke-lineær undermangfoldighed som approksimerer observationerne.Studiet af s˚adanne glatte mangfoldigheder, differential geometri, spiller en vigtig rollei sidstnævnte tilfælde, eller hvis rummet hvori observationerne ligger i sig selv er enmangfoldighed. Metoder til at analysere observationer p˚a en mangfoldighed udgørfeltet geometrisk statistik. I denne kontekst er det approksimerende rum, fundetvia dimensionsreduktion, naturligt en submangfoldighed af den givne mangfoldighed.Udgangspunktet for denne afhandling er geometrisk statistik for observationer p˚a ena priori kendt Riemannsk mangfoldighed, men dele af vores arbejde udgør et bidragselv i tilfældet med observationer i Euklidisk rum, Rd.Et vigtigt eksempel p˚a data p˚a en mangfoldighed er former, i vores tilfældediskrete kurver eller overflader. I evolutionsbiologi er forskere interesseret i at studeregrunde til og implikationer af morfologiske forskelle mellem arter. Former er ´en m˚adeat formalisere morfologi p˚a. Denne anvendelse motiverer det første hovedbidrag idenne afhandling. We generaliserer en metode til dimensionsreduktion brugt i evolutionsbiologi,phylogenetisk principal component analysis (P-PCA), til at virke for datap˚a en Riemannsk mangfoldighed - s˚a den kan anvendes til observationer af former. PPCAer en version af PCA for observationer som antages at være de yderste knuder iet phylogenetisk træ. Fra et statistisk synspunkt er den vigtige egenskab ved s˚adanneobservationer at de ikke nødvendigvis er uafhængige. We definerer og estimerer intrinsiskevægtede middelværdier og kovarianser p˚a en mangfoldighed, som tager højde fors˚adanne observationers afhængighed. Vi definerer derefter phylogenetisk PCA p˚a enmangfoldighed som egendekomposition af den vægtede kovarians i tanget-rummet tilden vægtede middelværdi. Vi viser at estimatoren af middelværdien som pt. bruges ievolutionsbiologi til at studere morfologi svarer til at tage kun et enkelt skridt af voresRiemannske gradient descent algoritme for den intrinsiske middelværdi, n˚ar formernerepræsenteres i Kendall´s form-mangfoldighed.Vores andet hovedbidrag er en ikke-parametrisk metode til dimensionsreduktionsom kan bruges til at approksimere et data sæt baseret p˚a en meget flexibel klasse afsubmangfoldigheder. Denne metode er ny ogs˚a i tilfældet med Euklidisk data. Metodenvirker ved at konstruere et under-bundt af tangentbundet p˚a datamangfoldighedenM via lokale PCA´er. Vi kalder dette underbundt principal underbundtet. Viobserverer at dette underbundt inducerer en sub-Riemannsk struktur p˚a M og vi viserat sub-Riemannske geodæter fra et givent punkt lokalt genererer en submangfoldighedsom radialt flugter med det estimerede subbundt, selv for ikke-integrable subbundter.Ved støjfyldt data forekommer ikke-integrabilitet med stor sandsynlighed, og voresmetode demonstrerer at sub-Riemannsk geometri er en naturlig tilgang til at h˚andteredette. Numeriske eksperimenter illustrerer styrkerne ved metoden ved at vise at denopn˚ar rekonstruktioner over store afstande, selv under høje niveauer af støj
Neeser, Rudolph. "A Comparison of Statistical and Geometric Reconstruction Techniques: Guidelines for Correcting Fossil Hominin Crania." Thesis, University of Cape Town, 2007. http://pubs.cs.uct.ac.za/archive/00000413/.
Dai, Xiaogang. "Score Test and Likelihood Ratio Test for Zero-Inflated Binomial Distribution and Geometric Distribution." TopSCHOLAR®, 2018. https://digitalcommons.wku.edu/theses/2447.
Xie, Weiyi. "A Geometric Approach to Visualization of Variability in Univariate and Multivariate Functional Data." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1500348052174345.
Prieto, Bernal Juan Carlos. "Multiparametric organ modeling for shape statistics and simulation procedures." Thesis, Lyon, INSA, 2014. http://www.theses.fr/2014ISAL0010/document.
