Tesi sul tema "Geometrisk statistik"

Nous proposons dans cette thèse de nouvelles méthodes de réduction de dimension fondées sur la géométrie différentielle. Il s'agit de trouver une représentation d'un ensemble d'observations dans un espace de dimension inférieure à l'espace d'origine des données. Les méthodes de réduction de dimension constituent la pierre angulaire des statistiques et ont donc un très large éventail d'applications. Dans les statistiques euclidiennes ordinaires, les données appartiennent à un espace vectoriel et l'espace de dimension inférieure peut être un sous-espace linéaire ou une sous-variété non linéaire approximant les observations. L'étude de telles variétés lisses, la géométrie différentielle, joue naturellement un rôle important dans ce dernier cas. Lorsque l'espace des données est lui-même une variété, l'espace approximant de dimension réduite est naturellement une sous-variété de la variété initiale. Les méthodes d'analyse de ce type de données relèvent du domaine des statistiques géométriques. Les statistiques géométriques pour des observations appartenant à une variété riemannienne sont le point de départ de cette thèse, mais une partie de notre travail apporte une contribution même dans le cas de données appartenant à l'espace euclidien, mathbb{R}^d.Les formes, dans notre cas des courbes ou des surfaces discrètes ou continues, sont un exemple important de données à valeurs dans les variétés. En biologie évolutive, les chercheurs s'intéressent aux raisons et aux implications des différences morphologiques entre les espèces. Cette application motive la première contribution principale de la thèse. Nous généralisons une méthode de réduction de dimension utilisée en biologie évolutive, l'analyse en composantes principales phylogénétiques (P-PCA), pour travailler sur des données à valeur dans une variété riemannienne - afin qu'elle puisse être appliquée à des données de forme. P-PCA est une version de PCA pour des observations qui sont les feuilles d'un arbre phylogénétique. D'un point de vue statistique, la propriété importante de ces données est que les observations ne sont pas indépendantes. Nous définissons et estimons des moyennes et des covariances intrinsèquement pondérées sur une variété qui prennent en compte cette dépendance des observations. Nous définissons ensuite l'ACP phylogénétique sur une variété comme la décomposition propre de la covariance pondérée dans l'espace tangent de la moyenne pondérée. Nous montrons que l'estimateur de moyenne actuellement utilisé en biologie évolutive pour étudier la morphologie correspond à ne prendre qu'une seule étape de notre algorithme de descente de gradient riemannien pour la moyenne intrinsèque, lorsque les observations sont représentées dans l'espace des formes de Kendall.Notre deuxième contribution principale est une méthode non paramétrique de réduction de dimension fondée sur une classe très flexible de sous-variétés qui est novatrice même dans le cas de données euclidiennes. Grâce à une PCA locale, nous construisons tout d'abord un sous-fibré du fibré tangent sur la variété des données que nous appelons le sous-fibré principal. Cette distribution (au sens géométrique) induit une structure sous riemannienne. Nous montrons que les géodésiques sous-riemanniennes correspondantes restent proches de l'ensemble des observations et que l'ensemble des géodésiques partant d'un point donné génèrent localement une sous-variété qui est radialement alignée avec le sous-fibré principal, même lorsqu'il est non intégrables, ce qui apparait lorsque les données sont bruitées. Notre méthode démontre que la géométrie sous-riemannienne est le cadre naturel pour traiter de tels problèmes. Des expériences numériques illustrent la puissance de notre cadre en montrant que nous pouvons réaliser des reconstructions d'une extension importante, même en présence de niveaux de bruit assez élevés
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

Recently geometric deep learning introduced a new way for machine learning algorithms to tackle point cloud data in its raw form. Pioneers like PointNet and many architectures building on top of its success realize the importance of invariance to initial data transformations. These include shifting, scaling and rotating the point cloud in 3D space. Similarly to our desire for image classifying machine learning models to classify an upside down dog as a dog, we wish geometric deep learning models to succeed on transformed data. As such, many models employ an initial data transform in their models which is learned as part of a neural network, to transform the point cloud into a global canonical space. I see weaknesses in this approach as they are not guaranteed to perform completely invariant to input data transformations, but rather approximately. To combat this I propose to use local deterministic transformations which do not need to be learned. The novelty layer of this project builds upon Edge Convolutions and is thus dubbed DirEdgeConv, with the directional invariance in mind. This layer is slightly altered to introduce another layer by the name of DirSplineConv. These layers are assembled in a variety of models which are then benchmarked against the same tasks as its predecessor to invite a fair comparison. The results are not quite as good as state of the art results, however are still respectable. It is also my belief that the results can be improved by improving the learning rate and its scheduling. Another experiment in which ablation is performed on the novel layers shows that the layers  main concept indeed improves the overall results.
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.

This Master Thesis is a product development project that has been carried out in cooperation with Scania CV AB. The purpose has been to find the relationship between geometric parameters and the swirl number for an XPI- cylinder head in order to improve the design specification. The increased focus on environmental issues from society leads to tighter emission legislations for the truck manufactures. In order to drive the development forward considerable resources are spent on research and development on diesel engines and its combustion process. The inlet air has a significant impact on the formation of emissions and the flow that is desired is called swirl. Swirl is achieved by making the air rotate around the vertical axis of the cylinder. Earlier cylinder heads that has been developed at Scania CV AB have shown clear relationships between geometric parameters of the cylinder head and the swirl number. With the new XPI- cylinder head observations have been made that other parameters also affect the swirl. By experience many parameters that affect the swirl and the airflow down the cylinder are known. Based on this knowledge and interviews with experts at Scania CV AB the parameters for the study have been developed and specified. After defining the geometric parameters, 33 in total, 120 cylinder heads were measured and swirl tested. The statistical investigation was carried out by Ekaterina Fetisova as a part of her bachelor degree project for the Department of Mathematical Statistics at the University of Stockholm. The investigation used multiple linear regression to develop six models that describes the relationships between the geometric parameters and the swirl number. The developed models provides an explanation rate between 54-60 % between the variance of the geometrical parameters and the variance of the swirl. A few parameters stand out as more important than others, some parameters more expected than others. Conclusions that are drawn are that the investigation probably is missing important parameters and that there is a great complexity between the geometric parameters and the swirl number. To get a better understanding of the impact that each channel has on the swirl number recommendations are made to swirl test one channel at a time and to find the relationship between the geometric parameters for each channel and the swirl number.
Detta examensarbete är ett produktutvecklingsarbete som har genomförts i samarbete med Scania CV AB. Syftet med arbetet har varit att finna ett samband mellan XPI- cylinderhuvudets geometriska parametrar och dess snurrtal för att förbättra konstruktionsunderlaget för cylinderhuvudet. Samhällets ökade fokus kring miljöfrågorna för med sig allt hårdare emissionslagstiftningar för lastbilstillverkarna. För att driva utvecklingen framåt läggs stora resurser på forskning och utveckling av dieselmotorn och dess förbränningsförlopp. Insugningsluftens strömning in i cylindern har en stor påverkan vid bildandet av emissioner och den typ av strömningen som söks kallas för snurr. Snurr fås då luften strömmar runt cylinderns vertikala axel inne i förbränningsrummet. På tidigare cylinderhuvuden som Scania CV AB tagit fram har klara samband setts mellan snurren och geometriska parametrar hos cylinderhuvudet men med det nya XPI- cylinderhuvudet ses att även andra parametrar är med och påverkar snurren. Av erfarenhet är det känt att flertalet geometrier påverkar snurren och flödet som fås ner i cylindern. Utifrån dessa kunskaper och intervjuer med sakkunniga på Scania CV AB har de parametrar som ska ingå i studien tagits fram. Efter definiering av de geometriska parametrarna, 33 stycken, har 120 cylinderhuvuden mätts upp och snurrtestats. Den statistiska undersökningen har genomförts av Ekaterina Fetisova som en del av hennes kandidatexamensprojekt för institutionen Matematisk Statistik på Stockholms Universitet. Undersökningen ledde till att sex modeller togs fram, med hjälp av multipel linjär regression, som representerar de geometriska parametrarnas påverkan på snurrtalet. De framtagna modellerna förklarar mellan 54-60 % av variationen hos snurren och ett fåtal geometriska parametrar sticker ut som viktigare än andra, vissa mer förväntade än andra. Slutsatser som dras är att viktiga parametrar troligtvis saknas i undersökningen och att det finns en stor komplexitet mellan de geometriska parametrarna och snurrtalet. För att få en bättre förståelse för kanalernas individuella påverkan på snurren rekommenderas att varje kanal snurrtestas var och en för sig och kopplas mot de geometriska parametrarna för respektive kanal.

