Дисертації з теми "Geometric statistics"

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.

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.

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

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

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.

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 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

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.
https://dc.etsu.edu/etsu_books/1064/thumbnail.jpg

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

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.

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.

[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

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.

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.

Estimating multiple geometric shapes such as tracks or surfaces creates significant mathematical challenges particularly in the presence of unknown data association. In particular, problems of this type have two major challenges. The first is typically the object of interest is infinite dimensional whilst data is finite dimensional. As a result the inverse problem is ill-posed without regularization. The second is the data association makes the likelihood function highly oscillatory. The focus of this thesis is on techniques to validate approaches to estimating problems in geometric statistical inference. We use convergence of the large data limit as an indicator of robustness of the methodology. One particular advantage of our approach is that we can prove convergence under modest conditions on the data generating process. This allows one to apply the theory where very little is known about the data. This indicates a robustness in applications to real world problems. The results of this thesis therefore concern the asymptotics for a selection of statistical inference problems. We construct our estimates as the minimizer of an appropriate functional and look at what happens in the large data limit. In each case we will show our estimates converge to a minimizer of a limiting functional. In certain cases we also add rates of convergence. The emphasis is on problems which contain a data association or classification component. More precisely we study a generalized version of the k-means method which is suitable for estimating multiple trajectories from unlabeled data which combines data association with spline smoothing. Another problem considered is a graphical approach to estimating the labeling of data points. Our approach uses minimizers of the Ginzburg-Landau functional on a suitably defined graph. In order to study these problems we use variational techniques and in particular I-convergence. This is the natural framework to use for studying sequences of minimization problems. A key advantage of this approach is that it allows us to deal with infinite dimensional and highly oscillatory functionals.

In this thesis we generalise the rigidity estimates of Friesecke et al. [2002] and Müller et al. [2014] to vector fields whose properties are constrained by both conditions on the support of their curl and the underlying discrete symmetries of the lattice Z2. These analytical estimates and other considerations are applied to a statistical model of a crystal containing defects based on work by Aumann [2015]. It is demonstrated in this thesis that we allow a finite density of defects. The main result is that regardless of crystal size, the ordering of the crystal, expressed via the L2-distance of a random vector field from the rotations, can be made arbitrarily small for sufficiently low temperature β-1.

