This master‘s thesis deals with the use of higher order cumulants for classification of cardiac cycles. Second-, third-, and fourth-order cumulants were calculated from ECG recorded in isolated rabbit hearts during experiments with repeated ischemia. Cumulants properties useful for the subsequent classification were verified on ECG segments from control and ischemic group. The results were statistically analyzed. Cumulants are then used as feature vectors for classification of ECG segments by means of artificial neural network.

Quelques développements récents en traitement du signal

In mathematical population genetics a multiple merger <i>n</i>-coalescent process, or <i>Λ</i> <i>n</i>-coalescent process, {<i>Πⁿ(t) t</i> ≥ 0} models the genealogical tree of a sample of size <i>n</i> (e.g. of DNA sequences) drawn from a large population of haploid individuals. We study various properties of <i>Λ</i> coalescents. Novel in our approach is that we introduce the partition lattice as well as cumulants into the study of functionals of coalescent processes. We illustrate the success of this approach on several examples. Cumulants allow us to reveal the relation between the tree height, <i>T_n</i>, respectively the total branch length, <i>L_n</i>, of the genealogical tree of Kingman’s <i>n</i>-coalescent, arguably the most celebrated coalescent process, and the Riemann zeta function. Drawing on results from lattice theory, we give a spectral decomposition for the generator of both the Kingman and the Bolthausen-Sznitman <i>n</i>-coalescent, the latter of which emerges as a genealogy in models of populations undergoing selection. Taking mutations into account, let <i>M_j</i> count the number of mutations that are shared by <i>j</i> individuals in the sample. The random vector (<i>M₁</i>,...,<i>M_n-1</i>), known as the site frequency spectrum, can be measured from genetical data and is therefore an important statistic from the point of view of applications. Fu worked out the expected value, the variance and the covariance of the marginals of the site frequency spectrum. Using the partition lattice we derive a formula for the cumulants of arbitrary order of the marginals of the site frequency spectrum. Following another line of research, we provide a law of large numbers for a family of <i>Λ</i> coalescents. To be more specific, we show that the process {<i>#Πⁿ(t), t</i> ≥ 0} recording the number <i>#Πⁿ(t)</i> of individuals in the coalescent at time <i>t</i>, coverges, after a suitable rescaling, towards a deterministic limit as the sample size <i>n</i> grows without bound. In the statistical physics literature this limit is known as a hydrodynamic limit. Up to date the hydrodynamic limit was known for Kingman’s coalescent, but not for other <i>Λ</i> coalescents. We work out the hydrodynamic limit for beta coalescents that come down from infinity, which is an important subclass of the <i>Λ</i> coalescents.

Cette thèse porte sur plusieurs questions liées aux mesures de Yang-Mills planaires et aux champs markoviens d'holonomies planaires. Les problèmes sont de deux sortes : étude des champs markoviens d'holonomies planaires pour un groupe de structure donné et l'étude asymptotique des mesures de Yang-Mills lorsque la dimension du groupe tend vers l'infini. On définit la notion de champs markoviens d'holonomies planaires qui axiomatise la notion de mesures de Yang-Mills planaires. En utilisant une nouvelle symétrie en théorie des probabilités, l'invariance par tresse, on construit, caractérise et classifie les champs markoviens d'holonomies planaires. Nous montrons que tout champ markovien d'holonomies planaire est associé à un processus de Lévy qui satisfait une condition de symétrie et vice-versa. Ceci nous permet de caractériser, pour les surfaces sphériques, les champs markoviens d'holonomies tels que définis précédemment par Thierry Lévy. Lorsque le groupe de structure est le groupe symétrique, on peut construire le champ markovien d'holonomies planaire associé grâce à un modèle de revêtements aléatoires. On prouve la convergence des monodromies de ce revêtement aléatoire en s'appuyant sur l'étude, développée dans cette thèse, de l'asymptotique des matrices aléatoires invariantes par conjugaison par le groupe symétrique. / This thesis focuses on planar Yang-Mills measures and planar Markovian holonomy fields. We consider two different questions : the study of planar Markovian holonomy fields with fixed structure group and the asymptotic study of the planar Yang-Mills measures when the dimension of the structure group grows. We define the notion of planar Markovian holonomy fields which generalizes the concept of planar Yang-Mills measures. We construct, characterize and classify the planar Markovian holonomy fields by introducing a new symmetry : the invariance under the action of braids. We show that there is a bijection between planar Markovian holonomy fields and some equivalent classes of Lévy processes. We use these results in order to characterize Markovian holonomy fields on spherical surfaces. The Markovian holonomy fields with the symmetric group as structure group can be constructed using random ramified coverings. We prove that the monodromies of these models of random ramified coverings converge as the number of sheets of the covering goes to infinity. To prove this, we develop general tools in order to study the limits of families of random matrices invariant by the symmetric group. This allows us to generalize ideas, developped by Thierry Lévy in order to study the planar Yang-Mills measure with the unitary structure group, to the setting where the structure group is the symmetric group.