La geometria differenziale è un insieme di strumenti che permette di compiere le tipiche operazioni di algebra e calcolo anche in spazi che non seguono le normali regole Euclidee degli spazi vettoriali, ad esempio come i punti di una superficie curva. Questo campo della matematica sta assumendo sempre maggiore rilevanza in vari ambiti, fra cui statistica e machine learning, a causa dell’enorme disponibilità di dati che appartengono a domini sempre più complessi. Un esempio di dominio di questo tipo è l’insieme delle matrici simmetriche e definite positive, ovvero le matrici di covarianza, che compaiono frequentemente nella diagnostica medica per immagini e sono spesso usate come spazio parametrico nei modelli statistici. Lo scopo di questa tesi è quello di raccogliere e organizzare la conoscenza sparsa sulla geometria Riemanniana delle matrici simmetriche e definite positive e di costruire delle tecniche pratiche, usando gli strumenti della geometria differenziale, che possano essere applicate direttamente in contesti di analisi statistica. Questo obiettivo è stato perseguito attraverso lo sviluppo di due metodi: il primo è un algoritmo di tipo quasi-Newton per l’ottimizzazione Riemanniana che può essere utilizzato in qualsiasi situazione in cui sia necessaria la massimizzazione di funzioni di matrici simmetriche e definite positive, come quelle che emergono nel contesto della inferenza di verosimiglianza e nella approssimazione variazionale. Il secondo è un algoritmo di registrazione Riemanniana per eseguire il pre-processamento di dati simmetrici e definiti positivi come quelli che si ottengono nella diagnostica medica per immagini o nelle interfacce cervello-computer. Questo algoritmo, fra le altre sue proprietà, fornisce una struttura teorica che consente di concentrare l’analisi sugli autovalori delle matrici analizzate, permettendo l’utilizzo di metodi Euclidei per l’inferenza statistica anche in un contesto Riemanniano.
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.

All the results of the present thesis have been obtained facing problems related to the study of the so called birth-and-growth stochastic processes, relevant in several real applications, like crystallization processes, tumour growth, angiogenesis, etc. We have introduced a Delta formalism, à la Dirac-Schwartz, for the description of random measures associated with random closed sets in R^d of lower dimensions, such that the usual Dirac delta at a point follows as particular case, in order to provide a natural framework for deriving evolution equations for mean densities at integer Hausdorff dimensions in terms of the relevant kinetic parameters associated to a given birth-and-growth process. In this context connections with the concepts of hazard functions and spherical contact distribution functions, together with local Steiner formulas at first order have been studied and, under suitable general conditions on the resulting random growing set, we may write evolution equations of the mean volume density in terms of the growing rate and of the mean surface density. To this end we have introduced definitions of discrete, continuous and absolutely continuous random closed set, which extend the standard well known definitions for random variables. Further, since in many real applications such as fibre processes, n-facets of random tessellations several problems are related to the estimation of such mean densities, in order to face such problems in the general setting of spatially inhomogeneous processes, we have analyzed an approximation of mean densities for sufficiently regular random closed sets, such that some known results in literature follow as particular cases.

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.

This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the 
Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots) 
and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South 
African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric 
Brownian motion time series should be characterised by the Hurst exponent of ½
. A value of a Hurst exponent different from that would indicate the presence of long memory or 
fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by 
assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the 
rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.

The thesis is focussed on the reconstruction of a damaged skull represented by a polygonal model. The reconstruction is based on a statistical shape model of the skull. The thesis covers the registration of skulls by using a thin-plate spline method, aligning polygonal models by generalized procrustes analysis, the identification of missing parts of a skull by means of statistical shape models outliers analysis. Finally, missing parts of the skull are reconstructed and the accuracy of the reconstruction is estimated.

Orientador: João Eloir Strapasson
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
Made available in DSpace on 2018-08-23T13:44:50Z (GMT). No. of bitstreams: 1 Porto_JuliannaPineleSantos_M.pdf: 2346170 bytes, checksum: 9f8b7284329ef1eb2f319c2e377b7a3c (MD5) Previous issue date: 2013
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

The aim of this thesis is to produce computational experiments related to a Central Limit Theorem (CLT) for random measures proved by Penrose in [4]. In particular we analyze some random measures associated with a segment process, a Voronoi tessellation and a Johnson-Mehl tessellation constructed on an underlying 2-dimensional point process. In the result by Penrose, this point process is supposed to be Poisson non-homogeneous (or Binomial, but we are not going to study this case). In this thesis, we also consider the case of an underlying Neyman-Scott cluster process. The main references are [4] and references therein; before this CLT, in [5] Penrose obtained a Law of Large Numbers for random measures. These results are an extension of [1], where Baryshnikov and Yukich proved similar theorems under stricter assumptions on the radius of stabilization of the considered random measure. There are also other results concerning CLTs on random measures(for example in [2]), but processes are always assumed to be homogeneous; through Baryshnikov, Yukich and Penrose’s approach it is possible to relax the homogeneity conditions, so that their hypotheses are more likely to be true for applications. The main difference between other CLTs for random measures and this new approach is that the limit is not considered as the window of observation increases (which is a problem, because we should have homogeneity conditions of the random process on the whole space), but as the intensity of the process increases (just one realization of a process with sufficiently large intensity gives information on the typical features of the process). In Baryshnikov, Yukich and Penrose’s works new theoretical results and some examples are presented, but simulations are not included; moreover, the hypothesis of Poisson distribution of the underlying point process is extended only to the case of a Binomial point process. This thesis is a computational approach to the theorem, which may lead to possible future generalizations from a theoretical point of view, including more general point processes, such as the cluster ones. In our work we analyze the behaviour of some particular random measures as the intensity of the underlying point process increases. We find some advantages and disadvantages of Penrose’s approach; we have a validation of the theorem through our simulations by observing a Gaussian trend of our data (and by 2 performing some Lilliefors tests), and we also obtain some Laws of Large Numbers. Our computational approach has been carried out for measures depending on homogeneous Poisson point processes, non-homogeneous ones and Neyman-Scott ones. References [1] Yu Baryshnikov, J.E.Yukich, Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 2005, 213-253. [2] V. Benes, J. Rataj, Stochastic Geometry: Selected Topics. Kluwer Academic Publishers, Boston, 2004.[3] V.Capasso, M.Burger, A.Micheletti, C.Salani, Mathematical models for polymer crystallization processes. In Mathematical Modelling for Polymer Processing (V. Capasso, Ed). Springer, Heidelberg, 2000, 167-242. [4] M. D. Penrose, Gaussian limits for random geometric measures. Electronic Journal of Probability 12, 2007, 989-1035. [5] M. D. Penrose, Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 2007, 1124-1150. 4

ABSTRACT It may be of no surprise that water quality data is right-skewed, but what appears to be overlooked by some is that the arithmetic mean and standard deviation most often fail as measures of central tendency in skewed data. When using the arithmetic mean and arithmetic standard deviation with nutrient data, one standard deviation about the arithmetic mean can capture nearly all of the data and extend into negative values. Representing nutrient data this way can be misleading to viewers who are using the statistics, and making assumptions, to understand the characteristics of those waters. Through an in-depth statistical analysis of Florida's nitrogen and phosphorus data, I have found the geometric mean and multiplicative standard deviation capture a better representation of the central region of skewed data. Including the geometric mean and multiplicative standard deviation in the descriptive statistics of nutrient data is relatively simple with today's tools and helps to better describe the data. Adding these statistics can contribute to more effective understanding of nutrient concentrations, better application of data, and the development of better data-derived policy. While the suggestions of this paper are by no means original, it is with added evidence provided by the study of the skewness, distributions, and central regions of 53 nutrient data sets that I intend to help reiterate the argument that a few additional descriptive statistics can greatly empower the communication of data, and because of the ease with which they can now be calculated, there is no excuse to ignore them.