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Comenzarnos realizando una aproximación al concepto de modelo estadístico desde el punto de vista geométrico, centrándonos principalmente en consideraciones sobre la introducción de distancias, y en particular estudiando la métrica informacional y sus propiedades. Dada una variedad paramétrica correspondiente a un modelo estadístico, hemos efectuado un estudio del espacio tangente y del espacio tangente dual en un punto a la variedad, introduciendo representaciones adecuadas de los mismos. Tales representaciones han permitido identificar a los elementos del espacio muestral con campos tensoriales covariantes de primer orden en la variedad, mientras que las variables aleatorias pueden ser identificados con campos tensoriales contravariantes también de primer orden. Hemos introducido dos definiciones de distancias, en sentido estricto pseudodistancias, entre valores muestrales basadas ambas en distancias en el espacio tangente dual entre formas lineales asociadas. La primera, a la que denominamos distancia inmediata, es definida a partir de la distancia euclídea en el espacio tangente dual. Se han obtenido expresiones explícitas para la distancia cuando los individuos estadísticos son muestras correspondientes a las distribuciones Poisson, Weibull, Gamma, Exponencial, Binomial, Binomial Negativa, Multinomial, Multinomial negativa, Wald, Logística, Normal univariante y Normal multivariante. Se han estudiarlo ciertas propiedades relacionadas con la distancia inmediata, entre las que destacamos su invarianza frente a cambios de la medida de referencia y transformaciones por estadísticas suficientes, y su no decrecimiento al aumentar el número de parámetros de las variedades. La distancia estructural es definida a partir de la distancia sobre el conjunto imagen del espacio muestral. Se demuestra que coincide con la distancia inmediata si el conjunto imagen es un conjunto convexo y también que dicho conjunto no es convexo si la dimensión del espacio muestral es uno y el número de parámetros de la variedad mayor o igual a dos. Se ha obtenido la expresión explícita para la distancia estructural entre muestras de tamaño uno correspondientes a una distribución normal univariante. Se han estudiado las aplicaciones de las distancias entre individuos a técnicas clásicas de inferencia estadística, definiendo nuevos procedimientos de estimación de parámetros y contraste de hipótesis desde el punto de vista geométrico. Se comprueba cómo utilizando la distancia inmediata se recuperan gran parte de los resultados clásicos, en particular las ecuaciones de verosimilitud y el contraste de hipótesis mediante el test de los multiplicadores de Lagrange. Hemos comprobado también como utilizando en estimación de parámetros la distancia estructural en un ejemplo en que éste difiere de la inmediata, se obtienen resultados que difieren respecto a la máxima verosimilitud clásica y que podemos considerar más acordes con resultados intuitivos al dejar indeterminada la estimación de la varianza trabajando con muestras de tamaño uno de una distribución Normal univariante. Se ha introducido una clase de funciones de densidad de probabilidad que pueden ser caracterizadas en una variedad paramétrica de dimensión finita. Se comprueba que las variedades resultantes son de curvatura constante y positiva. Se han obtenido las expresiones para las geodésicas y la distancia de Rao entre dos distribuciones. Hemos efectuado un estudio probabilístico en varios ejemplos y finalmente consideramos la aplicación de tales familias a la estimación no paramétrica de funciones de densidad gracias a su capacidad de adaptación. Se ha abordado el problema de la estimación de parámetros en las familias anteriormente citadas. Comprobamos los inconvenientes de la estimación máximo verosímil y para subsanarlos hemos propuesto un algoritmo tipo “stepwise” que toma en cuenta la significación de los incrementos de la verosimilitud al modificar el número de parámetros de las familias. Utilizamos diversas simulaciones para comprobar la bondad del algoritmo, obteniendo resultados satisfactorios tanto al trabajar con distribuciones clásicos como con las nuevas familias. Se han comparado los resultados con otros métodos clásicos de estimación no paramétrica, en particular con el método de los Kernel. También se ha estudiado el método de minimizar la esperanza del cuadrado de la distancia estructural entre individuos (MESD). Para poder llevar a cabo tal estudio se ha desarrollado una aproximación a la distinción Riemanniana y se han utilizado técnicas de minimización numérica de funciones de varias variables con restricciones. Se han obtenido algunos ejemplos que muestran un mejor comportamiento de la estimación MESD frente a la MLE. Finalmente se han considerado dos ejemplos prácticos consistentes en la estimación de una función de densidad bimodal a partir de unos datos en forma de histograma y en la clasificación de diversos patrones electroforéticos asimilándolos a funciones de densidad. En limbos ejemplos los resultados parecen validar completamente la metodología empleada.
We have studied the concept of statistical model from a geometric point of view considering particularly the information metric and the problem of introducing distances. Given a parametric manifold representing a statistical model and given a point of the manifold, we have defined two different distances between elements of sample space (statistical individuals) by means of a suitable representation of statistical individuals as linear forms of the dual tangent space to the manifold in the given point. Some properties have been studied and the explicit expressions for some examples have been obtained. Several techniques of statistical inference: parameter estimation, hypothesis tests, discrimination; have been studied in the light of the distances between elements of sample spaces. Some classical results have been recovered, in particular Iikelihood equations and Lagrange multipliers test. We have introduced a class of probability density functions that may be represented in finite dimensional manifolds. Geometrical properties of such manifolds have been studied and the Rao distance between two distributions has been obtained. We have considered several examples. We have also studied the problem of parameter estimation in the functions defined previously; we have developed a stepwise algorithm for nonparametric density estimation in order to some problems arising with classical maximum likelihood estimation when we handle a large number of parameters. We also present some examples applied lo biological data.