Geometric modeling has been one of the most researched areas in the medical domain. Today, there is not a well established methodology to model the shape of an organ. There are many approaches available and each one of them have different strengths and weaknesses. Most state of the art methods to model shape use surface information only. There is an increasing need for techniques to support volumetric information. Besides shape characterization, a technique to differentiate objects by shape is needed. This requires computing statistics on shape. The current challenge of research in life sciences is to create models to represent the surface, the interior of an object, and give statistical differences based on shape. In this work, we use a technique for shape modeling that is able to model surface and internal features, and is suited to compute shape statistics. Using this technique (s-rep), a procedure to model the human cerebral cortex is proposed. This novel representation offers new possibilities to analyze cortical lesions and compute shape statistics on the cortex. The second part of this work proposes a methodology to parameterize the interior of an object. The method is flexible and can enhance the visual aspect or the description of physical properties of an object. The geometric modeling enhanced with physical parameters is used to produce simulated magnetic resonance images. This image simulation approach is validated by analyzing the behavior and performance of classic segmentation algorithms for real images
Riou-Durand, Lionel. "Theoretical contributions to Monte Carlo methods, and applications to Statistics." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLG006/document.
The first part of this thesis concerns the inference of un-normalized statistical models. We study two methods of inference based on sampling, known as Monte-Carlo MLE (Geyer, 1994), and Noise Contrastive Estimation (Gutmann and Hyvarinen, 2010). The latter method was supported by numerical evidence of improved stability, but no theoretical results had yet been proven. We prove that Noise Contrastive Estimation is more robust to the choice of the sampling distribution. We assess the gain of accuracy depending on the computational budget. The second part of this thesis concerns approximate sampling for high dimensional distributions. The performance of most samplers deteriorates fast when the dimension increases, but several methods have proven their effectiveness (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). In the continuity of some recent works (Eberle et al., 2017; Cheng et al., 2018), we study some discretizations of the kinetic Langevin diffusion process and establish explicit rates of convergence towards the sampling distribution, that scales polynomially fast when the dimension increases. Our work improves and extends the results established by Cheng et al. for log-concave densities
Romon, Gabriel. "Contributions to high-dimensional, infinite-dimensional and nonlinear statistics." Electronic Thesis or Diss., Institut polytechnique de Paris, 2023. http://www.theses.fr/2023IPPAG013.
Three topics are explored in this thesis: inference in high-dimensional multi-task regression, geometric quantiles in infinite-dimensional Banach spaces and generalized Fréchet means in metric trees. First, we consider a multi-task regression model with a sparsity assumption on the rows of the unknown parameter matrix. Estimation is performed in the high-dimensional regime using the multi-task Lasso estimator. To correct for the bias induced by the penalty, we introduce a new data-driven object that we call the interaction matrix. This tool lets us develop normal and chi-square asymptotic distribution results, from which we obtain confidence intervals and confidence ellipsoids in sparsity regimes that are not covered by the existing literature. Second, we study the geometric quantile, which generalizes the classical univariate quantile to normed spaces. We begin by providing new results on the existence and uniqueness of geometric quantiles. Estimation is then conducted with an approximate M-estimator and we investigate its large-sample properties in infinite dimension. When the population quantile is not uniquely defined, we leverage the theory of variational convergence to obtain asymptotic statements on subsequences in the weak topology. When there is a unique population quantile, we show that the estimator is consistent in the norm topology for a wide range of Banach spaces including every separable uniformly convex space. In separable Hilbert spaces, we establish novel Bahadur-Kiefer representations of the estimator, from which asymptotic normality at the parametric rate follows. Lastly, we consider measures of central tendency for data that lives on a network, which is modeled by a metric tree. The location parameters that we study are called generalized Fréchet means: they obtained by relaxing the square in the definition of the Fréchet mean to an arbitrary convex nondecreasing loss. We develop a notion of directional derivative in the tree, which helps us locate and characterize the minimizers. We examine the statistical properties of the corresponding M-estimator: we extend the notion of stickiness to the setting of metrics trees, and we state a non-asymptotic sticky theorem, as well as a sticky law of large numbers. For the Fréchet median, we develop non-asymptotic concentration bounds and sticky central limit theorems
Helfgott, Michel. "Calculus of One Variable: An Eclectic Approach." Digital Commons @ East Tennessee State University, 2012. http://amzn.com/1477633871.
