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Martingales avec marginales spécifiesDavid, Baker 18 December 2012 (has links) (PDF)
Cette thèse décrit des méthodes de construction de martingales avec marginales spécifiées. La première collection de méthodes procède par quantization. C'est-à-dire en approximant une mesure par une autre mesure dont le support consiste en un nombre fini de points. Nous introduisons une méthode de quantization qui préserve l'ordre convexe. L'ordre convexe est un ordre partiel sur l'espace des mesures qui les compare en termes de leur dispertion relative. Cette nouvelle méthode de quantization présente l'avantage que si deux mesures admettent une transition de martingale alors les mesures quantisées en admettent aussi. Ceci n'est pas le cas pour la méthode de quantization habituellement utilisée en probabilités (la méthode de quantization L2). Pour les mesures quantifiés nous présentons plusieurs méthodes de construction de transition de martingale. La première méthode procède par programmation linéaire. La deuxième méthode procède par construction de matrices avec diagonale et spectre données. La troisième méthode procède par l'algorithme de Chacon et Walsh. Dans une seconde partie la thèse présente une nouvelle solution au problème du plongement de Skorokhod. Dans une troisième partie la thèse étudie la construction de martingales à temps continu avec marginales données. Des constructions sont données à l'aide du draps Brownien. D'autres constructions sont données en modifiant une méthode développée par Albin, les martingales construites ainsi possèdent une propriété de scaling.. Dans une partie annexe, certaines conséquences de cette théorie concernant le management du risque des options asiatiques, par rapport à leur sensibilité à la volatilité et à la maturité sont établies.
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Diagonals of Operators: Majorization, a Schur-Horn Theorem and Zero-Diagonal IdempotentsLoreaux, Jireh 03 October 2016 (has links)
No description available.
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Properties of greedy treesRazanajatovo Misanantenaina, Valisoa 12 1900 (has links)
Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: A greedy tree is constructed from a given degree sequence using a simple
greedy algorithm that assigns the highest degree to the root, the second,
the third, . . . , -highest degree to the root’s neighbours, etc. This particular
tree is the solution to numerous extremal problems among all trees with
given degree sequence. In this thesis, we collect results for some distancebased
graph invariants, the number of subtrees and the spectral radius
in which greedy trees play a major role. We show that greedy trees are
extremal for the aforementioned graph invariants by means of two different
approaches, one using level greedy trees and majorization, while the other
one is somewhat more direct. Finally, we prove some new results on greedy
trees for additive parameters with specific toll functions. / AFRIKAANSE OPSOMMING: ’n Gulsige boom word vanuit ’n gegewe graadry deur middel van ’n eenvoudige
gulsige algoritme gebou, wat die hoogste graad aan die wortel
toewys, die tweede-, derde-, . . . , -hoogste graad aan die wortel se bure,
ens. Hierdie spesifieke boom is die oplossing van ’n groot aantal ekstremale
probleme onder al die bome met gegewe graadry. In hierdie tesis
beskou ons ’n versameling van resultate oor afstand-gebaseerde grafiekinvariante,
die aantal subbome en die spektraalstraal waarin gulsige bome
’n belangrike rol speel. Ons bewys dat gulsige bome ekstremaal vir die
bogenoemde grafiekinvariante is deur van twee verskillende tegnieke gebruik
te maak: een met behulp van vlak-gulsige bome en majorering, en
’n ander metode wat effens meer direk is. Laastens bewys ons sommige
nuwe resultate oor gulsige bome vir additiewe parameters met spesifieke
tolfunksies.