In this dissertation, we propose and develop innovative data modeling and analysis methods for extracting meaningful and specific information about disease mechanisms from microarray gene expression data. To provide a high-level overview of gene expression data for easy and insightful understanding of data structure, we propose a novel statistical data clustering and visualization algorithm that is comprehensively effective for multiple clustering tasks and that overcomes some major limitations of existing clustering methods. The proposed clustering and visualization algorithm performs progressive, divisive hierarchical clustering and visualization, supported by hierarchical statistical modeling, supervised/unsupervised informative gene/feature selection, supervised/unsupervised data visualization, and user/prior knowledge guidance through human-data interactions, to discover cluster structure within complex, high-dimensional gene expression data. For the purpose of selecting suitable clustering algorithm(s) for gene expression data analysis, we design an objective and reliable clustering evaluation scheme to assess the performance of clustering algorithms by comparing their sample clustering outcome to phenotype categories. Using the proposed evaluation scheme, we compared the performance of our newly developed clustering algorithm with those of several benchmark clustering methods, and demonstrated the superior and stable performance of the proposed clustering algorithm. To identify the underlying active biological processes that jointly form the observed biological event, we propose a latent linear mixture model that quantitatively describes how the observed gene expressions are generated by a process of mixing the latent active biological processes. We prove a series of theorems to show the identifiability of the noise-free model. Based on relevant geometric concepts, convex analysis and optimization, gene clustering, and model stability analysis, we develop a robust blind source separation method that fits the model to the gene expression data and subsequently identify the underlying biological processes and their activity levels under different biological conditions. Based on the experimental results obtained on cancer, muscle regeneration, and muscular dystrophy gene expression data, we believe that the research work presented in this dissertation not only contributes to the engineering research areas of machine learning and pattern recognition, but also provides novel and effective solutions to potentially solve many biomedical research problems, for improving the understanding about disease mechanisms.
Ph. D.

Currently there is no complete face recognition system that is invariant to all facial expressions. Although humans find it easy to identify and recognise faces regardless of changes in illumination, pose and expression, producing a computer system with a similar capability has proved to be particularly di cult. Three dimensional face models are geometric in nature and therefore have the advantage of being invariant to head pose and lighting. However they are still susceptible to facial expressions. This can be seen in the decrease in the recognition results using principal component analysis when expressions are added to a data set. In order to achieve expression-invariant face recognition systems, we have employed a tensor algebra framework to represent 3D face data with facial expressions in a parsimonious space. Face variation factors are organised in particular subject and facial expression modes. We manipulate this using single value decomposition on sub-tensors representing one variation mode. This framework possesses the ability to deal with the shortcomings of PCA in less constrained environments and still preserves the integrity of the 3D data. The results show improved recognition rates for faces and facial expressions, even recognising high intensity expressions that are not in the training datasets. We have determined, experimentally, a set of anatomical landmarks that best describe facial expression e ectively. We found that the best placement of landmarks to distinguish di erent facial expressions are in areas around the prominent features, such as the cheeks and eyebrows. Recognition results using landmark-based face recognition could be improved with better placement. We looked into the possibility of achieving expression-invariant face recognition by reconstructing and manipulating realistic facial expressions. We proposed a tensor-based statistical discriminant analysis method to reconstruct facial expressions and in particular to neutralise facial expressions. The results of the synthesised facial expressions are visually more realistic than facial expressions generated using conventional active shape modelling (ASM). We then used reconstructed neutral faces in the sub-tensor framework for recognition purposes. The recognition results showed slight improvement. Besides biometric recognition, this novel tensor-based synthesis approach could be used in computer games and real-time animation applications.

Thesis (Ph. D.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2008.
Committee Chair: Allen Tannenbaum; Committee Member: Anthony Yezzi; Committee Member: Marc Niethammer; Committee Member: Patricio Vela; Committee Member: Yucel Altunbasak.

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.

We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This allows us to create an algebraic geometry--statistics dictionary. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification, in terms of primary decomposition of polynomial ideals, is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. Moreover, a complete algebraic classification, in terms of generating sets of polynomial ideals, is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed.
Ph. D.