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Riou-Durand, Lionel. "Theoretical contributions to Monte Carlo methods, and applications to Statistics." Electronic Thesis or Diss., Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLG006.
The first part of this thesis concerns the inference of un-normalized statistical models. We study two methods of inference based on sampling, known as Monte-Carlo MLE (Geyer, 1994), and Noise Contrastive Estimation (Gutmann and Hyvarinen, 2010). The latter method was supported by numerical evidence of improved stability, but no theoretical results had yet been proven. We prove that Noise Contrastive Estimation is more robust to the choice of the sampling distribution. We assess the gain of accuracy depending on the computational budget. The second part of this thesis concerns approximate sampling for high dimensional distributions. The performance of most samplers deteriorates fast when the dimension increases, but several methods have proven their effectiveness (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). In the continuity of some recent works (Eberle et al., 2017; Cheng et al., 2018), we study some discretizations of the kinetic Langevin diffusion process and establish explicit rates of convergence towards the sampling distribution, that scales polynomially fast when the dimension increases. Our work improves and extends the results established by Cheng et al. for log-concave densities
Slezak, Thomas Joseph. "Quantitative Morphological Classification of Planetary Craterforms Using Multivariate Methods of Outline-Based Shape Analysis." BYU ScholarsArchive, 2017. https://scholarsarchive.byu.edu/etd/6639.
Sánchez, Trigueros Fernando. "Geospatial patterns in the late pleistocene human settlement of the Sierra de Atapuerca (Burgos, Spain): spatial association, geometric probability and fuzzy statistics in the study of past land-use strategies." Doctoral thesis, Universitat Rovira i Virgili, 2013. http://hdl.handle.net/10803/125660.
Lauria, Gabriele. "The Human Biodiversity in the Middle of the Mediterranean. Study of native and settlers populations on the Sicilian context." Doctoral thesis, Universitat Politècnica de València, 2021. http://hdl.handle.net/10251/159789.
[ES] Durante los últimos 200.000 años, la especie humana se ha extendido por toda la Tierra, adaptando su morfología y fisiología a una amplia variedad de hábitats. Por tanto, el esqueleto humano ha registrado los principales efectos ambientales. Hoy las modernas técnicas de investigaciones cuantitativas de las principales características morfológicas nos permiten relacionarlas con la variabilidad genética. La posición geográfica de la Sicilia, su aislamiento y su larga y dinámica historia de colonización han creado un contexto peculiar que permite un estudio antropológico único, útil para extraer información importante sobre el "Flujo Migratorio" y "Influencia Población". Este proyecto se basa en la análisis antropológica de huesos humanos de diferentes poblaciones (indígenas y colonizadoras). Las técnicas de Análisis de Geometría Morfométrica y Análisis Estadístico Multivariante se han aplicado en tres caracteres esqueléticos diferentes (Dientes, Cráneos y Estatura). El objetivo del proyecto es realizar un análisis amplia de la Biodiversidad Humana Siciliana con el fin de: - Analizar datos odontométricos 2D con técnicas multivariadas para explorar las relaciones entre pueblos entre los siglos. - Utilizar modelos 3D y la morfometría craneofacial para estudiar la compleja variabilidad morfológica relacionada con los flujos migratorios. - Evaluar la tendencia secular de la estatura. - Utilizar estos tres caracteres para proporcionar una descripción general de la Biodiversidad Humana en Sicilia. Esto trabajo denota la confiabilidad de los métodos utilizados y, como en un estudio de la biodiversidad, varios caracteres son indispensables para comprender el proceso evolutivo. Los datos también demuestran la correlación entre los caracteres morfológicos y la influencia (no solo por factores ambientales) de los flujos humanos sobre el fenotipo. Los resultados muestran claramente que todos los caracteres evaluados están involucrados al mismo tiempo en el mismo proceso de diversificación. Las variaciones morfológicas muestran una disminución general del prognatismo maxilar y una ligera mesocefalilización con el cráneo que se convierte en más estrecho y ligeramente y menos alargado y la cara más ancha y corta. Siempre considerando la influencia de la relación tamaño/composición, de la muestra, tanto el análisis estadístico canónico como multivariado apoyan la teoría que la población del Paleolítico Superior de San Teodoro podría ser razonablemente la primera evidencia de colonización humana en Sicilia (esta teoría también es apoyada de la muestra Mesolítica que se agrupa separada). Son significativos los periodos de el Bronce y de la transición Bronce/Hierro