On étudie un système de particules en interactions. Deux types de processus sont utilisés pour modéliser le système. Tout d'abord des processus de Hawkes. On propose deux modèles pour lesquels on obtient l'existence et l'unicité d'une version stationnaire, ainsi qu'une construction graphique de la mesure stationnaire à l'aide d'une décomposition de type Kalikow et d'un algorithme de simulation parfaite.Le deuxième type de processus utilisés est un processus de Markov déterministe par morceaux (PDMP). On montre l'ergodicité de ce processus et propose un estimateur à noyau pour la fonction de taux de saut possédant une vitesse de convergence optimale dans L². / We work on interacting particles systems. Two different types of processes are studied. A first model using Hawkes processes, for which we state existence and uniqueness of a stationnary version. We also propose a graphical construction of the stationnary measure by the mean of a Kalikow-type decomposition and a perfect simulation algorithm.The second model deals with Piecewise deterministic Markov processes (PDMP). We state ergodicity and propose a Kernel estimator for the jump rate function having an optimal speed of convergence in L².

Orientador: João Francisco Galera Monico / Banca: José Tadeu Garcia Tommaselli / Banca: Edvaldo Simões da Fonseca Júnior / Resumo: Dados de receptores GPS de dupla freqüência são, em geral, processados utilizando a combinação ion-free, o que permite eliminar os efeitos de primeira ordem da ionosfera. Porém, os efeitos de segunda e terceira ordem, geralmente, são negligenciados no processamento de dados GPS. Nesse trabalho, esses efeitos foram levados em consideração no processamento dos dados. Foram investigados os modelos matemáticos associados a esses efeitos, as transformações envolvendo o campo magnético da Terra e a utilização do TEC advindo dos Mapas Globais da Ionosfera ou calculados a partir das pseudodistâncias. Numa outra investigação independente, os efeitos residuais de primeira ordem da ionosfera, resultantes da dupla diferença da pseudodistância e da fase da onda portadora, foram considerados como incógnitas no ajustamento. Porém, esses efeitos residuais foram tratados como pseudo-observações, associados aos processos aleatórios random walk e white noise e, adicionados ao algoritmo de filtro de Kalman. Dessa forma, o modelo matemático preserva a característica de número inteiro da ambigüidade da fase, facilitando a aplicação de algoritmos de solução da ambigüidade, que no caso desse trabalho, utilizou-se o método LAMBDA. Para o caso da consideração dos efeitos de segunda e terceira ordem da ionosfera, foram realizados processamentos de dados GPS envolvendo o modo relativo e o Posicionamento por Ponto Preciso. Os resultados mostraram que a não consideração desses efeitos no processamento dos dados GPS pode introduzir variações da ordem de três a quatro milímetros nas coordenadas das estações. / Abstract: Data from dual frequency receiver, in general, are processed using the ion-free combination that allows the elimination of the first order ionospheric effects. However, the second and third order ionospheric effects, generally, are neglected in the GPS data processing. In this work, these effects were taken into account in the GPS data processing. In this case, it was investigated the mathematical models associated with the second and third order effects, the transformations involving the Earth magnetic field and the use of TEC from Ionosphere Global Maps or calculated from the pseudoranges. The first order ionosphere residual effects, resulting from pseudorange and phase double difference, were taken into account as unknown in the adjustment. However, these effects were treated as pseudo-observables and it was associated with the random process random walk and white noise and added to the Kalman filter algorithm. Therefore, the mathematical model preserves the phase ambiguity "integerness", facilitating the application of ambiguity resolution approaches, which in the case of this work, it was used the LAMBDA method. / Mestre

In the paper we study sequences of random functions which are defined by some interpolation procedures for a given random function. We investigate the problem in what sense and under which conditions the sequences converge to the prescribed random function. Sufficient conditions for convergence of moment characteristics, of finite dimensional distributions and for weak convergence of distributions in spaces of continuous functions are given. The treatment of such questions is stimulated by an investigation of Monte Carlo simulation procedures for certain classes of random functions. In an appendix basic facts concerning weak convergence of probability measures in metric spaces are summarized.

