In this age, there are many applications of inverse problems to lots of areas ranging from astronomy, geoscience and so on. For example, image reconstruction and deblurring require the use of methods to solve inverse problems. Since the problems are subject to many factors and noise, we can't simply apply general inversion methods. Furthermore in the problems of interest, the number of unknown variables is huge, and some may depend nonlinearly on the data, such that we must solve nonlinear problems. It is quite different and significantly more challenging to solve nonlinear problems than linear inverse problems, and we need to use more sophisticated methods to solve these kinds of problems. / Master of Science / In various research areas, there are many required measurements which can't be observed due to physical and economical reasons. Instead, these unknown measurements can be recovered by known measurements. This phenomenon can be modeled and be solved by mathematics.

Time Difference Of Arrival (TDOA) based position tracking technique is one of the pinnacles of sports tracking technology. Using radio frequency com-munication, advanced filtering techniques and various computation methods, the position of a moving player in a virtually created sports arena can be iden-tified using MATLAB. It can also be related to player’s movement in real-time. For football in particular, this acts as a powerful tool for coaches to enhanceteam performance. Football clubs can use the player tracking data to boosttheir own team strengths and gain insight into their competing teams as well. This method helps to improve the success rate of Athletes and clubs by analyz-ing the results, which helps in crafting their tactical and strategic approach to game play. The algorithm can also be used to enhance the viewing experienceof audience in the stadium, as well as broadcast.In this thesis work, a typical football field scenario is assumed and an arrayof base stations (BS) are installed along perimeter of the field equidistantly.The player is attached with a radio transmitter which emits radio frequencythroughout the assigned game time. Using the concept of TDOA, the position estimates of the player are generated and the transmitter is tracked contin-uously by the BS. The position estimates are then fed to the Kalman filter, which filters and smoothens the position estimates of the player between the sample points considered. Different paths of the player as straight line, circu-lar, zig-zag paths in the field are animated and the positions of the player are tracked. Based on the error rate of the player’s estimated position, the perfor-mance of the Kalman filter is evaluated. The Kalman filter’s performance is analyzed by varying the number of sample points.

<p>We present a detailed study on application of factor<br />graphs and the belief propagation (BP) algorithm to the<br />power system state estimation (SE) problem. We start<br />from the BP solution for the linear DC model, for which<br />we provide a detailed convergence analysis. Using BPbased<br />DC model we propose a fast real-time state<br />estimator for the power system SE. The proposed<br />estimator is easy to distribute and parallelize, thus<br />alleviating computational limitations and allowing for<br />processing measurements in real time. The presented<br />algorithm may run as a continuous process, with each<br />new measurement being seamlessly processed by the<br />distributed state estimator. In contrast to the matrixbased<br />SE methods, the BP approach is robust to illconditioned<br />scenarios caused by significant differences<br />between measurement variances, thus resulting in a<br />solution that eliminates observability analysis. Using the<br />DC model, we numerically demonstrate the performance<br />of the state estimator in a realistic real-time system<br />model with asynchronous measurements. We note that<br />the extension to the non-linear SE is possible within the<br />same framework.<br />Using insights from the DC model, we use two different<br />approaches to derive the BP algorithm for the non-linear<br />model. The first method directly applies BP methodology,<br />however, providing only approximate BP solution for the<br />non-linear model. In the second approach, we make a key<br />further step by providing the solution in which the BP is<br />applied sequentially over the non-linear model, akin to<br />what is done by the Gauss-Newton method. The resulting<br />iterative Gauss-Newton belief propagation (GN-BP)<br />algorithm can be interpreted as a distributed Gauss-<br />Newton method with the same accuracy as the<br />centralized SE, however, introducing a number of<br />advantages of the BP framework. The thesis provides<br />extensive numerical study of the GN-BP algorithm,<br />provides details on its convergence behavior, and gives a<br />number of useful insights for its implementation.