This research develops a fault diagnosis method for complex systems in the presence of uncertainties and possibility of multiple solutions. Fault diagnosis is a challenging problem because data used in diagnosis contain random errors and often systematic errors as well. Furthermore, fault diagnosis is basically an inverse problem so that it inherits unfavorable characteristics of inverse problems: The existence and uniqueness of an inverse solution are not guaranteed and the solution may be unstable. The weighted least squares method and its variations are traditionally used for solving inverse problems. However, the existing algorithms often fail to identify multiple solutions if they are present. In addition, the existing algorithms are not capable of selecting variables systematically so that they generally use the full model in which may contain unnecessary variables as well as necessary variables. Ignoring this model uncertainty often gives rise to, so called, the smearing effect in solutions, because of which unnecessary variables are overestimated and necessary variables are underestimated. The proposed method solves the inverse problem using Bayesian inference. An engineering system can be parameterized using state variables. The probability of each state variable is inferred from observations made on the system. A bias in an observation is treated as a variable, and the probability of the bias variable is inferred as well. To take the uncertainty of model structure into account, multiple Bayesian models are created with various combinations of the state variables and the bias variables. The results from all models are averaged according to how likely each model is. Gibbs sampling is used for approximating updated probabilities. The method is demonstrated for two applications: the status matching of a turbojet engine and the fault diagnosis of an industrial gas turbine. In the status matching application only physical faults in the components of a turbojet engine are considered whereas in the fault diagnosis application sensor biases are considered as well as physical faults. The proposed method is tested in various faulty conditions using simulated measurements. Results show that the proposed method identifies physical faults and sensor biases simultaneously. It is also demonstrated that multiple solutions can be identified. Overall, there is a clear improvement in ability to identify correct solutions over the full model that contains all state and bias variables.

The research in distributed algorithms is linked with the developments of statistical inference in wireless sensor networks (WSNs) applications. Typically, distributed approaches process the collected signals from networked sensor nodes. That is to say, the sensors receive local observations and transmit information between each other. Each sensor is capable of combining the collected information with its own observations to improve performance. In this thesis, we propose novel distributed methods for the inference applications using wireless sensor networks. In particular, the efficient algorithms which are not computationally intensive are investigated. Moreover, we present a number of novel algorithms for processing asynchronous network events and robust state estimation. In the first part of the thesis, a distributed adaptive algorithm based on the component-wise EM method for decentralized sensor networks is investigated. The distributed component-wise Expectation-Maximization (EM) algorithm has been designed for application in a Gaussian density estimation. The proposed algorithm operates a component-wise EM procedure for local parameter estimation and exploit an incremental strategy for network updating, which can provide an improved convergence rate. Numerical simulation results have illustrated the advantages of the proposed distributed component-wise EM algorithm for both well-separated and overlapped mixture densities. The distributed component-wise EM algorithm can outperform other EM-based distributed algorithms in estimating overlapping Gaussian mixtures. In the second part of the thesis, a diffusion based EM gradient algorithm for density estimation in asynchronous wireless sensor networks has been proposed. Specifically, based on the asynchronous adapt-then-combine diffusion strategy, a distributed EM gradient algorithm that can deal with asynchronous network events has been considered. The Bernoulli model has been exploited to approximate the asynchronous behaviour of the network. Compared with existing distributed EM based estimation methods using a consensus strategy, the proposed algorithm can provide more accurate estimates in the presence of asynchronous networks uncertainties, such as random link failures, random data arrival times, and turning on or off sensor nodes for energy conservation. Simulation experiments have been demonstrated that the proposed algorithm significantly outperforms the consensus based strategies in terms of Mean-Square- Deviation (MSD) performance in an asynchronous network setting. Finally, the challenge of distributed state estimation in power systems which requires low complexity and high stability in the presence of bad data for a large scale network is addressed. A gossip based quasi-Newton algorithm has been proposed for solving the power system state estimation problem. In particular, we have applied the quasi-Newton method for distributed state estimation under the gossip protocol. The proposed algorithm exploits the Broyden- Fletcher-Goldfarb-Shanno (BFGS) formula to approximate the Hessian matrix, thus avoiding the computation of inverse Hessian matrices for each control area. The simulation results for IEEE 14 bus system and a large scale 4200 bus system have shown that the distributed quasi-Newton scheme outperforms existing algorithms in terms of Mean-Square-Error (MSE) performance with bad data.