La modélisation géométrique a été l'un des sujets les plus étudiés pour la représentation des structures anatomiques dans le domaine médical. Aujourd'hui, il n'y a toujours pas de méthode bien établie pour modéliser la forme d'un organe. Cependant, il y a plusieurs types d'approches disponibles et chaque approche a ses forces et ses faiblesses. La plupart des méthodes de pointe utilisent uniquement l'information surfacique mais un besoin croissant de modéliser l'information volumique des objets apparaît. En plus de la description géométrique, il faut pouvoir différencier les objets d'une population selon leur forme. Cela nécessite de disposer des statistiques sur la forme dans organe dans une population donné. Dans ce travail de thèse, on utilise une représentation capable de modéliser les caractéristiques surfaciques et internes d'un objet. La représentation choisie (s-rep) a en plus l'avantage de permettre de déterminer les statistiques de forme pour une population d'objets. En s'appuyant sur cette représentation, une procédure pour modéliser le cortex cérébral humain est proposée. Cette nouvelle modélisation offre de nouvelles possibilités pour analyser les lésions corticales et calculer des statistiques de forme sur le cortex. La deuxième partie de ce travail propose une méthodologie pour décrire de manière paramétrique l'intérieur d'un objet. La méthode est flexible et peut améliorer l'aspect visuel ou la description des propriétés physiques d'un objet. La modélisation géométrique enrichie avec des paramètres physiques volumiques est utilisée pour la simulation d'image par résonance magnétique pour produire des simulations plus réalistes. Cette approche de simulation d'images est validée en analysant le comportement et les performances des méthodes de segmentations classiquement utilisées pour traiter des images réelles du cerveau
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

La première partie de cette thèse concerne l'inférence de modèles statistiques non normalisés. Nous étudions deux méthodes d'inférence basées sur de l'échantillonnage aléatoire : Monte-Carlo MLE (Geyer, 1994), et Noise Contrastive Estimation (Gutmann et Hyvarinen, 2010). Cette dernière méthode fut soutenue par une justification numérique d'une meilleure stabilité, mais aucun résultat théorique n'avait encore été prouvé. Nous prouvons que Noise Contrastive Estimation est plus robuste au choix de la distribution d'échantillonnage. Nous évaluons le gain de précision en fonction du budget computationnel. La deuxième partie de cette thèse concerne l'échantillonnage aléatoire approché pour les distributions de grande dimension. La performance de la plupart des méthodes d’échantillonnage se détériore rapidement lorsque la dimension augmente, mais plusieurs méthodes ont prouvé leur efficacité (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). Dans la continuité de certains travaux récents (Eberle et al., 2017 ; Cheng et al., 2018), nous étudions certaines discrétisations d’un processus connu sous le nom de kinetic Langevin diffusion. Nous établissons des vitesses de convergence explicites vers la distribution d'échantillonnage, qui ont une dépendance polynomiale en la dimension. Notre travail améliore et étend les résultats de Cheng et al. pour les densités log-concaves
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

Trois problèmes sont abordés dans cette thèse: l'inférence en régression multi-tâche de grande dimension, les quantiles géométriques dans les espaces normés de dimension infinie, et les moyennes de Fréchet généralisées dans les arbres métriques. Premièrement, nous considérons un modèle de régression multi-tâche avec une hypothèse de sparsité sur les lignes de la matrice paramètre. L'estimation est faite en haute dimension avec l'estimateur Lasso multi-tâche. Afin de corriger le biais induit par la pénalité, nous introduisons un nouvel objet dépendant uniquement des données que nous appelons matrice d'interaction. Cet outil nous permet d'établir des résultats asymptotiques avec des lois limites normales ou chi². Il en découle des intervalles de confiance et des ellipsoïdes de confiance, qui sont valides dans des régimes de sparsité qui ne sont pas couverts par la littérature existante. Deuxièmement, nous étudions le quantile géométrique, qui généralise le quantile classique au cadre des espaces normés. Nous commençons par fournir de nouveaux résultats sur l'existence et l'unicité des quantiles géométriques. L'estimation est effectuée avec un M-estimateur approché et nous examinons ses propriétés asymptotiques en dimension infinie. Quand le quantile théorique n'est pas unique, nous utilisons la théorie de la convergence variationnelle pour obtenir des résultats asymptotiques sur les sous-suites dans la topologie faible. Quand le quantile théorique est unique, nous montrons que l'estimateur est consistant pour la topologie de la norme dans une large classe d'espaces de Banach, en particulier dans les espaces séparables et uniformément convexes. Dans les Hilbert séparables nous démontrons des représentations de Bahadur-Kiefer de l'estimateur, dont découle immédiatement la normalité asymptotique à la vitesse paramétrique. Finalement, nous considérons des mesures de tendance centrale pour des données vivant sur un réseau, qui est modélisé par un arbre métrique. Les paramètres de localisation que nous étudions sont appelés moyennes de Fréchet généralisées: elles sont obtenues en remplaçant le carré dans la définition de la moyenne de Fréchet par une fonction de perte convexe et croissante. Nous élaborons une notion de dérivée directionnelle dans l'arbre, ce qui nous aide à localiser et caractériser les minimiseurs. Nous examinons les propriétés statistiques du M-estimateur correspondant: nous étendons le concept de moyenne collante au contexte des arbres métriques, puis nous obtenons un théorème collant non-asymptotique et une loi des grands nombres collante. Pour la médiane de Fréchet, nous établissons des bornes de concentration non-asymptotiques et des théorèmes central limite collants
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

Dans le cadre de l'apprentissage automatique, l'utilisation de représentations alternatives, ou descripteurs, pour les données est un problème fondamental permettant d'améliorer sensiblement les résultats des algorithmes. Parmi eux, les descripteurs topologiques calculent et encodent l'information de nature topologique contenue dans les données géométriques. Ils ont pour avantage de bénéficier de nombreuses bonnes propriétés issues de la topologie, et désirables en pratique, comme par exemple leur invariance aux déformations continues des données. En revanche, la structure et les opérations nécessaires à de nombreuses méthodes d'apprentissage, comme les moyennes ou les produits scalaires, sont souvent absents de l'espace de ces descripteurs. Dans cette thèse, nous étudions en détail les propriétés métriques et statistiques des descripteurs topologiques les plus fréquents, à savoir les diagrammes de persistance et Mapper. En particulier, nous montrons que le Mapper, qui est empiriquement un descripteur instable, peut être stabilisé avec une métrique appropriée, que l'on utilise ensuite pour calculer des régions de confiance et pour régler automatiquement ses paramètres. En ce qui concerne les diagrammes de persistance, nous montrons que des produits scalaires peuvent être utilisés via des méthodes à noyaux, en définissant deux noyaux, ou plongements, dans des espaces de Hilbert en dimension finie et infinie
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

The main purpose of this thesis is to compare the performance of the score test and the likelihood ratio test by computing type I errors and type II errors when the tests are applied to the geometric distribution and inflated binomial distribution. We first derive test statistics of the score test and the likelihood ratio test for both distributions. We then use the software package R to perform a simulation to study the behavior of the two tests. We derive the R codes to calculate the two types of error for each distribution. We create lots of samples to approximate the likelihood of type I error and type II error by changing the values of parameters. In the first chapter, we discuss the motivation behind the work presented in this thesis. Also, we introduce the definitions used throughout the paper. In the second chapter, we derive test statistics for the likelihood ratio test and the score test for the geometric distribution. For the score test, we consider the score test using both the observed information matrix and the expected information matrix, and obtain the score test statistic zO and zI . Chapter 3 discusses the likelihood ratio test and the score test for the inflated binomial distribution. The main parameter of interest is w, so p is a nuisance parameter in this case. We derive the likelihood ratio test statistics and the score test statistics to test w. In both tests, the nuisance parameter p is estimated using maximum likelihood estimator pˆ. We also consider the score test using both the observed and the expected information matrices. Chapter 4 focuses on the score test in the inflated binomial distribution. We generate data to follow the zero inflated binomial distribution by using the package R. We plot the graph of the ratio of the two score test statistics for the sample data, zI /zO , in terms of different values of n0, the number of zero values in the sample. In chapter 5, we discuss and compare the use of the score test using two types of information matrices. We perform a simulation study to estimate the two types of errors when applying the test to the geometric distribution and the inflated binomial distribution. We plot the percentage of the two errors by fixing different parameters, such as the probability p and the number of trials m. Finally, we conclude by briefly summarizing the results in chapter 6.