en los que asistimos a importantes cambios morfológicos (Dientes, Cráneos y Estatura) debido a los constantes y numéricamente significativos "Flujos Migratorios". Esta variación coincide exactamente con los primeros "Flujos de Población" estables como consecuencia de las migraciones humanas desde el continente. Sin embargo, las muestras Prehistóricas de algunas poblaciones conservan algunas características arcaicas incluso después de la Edad del Hierro (Era Histórica) mientras la "Continuidad de la Población" (resultante de la coexistencia y alternancia de la colonización) desde la Antigüedad hasta la Edad Media produjo una mayor progresiva variabilidad sin pero mayor variación entre Eignevalue y Componentes Principales. La ausencia de relación interna causada por el intrincado período de colonización está presente en la muestra prehistórica en la que hay una clara variación entre las Componentes Principales. Las correlaciones entre la "Afluencia de Población" y la Variabilidad también se pueden observar en la influencia de los colonos Islámicos sobre los indígenas durante la Edad Media. Sin embargo, la amplia variabilidad y el morfoespacio homogéneo muestran que poblaciones bien definidas no son reconocibles después de estos grupos (hasta los contemporáneos).
[EN] During the last 200,000 years, human species has spread throughout Earth, adapting their morphology and physiology to a wide range of habitats. The human skeleton has therefore, recorded the main environmental effects. Nowadays modern quantitative investigations of the main morphological features permit us to relate them with the genetic variability. The Sicilian geographic position, isolation and its long and dynamic history of colonization) made a peculiar context that allows a unique anthropological study, useful to sign-out important information about the "Migratory Flow" and the consequent "Populations Influx". This project is based on the Anthropological Analysis of the human bones coming from different populations distributed from Paleolithic to the Contemporary Age. The techniques of Morphometric Geometric analysis and Multivariate Statistic Analysis were applied over three different catchers (Teeth, Skulls and Stature). The project aim is to perform a wide analysis of the Sicilian Human Biodiversity in order to: - Analyze 2D odontometrics data with multivariate techniques to explore the relationships between the peoples over the centuries. - Use 3D models and skull-facial morphometry to study the complex morphological variability concerning the "Populations". - Evaluate the "Stature's Secular Trend". - Use these three characters to provide a general overview of the human biodiversity in Sicily. Our work denotes the reliable of the methods employed underlying as in a study of biodiversity several characters are indispensable to understand the evolutionary process. Data also provided to demonstrate the correlation between the morphological characters and the influence carried (not only by the environmental factors) by the human flow on the phenotype. Results clearly shows as all the characters evaluated are at the same time involved in the same process of diversification. Morphological variations show a general decrease of Maxilla Prognathism and a soft Mesocephalization with the skull that becomes tighter and slightly and less elongated and the face that become wider and shorter. Always considering simple size/composition both Canonical and Multivariate Statistics Analysis display, as the Upper-Paleolithic Würm-Settlers of San Teodoro could reasonably be the first evidence of human colonization in Sicily (this theory is supported by the Mesoltitch Hunter-Gatherers specimens clustered separated from the first one). Meaningful is the periods of Bronze/Iron transition in we assist to the prime plainness of morphological changes (teeth, skulls and statures) due to the constant and numerically significative "Migratory Flows". This variation exactly coincides with the first "Population Influx" consequent of the human migrations from the continent. Instead, Prehistorical samples of some populations, keep some archaic characters after Iron Age (Historical Era) the "Population Continuity" (consequent of the cohabitation and alternations of the several Mediterranean populations) from Antiquity to Middle Ages produced a progressive increase of variability without big variation among Eigenvalue and Principal Component. The absence of internal relationship caused by the intricate colonization period is on the contrary present on Prehistorichal sample on which we can find a clear variation between the PC. Correlations between "Population Influx" and Variability are also observable on the influence of Islamic settlers on the Indigenous during the Middle Ages. However, the wide variability and the homogenous morphospace showed by these groups and the Contemporary resulted in no well-defined populations.