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.

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.

In order for computers to interact with and understand the visual world, they must be equipped with reasoning systems that include high–level quantities such as objects, actions, and scenes. This thesis is concerned with extracting such representations of the world from visual input. The first part of this thesis describes an approach to scene understanding in which texture characteristics of the visual world are used to infer scene categories. We show that in the context of a moving camera, it is common to observe images containing very few individually salient image regions, yet overall texture structure often allows our system to derive powerful contextual cues about the environment. Our approach builds on ideas from texture recognition, and we show that our algorithm out–performs the well–known Gist descriptor on several classification tasks. In the second part of this thesis we we are interested in scene understanding in the context of multiple calibrated views of a scene, as might be obtained from a Structure–from–Motion or Simultaneous Localization and Mapping (SLAM) system. Though such systems are capable of localizing the camera robustly and efficiently, the maps produced are typically sparse point-clouds that are difficult to interpret and of little use for higher–level reasoning tasks such as scene understanding or human-machine interaction. In this thesis we begin to address this deficiency, presenting progress towards modeling scenes using semantically meaningful primitives such as floor, wall, and ceiling planes. To this end we adopt the indoor Manhattan representation, which was recently proposed for single–view reconstruction. This thesis presents the first in–depth description and analysis of this model in the literature. We describe a probabilistic model relating photometric features, stereo photo–consistencies, and 3D point clouds to Manhattan scene structure in a Bayesian framework. We then present a fast dynamic programming algorithm that solves exact MAP inference in this model in time linear in image size. We show detailed comparisons with the state–of–the art in both the single– and multiple–view contexts. Finally, we present a framework for learning within the indoor Manhattan hypothesis class. Our system is capable of extrapolating from labelled training examples to predict scene structure for unseen images. We cast learning as a structured prediction problem and show how to optimize with respect to two realistic loss functions. We present experiments in which we learn to recover scene structure from both single and multiple views — from the perspective of our learning algorithm these problems differ only by a change of feature space. This work constitutes one of the most complicated output spaces (in terms of internal constraints) yet considered within a structure prediction framework.

3D shape modelling is a fundamental component in computer vision and computer graphics. Applications include shape interpolation and extrapolation, shape reconstruction, motion capture and mesh editing, etc. By "modelling" we mean the process of learning a parameter-driven model. This thesis focused on the scope of statistical modelling for 3D non-rigid shapes, such as human faces and bodies. The problem is challenging due to highly non-linear deformations, high dimensionality, and data sparsity. Several new algorithms are proposed for 3D shape modelling, 3D shape matching (computing dense correspondence) and applications. First, we propose a variant of Principal Component Analysis called "Shell PCA" which provides a physically-inspired statistical shape model. This is our first attempt to use a physically plausible metric (specifically, the discrete shell model) for statistical shape modelling. Second, we further develop this line of work into a fully Riemannian approach called "Shell PGA". We demonstrate how to perform Principal Geodesic Analysis in the space of discrete shells. To achieve this, we present an alternate formulation of PGA which avoids working in the tangent space and deals with shapes lying on the manifold directly. Unlike displacement-based methods, Shell PGA is invariant to rigid body motion, and therefore alignment preprocessing such as Procrustes analysis is not needed. Third, we propose a groupwise shape matching method using functional map representation. Targeting at near-isometric deformations, we consider groupwise optimisation of consistent functional maps over a product of Stiefel manifolds, and optimise over a minimal subset of the transformations for efficiency. Last, we show that our proposed shape model achieves state-of-the-art performance in two very challenging applications: handle-based mesh editing, and model fitting using motion capture data. We also contribute a new algorithm for human body shape estimation using clothed scan sequence, along with a new dataset "BUFF" for evaluation.