La première partie de cette thèse concerne l'inférence de modèles statistiques non normalisés. Nous étudions deux méthodes d'inférence basées sur de l'échantillonnage aléatoire : Monte-Carlo MLE (Geyer, 1994), et Noise Contrastive Estimation (Gutmann et Hyvarinen, 2010). Cette dernière méthode fut soutenue par une justification numérique d'une meilleure stabilité, mais aucun résultat théorique n'avait encore été prouvé. Nous prouvons que Noise Contrastive Estimation est plus robuste au choix de la distribution d'échantillonnage. Nous évaluons le gain de précision en fonction du budget computationnel. La deuxième partie de cette thèse concerne l'échantillonnage aléatoire approché pour les distributions de grande dimension. La performance de la plupart des méthodes d’échantillonnage se détériore rapidement lorsque la dimension augmente, mais plusieurs méthodes ont prouvé leur efficacité (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). Dans la continuité de certains travaux récents (Eberle et al., 2017 ; Cheng et al., 2018), nous étudions certaines discrétisations d’un processus connu sous le nom de kinetic Langevin diffusion. Nous établissons des vitesses de convergence explicites vers la distribution d'échantillonnage, qui ont une dépendance polynomiale en la dimension. Notre travail améliore et étend les résultats de Cheng et al. pour les densités log-concaves
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

Studien undersöker vilka digitala resurser som matematiklärare använder i matematikundervisningen i grundskolan, möjligheter samt hinder. Då digitala resurser kan nyttjas på olika nivåer analyseras även vilken nivå som undervisningen ligger på. Fokus har varit att identifiera lärare som använder dynamiska programvaror och/eller digital teknik. Studien är en fallstudie, genomförd genom en inledande enkät för att identifiera lärare att intervjua. Efter intervjuer har de aktiviteter som lärare anger att de använder digitala resurser till eller ser möjligheter i analyserats för att se på vilken nivå de kan anses utveckla och förnya undervisningen. Studien visar att lärare i vissa fall använder digitala resurser i form av dynamisk programvara på en hög nivå, men att flertalet aktiviteter har potential att utvecklas. Utbildning av lärare i användandet av dynamisk programvara är en framgångsfaktor och krävs för att matematiklärare ska nå högre nivå på undervisningen.

Matematik

The study of human evolution centres, to a large extent, around the study of fossil morphology, including the comparison and interpretation of these remains within the context of what is known about morphological variation within living species. However, many fossils suffer from environmentally caused damage (taphonomic distortion) which hinders any such interpretation: fossil material may be broken and fragmented while the weight and motion of overlaying sediments can cause their plastic distortion. To date, a number of studies have focused on the reconstruction of such taphonomically damaged specimens. These studies have used myriad approaches to reconstruction, including thin plate spline methods, mirroring, and regression-based approaches. The efficacy of these techniques remains to be demonstrated, and it is not clear how different parameters (e.g., sample sizes, landmark density, etc.) might effect their accuracy. In order to partly address this issue, this thesis examines three techniques used in the virtual reconstruction of fossil remains by statistical or geometrical means: mean substitution, thin plate spline warping (TPS), and multiple linear regression. These methods are compared by reconstructing the same sample of individuals using each technique. Samples drawn from Homo sapiens, Pan troglodytes, Gorilla gorilla, and various hominin fossils are reconstructed by iteratively removing then estimating the landmarks. The testing determines the methods' behaviour in relation to the extant of landmark loss (i.e., amount of damage), reference sample sizes (this being the data used to guide the reconstructions), and the species of the population from which the reference samples are drawn (which may be different to the species of the damaged fossil). Given a large enough reference sample, the regression-based method is shown to produce the most accurate reconstructions. Various parameters effect this: when using small reference samples drawn from a population of the same species as the damaged specimen, thin plate splines is the better method, but only as long as there is little damage. As the damage becomes severe (missing 30% of the landmarks, or more), mean substitution should be used instead: thin plate splines are shown to have a rapid error growth in relation to the amount of damage. When the species of the damaged specimen is unknown, or it is the only known individual of its species, the smallest reconstruction errors are obtained with a regression-based approach using a large reference sample drawn from a living species. Testing shows that reference sample size (combined with the use of multiple linear regression) is more important than morphological similarity between the reference individuals and the damaged specimen. The main contribution of this work are recommendations to the researcher on which of the three methods to use, based on the amount of damage, number of reference individuals, and species of the reference individuals.

Gegenstand der Arbeit sind zufällige Mengen $Xi$ aus dem erweiterten Konvexring und zugehörige markierte Punktprozesse $Psi$ in $mathbb{R}^d$ mit Marken aus dem System der konvexen Körper. Es wird gezeigt, unter welchen Voraussetzungen an $Psi$ die zweite Produktdichte $varrho_S^{(2)}$ des durch $Xi$ induzierten zufälligen Oberflächenmaßes $S_{Xi}$ existiert und eine klassische Beziehung zwischen der Intensitätsfunktion von $S_{Xi}$ und der Ableitung der sphärischen Kontaktverteilungsfunktion von $Xi$ bei Null auf entsprechende Größen zweiter Ordnung übertragen werden kann. Mit Hilfe des so erhaltenen Zugangs wird $varrho_S^{(2)}$ für einige Beispiele bestimmt. Desweiteren werden spezielle markierte Punktprozesse $Psi$ betrachtet, die durch Verdünnung gemäß einer Methode nach Matérn aus einem markierten Poisson-Prozess hervorgehen. Als praktische Anwendung wird für zwei Proben eines Feuerbetons mit kugelförmigen Einschlüssen untersucht, welche Modelle für zufällige Systeme harter Kugeln zur Beschreibung geeignet sind.