Le travail effectué dans cette thèse concerne les domaines de la théorie des matrices aléatoires et des probabilités libres, dont on connaît les riches connexions depuis le début des années 90. Les résultats s'organisent principalement en deux parties : la première porte sur la liberté infinitésimale, la seconde sur les matrices aléatoires déformées. Plus précisément, on jette les bases d'une théorie combinatoire de la liberté infinitésimale, au premier ordre d'abord, telle que récemment introduite par Belinschi et Shlyakhtenko, puis aux ordres supérieurs. On en donne un cadre simple et général, et on introduit des fonctionnelles de cumulants non-croisés, caractérisant la liberté infinitésimale. L'accent est mis sur la combinatoire et les idées d'essence différentielle qui sous-tendent cette notion. La seconde partie poursuit l'étude des déformations de modèles matriciels, qui a été ces dernières années un champ de recherche très actif. Les résultats présentés sont originaux en ce qu'ils concernent des perturbations déterministes Hermitiennes de rang non nécessairement fini de matrices de Wigner et de Wishart. En outre, un apport de ce travail est la mise en lumière du lien entre la convergence des valeurs propres de ces modèles et les probabilités libres, plus particulièrement le phénomène de subordination pour la convolution libre. Ce lien donne une illustration de la puissance des idées des probabilités libres dans les problèmes de matrices aléatoires.

Cette thèse présente des méthodes de séparation aveugle de sources pour un modèle de mélange non-linéaire particulier, le cas quadratique avec auto-termes et termes croisés. Dans la première partie, nous présentons la structure de séparation étudiée qui se décline sous deux formes : étendue ou basique. Les propriétés de ce réseau récurrent sont ensuite analysées (points d'équilibre, stabilité locale). Nous proposons alors deux méthodes de séparation aveugle de sources. La première exploite les cumulants des observations en un bloc placé en amont de la structure récurrente. La deuxième méthode est basée sur une approche par maximum de vraisemblance. Le tout est validé par des simulations numériques.

Ce manuscrit de thèse porte sur l’analyse mathématique de modèles intégro-différentiels issus de la génétique des populations. Les deux modèles étudiés sont des équations de réaction-dispersion de type ∂tp(t,m) = UD[p](t,m) + f[p](t,m). Ils décrivent la dynamique de la distribution de la fitness (ou valeur sélective) dans une population asexuée sous l’effet des mutations et de la sélection représentées respectivement par les termes non locaux UD[p](t,m) et par f[p](t,m). La différence entre les deux modèles se situe au niveau du terme de mutation. En effet, dans le premier modèle, les effets des mutations sur la fitness ne dépendent pas de la fitness du parent, cela se traduit donc par un terme de convolution classique : D[p](t,m) =RR J(m−y)p(t,y)dy−p(t,m). Lorsqu’une mutation a lieu, la fonction J(m−y) représente la densité de probabilité pour un individu de fitness y d’avoir un descendant de fitness m. Le taux de mutation est donné par la constante U. Dans le second modèle, les effets des mutations sur la fitness dépendent aussi de la fitness du parent. Dans ce cas, un individu de fitness y a un descendant de fitness m avec la densité de probabilité Jy(m−y). Ce type de dépendance apparaît naturellement lorsque l’on suppose qu’il existe une fitness optimale (ou encore un optimum phénotypique). Pour chacun des deux modèles, nous établissons dans un premier temps des résultats d’existence et d’unicité ainsi que des propriétés de décroissance de la solution. Cette décroissance permet de définir la fonction génératrice des cumulants (CGF) associée à la distribution de fitness. La CGF est la solution d’une équation de transport non locale. Pour le premier modèle, l’étude de cette équation permet d’obtenir une solution analytique et donc d’obtenir une description complète de la distribution p(t,m) via ses moments. Nous étudions ensuite les états stationnaires pour chacun des deux modèles, et établissons des conditions suffisantes pour l’existence et la non-existence de phénomènes de concentration, correspondant à une accumulation d’individus de phénotypes optimaux. Nos résultats sont comparés à des sorties de modèles stochastiques individu-centrés représentant le même type de dynamiques évolutives. / This manuscript is devoted to the mathematical analysis of integro-differential models from population genetics. Both models are reaction-dispersion equations of the form ∂tp(t,m) = UD[p](t,m)+ f[p](t,m). They describe the dynamics of fitness distribution in an asexual population under the effect of mutation and selection. These two processes are represented by the nonlocal terms UD[p](t,m) and by f[p](t,m) respectively. The difference between the models rests on the mutation term. Indeed, in the first model, the mutation effects on fitness do not depend on the fitness of the parent. Thus, the mutation term is a standard convolution product: D[p](t,m) =RR J(m−y)p(t,y)dy −p(t,m). When a mutation occurs, the function J(m − y) represents the density of probability for an individual with fitness y to have an offspring with fitness m. The mutation rate is given by the constant U. In the second model, the mutation effects on fitness depend on the fitness of the parent. In this case, an individual with fitness y has an offspring with fitness m with a probability density Jy(m−y). This type of dependence naturally arises when the existence of an optimal fitness (or a phenotypic optimum) is assumed. For both models, we first establish existence and uniqueness results as well as decay properties of the solution. The decay property allows us to define the cumulant generating function (CGF). The CGF obeys a nonlocal transport equation. In the first model, we compute the analytical solution of this transport equation and thus, we obtain a complete description of the distribution p(t,m) through its moments. Then, we study the stationary states for both models, and establish sufficient conditions for the existence and non-existence of a concentration phenomenon corresponding to an accumulation of individuals with best possible phenotype. The results are compared to the results of stochastic individual based models which represent the same kind of evolutionary dynamics.