Na estatística experimental, especificamente quando se faz análise de variância, os testes de hipóteses têm sido amplamente utilizados para se concluir a respeito das fontes de variação consideradas nos modelos lineares. Para tanto, é comum a utilização de sistemas estatísticos que fornecem análises de variância e a estatística F, entre outras, para a tomada de decisões. Entretanto, o teste F numa análise de variância para tratamentos com mais de um grau de liberdade proporciona informações gerais, relacionadas com o comportamento médio dos tratamentos. Por essa razão, deve-se planejar comparações objetivas, fazendo-se desdobramentos dos graus de liberdade de tratamentos para obter informações mais específicas. Nesse sentido, uma técnica usada para esses desdobramentos baseia-se na utilização de contrastes, sendo necessário que cada componente seja explicado por um contraste, com todos os contrastes sendo ortogonais entre si, para que as comparações sejam independentes. Entretanto, essa técnica torna-se complexa à medida que o número de tratamentos aumenta. Frente a isso, utilizando-se os dados provenientes de um experimento de competição entre dois grupos de variedades de cana-de-açúcar, inteiramente ao acaso com seis tratamentos e cinco repetições, e também nos dados obtidos de um experimento fictício de competição entre híbridos de milho no delineamento blocos casualizados, propôs-se uma técnica, empregando variáveis auxiliares, para facilitar o desdobramento ortogonal dos graus de liberdade de tratamentos, procurando-se evidenciar que essa técnica facilita o desdobramento ortogonal dos graus de liberdade de tratamentos e tem resultados equivalentes aos obtidos utilizando-se a função CONTRAST do PROC GLM do SAS. Outro problema refere-se à análise de experimentos fatoriais com desbalanceamento das amostras, tendo em vista que as técnicas de estimação de parcelas perdidas não resolvem satisfatoriamente o problema, principalmente se existem muitas parcelas perdidas. Quando os dados são desbalanceados, há necessidade de se conhecer que hipóteses estão sendo testadas e se estas são de interesse do pesquisador, devido à complexidade dessas hipóteses, principalmente em presença de caselas vazias. Além disso, muito têm sido escrito sobre os diferentes resultados da análise de variância apresentados por sistemas estatísticos para dados desbalanceados com caselas vazias, o que tem gerado confusão entre os pesquisadores. Com a finalidade de propor um método alternativo para a obtenção de hipóteses de interesse, utilizaram-se os resultados de um experimento fatorial 2x3, inteiramente ao acaso, com quatro repetições, para testar os efeitos de três reguladores de crescimento (hormônios), sobre a propagação “in vitro” de dois porta-enxertos (cultivares) de macieira. Assim, diante do fato que testar uma hipótese é equivalente a impor uma restrição estimável aos parâmetros do modelo, utilizaram-se restrições paramétricas estimáveis como um critério alternativo para realizar testes de hipóteses de interesse em modelos lineares com dados desbalanceados. Os resultados mostram que esse método permite que o pesquisador teste diretamente hipóteses de seu interesse, com resultados equivalentes aos encontrados com a função CONTRAST do PROC GLM do SAS. / For experimental designs, it is usually necessary to do tests of hypotheses to conclude about effects considered in the linear models. In these cases, it is common to use statistical softwares that supply the analyses of variance and F statistics, among others, for taking decisions. However, the test F in an analysis of variance for sources of variation with more than a degree of freedom provides general information, about significant differences of levels of the factor. Therefore, it should be planned objective comparisons, making orthogonal decompositions of the degrees of the effects of interest to get more specific information. One technique used frequently based on the orthogonal contrasts, so that the comparisons are independent. However, this technique becomes complex as the number of levels of the factor increases. To study alternative methods to do these comparisons, we use data from a yield trail experiment considering two groups of varieties of sugarcane, in a complete randomized design with 6 treatments and 5 repetitions. Also, we use data from a fictitious experiment comparing hybrids of maize in the randomized complete block design. The technique of analysis using dummy variables to facilitate the orthogonal decomposition of degrees of freedom of treatments was proposed. This technique facilitates the orthogonal decomposition and has the same results of those obtained the function CONTRAST of PROC GLM of SAS. Another situation considered