<br />Finally, we define the bad data test based on the BP<br />algorithm for the non-linear model. The presented model<br />establishes local criteria to detect and identify bad data<br />measurements. We numerically demonstrate that the<br />BP-based bad data test significantly improves the bad<br />data detection over the largest normalized residual test.</p> / <p>Glavni rezultati ove teze su dizajn i analiza novih<br />algoritama za re&scaron;avanje problema estimacije stanja<br />baziranih na faktor grafovima i &bdquo;Belief Propagation&ldquo; (BP)<br />algoritmu koji se mogu primeniti kao centralizovani ili<br />distribuirani estimatori stanja u elektroenergetskim<br />sistemima. Na samom početku, definisan je postupak za<br />re&scaron;avanje linearnog (DC) problema kori&scaron;ćenjem BP<br />algoritma. Pored samog algoritma data je analiza<br />konvergencije i predloženo je re&scaron;enje za unapređenje<br />konvergencije. Algoritam se može jednostavno<br />distribuirati i paralelizovati, te je pogodan za estimaciju<br />stanja u realnom vremenu, pri čemu se informacije mogu<br />prikupljati na asinhroni način, zaobilazeći neke od<br />postojećih rutina, kao npr. provera observabilnosti<br />sistema. Pro&scaron;irenje algoritma za nelinearnu estimaciju<br />stanja je moguće unutar datog modela.<br />Dalje se predlaže algoritam baziran na probabilističkim<br />grafičkim modelima koji je direktno primenjen na<br />nelinearni problem estimacije stanja, &scaron;to predstavlja<br />logičan korak u tranziciji od linearnog ka nelinearnom<br />modelu. Zbog nelinearnosti funkcija, izrazi za određenu<br />klasu poruka ne mogu se dobiti u zatvorenoj formi, zbog<br />čega rezultujući algoritam predstavlja aproksimativno<br />re&scaron;enje. Nakon toga se predlaže distribuirani Gaus-<br />Njutnov metod baziran na probabilističkim grafičkim<br />modelima i BP algoritmu koji postiže istu tačnost kao i<br />centralizovana verzija Gaus-Njutnovog metoda za<br />estimaciju stanja, te je dat i novi algoritam za otkrivanje<br />nepouzdanih merenja (outliers) prilikom merenja<br />električnih veličina. Predstavljeni algoritam uspostavlja<br />lokalni kriterijum za otkrivanje i identifikaciju<br />nepouzdanih merenja, a numerički je pokazano da<br />algoritam značajno pobolj&scaron;ava detekciju u odnosu na<br />standardne metode.</p>

Submitted by (ana.araujo@ufrpe.br) on 2016-08-04T13:26:23Z No. of bitstreams: 1 Magali Teresopolis Reis Amaral.pdf: 5438608 bytes, checksum: a3ca949533ae94adaf7883fd465a627a (MD5) / Made available in DSpace on 2016-08-04T13:26:23Z (GMT). No. of bitstreams: 1 Magali Teresopolis Reis Amaral.pdf: 5438608 bytes, checksum: a3ca949533ae94adaf7883fd465a627a (MD5) Previous issue date: 2008-08-18 / The adjustment of the weight-age growth curves for animals plays an important role in animal production planning. These adjusted growth curves must be coherent with the biological interpretation of animal growth, which often demands imposition of constraints on model parameters.The inference of the parameters of nonlinear models with constraints, using classical techniques, presents various difficulties. In order to bypass those difficulties, a bayesian approach for adjustment of the growing curves is proposed. In this respect the bayesian proposed approach introduces restrictions on model parameters through choice of the prior density. Due to the nonlinearity, the posterior density of those parameters does not have a kernel that can be identified among the traditional distributions, and their moments can only be obtained using numerical techniques. In this work the MCMC simulation (Monte Carlo chain Markov) was implemented to obtain a summary of the posterior density. Besides, selection model criteria were used for the observed data, based on generated samples of the posterior density.The main purpose of this work is to show that the bayesian approach can be of practical use, and to compare the bayesian inference of the estimated parameters considering noninformative prior density (from Jeffreys), with the classical inference obtained by the Gauss-Newton method. Therefore it was possible to observe that the calculation of the confidence intervals based on the asymptotic theory fails, indicating non significance of certain parameters of some models, while in the bayesian approach the intervals of credibility do not present this problem. The programs in this work were implemented in R language,and to illustrate the utility of the proposed method, analysis of real data was performed, from an experiment of evaluation of system of crossing among cows from different herds, implemented by Embrapa Pecuária Sudeste. The data correspond to 12 measurements of weight of animals between 8 and 19 months old, from the genetic groups of the races Nelore and Canchim, belonging to the genotype AALLAB (Paz 2002). The