Estudamos o diagrama de fases e o ponto tricrítico da transição de colapso em um modelo de animais na rede quadrada, a partir da expansão em série da compressibilidade isotérmica KT do sistema. Como função das variáveis x (fugacidade) e y = e1/T (T é a temperatura reduzida), a serie KT é analisada utilizando-se a técnica dos aproximantes diferenciais parciais. Determinamos o padrão de fluxo das trajetórias características de um típico aproximante diferencial parcial com ponto fixo estável. Obtemos estimativas satisfatórias para a fugacidade tricrítica Xt = 0.024 &#177 0.005 e a temperatura tricritica Tt = 0.54 &#177 0.04. Considerando somente campos de escala lineares, obtemos também o expoente de escala &#947 = 1.4 &#177 0.2 e o expoente \"crossover\" &#934 = 0.66 &#177 0.08. Nossos resultados estão em boa concordância com estimativas prévias obtidas por outros métodos. Também estudamos um processo de polimerização ramifIcada através de simulações computacionais na rede quadrada baseadas em um modelo de crescimento cinético generalizado para se incorporar ramifIcações e impurezas. A configuração do polímero e identificada com uma árvore-ligação (\"bond tree\") a fim de se examinar os aspectos topológicos. As dimensões fractais dos aglomerados (\"clusters\") são obtidas na criticalidade. As simulações também permitem o estudo da evolução temporal dos aglomerados bem como a determinação das auto-correlações temporais e expoentes críticos dinâmicos. Com relação aos efeitos de tamanho finito, uma técnica de cumulantes de quarta ordem e empregada para se estimar a probabilidade de ramificação critica bc e os expoentes críticos v e &#946. Na ausência de impurezas, a rugosidade da superfície e descrita em termos dos expoentes de Hurst. Finalmente, simulamos este modelo de crescimento cinético na rede quadrada utilizando um método de Monte Carlo para estudar a polimerização ramificada com interações atrativas de curto alcance entre os monômeros. O diagrama de fases que separa os regimes de crescimento finito e infinito e obtido no plano (T,b) (T é a temperatura reduzida e b é a probabilidade de ramificação). No limite termodinâmico, extrapolamos a temperatura T&#8727 = 0.102 &#177 0.005 abaixo da qual a fase e sempre infinita. Observamos também a ocorrência de uma transição de rugosidade na superfície do polímero. / The phase diagram and the tricritical point of a collapsing lattice animal are studied through an extended series expansion of the isothermal compressibility KT on a square lattice. As a function of the variables x (fugacity) and y = e1/T (T is the reduced temperature), this series KT is investigated using the partial differential approximants technique. The characteristic flow pattern of partial differential approximant trajectories is determined for a typical stable fixed point. We obtain satisfactory estimates for the tricritical fugacity Xt = 0.024 &#177 0.005and temperature Tt = 0.54 &#177 0.04.Taking into account only linear scaling fields we are also able to get the scaling exponent &#947 = 1.4 &#177 0.2 and the crossover exponent &#934 = 0.66 &#177 0.08. Our results are in good agreement with previous estimates from other methods. We also study ramified polymerization through computational simulations on the square lattice of a kinetic growth model generalized to incorporate branching and impurities. The polymer configuration is identified with a bond tree in order to examine its topology. The fractal dimensions of clusters are obtained at criticality. Simulations also allow the study of time evolution of clusters as well as the determination of time autocorrelations and dynamical critical exponents. In regard to finite size effects, a fourth-order cumulant technique is employed to estimate the critical branching probability be and the critical exponents v and &#946. In the absence of impurities, the surface roughness is described in terms of the Hurst exponents. Finally we simulate this kinetic growth model on the square lattice using a Monte Carlo approach in order to study ramified polymerization with short distance attractive interactions between monomers. The phase boundary separating finite from infinite growth regimes is obtained in the (T,b) space (T is the reduced temperature and b is the branching probability). In the thermodynamic limit, we extrapolate the temperature T = 0.102 &#177 0.005 below which the phase is found to be always infinite. We also observe the occurrence of a roughening transition at the polymer surface.