Cette thèse porte sur le problème d'estimation à travers les échelles pour un processus stochastique. Nous étudions comment le choix du pas d'échantillonnage impacte les procédures statistiques. Nous nous intéressons à l'estimation de processus à sauts à partir de l'observation d'une trajectoire discrétisée sur [0, T]. Lorsque la longueur de l'intervalle d'observation T va à l'infini, le pas d'échantillonnage tend soit vers 0 (échelle microscopique), vers une constante positive (échelle intermédiaire) ou encore vers l'infini (échelle macroscopique). Dans chacun de ces régimes nous supposons que le nombre d'observations tend vers l'infini. Dans un premier temps le cas particulier d'un processus de Poisson composé d'intensité inconnue avec des sauts symétriques {-1,1} est étudié. Le Chapitre 2 illustre la notion d'estimation statistique dans les trois échelles définies ci-dessus. Dans ce modèle, on s'intéresse aux propriétés des expériences statistiques. On montre la propriété de Normalité Asymptotique Locale dans les trois échelles microscopiques, intermédiaires et macroscopiques. L'information de Fisher est alors connue pour chacun de ces régimes. Ensuite nous analysons comment se comporte une procédure d'estimation de l'intensité qui est efficace (de variance minimale) à une échelle donnée lorsqu'on l'applique à des observations venant d'une échelle différente. On regarde l'estimateur de la variation quadratique empirique, qui est efficace dans le régime macroscopique, et on l'utilise sur des données provenant des régimes intermédiaire ou microscopique. Cet estimateur reste efficace dans les échelles microscopiques, mais montre une perte substantielle d'information aux échelles intermédiaires. Une procédure unifiée d'estimation est proposée, elle est efficace dans tous les régimes. Les Chapitres 3 et 4 étudient l'estimation non paramétrique de la densité de saut d'un processus renouvellement composé dans les régimes microscopiques, lorsque le pas d'échantillonnage tend vers 0. Un estimateur de cette densité utilisant des méthodes d'ondelettes est construit. Il est adaptatif et minimax pour des pas d'échantillonnage qui décroissent en T^{-alpha}, pour alpha>0. La procédure d'estimation repose sur l'inversion de l'opérateur de composition donnant la loi des incréments comme une transformation non linéaire de la loi des sauts que l'on cherche à estimer. L'opérateur inverse est explicite dans le cas du processus de Poisson composé (Chapitre 3), mais n'a pas d'expression analytique pour les processus de renouvellement composés (Chapitre 4). Dans ce dernier cas, il est approché via une technique de point fixe. Le Chapitre 5 étudie le problème de perte d'identifiabilité dans les régimes macroscopiques. Si un processus à sauts est observé avec un pas d'échantillonnage grand, certaines approximations limites, telles que l'approximation gaussienne, deviennent valides. Ceci peut entraîner une perte d'identifiabilité de la loi ayant généré le processus, dès lors que sa structure est plus complexe que celle étudiée dans le Chapitre 2. Dans un premier temps un modèle jouet à deux paramètres est considéré. Deux régimes différents émergent de l'étude : un régime où le paramètre n'est plus identifiable et un où il reste identifiable mais où les estimateurs optimaux convergent avec des vitesses plus lentes que les vitesses paramétriques habituelles. De l'étude de cas particulier, nous dérivons des bornes inférieures montrant qu'il n'existe pas d'estimateur convergent pour les processus de Lévy de saut pur ou pour les processus de renouvellement composés dans les régimes macroscopiques tels que le pas d'échantillonnage croît plus vite que racine de T. Enfin nous identifions des régimes macroscopiques où les incréments d'un processus de Poisson composé ne sont pas distinguables de variables aléatoires gaussiennes, et des régimes où il n'existe pas d'estimateur convergent pour les processus de Poisson composés dépendant de trop de paramètres / This thesis studies the problem of statistical inference across time scales for a stochastic process. More particularly we study how the choice of the sampling parameter affects statistical procedures. We narrow down to the inference of jump processes from the discrete observation of one trajectory over [0,T]. As the length of the observation interval T tends to infinity, the sampling rate either goes to 0 (microscopic scale) or to some positive constant (intermediate scale) or grows to infinity (macroscopic scale). We set in a case where there are infinitely many observations. First we specialise in a toy model: a compound Poisson process of unknown intensity with symmetric Bernoulli jumps. Chapter 2 highlights the concept of statistical estimation in the three regimes defined above and the phenomena at stake. We study the properties of the statistical experiments in each regime, we show that the Local Asymptotic Normality property holds in every regimes (microscopic, intermediate and macroscopic). We also provide the formula of the associated Fisher information in each regime. Then we study how a statistical procedure which is optimal (of minimal variance) at a given scale is affected when we use it on data coming from another scale. We focus on the empirical quadratic variation estimator, it is an optimal procedure at macroscopic scales. We apply it on data coming from intermediate and microscopic regimes. Although the estimator remains efficient at microscopic scales, it shows a substantial loss of information when used on data coming from an intermediate regime. That loss can be explicitly related to the sampling rate. We provide an unified procedure, efficient in all regimes. Chapters 3 and 4 focus on microscopic regimes, when the sampling rate decreases to 0. The nonparametric estimation of the jump density of a renewal reward process is studied. We propose an adaptive wavelet threshold density estimator. It achieves minimax rates of convergence for sampling rates that vanish polynomially with T, namely in T^{-alpha} for alpha>0. The estimation procedure is based on the inversion of the compounding operator in the same spirit as Buchmann and Grübel (2003), which specialiase in the study of discrete compound laws. The inverse operator is explicit in the case of a compound Poisson process (see Chapter 3), but has no closed form expression for renewal reward processes (see Chapter 4). In that latter case the inverse operator is approached with a fixed point technique. Finally Chapter 5 studies at which rate identifiability is lost in macroscopic regimes. Indeed when a jump process is observed at an arbitrarily large sampling rate, limit approximations, like Gaussian approximations, become valid and the specificities of the jumps may be lost, as long as the structure of the process is more complex than the one introduced in Chapter 2. First we study a toy model depending on a 2-dimensional parameter. We distinguish two different regimes: fast (macroscopic) regimes where all information on the parameter is lost and slow regimes where the parameter remains identifiable but where optimal estimators converge with slower rates than the expected usual parametric ones. From this toy model lower bounds are derived, they ensure that consistent estimation of Lévy processes or renewal reward processes is not possible when the sampling rate grows faster than the square root of T. Finally we identify regimes where an experiment consisting in increments of a compound Poisson process is asymptotically equivalent to an experiment consisting in Gaussian random variables. We also give regimes where there is no consistent estimator for compound Poisson processes depending on too many parameters