results reveal excellent applicability of the bayesian method, where the model of Richard presented difficulties of convergence both in the classical and in the bayesian approach (with non informative prior). On the other hand the logistic model provided the best adjustment of the data for both methodologies when opting for non informative and informative prior density. / O ajuste de curva de crescimento peso-idade para animais tem um papel importante no planejamento da produção animal. No entanto, as curvas de crescimento ajustadas devem ser coerentes com as interpretações biológicas do crescimento do animal, o que exige muitas vezes que sejam impostas restrições aos parâmetros desse modelo.A inferência de parâmetros de modelos não lineares sujeito a restrições, utilizando técnicas clássicas apresenta diversas dificuldades. Para contornar estas dificuldades, foi proposta uma abordagem bayesiana para ajuste de curvas de crescimento. Neste sentido,a abordagem bayesiana proposta introduz as restrições nos parâmetros dos modelos através das densidades de probabilidade a priori adotadas. Devido à não linearidade, as densidades a posteriori destes parâmetros não têm um núcleo que possa ser identificado entre as distribuições tradicionalmente conhecidas e os seus momentos só podem ser obtidos numericamente. Neste trabalho, as técnicas de simulação de Monte Carlo Cadeia de Markov (MCMC) foram implementadas para obtenção de um sumário das densidades a posteriori. Além disso, foram utilizados critérios de seleção do melhor modelo para um determinado conjunto de dados baseados nas amostras geradas das densidades a posteriori.O objetivo principal deste trabalho é mostrar a viabilidade da abordagem bayesiana e comparar a inferência bayesiana dos parâmetros estimados, considerando-se densidades a priori não informativas (de Jeffreys), com a inferência clássica das estimativas obtidas pelo método de Gauss-Newton. Assim, observou-se que o cálculo de intervalos de confiança, baseado na teoria assintótica, falha, levando a não significância de certos parâmetros de alguns modelos. Enquanto na abordagem bayesiana os intervalos de credibilidade não apresentam este problema. Os programas utilizados foram implementados no R e para ilustração da aplicabilidade do método proposto, foram realizadas análises de dados reais oriundos de um experimento de avaliação de sistema de cruzamento entre raças bovinas de corte, executado na Embrapa Pecuária Sudeste. Os dados correspondem a 12 mensurações de peso dos 8 aos 19 meses de idade do grupo genético das raças Nelore e Canchim, pertencente ao grupo de genotípico AALLAB, ver (Paz 2002). Os resultados revelaram excelente aplicabilidade do método bayesiano, destacando que o modelo de Richard apresentou dificuldades de convergência tanto na abordagem clássica como bayesiana (com priori não informativa). Por outro lado o modelo Logístico foi quem melhor se ajustou aos dados em ambas metodologias quando se optou por densidades a priori não informativa e informativa.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-03-05T14:28:50Z No. of bitstreams: 2 Dissertação - Ademir Alves Aguiar - 2014.pdf: 1975016 bytes, checksum: 31320b5840b8b149afedc97d0e02b49b (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-03-06T10:38:03Z (GMT) No. of bitstreams: 2 Dissertação - Ademir Alves Aguiar - 2014.pdf: 1975016 bytes, checksum: 31320b5840b8b149afedc97d0e02b49b (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-03-06T10:38:03Z (GMT). No. of bitstreams: 2 Dissertação - Ademir Alves Aguiar - 2014.pdf: 1975016 bytes, checksum: 31320b5840b8b149afedc97d0e02b49b (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-12-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation we present a semi-local convergence analysis for the Gauss-Newton method to solve a special class of systems of non-linear equations, under the hypothesis that the derivative of the non-linear operator satisfies a majorant condition. The proofs and conditions of convergence presented in this work are simplified by using a simple majorant condition. Another tool of demonstration that simplifies our study is to identify regions where the iteration of Gauss-Newton is “well-defined”. Moreover, special cases of the general theory are presented as applications. / Nesta dissertação apresentamos uma análise de convergência semi-local do método de Gauss-Newton para resolver uma classe especial de sistemas de equações não-lineares, sob a hipótese que a derivada do operador não-linear satisfaz uma condição majorante. As demonstrações e condições de convergência apresentadas neste trabalho são simplificadas pelo uso de uma simples condição majorante. Outra ferramenta de demonstração que simplifica o nosso estudo é a identificação de regiões onde a iteração de Gauss-Newton está “bem-definida”. Além disso, casos especiais da teoria geral são apresentados como aplicações.