Estudamos o diagrama de fases e o ponto tricrítico da transição de colapso em um modelo de animais na rede quadrada, a partir da expansão em série da compressibilidade isotérmica KT do sistema. Como função das variáveis x (fugacidade) e y = e1/T (T é a temperatura reduzida), a serie KT é analisada utilizando-se a técnica dos aproximantes diferenciais parciais. Determinamos o padrão de fluxo das trajetórias características de um típico aproximante diferencial parcial com ponto fixo estável. Obtemos estimativas satisfatórias para a fugacidade tricrítica Xt = 0.024 &#177 0.005 e a temperatura tricritica Tt = 0.54 &#177 0.04. Considerando somente campos de escala lineares, obtemos também o expoente de escala &#947 = 1.4 &#177 0.2 e o expoente \"crossover\" &#934 = 0.66 &#177 0.08. Nossos resultados estão em boa concordância com estimativas prévias obtidas por outros métodos. Também estudamos um processo de polimerização ramifIcada através de simulações computacionais na rede quadrada baseadas em um modelo de crescimento cinético generalizado para se incorporar ramifIcações e impurezas. A configuração do polímero e identificada com uma árvore-ligação (\"bond tree\") a fim de se examinar os aspectos topológicos. As dimensões fractais dos aglomerados (\"clusters\") são obtidas na criticalidade. As simulações também permitem o estudo da evolução temporal dos aglomerados bem como a determinação das auto-correlações temporais e expoentes críticos dinâmicos. Com relação aos efeitos de tamanho finito, uma técnica de cumulantes de quarta ordem e empregada para se estimar a probabilidade de ramificação critica bc e os expoentes críticos v e &#946. Na ausência de impurezas, a rugosidade da superfície e descrita em termos dos expoentes de Hurst. Finalmente, simulamos este modelo de crescimento cinético na rede quadrada utilizando um método de Monte Carlo para estudar a polimerização ramificada com interações atrativas de curto alcance entre os monômeros. O diagrama de fases que separa os regimes de crescimento finito e infinito e obtido no plano (T,b) (T é a temperatura reduzida e b é a probabilidade de ramificação). No limite termodinâmico, extrapolamos a temperatura T&#8727 = 0.102 &#177 0.005 abaixo da qual a fase e sempre infinita. Observamos também a ocorrência de uma transição de rugosidade na superfície do polímero. / The phase diagram and the tricritical point of a collapsing lattice animal are studied through an extended series expansion of the isothermal compressibility KT on a square lattice. As a function of the variables x (fugacity) and y = e1/T (T is the reduced temperature), this series KT is investigated using the partial differential approximants technique. The characteristic flow pattern of partial differential approximant trajectories is determined for a typical stable fixed point. We obtain satisfactory estimates for the tricritical fugacity Xt = 0.024 &#177 0.005and temperature Tt = 0.54 &#177 0.04.Taking into account only linear scaling fields we are also able to get the scaling exponent &#947 = 1.4 &#177 0.2 and the crossover exponent &#934 = 0.66 &#177 0.08. Our results are in good agreement with previous estimates from other methods. We also study ramified polymerization through computational simulations on the square lattice of a kinetic growth model generalized to incorporate branching and impurities. The polymer configuration is identified with a bond tree in order to examine its topology. The fractal dimensions of clusters are obtained at criticality. Simulations also allow the study of time evolution of clusters as well as the determination of time autocorrelations and dynamical critical exponents. In regard to finite size effects, a fourth-order cumulant technique is employed to estimate the critical branching probability be and the critical exponents v and &#946. In the absence of impurities, the surface roughness is described in terms of the Hurst exponents. Finally we simulate this kinetic growth model on the square lattice using a Monte Carlo approach in order to study ramified polymerization with short distance attractive interactions between monomers. The phase boundary separating finite from infinite growth regimes is obtained in the (T,b) space (T is the reduced temperature and b is the branching probability). In the thermodynamic limit, we extrapolate the temperature T = 0.102 &#177 0.005 below which the phase is found to be always infinite. We also observe the occurrence of a roughening transition at the polymer surface.