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Factor analysis of dynamic PET imagesCruz Cavalcanti, Yanna 31 October 2018 (has links) (PDF)
Thanks to its ability to evaluate metabolic functions in tissues from the temporal evolution of a previously injected radiotracer, dynamic positron emission tomography (PET) has become an ubiquitous analysis tool to quantify biological processes. Several quantification techniques from the PET imaging literature require a previous estimation of global time-activity curves (TACs) (herein called \textit{factors}) representing the concentration of tracer in a reference tissue or blood over time. To this end, factor analysis has often appeared as an unsupervised learning solution for the extraction of factors and their respective fractions in each voxel. Inspired by the hyperspectral unmixing literature, this manuscript addresses two main drawbacks of general factor analysis techniques applied to dynamic PET. The first one is the assumption that the elementary response of each tissue to tracer distribution is spatially homogeneous. Even though this homogeneity assumption has proven its effectiveness in several factor analysis studies, it may not always provide a sufficient description of the underlying data, in particular when abnormalities are present. To tackle this limitation, the models herein proposed introduce an additional degree of freedom to the factors related to specific binding. To this end, a spatially-variant perturbation affects a nominal and common TAC representative of the high-uptake tissue. This variation is spatially indexed and constrained with a dictionary that is either previously learned or explicitly modelled with convolutional nonlinearities affecting non-specific binding tissues. The second drawback is related to the noise distribution in PET images. Even though the positron decay process can be described by a Poisson distribution, the actual noise in reconstructed PET images is not expected to be simply described by Poisson or Gaussian distributions. Therefore, we propose to consider a popular and quite general loss function, called the $\beta$-divergence, that is able to generalize conventional loss functions such as the least-square distance, Kullback-Leibler and Itakura-Saito divergences, respectively corresponding to Gaussian, Poisson and Gamma distributions. This loss function is applied to three factor analysis models in order to evaluate its impact on dynamic PET images with different reconstruction characteristics.
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Parametric classification and variable selection by the minimum integrated squared error criterionJanuary 2012 (has links)
This thesis presents a robust solution to the classification and variable selection problem when the dimension of the data, or number of predictor variables, may greatly exceed the number of observations. When faced with the problem of classifying objects given many measured attributes of the objects, the goal is to build a model that makes the most accurate predictions using only the most meaningful subset of the available measurements. The introduction of [cursive l] 1 regularized model titling has inspired many approaches that simultaneously do model fitting and variable selection. If parametric models are employed, the standard approach is some form of regularized maximum likelihood estimation. While this is an asymptotically efficient procedure under very general conditions, it is not robust. Outliers can negatively impact both estimation and variable selection. Moreover, outliers can be very difficult to identify as the number of predictor variables becomes large. Minimizing the integrated squared error, or L 2 error, while less efficient, has been shown to generate parametric estimators that are robust to a fair amount of contamination in several contexts. In this thesis, we present a novel robust parametric regression model for the binary classification problem based on L 2 distance, the logistic L 2 estimator (L 2 E). To perform simultaneous model fitting and variable selection among correlated predictors in the high dimensional setting, an elastic net penalty is introduced. A fast computational algorithm for minimizing the elastic net penalized logistic L 2 E loss is derived and results on the algorithm's global convergence properties are given. Through simulations we demonstrate the utility of the penalized logistic L 2 E at robustly recovering sparse models from high dimensional data in the presence of outliers and inliers. Results on real genomic data are also presented.
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Largest Laplacian Eigenvalue and Degree Sequences of TreesBiyikoglu, Türker, Hellmuth, Marc, Leydold, Josef January 2008 (has links) (PDF)
We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics
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Exact D-optimal Designs for First-order Trigonometric Regression Models on a Partial CircleSun, Yi-Ying 24 June 2011 (has links)
Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-order trigonometric regression models without intercept on a partial circle. We investigate the intricate geometry of the non-convex exact trigonometric moment set and provide characterizations of its boundary. Building on these results we obtain a complete solution of the exact D-optimal design problem. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of observations.
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Graphs with given degree sequence and maximal spectral radiusBiyikoglu, Türker, Leydold, Josef January 2008 (has links) (PDF)
We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spectral radius in such classes of trees is strictly monotone with respect to majorization. This paper is the revised final version of the preprint no. 35 of this research report series. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics
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Admissible Unbiased Quantizations: Distributions with Linear ComponentsPötzelberger, Klaus January 2000 (has links) (PDF)
We show that results on the characterization of admissible quantizations, which have been derived in Potzelberger [3], have to be modified in case the probability distribution has linear components. Furthermore, we provide an example, where the limit of optimal quantizations is not admissible. (author's abstract) / Series: Forschungsberichte / Institut für Statistik
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Admissible Unbiased Quantizations: Distributions without Linear ComponentsPötzelberger, Klaus January 2000 (has links) (PDF)
Let P be a Borel probability measure on Rd. We characterize the maximal elements p E M(P,m) with respect to the Bishop-De Leeuw order, where p E M(P, m) if and only if p P and [supp(p)] m. The results obtained have important consequences for statistical inference, such as tests of homogeneity or multivariate cluster analysis and for the theory of comparison of experiments. (author's abstract) / Series: Forschungsberichte / Institut für Statistik
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