involves experiments with unbalanced data. In this case, it is possible to suppose that there is the necessity of knowing what hypotheses are being tested and if they are useful. Much has been written on the different results of analysis of variance presented by statistical software for unbalanced data. This can create confusion to the researcher. To illustrate, we used the results of an 2x3 factorial experiment with 4 replicates, to test the effect of 3 hormones, on the propagation of 2 in vitro cultivars of apple trees. Thus, considering that to test a hypotheses is equivalent to impose an estimable restriction to the parameters of the model, we use these restrictions as an alternative criteria to directly carry out tests of hypotheses in linear models with unbalanced data. The results showed that this procedure is equivalent of that used by the function CONTRAST of PROC GLM/SAS.

This thesis presents methods for the inference of system state and the learning of model structure for a number of hidden-state time series models, within a Bayesian probabilistic framework. Motivating examples are taken from application areas including finance, physical object tracking and audio restoration. The work in this thesis can be broadly divided into three themes: system and parameter estimation in linear jump-diffusion systems, non-parametric model (system) estimation and batch audio restoration. For linear jump-diffusion systems, efficient state estimation methods based on the variable rate particle filter are presented for the general linear case (chapter 3) and a new method of parameter estimation based on Particle MCMC methods is introduced and tested against an alternative method using reversible-jump MCMC (chapter 4). Non-parametric model estimation is examined in two settings: the estimation of non-parametric environment models in a SLAM-style problem, and the estimation of the network structure and forms of linkage between multiple objects. In the former case, a non-parametric Gaussian process prior model is used to learn a potential field model of the environment in which a target moves. Efficient solution methods based on Rao-Blackwellized particle filters are given (chapter 5). In the latter case, a new way of learning non-linear inter-object relationships in multi-object systems is developed, allowing complicated inter-object dynamics to be learnt and causality between objects to be inferred. Again based on Gaussian process prior assumptions, the method allows the identification of a wide range of relationships between objects with minimal assumptions and admits efficient solution, albeit in batch form at present (chapter 6). Finally, the thesis presents some new results in the restoration of audio signals, in particular the removal of impulse noise (pops and clicks) from audio recordings (chapter 7).

Orientadores: Caio Lucidius Naberezny Azevedo, Filidor Edilfonso Vilca Labra / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T22:33:37Z (GMT). No. of bitstreams: 1 Chaves_NathaliaLima_M.pdf: 3044792 bytes, checksum: 8fea3cd9d074997b605026a7a4526c35 (MD5) Previous issue date: 2015 / Resumo: A classe de modelos Birnbaum-Saunders (BS) foi desenvolvida a partir de problemas que surgiram na área de confiabilidade de materiais. Tais problemas, em geral, são ligados ao estudo de fadiga de materiais. No entanto, nos últimos tempos, essa classe de modelos tem sido aplicada em áreas fora do referido contexto como, por exemplo, em ciências da saúde, ambiental, florestal, demográficas, atuariais, financeira, entre outras, devido à sua grande versatilidade. Neste trabalho desenvolvemos a distribuição Birnbaum-Saunders (BS) baseada na normal assimétrica padrão sob a parametrização centrada (BSNAC) que, além de representar uma extensão da distribuição BS usual, apresenta diversas vantagens em relação à distribuição BS baseada na distribuição normal assimétrica sob a parametrização usual. Desenvolvemos também um modelo de regressão linear log-Birnbaum-Saunders. Apresentamos, tanto para a distribuição BSNAC quanto para o respectivo modelo de regressão, diversas propriedades. Desenvolvemos procedimentos de estimação sob os enfoques frenquentista e bayesiano, bem como ferramentas de diagnóstico para os modelos propostos, contemplando análise residual e medidas de influência. Realizamos estudos de simulação, considerando diferentes cenários, com o intuito de comparar as estimativas frequentistas e bayesianas, bem como avaliar o desempenho das medidas de diagnóstico. A metodologia aqui proposta foi ilustrada tanto com dados provenientes de estudos de simulação, quanto com conjuntos de dados reais / Abstract: The class of Birnbaum-Saunders (BS) models was developed from problems that arose in the