<p>In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D).</p><p>In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively.</p><p>In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels.</p><p>Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity.</p><p>These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.</p>

In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D). In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively. In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels. Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity. These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.

In this work, I will describe a new statistical tool: the canonical bicoherence, which is a combination of the canonical coherence and the bicoherence. I will provide its definitions, properties, estimation by multitaper methods and statistics, and estimate the variance of the estimates by the weighted jackknife method. I will discuss its applicability and usefulness in nonlinear quadratic phase coupling detection and analysis for multivariate random processes. Furthermore, I will develop the time-varying canonical bicoherence for the nonlinear analysis of non-stationary random processes. In this thesis, the canonical bicoherence is mainly applied in two types of data: a) three-component geomagnetic field data, and b) high-dimensional brain electroencephalogram data. Both results obtained will be linked with physical or physiological interpretations. In particular, this thesis is the first work where the novel method of ``canonical bicoherence'' is introduced and applied to the nonlinear quadratic phase coupling detection and analysis for multivariate random processes. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-10-31 15:03:57.596

When the human mind searches concept space for solutions to a given condition we have a choice between conventional and creative thinking. But what are the probabilities of improving a given situation using creative thinking compared with conventional thinking? To answer this question we are extending the meaning of creativity beyond human creativity. We view creativity as an optimised search strategy applicable to the larger set of all teleonomic systems and term this creativity teleonomic creativity. We argue that an analog process is common to all manifestations of creativity within teleonomic systems and describe this process and its cause. In order to show this process and to make quantitative comparisons, we utilise the metaphor of an adaptive fitness landscape and simple statistical techniques. The term fitness in our case describes the condition of a well-defined property being suitable for a purpose, rather than an overall evaluation of many complex interactions measuring reproductive success. We define creativity as the successful attempt of either individuals or populations to gain higher fitness via exploration of global fitness peaks as opposed to the exploitation of a currently occupied local peak. We then show mathematically how the inclusion of creativity in a search can dramatically increase the chances of finding appropriate solutions. We also recognise that creative behaviour is most successful when the environmentis unstable. We note the existence of a strategic meta-parameter that allows self-adaptation when tuned via a feedback loop from the environment. We show that creativity can be understood as a random process with an optimal setting for the standard deviation that maximises the probability of hitting a target of higher fitness. We support our claims with computer simulations and observe several occurrences of teleonomic creativity in nature. In addition we measure the entropy of a teleonomic system via the phase-space of internal variables and observe a sudden entropy increase during the onset of creative behaviour in a teleonomic system. Our investigations also enable us to rationalise the processes, conditions and phenomena surrounding human creativity such as mistakes, madness, serendipity, humor, analogy making and interpret the function of creativity promoters and inhibitors. Our findings may also allow us to incorporate creativity into artificial computer models. We speculate that creativity is an emerging property of any teleonomic system and as such ubiquitous in nature.