The dynamics of a complex system can be understood by analyzing the phase space structure of that system. We apply geometric and statistical techniques to two Hamiltonian systems to find and exploit structure in the phase space that helps us get qualitative and quantitative results about the phase space transport. While the structure can be revealed by the study of invariant manifolds of fixed points and periodic orbits in the first system, there do not exist any fixed points (and hence invariant manifolds) in the second system. The use of statistical (or measure theoretic) and topological methods reveals the phase space structure even in the absence of fixed points or stable and unstable invariant manifolds. The first problem we study is the four-body problem in the context of a spacecraft in the presence of a planet and two of its moons, where we exploit the phase space structure of the problem to devise an intelligent control strategy to achieve mission objectives. We use a family of analytically derived controlled Keplerian Maps in the Patched-Three-Body framework to design fuel efficient trajectories with realistic flight times. These maps approximate the dynamics of the Planar Circular Restricted Three Body Problem (PCR3BP) and we patch solutions in two different PCR3BPs to form the desired trajectories in the four body system. The second problem we study concerns phase space mixing in a two-dimensional time dependent Stokes flow system. Topological analysis of the braiding of periodic points has been recently used to find lower bounds on the complexity of the flow via the Thurston-Nielsen classification theorem (TNCT). We extend this framework by demonstrating that in a perturbed system with no apparent periodic points, the almost-invariant sets computed using a transfer operator approach are the natural objects on which to pin the TNCT.
Ph. D.

Durant les trois années de mon projet de doctorat, j'ai développé plusieurs méthodes complémentaires pour l'analyse de données de type -omique, dont: (i) un modèle pour la génomique intégrative dans lequel toutes les sortes d'informations qui peuvent être obtenues sur un génome sont modélisées d'une manière probabiliste unifiée, permettant ainsi d'analyser les corrélations entre des données hétérogènes à l'échelle du génome, (ii) un test statistique ayant pour critère l'amplification de l'expression pour l'identification de gènes différentiellement et similairement exprimés entre deux conditions biologiques, et permettant la détermination d'intervalles de confiance concernant l'amplification, (iii) de nouvelles méthodes de réduction de dimensionnalité qui surpassent les autres méthodes existantes et produisant des représentations géométriques plus facilement interprétables dans le contexte de grands ensembles de données. Ces méthodes ont été appliquées à plusieurs nalyses et études biologiques dans le cadre de collaborations scientifiques: (i) afin d'identifier des domaines fonctionnels dans les régions promotrices de gènes candidats impliqués dans le pseudohypoaldostéronisme. (ii) pour découvrir les réponses transcriptionnelles qui sous-tendent les différences entre les virus pulmonaires faiblement et fortement pathogènes basé sur un ensemble de réponses transcriptomiques. (iii) afin d'étudier la progression du virus de l'hépatite C chez des patients infectés ayant subi une transplantation hépatique (iv) afin d'analyser une banque de marqueur de séquences exprimées obtenues à partir de cellules de sang périphérique de singes verts africains infectés ou non par le SIV
During the three years of my Ph. D. Project, I developed several complementary methods and frameworks for the analysis of -omics data, such as: (i) a framework for integrative genomics in which every kind of information that can be obtained about the genomic processes and features are modeled in a common probabilistic manner, allowing then to analyze the correlations among the heterogeneous genome-wide information, (ii) a fold-change based statistical test for the identification of differentially and similarly expressed genes between two biological conditions, allowing also the determination of confidence intervals of specific confidence levels for the fold-change. (iii) novel dimensionality reduction methods that outperform other related existing methods and provide more interpretable geometrical representations in the context of large dataset of-omics data. These methods have been applied to several biological analyses and studies as part of different scientific collaborations: (i) to identify functional Glucocorticoid Response Elements in the promoter regions of specific candidate genes involved in Type 1 Pseudohypoaldosteronism. (ii) to uncover the host transcriptional responses underlying differences between low- and high- pathogenic pulmonary viruses based on a compendium of host transcription responses of infected cells from mouse lungs. (iii) to study the progression of the hepatitis C virus in infected patients who underwent orthotopic liver transplantation, based on a cohort of transcriptome profiles for liver biopsy specimens, (iv) to analyze an Expression Sequence Tag library obtained from PBMC of African green monkeys infected or not by the SIV

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.