field of material reliability. These problems generally are related to the study of material fatigue. However, in the last years, this class of models has been applied in areas outside that context, such as in health sciences, environmental, forestry, demographic, actuarial, financial, among others, due to its great versatility. In this work, we developed the skew-normal Birnbaum-Saunders distribution under the centered parameterization (BSNAC), which also represents an extension of the usual BS distribution and presents several advantages over the BS distribution based on the skew-normal distribution under the usual parameterization. We also developed a log-Birnbaum-Saunders linear regression model. We present several properties of both BSNAC distribution and the related regression model. We develop estimation procedures under the frequentist and Bayesian approaches, as well as diagnostic tools for the proposed models, contemplating residual analysis and measures of influence. We conducted simulation studies considering different scenarios, in order to compare the frequentist and Bayesian estimates and evaluate the performance of diagnostic measures. The methodology proposed here is illustrated with data sets from both simulation studies and real data sets / Mestrado / Estatistica / Mestra em Estatística

In this thesis, we hope to dispel some confusion regarding the so-called post hoc power, i.e. power computed making the assumption that the estimated sample effect is equal to the population effect size. In previous research, it has been shown that post hoc power is a function of the p-value, making it redundant as a tool of analysis. We go further, arguing for it to never be reported, since it is a source of confusion and potentially harmful incentives. We also conduct a Monte Carlo simulation to illustrate our points of view. Previous research is confirmed by the results of this study.

<div>In this thesis, we focus on the problem of estimating an unknown quantity of interest, when the information required to do so is dispersed over a network of agents. In particular, each agent in the network receives sequential observations generated by the unknown quantity, and the collective goal of the network is to eventually learn this quantity by means of appropriately crafted information diffusion rules. The abstraction described above can be used to model a variety of problems ranging from environmental monitoring of a dynamical process using autonomous robot teams, to statistical inference using a network of processors, to social learning in groups of individuals. The limited information content of each agent, coupled with dynamically changing networks, the possibility of adversarial attacks, and constraints imposed by the communication channels, introduce various unique challenges in addressing such problems. We contribute towards systematically resolving some of these challenges.</div><div><br></div><div>In the first part of this thesis, we focus on tracking the state of a dynamical process, and develop a distributed observer for the most general class of LTI systems, linear measurement models, and time-invariant graphs. To do so, we introduce the notion of a multi-sensor observable decomposition - a generalization of the Kalman observable canonical decomposition for a single sensor. We then consider a scenario where certain agents in the network are compromised based on the classical Byzantine adversary model. For this worst-case adversarial setting, we identify certain fundamental necessary conditions that are a blend of system- and network-theoretic requirements. We then develop an attack-resilient, provably-correct, fully distributed state estimation algorithm. Finally, by drawing connections to the concept of age-of-information for characterizing information freshness, we show how our framework can be extended to handle a broad class of time-varying graphs. Notably, in each of the cases above, our proposed algorithms guarantee exponential convergence at any desired convergence rate.</div><div><br></div><div>In the second part of the thesis, we turn our attention to the problem of distributed hypothesis testing/inference, where each agent receives a stream of stochastic signals generated by an unknown static state that belongs to a finite set of hypotheses. To enable each agent to uniquely identify the true state, we develop a novel distributed learning rule that employs a min-protocol for data-aggregation, as opposed to the large body of existing techniques that rely on "belief-averaging". We establish consistency of our rule under minimal requirements on the observation model and the network structure, and prove that it guarantees exponentially fast convergence to the truth with probability 1. Most importantly, we establish that the learning rate of our algorithm is network-independent, and a strict improvement over all existing approaches. We also develop a simple variant of our learning algorithm that can account for misbehaving agents. As the final contribution of this work, we develop communication-efficient rules for distributed hypothesis testing. Specifically, we draw on ideas from event-triggered control to reduce the number of communication rounds, and employ an adaptive quantization scheme that guarantees exponentially fast learning almost surely, even when just 1 bit is used to encode each hypothesis. </div>