This textbook is intended for a two-semester course on calculus of one variable. The target audience is comprised of first-year students in biology, chemistry, physics and other related disciplines. The title of the book reflects the fact that it is not limited to one single approach to calculus. Rather, we use graphing calculators or applications whenever they are necessary to introduce certain topics. Nonetheless, as expected, a conceptual framework permeates the whole book. A distinctive characteristic of the book is the early introduction of sequences and geometric series, and a gradual development of simple differential equations, as well as the use of linear regression to analyze data. The core of the book is to be found in the first three chapters, in which examples from biology, chemistry and physics are analyzed with care, emphasizing the close links between calculus and the natural sciences. The last two chapters, or sections thereof, can be used as a sort of capstone in order to show how mathematics helps in the understanding of enzyme kinetics and transport across cell membranes.
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[IT] Negli ultimi 200.000 anni, la specie umana si è diffusa in tutta la Terra, adattando la sua morfologia e fisiologia a un'ampia gamma di habitat. Lo scheletro umano ha quindi registrato i principali effetti ambientali e di conseguenza i reperti scheletrici assumono grande importanza nell'indagine dei processi evolutivi. Oggi le moderne tecniche di indagini quantitative delle principali caratteristiche morfologiche consentono di metterle in relazione con la variabilità genetica. La posizione geografica della Sicilia, l'isolamento e la sua lunga e dinamica storia di colonizzazione (diversi e numerosi contributi culturali e biologici) hanno creato un contesto peculiare che consente uno studio antropologico unico, utile per sottrarre informazioni importanti sul “Flusso Migratorio” e il conseguente "Influenza delle Popolazioni" sui resti scheletrici umani. Questo progetto si basa sull'analisi antropologica delle ossa umane provenienti da diverse popolazioni (indigene e colonizzatori) distribuite dal Paleolitico all'Età Contemporanea. Le più moderne tecniche di Analisi Geometria Morfometrica (ricostruzione 3D) e di Analisi Statistica Multivariata sono state applicate su tre diversi caratteri scheletrici (Denti, Crani e Statura). L'obiettivo del progetto è quello di eseguire un'ampia analisi della Biodiversità Umana Siciliana al fine di: - Analizzare i dati odontometrici 2D con tecniche multivariate per esplorare le relazioni tra i popoli nel corso dei secoli. - Usare modelli 3D e la morfometria cranio-facciale per studiare la complesso variabilità morfologica relativa alle influenze dei flussi migratori. - Valutare il Secular Trend della Statura. - Usare questi tre caratteri per fornire una panoramica generale della Biodiversità Umana in Sicilia. Il nostro lavoro denota l'affidabilità dei metodi impiegati e come in uno studio sulla biodiversità diversi caratteri sono indispensabili per comprendere il processo evolutivo. I dati forniti dimostrano anche la correlazione tra i caratteri morfologici XI e l'influenza esercitata (non solo dai fattori ambientali) dal flusso umano sul fenotipo. I risultati mostrano chiaramente come tutti i caratteri valutati siano coinvolti allo stesso tempo nello stesso processo di diversificazione. Le variazioni morfologiche mostrano una generale diminuzione del prognatismo mascellare e una leggera mesocefalizzazione con il cranio che diventa più stretto e leggermente e meno allungato e il viso che diventa più largo e più corto. Considerando sempre l’influenza del rapporto dimensione/composizione sia l'analisi statistica canonica che quella multivariata, supportano la teoria che i coloni del Paleolitico superiore di San Teodoro potrebbero ragionevolmente essere la prima prova di colonizzazione umana in Sicilia (questa teoria è anche supportata dai campioni Mesolitici che clusterizzano separati dai primi). Significativi sono i periodi del Bronzo della transizione Bronzo/Ferro nei quali assistiamo ad importanti cambiamenti morfologici (Denti, Crani e Stature) dovuti a “Flussi Migratori” costanti e numericamente significativi. Questa variazione coincide esattamente con i primi “Afflussi di Popolazione” stabili conseguenti alle migrazioni umane dal continente. Tuttavia i campioni preistorici di alcune popolazioni, conservano alcuni caratteri arcaici anche dopo l'Età del Ferro (Era Storica) mentre la "Continuità di Popolazione" (conseguente alla convivenza e agli alternamenti delle diverse colonizzazioni) dall'Antichità al Medioevo ha prodotto un progressivo aumento della variabilità senza grandi variazione tra Eignevalue e Componenti Principali. L'assenza di relazione interna causata dall'intricato periodo di colonizzazione è invece presente sul campione preistorico sul quale si riscontra una netta variazione tra i PC. Le correlazioni tra "Afflusso di popolazione" e Variabilità sono osservabili anche nell'influenza dei coloni islamici sugli indigeni durante il Medioevo. Tuttavia, l'ampia variabilità e il morfospazio omogeneo mostrano che dopo questi gruppi (fino ai Contemporanei) sono riconoscibili popolazioniben definite.
[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

La presente tesis doctoral se centra en el diseño, implementación y aplicación de técnicas probalbilísticas y estadísticas para el análisis espacial y la evaluación de la incertidumbre, en el contexto de un problema de palaeoecología humana. La razón técnica de tal estrategia de investigación se debe a que los patrones de datos arqueológicos han sido afectados por procesos tafonómicos y por restricciones a la captura de datos, los cuales pueden identificarse pero no reconstruirse con total certidumbre. En la definición de problemáticas específicas a partir de tales cuestiones fijamos tres objetivos principales para esta tesis.

Sviluppo di un modello che fornisce l'analisi numerica dei parametri che definiscono il profilo del laminato a caldo piano e il relativo output grafico. Il modello è utilizzato per indagare l'incidenza del difetto "canale di laminazione" e per identificare le specifiche di acquisto della materia prima per l'azienda. Vengono infine proposte alcune tecniche di controllo statistico di processo, per tenere monitorate le caratteristiche della materia prima in ingresso.

La these comporte deux parties independantes. La premiere partie traite de l'utilisation de la geometrie differentielle en statistique. Dans les chapitres i et ii sont rappelees les notions fondamentales de geometrie differentielle et de statistique. Le chapitre iii est consacre a la theorie des "strings" dont de nombreux exemples apparaissent en statistique inferentielle. Ce nouveau concept a ete introduit et etudie par m. M. O. E. Barndorff nielsen et p. Blaesild. Nous en donnons ici une nouvelle definition, purement mathematique, basee sur un concept de differenciation d'ordre superieur, les objets differencies etant des fonctions, champs de vecteurs tangents, contangents ou jets. Enfin, dans le chapitre iv, a partir de structures geometriques specifiques definies sur des modeles statistiques parametriques reguliers et basees sur un point de vue de conditionnement pour une statistique ancillaire donnee, nous elaborons des developpements asymptotiques pour les lois du vecteur score et de l'estimateur du maximum de vraisemblance. La seconde partie concerne les familles exponentielles naturelles k-dimensionnelles et les fonctions-variance qui les caracterisent. Dans ce contexte nous etablissons dans le chapitre v un theoreme de convergence qui montre que l'ensemble des fonctions variances est ferme pour la convergence uniforme sur tout compact

Craters formed by impact and volcanic processes are among the most fundamental planetary landforms. This study examines the morphology of diverse craterforms on Io, the Moon, Mars, and Earth using quantitative, outline-based shape analysis and multivariate statistical methods to evaluate the differences between different types of. Ultimately, this should help establish relationships between the form and origin of craterforms. Developed in the field of geometric morphometrics by paleontological and biological sciences communities, these methods were used for the analysis of the shapes of crater outlines. The shapes of terrestrial ash-flow calderas, terrestrial basaltic shield calderas, martian calderas, Ionian paterae, and lunar impact craters were quantified and compared. Specifically, we used circularity, ellipticity, elliptic Fourier analysis (EFA), Zahn and Roskies (Z-R) shape function, and diameter. Quantitative shape descriptors obtained from EFA yield coefficients from decomposition of the Fourier series that separates the vertical and horizontal components among the outline points for each shape. The shape descriptors extracted from Z-R analysis represent the angular deviation of the shapes from a circle. These quantities were subjected to multivariate statistical analysis including principal component analysis (PCA) and discriminant analysis, to examine maximum differences between each a priori established group. Univariate analyses of morphological quantities including diameter, circularity, and ellipticity, as well as multivariate analyses of elliptic Fourier coefficients and Z-R shape function angular quantities show that ash-flow calderas and paterae on Io, as well as basaltic shield calderas and martian calderas, are most similar in shape. Other classes of craters are also shown to be statistically distinct from one another. Multivariate statistical models provide successful classification of different types of craters. Three classification models were built with overall successful classification rates ranging from 90% to 75%, each conveying different shape information. The EFA model including coefficients from the 2nd to 10th harmonic was the most successful supervised model with the highest overall classification rate and most successful predictive group membership assignments for the population of examined craterforms. Multivariate statistical methods and classification models can be effective tools for analyzing landforms on planetary surfaces and geologic morphology. With larger data sets used to enhance supervision of the model, more successful classification by the supervised model could likely reveal clues to the formation and variables involved in the genesis of landforms.

Stochastic hybrid dynamic systems that incorporate both continuous and discrete dynamics have been an area of great interest over the recent years. In view of applications, stochastic hybrid dynamic systems have been employed to diverse fields of studies, such as communication networks, air traffic management, and insurance risk models. The aim of the present study is to investigate properties of some classes of stochastic hybrid dynamic systems. The class of stochastic hybrid dynamic systems investigated has random jumps driven by a non-homogeneous Poisson process and deterministic jumps triggered by hitting the boundary. Its real-valued continuous dynamic between jumps is described by stochastic differential equations of the It\^o-Doob type. Existing results of piecewise deterministic models are extended to obtain the infinitesimal generator of the stochastic hybrid dynamic systems through a martingale approach. Based on results of the infinitesimal generator, some stochastic stability results are derived. The infinitesimal generator and stochastic stability results can be used to compute the higher moments of the solution process and find a bound of the solution. Next, the study focuses on a class of multidimensional stochastic hybrid dynamic systems. The continuous dynamic of the systems under investigation is described by a linear non-homogeneous systems of It\^o-Doob type of stochastic differential equations with switching coefficients. The switching takes place at random jump times which are governed by a non-homogeneous Poisson process. Closed form solutions of the stochastic hybrid dynamic systems are obtained. Two important special cases for the above systems are the geometric Brownian motion process with jumps and the Ornstein-Uhlenbeck process with jumps. Based on the closed form solutions, the probability distributions of the solution processes for these two special cases are derived. The derivation employs the use of the modal matrix and transformations. In addition, the parameter estimation problem for the one-dimensional cases of the geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps are investigated. Through some existing and modified methods, the estimation procedure is presented by first estimating the parameters of the discrete dynamic and subsequently examining the continuous dynamic piecewisely. Finally, some simulated stochastic hybrid dynamic processes are presented to illustrate the aforementioned parameter-estimation methods. One simulated insurance example is given to demonstrate the use of the estimation and simulation techniques to obtain some desired quantities.