Understanding the complex dynamics that underpin the response of structures in the occurrence of earthquakes is of paramount importance in ensuring community resilience. The operational continuity of structures is influenced by the performance of nonstructural components, also known as secondary structures. Inherent vulnerability characteristics, nonlinearities and uncertainties in their properties or in the excitation pose challenges that render their response determination as a non-straightforward task. This dissertation settles in the context of mathematical modelling and response quantification of seismically driven secondary systems. The case of bilinear hysteretic, rigid-plastic and free-standing rocking oscillators is first considered, as a representative class of secondary systems of distinct behaviour excited at a single point in the primary structure. The equations governing their full dynamic interaction with linear primary oscillators are derived with the purpose of assessing the appropriateness of simplified analysis methods where the secondary-primary feedback action is not accounted for. Analyses carried out in presence of pulse-type excitation have shown that the cascade approximation can be considered satisfactory for bilinear systems provided the secondary-primary mass ratio is adequately low and the system does not approach resonance. For the case of sliding and rocking systems, much lighter secondary systems need to be considered if the cascade analysis is to be adopted, with the validity of the approximation dictated by the selection of the input parameters. Based on the premise that decoupling is permitted, new analytical solutions are derived for the pulse driven nonlinear oscillators considered, conveniently expressing the seismic response as a function of the input parameters and the relative effects are quantified. An efficient numerical scheme for a general-type of excitation is also presented and is used in conjunction with an existing nonstationary stochastic far-field ground motion model to determine the seismic response spectra for the secondary oscillators at given site and earthquake characteristics. Prompted by the presence of uncertainty in the primary structure, and in line with the classical modal analysis, a novel approach for directly characterising uncertainty in the modal shapes, frequencies and damping ratios of the primary structure is proposed. A procedure is then presented for the identification of the model parameters and demonstrated with an application to linear steel frames with uncertain semi-rigid connections. It is shown that the proposed approach reduces the number of the uncertain input parameters and the size of the dynamic problem, and is thus particularly appealing for the stochastic assessment of existing structural systems, where partial modal information is available e.g. through operational modal analysis testing. Through a numerical example, the relative effect of stochasticity in a bi-directional seismic input is found to have a more prominent role on the nonlinear response of secondary oscillators when compared to the uncertainty in the primary structure. Further extending the analyses to the case of multi-attached linear secondary systems driven by deterministic seismic excitation, a convenient variant of the component-mode synthesis method is presented, whereby the primary-secondary dynamic interaction is accounted for through the modes of vibration of the two components. The problem of selecting the vibrational modes to be retained in analysis is then addressed for the case of secondary structures, which may possess numerous low frequency modes with negligible mass, and a modal correction method is adopted in view of the application for seismic analysis. The influence of various approaches to build the viscous damping matrix of the primary-secondary assembly is also investigated, and a novel technique based on modal damping superposition is proposed. Numerical applications are demonstrated through a piping secondary system multi-connected on a primary frame exhibiting various irregularities in plan and elevation, as well as a multi-connected flexible secondary system. Overall, this PhD thesis delivers new insights into the determination and understanding of the response of seismically driven secondary structures. The research is deemed to be of academic and professional engineering interest spanning several areas including seismic engineering, extreme events, structural health monitoring, risk mitigation and reliability analysis.

Multilevel Random Pulse Width Modulation (RPWM) schemes have drawn increasing attention in the past few years. Multilevel topologies provide high voltage and high power capabilities and random PWM schemes offer reduction in discrete harmonics spectral. This dissertation provides a generalized theory and analysis methods of the standard five-level Weighted RPWM (WRPWM). Equations have been derived to analyze the spectral performance and average switching frequency of the WRPWM output waveform using statistical approach. A modified WRPWM scheme has been proposed. The modified WRPWM scheme is then analyzed with the equations derived from the same approach. The analyzed theoretical spectrum of the standard five-level WRPWM is then compared with the three-level WRPWM scheme and the conventional carrier based PWM scheme. A scaled laboratory prototype diode clamping five-level inverter has been built for verification of the standard and the proposed modified WRPWM schemes. It can be seen that the experimental measurements and the theoretical analyzed results are all in good agreement. Results show the two five-level WRPWM schemes offers significant improvements on the spectrum content than the conventional carrier based PWM scheme. It was found that the five-level WRPWM schemes have successfully suppress the magnitude of third harmonic below 5% of the magnitude of fundamental component and even less for the higher order harmonic components. Research contributions made by the dissertation are: - The proposed modified multilevel WRPWM scheme which utilizing the switching decision redundancy of multilevel inverter to manipulate the harmonic content of the output signal. - The derived mathematical equations of the standard and modified five-level WRPWM scheme for analytical purposes. / Dissertation (MEng (Electrical Engineering))--University of Pretoria, 2005. / Electrical, Electronic and Computer Engineering / unrestricted