Cette thèse construit un modèle théorique qui renouvelle l’approche traditionnelle de l’équilibre du marché. En introduisant dans le paradigme néo-classique le principe de préférence pour la quantité, il génère de façon optimale des stocks dans un marché concurrentiel. Les résultats sont très importants, car ils expliquent à la fois l’émergence des invendus et l’existence de cycles économiques. En outre, il étudie le comportement optimal du monopole dont la puissance de marché dépend non seulement de la quantité de biens étalés, mais aussi de celle de biens achetés. Contrairement à l’hypothèse traditionnelle selon laquelle le monopoleur choisit le prix ou la quantité qui maximise son profit, il attire, via un indice de Lerner généralisé la demande à la fois par le prix et la quantité de biens exposés. Quelle que soit la structure du marché, le phénomène d’accumulation des stocks de biens apparaît dans l’économie. De plus, il a l’avantage d’expliquer explicitement les achats impulsifs non encore traités par la théorie économique. Pour vérifier la robustesse des résultats du modèle théorique, ils sont testés sur des données américaines. En raison de leur non-linéarité, la méthode de Gauss-Newton est appropriée pour analyser l’impact de la préférence pour la quantité sur la production et l’accumulation de biens, et par conséquent sur les prévisions de PIB. Enfin, cette thèse construit un modèle à générations imbriquées à deux pays qui étend l’équilibre dynamique à un gamma-équilibre dynamique sans friction. Sur la base de la contrainte de détention préalable d’encaisse, il ressort les conditions de sur-accumulation du capital et les conséquences de la mobilité du capital sur le bien-être dans un contexte d’accumulation du stock d’invendus / This thesis constructs a theoretical model that renews the traditional approach of the market equilibrium. By introducing into the neoclassical paradigm the principle of preference for quantity, it optimally generates inventories within a competitive market. The results are very important since they explain both the emergence of unsold goods and the existence of economic cycles. In addition, it studies the optimal behavior of a monopolist whose the market power depends not only on the quantity of displayed goods but also that of goods that the main consumer is willing to buy. Contrary to the traditional assumption that the monopolist chooses price or quantity that maximizes its profit, through a generalized Lerner index (GLI) it attracts customers’ demand by both the price and the quantity of displayed goods. Whatever the market structure, the phenomenon of inventory accumulation appears in the economy. Furthermore, it has the advantage of explicitly explaining impulse purchases untreated by economics. To check the robustness of the results,the theoretical model is fitted to U.S. data. Due to its nonlinearity, the Gauss-Newtonmethod is appropriate to highlight the impact of consumers’ preference for quantity on production and accumulation of goods and consequently GDP forecast. Finally, this thesis builds a two-country overlapping generations (OLG) model which extends the dynamic OLG equilibrium to a frictionless dynamic OLG gamma-equilibrium. Based on the cash-inadvance constraint, it highlights the conditions of over-accumulation of capital and welfare implications of capital mobility in a context of accumulation of stock of unsold goods.