At the core of successful manipulation and computation over large geometric data is the notion of approximation, both structural and computational. The focus of this thesis will be on the combinatorial and algorithmic aspects of approximations of point-set data P in d-dimensional Euclidean space. It starts with a study of geometric data depth where the goal is to compute a point which is the 'combinatorial center' of P. Over the past 50 years several such measures of combinatorial centers have been proposed, and we will re-examine several of them: Tukey depth, Simplicial depth, Oja depth and Ray-Shooting depth. This can be generalized to approximations with a subset, leading to the notion of epsilon-nets. There we will study the problem of approximations with respect to convexity. Along the way, this requires re-visiting and generalizing some basic theorems of convex geometry, such as the Caratheodory's theorem. Finally we will turn to the algorithmic aspects of these problems. We present a polynomial-time approximation scheme for computing hitting-sets for disks in the plane. Of separate interest is the technique, an analysis of local-search via locality graphs. A further application of this technique is then presented in computing independent sets in intersection graphs of rectangles in the plane.

Estimativa e simulação baseados na estatística de dois pontos têm sido usadas desde a década de 1960 na análise geoestatístico. Esses métodos dependem do modelo de correlação espacial derivado da bem conhecida função semivariograma. Entretanto, a função semivariograma não pode descrever a heterogeneidade geológica encontrada em depósitos minerais e reservatórios de petróleo. Assim, ao invés de usar a estatística de dois pontos, a geoestatística multi-pontos, baseada em distribuições de probabilidade de múltiplo pontos, tem sido considerada uma alternativa confiável para descrição da heterogeneidade geológica. Nessa tese, o algoritmo multi-ponto é revisado e uma nova solução é proposta. Essa solução é muito melhor que a original, pois evita usar as probabilidades marginais quando um evento que nunca ocorre é encontrado no template. Além disso, para cada realização a zona de incerteza é ressaltada. Uma base de dados sintética foi gerada e usada como imagem de treinamento. A partir dessa base de dados completa, uma amostra com 25 pontos foi extraída. Os resultados mostram que a aproximação proposta proporciona realizações mais confiáveis com zonas de incerteza menores.
Estimation and simulation based on two-point statistics have been used since 1960\'s in geostatistical analysis. These methods depend on the spatial correlation model derived from the well known semivariogram function. However, the semivariogram function cannot describe the geological heterogeneity found in mineral deposits and oil reservoirs. Thus, instead of using two-point statistics, multiple-point geostatistics based on probability distributions of multiple-points has been considered as a reliable alternative for describing the geological heterogeneity. In this thesis, the multiple-point algorithm is revisited and a new solution is proposed. This solution is much better than the former one because it avoids using marginal probabilities when a never occurring event is found in a template. Moreover, for each realization the uncertainty zone is highlighted. A synthetic data base was generated and used as training image. From this exhaustive data set, a sample with 25 points was drawn. Results show that the proposed approach provides more reliable realizations with smaller uncertainty zones.

Cette thèse est consacrée à un nouveau modèle d'apparence pour la segmentation d'images basée modèle. Ce modèle, dénommé Multimodal Prior Appearance Model (MPAM), est construit à partir d'une classification EM de profils d'intensité combinée avec une méthode automatique pour déterminer le nombre de classes. Contrairement aux approches classiques basées ACP, les profils d'intensité sont classifiés pour chaque maillage et non pour chaque sommet. Tout d'abord, nous décrivons la construction du MPAM à partir d'un ensemble de maillages et d'images. La classification de profils d'intensité et la détermination du nombre de régions par un nouveau critère de sélection sont expliquées. Une régularisation spatiale pour lisser la classification est présentée et la projection de l'information d'apparence sur un maillage de référence est décrite. Ensuite, nous présentons une classification de type spectrale dont le but est d'optimiser la classification des profils pour la segmentation. La représentation de la similitude entre points de données dans l'espace spectral est expliquée. Des résultats comparatifs sur des profils d'intensité du foie à partir d'images tomodensitométriques montrent que notre approche surpasse les modèles basés ACP. Finalement, nous présentons des méthodes d'analyse pour les structures des membres inférieurs à partir d'images IRM. D'abord, notre technique pour créer des modèles spécifiques aux sujets pour des simulations cinématiques des membres inférieurs est décrite. Puis, la performance de modèles statistiques est comparée dans un contexte de segmentation des os lorsqu'un faible ensemble de données est disponible.

Cette thèse porte sur l'étude d'algorithmes stochastiques en grande dimension ainsi qu'à leur application en statistique robuste. Dans la suite, l'expression grande dimension pourra aussi bien signifier que la taille des échantillons étudiés est grande ou encore que les variables considérées sont à valeurs dans des espaces de grande dimension (pas nécessairement finie). Afin d'analyser ce type de données, il peut être avantageux de considérer des algorithmes qui soient rapides, qui ne nécessitent pas de stocker toutes les données, et qui permettent de mettre à jour facilement les estimations. Dans de grandes masses de données en grande dimension, la détection automatique de points atypiques est souvent délicate. Cependant, ces points, même s'ils sont peu nombreux, peuvent fortement perturber des indicateurs simples tels que la moyenne ou la covariance. On va se concentrer sur des estimateurs robustes, qui ne sont pas trop sensibles aux données atypiques. Dans une première partie, on s'intéresse à l'estimation récursive de la médiane géométrique, un indicateur de position robuste, et qui peut donc être préférée à la moyenne lorsqu'une partie des données étudiées est contaminée. Pour cela, on introduit un algorithme de Robbins-Monro ainsi que sa version moyennée, avant de construire des boules de confiance non asymptotiques et d'exhiber leurs vitesses de convergence $L^{p}$ et presque sûre.La deuxième partie traite de l'estimation de la "Median Covariation Matrix" (MCM), qui est un indicateur de dispersion robuste lié à la médiane, et qui, si la variable étudiée suit une loi symétrique, a les mêmes sous-espaces propres que la matrice de variance-covariance. Ces dernières propriétés rendent l'étude de la MCM particulièrement intéressante pour l'Analyse en Composantes Principales Robuste. On va donc introduire un algorithme itératif qui permet d'estimer simultanément la médiane géométrique et la MCM ainsi que les $q$ principaux vecteurs propres de cette dernière. On donne, dans un premier temps, la forte consistance des estimateurs de la MCM avant d'exhiber les vitesses de convergence en moyenne quadratique.Dans une troisième partie, en s'inspirant du travail effectué sur les estimateurs de la médiane et de la "Median Covariation Matrix", on exhibe les vitesses de convergence presque sûre et $L^{p}$ des algorithmes de gradient stochastiques et de leur version moyennée dans des espaces de Hilbert, avec des hypothèses moins restrictives que celles présentes dans la littérature. On présente alors deux applications en statistique robuste: estimation de quantiles géométriques et régression logistique robuste.Dans la dernière partie, on cherche à ajuster une sphère sur un nuage de points répartis autour d'une sphère complète où tronquée. Plus précisément, on considère une variable aléatoire ayant une distribution sphérique tronquée, et on cherche à estimer son centre ainsi que son rayon. Pour ce faire, on introduit un algorithme de gradient stochastique projeté et son moyenné. Sous des hypothèses raisonnables, on établit leurs vitesses de convergence en moyenne quadratique ainsi que la normalité asymptotique de l'algorithme moyenné
This thesis focus on stochastic algorithms in high dimension as well as their application in robust statistics. In what follows, the expression high dimension may be used when the the size of the studied sample is large or when the variables we consider take values in high dimensional spaces (not necessarily finite). In order to analyze these kind of data, it can be interesting to consider algorithms which are fast, which do not need to store all the data, and which allow to update easily the estimates. In large sample of high dimensional data, outliers detection is often complicated. Nevertheless, these outliers, even if they are not many, can strongly disturb simple indicators like the mean and the covariance. We will focus on robust estimates, which are not too much sensitive to outliers.In a first part, we are interested in the recursive estimation of the geometric median, which is a robust indicator of location which can so be preferred to the mean when a part of the studied data is contaminated. For this purpose, we introduce a Robbins-Monro algorithm as well as its averaged version, before building non asymptotic confidence balls for these estimates, and exhibiting their $L^{p}$ and almost sure rates of convergence.In a second part, we focus on the estimation of the Median Covariation Matrix (MCM), which is a robust dispersion indicator linked to the geometric median. Furthermore, if the studied variable has a symmetric law, this indicator has the same eigenvectors as the covariance matrix. This last property represent a real interest to study the MCM, especially for Robust Principal Component Analysis. We so introduce a recursive algorithm which enables us to estimate simultaneously the geometric median, the MCM, and its $q$ main eigenvectors. We give, in a first time, the strong consistency of the estimators of the MCM, before exhibiting their rates of convergence in quadratic mean.In a third part, in the light of the work on the estimates of the median and of the Median Covariation Matrix, we exhibit the almost sure and $L^{p}$ rates of convergence of averaged stochastic gradient algorithms in Hilbert spaces, with less restrictive assumptions than in the literature. Then, two applications in robust statistics are given: estimation of the geometric quantiles and application in robust logistic regression.In the last part, we aim to fit a sphere on a noisy points cloud spread around a complete or truncated sphere. More precisely, we consider a random variable with a truncated spherical distribution, and we want to estimate its center as well as its radius. In this aim, we introduce a projected stochastic gradient algorithm and its averaged version. We establish the strong consistency of these estimators as well as their rates of convergence in quadratic mean. Finally, the asymptotic normality of the averaged algorithm is given