[CA] Durant els últims 200.000 anys, l'espècie humana s'ha estés per tota la Terra, adaptant la seua morfologia i fisiologia a una àmplia varietat d'hàbitats. Per tant, l'esquelet humà ha registrat els principals efectes ambientals. Hui les modernes tècniques d'investigacions quantitatives de les principals característiques morfològiques ens permeten relacionar-les amb la variabilitat genètica. La posició geogràfica de la Sicília, el seu aïllament i la seua llarga i dinàmica història de colonització han creat un context peculiar que permet un estudi antropològic únic, útil per a extraure informació important sobre el "Flux Migratori" i "Influència Població". Aquest projecte es basa en l'anàlisi antropològica d'ossos humans de diferents poblacions des del Paleolític fins a l'Edat Contemporània. Les tècniques d'Anàlisis de Geometria Morfomètrica (reconstrucció 3D) i Anàlisi Estadística Multivariante s'han aplicat en tres caràcters esquelètics diferents (Dents, Cranis i Alçada). L'objectiu del projecte és realitzar una anàlisi àmplia de la Biodiversitat Humana Siciliana amb la finalitat de: - Analitzar dades odontométricos 2D amb tècniques multivariades per a explorar les relacions entre pobles entre els segles. - Utilitzar models 3D i la morfometria craniofacial per a estudiar la complexa variabilitat morfològica relacionada amb els fluxos migratoris. - Avaluar la tendència secular de l'alçada. - Utilitzar aquests tres caràcters per a proporcionar una descripció general de la Biodiversitat Humana a Sicília. Això treball denota la confiabilitat dels mètodes utilitzats i, com en un estudi de la biodiversitat, diversos caràcters són indispensables per a comprendre el procés evolutiu. Les dades també demostren la correlació entre els caràcters morfològics i la influència (no sols per factors ambientals) dels fluxos humans sobre el fenotip. Els resultats mostren clarament que tots els caràcters avaluats estan involucrats al mateix temps en el mateix procés de diversificació. Les variacions morfològiques mostren una disminució general del prognatisme maxil·lar i una lleugera mesocefalilización amb el crani que es converteix en més estret i lleugerament i menys allargat i la cara més ampla i tala. Sempre considerant la influència de la relació grandària/composició, de la mostra, tant l'anàlisi estadística canònica com multivariat donen suport a la teoria que la poblacion del Paleolític Superior de Sant Teodoro podria ser raonablement la primera evidència de colonització humana a Sicília (aquesta teoria també és secundada de la mostra Mesolítica que s'agrupa separada). Són significatius els períodes del Bronze i de la transició Bronze/Ferro en els quals assistim a importants canvis morfològics (Dents, Cranis i Alçada) a causa dels constants i numèricament significatius "Fluxos Migratoris". Aquesta variació coincideix exactament amb els primers "Fluxos de Població" estables com a conseqüència de les migracions humanes des del continent. No obstant això, les mostres Prehistòriques d'algunes poblacions conserven algunes característiques arcaiques fins i tot després de l'Edat del Ferro (Era Històrica) mentre la "Continuïtat de la Població" (resultant de la coexistència i alternança de la colonizacion) des de l'Antiguitat fins a l'Edat mitjana va produir una major progressiva variabilitat sense però major variació entre Eignevalue i Components Principals. L'absència de relació interna causada per l'intricat període de colonització està present, en contrero, en la mostra prehistòrica en la qual hi ha una clara variació entre les Components Principals. Les correlacions entre l' "Afluència de Població" i la Variabilitat també es poden observar en la influència dels colons Islàmics sobre els indígenos durant l'Edat mitjana. No obstant això, l'àmplia variabilitat i el morfoespacio homogeni mostren que poblacions ben definides no són recognoscibles després d'aquests grups (fins als contemporanis).