Le sujet de cette thèse est l'inférence statistique de processus d'Ornstein-Uhlenbeck multi-dimensionnels. Dans une première partie, nous introduisons un modèle de graphes stochastiques définis comme observations binaires de trajectoires. Nous montrons alors qu'il est possible de déduire la dynamique de la trajectoire sous-jacente à partir des observations binaires. Pour ceci, nous construisons des statistiques à partir du graphe et montrons de nouvelles propriétés de convergence dans le cadre d'une observation en temps long et en haute fréquence. Nous analysons aussi les propriétés des graphes stochastiques du point de vue des réseaux évolutifs. Dans une deuxième partie, nous travaillons sous l'hypothèse d'information complète et en temps continu et ajoutons une hypothèse de sparsité concernant le paramètre de textit{drift} du processus d'Ornstein-Uhlenbeck. Nous montrons alors des propriétés d'oracle pointues de l'estimateur Lasso, prouvons une borne inférieure sur l'erreur d'estimation au sens minimax et démontrons des propriétés d'optimalité asymptotique de l'estimateur Lasso Adaptatif. Nous appliquons ensuite ces méthodes pour estimer la vitesse de retour à la moyenne des retours journaliers d'actions américaines ainsi que des prix de futures de dividendes pour l'indice EURO STOXX 50. / The subject if this thesis is the statistical inference of multi-dimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs, defined as binary observations of a trajectory. We show then that it is possible to retrieve the dynamic of the underlying trajectory from the binary observations. For this, we build statistics of the stochastic graph and prove new results on their convergence in the long-time, high-frequency setting. We also analyse the properties of the stochastic graph from the point of view of evolving networks. In a second part, we work in the setting of complete information and continuous time. We add then a sparsity assumption applied to the drift matrix coefficient of the Ornstein-Uhlenbeck process. We prove sharp oracle inequalities for the Lasso estimator, construct a lower bound on the estimation error for sparse estimators and show optimality properties of the Adaptive Lasso estimator. Then, we apply the methods to estimate mean-return properties of real-world financial datasets: daily returns of SP500 components and EURO STOXX 50 Dividend Future prices.

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The context for this research is statistical inference, the process of making predictions or inferences about a population from observation and analyses of a sample. In this context, many researchers want to grasp what inferences can be made that are valid, in the sense of being able to uphold or justify by argument or evidence. Another pressing question among users of statistical methods is: how can spurious relationships be distinguished from genuine ones? Underlying both of these issues is the concept of evidence. In response to these (and similar) questions, two questions I work on in this essay are: (1) what is a genuine principle of evidence? and (2) do error probabilities have more than a long-run role? Concisely, I propose that felicitous genuine principles of evidence should provide concrete guidelines on precisely how to examine error probabilities, with respect to a test's aptitude for unmasking pertinent errors, which leads to establishing sound interpretations of results from statistical techniques. The starting point for my definition of genuine principles of evidence is Allan Birnbaum's confidence concept, an attempt to control misleading interpretations. However, Birnbaum's confidence concept is inadequate for interpreting statistical evidence, because using only pre-data error probabilities would not pick up on a test's ability to detect a discrepancy of interest (e.g., "even if the discrepancy exists" with respect to the actual outcome. Instead, I argue that Deborah Mayo's severity assessment is the most suitable characterization of evidence based on my definition of genuine principles of evidence. / Master of Arts