Computer vision is the science that studies how machines understand scenes and automatically make decisions based on meaningful information extracted from an image or multi-dimensional data of the scene, like human vision. One common and well-studied field of computer vision is visual tracking. It is challenging and active research area in the computer vision community. Visual tracking is the task of continuously estimating the pose of an object of interest from the background in consecutive frames of an image sequence. It is a ubiquitous task and a fundamental technology of computer vision that provides low-level information used for high-level applications such as visual navigation, human-computer interaction, and surveillance system. The focus of the research in this thesis is visual tracking and its applications. More specifically, the object of this research is to design a reliable tracking algorithm for a deformable object that is robust to clutter and capable of occlusion handling and target reacquisition in realistic tracking scenarios by using statistical and geometric methods. To this end, the approaches developed in this thesis make extensive use of region-based active contours and particle filters in a variational framework. In addition, to deal with occlusions and target reacquisition problems, we exploit the benefits of coupling 2D and 3D information of an image and an object. In this thesis, first, we present an approach for tracking a moving object based on 3D range information in stereoscopic temporal imagery by combining particle filtering and geometric active contours. Range information is weighted by the proposed Gaussian weighting scheme to improve segmentation achieved by active contours. In addition, this work present an on-line shape learning method based on principal component analysis to reacquire track of an object in the event that it disappears from the field of view and reappears later. Second, we propose an approach to jointly track a rigid object in a 2D image sequence and to estimate its pose in 3D space. In this work, we take advantage of knowledge of a 3D model of an object and we employ particle filtering to generate and propagate the translation and rotation parameters in a decoupled manner. Moreover, to continuously track the object in the presence of occlusions, we propose an occlusion detection and handling scheme based on the control of the degree of dependence between predictions and measurements of the system. Third, we introduce the fast level-set based algorithm applicable to real-time applications. In this algorithm, a contour-based tracker is improved in terms of computational complexity and the tracker performs real-time curve evolution for detecting multiple windows. Lastly, we deal with rapid human motion in context of object segmentation and visual tracking. Specifically, we introduce a model-free and marker-less approach for human body tracking based on a dynamic color model and geometric information of a human body from a monocular video sequence. The contributions of this thesis are summarized as follows: 1. Reliable algorithm to track deformable objects in a sequence consisting of 3D range data by combining particle filtering and statistics-based active contour models. 2. Effective handling scheme based on object's 2D shape information for the challenging situations in which the tracked object is completely gone from the image domain during tracking. 3. Robust 2D-3D pose tracking algorithm using a 3D shape prior and particle filters on SE(3). 4. Occlusion handling scheme based on the degree of trust between predictions and measurements of the tracking system, which is controlled in an online fashion. 5. Fast level set based active contour models applicable to real-time object detection. 6. Model-free and marker-less approach for tracking of rapid human motion based on a dynamic color model and geometric information of a human body.