L’un des principaux objectifs de la physique des réacteurs nucléaires est de caractériser la répartition aléatoire de la population de neutrons au sein d’un réacteur. Les fluctuations de cette population sont liées à la nature stochastique des interactions des neutrons avec les noyaux fissiles du milieu : diffusion, capture stérile, ou encore émission de plusieurs neutrons lors de la fission d’un noyau. L’ensemble de ces mécanismes physiques confère une structure aléatoire branchante à la trajectoire des neutrons, alors modélisée par des marches aléatoires. Avec environs 10⁸ neutrons par centimètre cube dans un réacteur de type REP à pleine puissance en conditions stationnaires, les grandeurs physiques du système (flux, taux de réaction, énergie déposée) sont, en première approximation, bien représentées par leurs valeurs moyennes respectives. Ces observables physiques moyennes obéissent alors à l’équation de transport linéaire de Boltzmann. Au cours de ma thèse, je me suis penchée sur deux aspects du transport qui ne sont pas décrits par cette équation, et pour lesquels je me suis appuyée sur des outils issus de la théorie des marches aléatoires. Tout d’abord, grâce au formalisme de Feynman-Kac, j’ai étudié les fluctuations statistiques de la population de neutrons, et plus particulièrement le phénomène de « clustering neutronique », qui a été mis en évidence numériquement pour de faibles densités de neutrons (typiquement un réacteur au démarrage). Je me suis ensuite intéressée à différentes propriétés de la statistique d’occupation des neutrons effectuant un transport anormal (càd non-exponentiel). Ce type de transport permet de modéliser le transport dans des matériaux fortement hétérogènes et désordonnés, tel que les réacteurs à lit de boulets. L’un des aspects novateurs de ce travail est la prise en compte de la présence de bords. En effet, bien que les systèmes réels soient de taille finie, la plupart des résultats théoriques pré-existants sur ces thématiques ont été obtenus sur des systèmes de taille infinie
One of the key goals of nuclear reactor physics is to determine the distribution of the neutron population within a reactor core. This population indeed fluctuates due to the stochastic nature of the interactions of the neutrons with the nuclei of the surrounding medium: scattering, emission of neutrons from fission events and capture by nuclear absorption. Due to these physical mechanisms, the stochastic process performed by neutrons is a branching random walk. For most applications, the neutron population considered is very large, and all physical observables related to its behaviour, such as the heat production due to fissions, are well characterised by their average values. Generally, these mean quantities are governed by the classical neutron transport equation, called linear Boltzmann equation. During my PhD, using tools from branching random walks and anomalous diffusion, I have tackled two aspects of neutron transport that cannot be approached by the linear Boltzmann equation. First, thanks to the Feynman-Kac backward formalism, I have characterised the phenomenon of “neutron clustering” that has been highlighted for low-density configuration of neutrons and results from strong fluctuations in space and time of the neutron population. Then, I focused on several properties of anomalous (non-exponential) transport, that can model neutron transport in strongly heterogeneous and disordered media, such as pebble-bed reactors. One of the novel aspects of this work is that problems are treated in the presence of boundaries. Indeed, even though real systems are finite (confined geometries), most of previously existing results were obtained for infinite systems

Cette thèse s'inscrit dans un partenariat entre l'INRA et l'INRIA et dans le champs de l'extraction de connaissances à partir de bases de données spatiales. La problématique porte sur la caractérisation et la simulation de paysages agricoles. Plus précisément, nous nous concentrons sur des lignes qui structurent le paysage agricole, telles que les routes, les fossés d'irrigation et les haies. Notre objectif est de modéliser les haies en raison de leur rôle dans de nombreux processus écologiques et environnementaux. Nous étudions les moyens de caractériser les structures de haies sur deux paysages agricoles contrastés, l'un situé dans le sud-Est de la France (majoritairement composé de vergers) et le second en Bretagne (Ouest de la France, de type bocage). Nous déterminons également si, et dans quelles circonstances, la répartition spatiale des haies est structurée par la position des éléments linéaires plus pérennes du paysage tels que les routes et les fossés et l'échelle de ces structures. La démarche d'extraction de connaissances à partir de base de données (ECBD) mise en place comporte différentes étapes de prétraitement et de fouille de données, alliant des méthodes mathématiques et informatiques. La première partie du travail de thèse se concentre sur la création d'un indice spatial statistique, fondé sur une notion géométrique de voisinage et permettant la caractérisation des structures de haies. Celui-Ci a permis de décrire les structures de haies dans le paysage et les résultats montrent qu'elles dépendent des éléments plus pérennes à courte distance et que le voisinage des haies est uniforme au-Delà de 150 mètres. En outre différentes structures de voisinage ont été mises en évidence selon les principales orientations de haies dans le sud-Est de la France, mais pas en Bretagne. La seconde partie du travail de thèse a exploré l'intérêt du couplage de méthodes de linéarisation avec des méthodes de Markov. Les méthodes de linéarisation ont été introduites avec l'utilisation d'une variante des courbes de Hilbert : les chemins de Hilbert adaptatifs. Les données spatiales linéaires ainsi construites ont ensuite été traitées avec les méthodes de Markov. Ces dernières ont l'avantage de pouvoir servir à la fois pour l'apprentissage sur les données réelles et pour la génération de données, dans le cadre, par exemple, de la simulation d'un paysage. Les résultats montrent que ces méthodes couplées permettant un apprentissage et une génération automatique qui capte des caractéristiques des différents paysages. Les premières simulations sont encourageantes malgré le besoin d'un post-Traitement. Finalement, ce travail de thèse a permis la création d'une méthode d'exploration de données spatiales basée sur différents outils et prenant en charge toutes les étapes de l'ECBD classique, depuis la sélection des données jusqu'à la visualisation des résultats. De plus, la construction de cette méthode est telle qu'elle peut servir à son tour à la génération de données, volet nécessaire pour la simulation de paysage
This thesis is part of a partnership between INRA and INRIA in the field of knowledge extraction from spatial databases. The study focuses on the characterization and simulation of agricultural landscapes. More specifically, we focus on linears that structure the agricultural landscape, such as roads, irrigation ditches and hedgerows. Our goal is to model the spatial distribution of hedgerows because of their role in many ecological and environmental processes. We more specifically study how to characterize the spatial structure of hedgerows in two contrasting agricultural landscapes, one located in south-Eastern France (mainly composed of orchards) and the second in Brittany (western France, \emph{bocage}-Type). We determine if the spatial distribution of hedgerows is structured by the position of the more perennial linear landscape features, such as roads and ditches, or not. In such a case, we also detect the circumstances under which this spatial distribution is structured and the scale of these structures. The implementation of the process of Knowledge Discovery in Databases (KDD) is comprised of different preprocessing steps and data mining algorithms which combine mathematical and computational methods. The first part of the thesis focuses on the creation of a statistical spatial index, based on a geometric neighborhood concept and allowing the characterization of structures of hedgerows. Spatial index allows to describe the structures of hedgerows in the landscape. The results show that hedgerows depend on more permanent linear elements at short distances, and that their neighborhood is uniform beyond 150 meters. In addition different neighborhood structures have been identified depending on the orientation of hedgerows in the South-East of France but not in Brittany. The second part of the thesis explores the potential of coupling linearization methods with Markov methods. The linearization methods are based on the use of alternative Hilbert curves: Hilbert adaptive paths. The linearized spatial data thus constructed were then treated with Markov methods. These methods have the advantage of being able to serve both for the machine learning and for the generation of new data, for example in the context of the simulation of a landscape. The results show that the combination of these methods for learning and automatic generation of hedgerows captures some characteristics of the different study landscapes. The first simulations are encouraging despite the need for post-Processing. Finally, this work has enabled the creation of a spatial data mining method based on different tools that support all stages of a classic KDD, from the selection of data to the visualization of results. Furthermore, this method was constructed in such a way that it can also be used for data generation, a component necessary for the simulation of landscapes