Thanks to: Dr. Francesa Spatafora (Archaeological Museum “A. Salinas”, Palermo, Sicily, Italy), Dr. Maria Grazia Griffo (Archaeological Museum “Baglio Anselmi”. Marsala, Sicily, Italy) , Dr. Maria Amalia Mastelloni (Archaeological Museum “Bernabò-Brea", Aeolian Island, Sicily, Italy) and Dr. Carolina Di Patti (Geological University Museum of Palermo “Gemmellaro”, Sicily, Italy) for the authorization to study the materials. Thanks to Prof. Armando González Martín, Prof. Oscar Cambra-Moo Laboratorio de Poblaciones del Pasado (LAPP), Universidad Autónoma de Madrid (UAM), Madrid, Spain) for the invaluable help. Thanks to the Laboratorio de Ecología Evolutiva Humana (LEEH) - Universidad Nacional del Centro de la Provincia de Buenos Aires (UNCPBA), Buenos Aires, Argentina), to all the laboratoy directors to provide the modern specimens of reference sample. Thanks to the Mayor Mr. Domenico Giannopolo, the council member of cultural heritage Mrs. Nieta Gennuso and Dr. Filippo Ianni (Municipality of Caltavuturo, Sicily, Italy) for the excavation seasons and the authorization to study the materials. To the Museum of Mozia. (Sicily, Italy), The Whitaker foundation and Dr. Pamela Toti for the authorization to study the materials. To the Soprintendenza BB.CC.AA. di Palermo. (Sicily, Italy) Dr. Stefano Vassallo and Dr. Maria Grazia Cucco for the opportunity of the excavations in Caltavuturo and authorization to study the materials. Thanks to the Soprintendenza BB.CC.AA. di Trapani. (Sicily, Italy), Soprintendente and Dr. Rossella Giglio, Township Museum of Mussomeli. (Sicily, Italy), and Prof. Erich Kistler and Dr. Nicole Mölk (University of Innsbruck, Austria, Innsbruck) for the authorization to study the materials.
Lauria, G. (2020). The Human Biodiversity in the Middle of the Mediterranean. Study of native and settlers populations on the Sicilian context [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/159789
TESIS
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We have studied the concept of statistical model from a geometric point of view considering particularly the information metric and the problem of introducing distances. Given a parametric manifold representing a statistical model and given a point of the manifold, we have defined two different distances between elements of sample space (statistical individuals) by means of a suitable representation of statistical individuals as linear forms of the dual tangent space to the manifold in the given point. Some properties have been studied and the explicit expressions for some examples have been obtained. Several techniques of statistical inference: parameter estimation, hypothesis tests, discrimination; have been studied in the light of the distances between elements of sample spaces. Some classical results have been recovered, in particular Iikelihood equations and Lagrange multipliers test. We have introduced a class of probability density functions that may be represented in finite dimensional manifolds. Geometrical properties of such manifolds have been studied and the Rao distance between two distributions has been obtained. We have considered several examples. We have also studied the problem of parameter estimation in the functions defined previously; we have developed a stepwise algorithm for nonparametric density estimation in order to some problems arising with classical maximum likelihood estimation when we handle a large number of parameters. We also present some examples applied lo biological data.
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Ph. D.