This thesis addresses different aspects of spatial data handling in connection with investigations of large rockslides. As such, most of the research was carried out in a cross disciplinary and highly applied context. The focus of the thesis is on spatial data handling methodology which directly or indirectly can be used to support in rockslide investigations. Rockslide investigation is a comprehensive term covering all aspects of the evaluation process; from the initial planning of field investigations to data analysis and communication of final results. Central topics addressed in this thesis are; a) How data reduction affect the geometrical accuracy of digital terrain models b) How interactive geometric modelling and geovisualization can be used in complex rockslide investigations and c) How statistical analyses can be used to evaluate displacement measurements of unstable rock slopes. Digital terrain modelling forms an important component of the methodology used for rockslide investigations. The first subtopic addressed in this thesis is related to the construction of Triangulated Irregular Networks (TINs) from Light Detection and Ranging (LIDAR) data. As the LIDAR technology tends to generate large data volumes, the resulting terrain models are generally too large to be efficiently handled by ordinary workstations. Therefore, comparisons of various data reduction (decimation) methods were conducted. Their performances were evaluated by means of deviations from terrain models constructed from full datasets. Evaluation criteria included deviations in volume, surface area and elevation. The results showed that the method using a vertical point selection threshold combined with a data dependent triangulation had the overall best performance when tested on 30 different test datasets. The main objective of the geovisualization part of this thesis was to determine the geometric shapes and locations of potential basal sliding surfaces, for the Åknes rockslide in western Norway, along with the volumes of unstable rock associated with different sliding scenarios. The Åknes rockslide is one of the world's most investigated rockslides due to its potentially catastrophic consequences. A custom written geovisualization application for the Åknes investigation provided the visual context needed for data interpretation and interactive geometric modelling of sliding surfaces. This geovisualization approach enabled geoscientists to develop different sliding scenarios. A scenario putting the basal sliding surface at a depth of 105m to 115m below the topographic surface, delineating an unstable rock volume of 43 million m3, was considered as the most realistic. Statistical approaches for analyzing displacement measurements were also addressed in this thesis. Several methods including regression analysis, spectral analysis and hypothesis testing were demonstrated to measurements obtained from Global Positioning System (GPS), total stations and extensometers at the Åknes rockslide. Displacement measurements obtained from lasers and crackmeters at the Nordnes rockslide in Northern Norway were also analysed. As with the Åknes rockslide, the Nordnes rockslide has the potential for devastating consequences in terms tsunami generation. Consequently, thorough statistical analyses of the available displacement data are crucial in order to obtain accurate estimates for the displacement rates as well as for gaining insight into the sliding processes. Displacement data from both sites clearly showed seasonal variations but the overall long term displacements were regarded constant. Prediction intervals were derived from the current monitoring data from the Nordnes site. These prediction intervals are considered useful for evaluation of future displacement measurements.

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 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.

Analysis of brain tissues such as white matter (WM), gray matter (GM), cerebrospinal fluid (CSF), and pathological regions from magnetic resonance imaging (MRI) scans of normal adults and patients with neurodegenerative diseases such as multiple sclerosis (MS) allows for improved understanding of disease progression in vivo. As images are often confounded by acquisition noise and partial-volume effects, developing an automatic, robust, and efficient segmentation is essential to the accurate quantification of disease severity. Existing methods often require subjective parameter tuning, anatomical atlases, and training, which are impractical and undesirable. The contributions of this thesis are three-fold. First, a 3D deformable model was explored by integrating statistical and geometric information into a novel hybrid feature to provide robust regularization of the evolving contours. Second, to improve efficiency and noise resiliency, a 3D region-based hidden Markov model (rbHMM) was developed. The novelty of this model lies in subdividing an image into irregularly-shaped regions to reduce the problem dimensionality. A tree-structured estimation algorithm, based on Viterbi decoding, then enabled rotationally invariant estimation of the underlying discrete tissue labels given noisy observations. Third, estimation of partial volumes was incorporated in a 3D fuzzy rbHMM (frbHMM) for analyzing images suffering from acquisition-related resolution limitation by incorporating forward-backward estimations. These methods were successfully applied to the segmentation of WM, GM, CSF, and white matter lesions. Extensive qualitative and quantitative validations were performed on both synthetic 3D geometric shapes and simulated brain MRIs before applying to clinical scans of normal adults and MS patients. These experiments demonstrated 40% and 10% improvement in segmentation efficiency and accuracy, respectively, over state-of-the-art approaches under noise. When modeling partial-volume effects, an additional 30% reduction in segmentation errors was observed. Furthermore, the rotational invariance property introduced is especially valuable as segmentation should be invariant to subject positioning in the scanner to minimize analysis variability. Given such improvement in the quantification of tissue volumes, these methods could potentially be extended to the studies of other neurodegenerative diseases such as Alzheimer’s. Furthermore, the methods developed in this thesis are general and can potentially be adopted in other computer vision-related segmentation applications in the future.