The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale.

Cette thèse s'inscrit dans un partenariat entre l'INRA et l'INRIA et dans le champs de l'extraction de connaissances à partir de bases de données spatiales. La problématique porte sur la caractérisation et la simulation de paysages agricoles. Plus précisément, nous nous concentrons sur des lignes qui structurent le paysage agricole, telles que les routes, les fossés d'irrigation et les haies. Notre objectif est de modéliser les haies en raison de leur rôle dans de nombreux processus écologiques et environnementaux. Nous étudions les moyens de caractériser les structures de haies sur deux paysages agricoles contrastés, l'un situé dans le sud-Est de la France (majoritairement composé de vergers) et le second en Bretagne (Ouest de la France, de type bocage). Nous déterminons également si, et dans quelles circonstances, la répartition spatiale des haies est structurée par la position des éléments linéaires plus pérennes du paysage tels que les routes et les fossés et l'échelle de ces structures. La démarche d'extraction de connaissances à partir de base de données (ECBD) mise en place comporte différentes étapes de prétraitement et de fouille de données, alliant des méthodes mathématiques et informatiques. La première partie du travail de thèse se concentre sur la création d'un indice spatial statistique, fondé sur une notion géométrique de voisinage et permettant la caractérisation des structures de haies. Celui-Ci a permis de décrire les structures de haies dans le paysage et les résultats montrent qu'elles dépendent des éléments plus pérennes à courte distance et que le voisinage des haies est uniforme au-Delà de 150 mètres. En outre différentes structures de voisinage ont été mises en évidence selon les principales orientations de haies dans le sud-Est de la France, mais pas en Bretagne. La seconde partie du travail de thèse a exploré l'intérêt du couplage de méthodes de linéarisation avec des méthodes de Markov. Les méthodes de linéarisation ont été introduites avec l'utilisation d'une variante des courbes de Hilbert : les chemins de Hilbert adaptatifs. Les données spatiales linéaires ainsi construites ont ensuite été traitées avec les méthodes de Markov. Ces dernières ont l'avantage de pouvoir servir à la fois pour l'apprentissage sur les données réelles et pour la génération de données, dans le cadre, par exemple, de la simulation d'un paysage. Les résultats montrent que ces méthodes couplées permettant un apprentissage et une génération automatique qui capte des caractéristiques des différents paysages. Les premières simulations sont encourageantes malgré le besoin d'un post-Traitement. Finalement, ce travail de thèse a permis la création d'une méthode d'exploration de données spatiales basée sur différents outils et prenant en charge toutes les étapes de l'ECBD classique, depuis la sélection des données jusqu'à la visualisation des résultats. De plus, la construction de cette méthode est telle qu'elle peut servir à son tour à la génération de données, volet nécessaire pour la simulation de paysage
This thesis is part of a partnership between INRA and INRIA in the field of knowledge extraction from spatial databases. The study focuses on the characterization and simulation of agricultural landscapes. More specifically, we focus on linears that structure the agricultural landscape, such as roads, irrigation ditches and hedgerows. Our goal is to model the spatial distribution of hedgerows because of their role in many ecological and environmental processes. We more specifically study how to characterize the spatial structure of hedgerows in two contrasting agricultural landscapes, one located in south-Eastern France (mainly composed of orchards) and the second in Brittany (western France, \emph{bocage}-Type). We determine if the spatial distribution of hedgerows is structured by the position of the more perennial linear landscape features, such as roads and ditches, or not. In such a case, we also detect the circumstances under which this spatial distribution is structured and the scale of these structures. The implementation of the process of Knowledge Discovery in Databases (KDD) is comprised of different preprocessing steps and data mining algorithms which combine mathematical and computational methods. The first part of the thesis focuses on the creation of a statistical spatial index, based on a geometric neighborhood concept and allowing the characterization of structures of hedgerows. Spatial index allows to describe the structures of hedgerows in the landscape. The results show that hedgerows depend on more permanent linear elements at short distances, and that their neighborhood is uniform beyond 150 meters. In addition different neighborhood structures have been identified depending on the orientation of hedgerows in the South-East of France but not in Brittany. The second part of the thesis explores the potential of coupling linearization methods with Markov methods. The linearization methods are based on the use of alternative Hilbert curves: Hilbert adaptive paths. The linearized spatial data thus constructed were then treated with Markov methods. These methods have the advantage of being able to serve both for the machine learning and for the generation of new data, for example in the context of the simulation of a landscape. The results show that the combination of these methods for learning and automatic generation of hedgerows captures some characteristics of the different study landscapes. The first simulations are encouraging despite the need for post-Processing. Finally, this work has enabled the creation of a spatial data mining method based on different tools that support all stages of a classic KDD, from the selection of data to the visualization of results. Furthermore, this method was constructed in such a way that it can also be used for data generation, a component necessary for the simulation of landscapes

The advantage of the geometric approach to linear model and its applications is known to many authors. In spite of that, it still remains to be rather unpopular in teaching statistics around the world and is almost missing in the Czech Republic. In this work, we use geometry of multidimensional vector spaces to derive some well-known properties of the linear model and to explain some of the most familiar statistical methods to show usefulness of this approach, also known as "free-coordinate". Besides, historical background including selected results of R. A. Fisher is briefly discussed; it follows that the geometry approach to linear model is justifiable from the historical point of view, too. Powered by TCPDF (www.tcpdf.org)

In this thesis we investigate 1-skeleta and their associated cohomology rings. 1-skeleta arise from the 0- and 1-dimensional orbits of a certain class of manifold admitting a compact torus action and many questions that arise in the theory of 1-skeleta are rooted in the geometry and topology of these manifolds. The three main results of this work are: a lifting result for 1-skeleta (related to extending torus actions on manifolds), a classification result for certain 1-skeleta which have the Morse package (a property of 1-skeleta motivated by Morse theory for manifolds) and two constructions on 1-skeleta which we show preserve the Lefschetz package (a property of 1-skeleta motivated by the hard Lefschetz theorem in algebraic geometry). A corollary of this last result is a conceptual proof (applicable in certain cases) of the fact that the coinvariant ring of a finite reflection group has the strong Lefschetz property.