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During the three years of my Ph. D. Project, I developed several complementary methods and frameworks for the analysis of -omics data, such as: (i) a framework for integrative genomics in which every kind of information that can be obtained about the genomic processes and features are modeled in a common probabilistic manner, allowing then to analyze the correlations among the heterogeneous genome-wide information, (ii) a fold-change based statistical test for the identification of differentially and similarly expressed genes between two biological conditions, allowing also the determination of confidence intervals of specific confidence levels for the fold-change. (iii) novel dimensionality reduction methods that outperform other related existing methods and provide more interpretable geometrical representations in the context of large dataset of-omics data. These methods have been applied to several biological analyses and studies as part of different scientific collaborations: (i) to identify functional Glucocorticoid Response Elements in the promoter regions of specific candidate genes involved in Type 1 Pseudohypoaldosteronism. (ii) to uncover the host transcriptional responses underlying differences between low- and high- pathogenic pulmonary viruses based on a compendium of host transcription responses of infected cells from mouse lungs. (iii) to study the progression of the hepatitis C virus in infected patients who underwent orthotopic liver transplantation, based on a cohort of transcriptome profiles for liver biopsy specimens, (iv) to analyze an Expression Sequence Tag library obtained from PBMC of African green monkeys infected or not by the SIV
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Differential geometry is the set of tools that allows to perform the usual mathematical tasks of algebra and calculus on spaces that do not behave like Euclidean vector spaces, for instance points on a curved surface. This field of mathematics is becoming more and more relevant in multiple fields, statistics and machine learning among those, due to the enormous availability of data belonging to increasingly complex domains. An example among many of such complex domains is the set of Symmetric and Positive Definite matrices, i.e. the set of covariance matrices, that appears frequently in medical imaging but is also used often as parameter space in statistical modeling scenarios. The aim of this thesis is to collect and organize the scattered knowledge on the Riemannian geometry of the symmetric and positive definite matrices, and to build practical techniques using the tools of differential geometry that can be readily applied within a pipeline of statistical analysis. This has been achieved with two different methods: the first is a quasi-Newton algorithm for Riemannian optimization that can be plugged in any situation in which maximization of a function of symmetric and positive definite matrices is required, such as those that arise in the context of likelihood inference and variational approximation. The second is a Riemannian registration algorithm to perform a pre-processing of symmetric and positive definite data such as those arising from medical imaging or brain computer interface. This algorithm, among other properties, provides a theoretical framework to focus the analysis on the eigenvalues of the analyzed matrices, allowing the employment of Euclidean methods for statistical inference also in a Riemannian context.
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Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: A Geometria da Informação é uma área da matemática que utiliza ferramentas geométricas no estudo de modelos estatísticos. Em 1945, Rao introduziu uma métrica Riemanniana no espaço das distribuições de probabilidade usando a matriz de informação, dada por Ronald Fisher em 1921. Com a métrica associada a essa matriz, define-se uma distância entre duas distribuições de probabilidade (distância de Rao), geodésicas, curvaturas e outras propriedades do espaço. Desde então muitos autores veem estudando esse assunto, que está naturalmente ligado a diversas aplicações como, por exemplo, inferência estatística, processos estocásticos, teoria da informação e distorção de imagens. Neste trabalho damos uma breve introdução à geometria diferencial e Riemanniana e fazemos uma coletânea de alguns resultados obtidos na área de Geometria da Informação. Mostramos a distância de Rao entre algumas distribuições de probabilidade e damos uma atenção especial ao estudo da distância no espaço formado por distribuições Normais Multivariadas. Neste espaço, como ainda não é conhecida uma fórmula fechada para a distância e nem para a curva geodésica, damos ênfase ao cálculo de limitantes para a distância de Rao. Conseguimos melhorar, em alguns casos, o limitante superior dado por Calvo e Oller em 1990
Abstract: Information Geometry is an area of mathematics that uses geometric tools in the study of statistical models. In 1945, Rao introduced a Riemannian metric on the space of the probability distributions using the information matrix provided by Ronald Fisher in 1921. With the metric associated with this matrix, we define a distance between two probability distributions (Rao's distance), geodesics, curvatures and other properties. Since then, many authors have been studying this subject, which is associated with various applications, such as: statistical inference, stochastic processes, information theory, and image distortion. In this work we provide a brief introduction to Differential and Riemannian Geometry and a survey of some results obtained in Information Geometry. We show Rao's distance between some probability distributions, with special atention to the study of such distance in the space of multivariate normal distributions. In this space, since closed forms for the distance and for the geodesic curve are not known yet, we focus on the calculus of bounds for Rao's distance. In some cases, we improve the upper bound provided by Calvo and Oller in 1990
Mestrado
Matematica Aplicada
Mestra em Matemática Aplicada
Studeny, Angelika C. "Quantifying biodiversity trends in time and space." Thesis, University of St Andrews, 2012. http://hdl.handle.net/10023/3414.
Huang, Ko-Kai Albert. "Novel statistical and geometric models for automated brain tissue labeling in magnetic resonance images." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/23707.