Programa de Pós-Graduação em Matemática Aplicada e Estatística
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Master Thesis Análise de modelos de regressão para taxas estaduais de mortalidade por Covid-19 no Brasil(Universidade Federal do Rio Grande do Norte, 2025-05-20) Vieira, Mirela da Silva; Lemonte, Artur José; https://orcid.org/0000-0002-0249-7474; http://lattes.cnpq.br/4283549028869521; http://lattes.cnpq.br/5028431925410574; Costa, Eliardo Guimarães da; http://lattes.cnpq.br/3160805152538713; Magalhães, Tiago Maia; https://orcid.org/0000-0003-3814-9532; http://lattes.cnpq.br/7953363504273397Master Thesis Inferência bayesiana para modelos binomiais com priori conjugada baseada em generalizações da distribuição beta(Universidade Federal do Rio Grande do Norte, 2025-03-20) Leite, Luana Mayara Lucas; Pereira, Marcelo Bourguignon; https://orcid.org/0000-0002-1182-5193; http://lattes.cnpq.br/9358366674842900; http://lattes.cnpq.br/0862072928793748; Costa, Eliardo Guimarães da; Costa, José Mir Justino daThe present dissertation aims to propose alternative conjugate prior distributions for the binomial model based on generalizations of the beta distribution. In this context, the developed methodology seeks to estimate the proportion parameter π of the binomial distribution using a Bayesian approach, employing generalizations of the beta distribution as priors in such a way that the posterior distribution also belongs to the same class as the prior. Additionally, the properties of these distributions will be studied in detail, with simulations conducted through random number generation. Their advantages over the beta distribution will be assessed, as the aim is to achieve better fitting results than those obtained by the beta distribution when compared.Master Thesis Inferência bayesiana para modelos Poisson com priori conjugada baseada em misturas da distribuição Gama(Universidade Federal do Rio Grande do Norte, 2025-03-17) Araújo, Karine dos Santos; Pereira, Marcelo Bourguignon; https://orcid.org/0000-0002-1182-5193; http://lattes.cnpq.br/9358366674842900; http://lattes.cnpq.br/8535082485035743; Nascimento, Fernando Ferraz do; Souza Filho, Nelson Lima deBayesian inference is a statistical methodology that combines prior information about model parameters with observational data to estimate the posterior distribution of unknown parameters. One of the advantages of using conjugate priors is that the resulting posterior distribution remains within the same family of distributions as the prior, which facilitates both the calculations and the intuitive interpretation of the posterior parameters. In this study, we adopted mixtures of Gamma distributions as conjugate priors for the λ parameter of the Poisson distribution, which provides a more flexible approach capable of better adjusting to different data characteristics. The mixtures of Gamma distributions explored in this work include the generalized Lindley distribution 1 (ABOUAMMOH; ALSHANGITI; RAGAB, 2015), the generalized Lindley distribution 2 (RAMOS; LOUZADA; MOALA, 2021) and the generalized Lindley distribution 3 (ZAKERZADEH; DOLATI, 2009). These distributions are extensions of the classic Lindley distribution. To illustrate the practical application of the proposed methodology, we carried out a study with real data.Master Thesis Modelando eventos extremos de precipitação no semiárido nordestino utilizando a Distribuição ZIGEV(Universidade Federal do Rio Grande do Norte, 2025-02-21) Araújo, Iarythssa Duarte de; Medeiros, Francisco Moisés Cândido de; https://orcid.org/0000-0001-6751-2666; http://lattes.cnpq.br/2662558366496381; http://lattes.cnpq.br/7633642310017371; Rodrigues, Daniele Tôrres; Costa, Eliardo Guimarães da; Morales, Fidel Ernesto CastroThe semiárido region of Northeast Brazil (NEB) is characterized by irregular rainfall patterns and high evapotranspiration rates, leading to water scarcity. Additionally, the region is prone to extreme precipitation events, which can trigger socio-environmental disasters with severe impacts. Despite the development of various models for modeling extreme events, their application in the semi-arid region of the NEB is limited by the lack of detailed data and methodologies tailored to its local characteristics. This study aims to model extreme precipitation events in this region using the Zero-Inflated Generalized Extreme Value (ZIGEV) distribution. The methodological approach is based on Bayesian inference, which incorporates uncertainties and prior information into the parameter estimation process. The analysis focuses on a sub-region of the NEB semi-arid zone, marked by extremely low annual precipitation and high susceptibility to extreme precipitation events. Daily precipitation data from 16 (sixteen) localities within the study area were used to fit the model and estimate the return levels of extreme events. The results demonstrated the effectiveness of the ZIGEV model in predicting extreme precipitation events, providing reliable parameter estimates and outperforming the standard GEV model, which faced convergence issues in many datasets. The ZIGEV model proved suitable for data with a high proportion of zeros and also converged to the GEV model when the proportion of zeros was lower. Additionally, the estimated return levels for each locality provided valuable insights into the occurrence and intensity patterns of these events, contributing to water resource management and the development of disaster mitigation strategies for the NEB semi-arid region.Master Thesis Existência de soluções para uma classe de equações de Kirchhoff-Schrödinger com crescimento crítico de Sobolev(Universidade Federal do Rio Grande do Norte, 2025-02-14) Silva, Cristiano Victor Medeiros da; Souza, Diego Ferraz de; https://orcid.org/0000-0002-8605-0046; http://lattes.cnpq.br/4757061859245287; http://lattes.cnpq.br/0991482396568693; Silva, Esteban Pereira da; https://orcid.org/0000-0001-9002-4919; http://lattes.cnpq.br/8085933568620489; Duarte, Ronaldo César; https://orcid.org/0000-0002-5611-1901; http://lattes.cnpq.br/0322664441555339; Parra, Juan Luis Miguel Arratia; https://orcid.org/0000-0003-1025-017X; http://lattes.cnpq.br/1882239439174348; López, Pedro Eduardo Ubilla; https://orcid.org/0000-0001-8300-7205; http://lattes.cnpq.br/1086329020331718; Clemente, Rodrigo Genuino; http://orcid.org/0000-0001-9941-8199; http://lattes.cnpq.br/4351609162717260In this work, we establish the existence of positive solutions for a class of stationary Kirchhoff-Schrödinger equations defined in the whole R3 with nonlinearity with critical growth in the Sobolev sense and nonnegative potential that can decay to zero at infinity. For this purpose, we use the variational method of critical point theory, which consists of associating the solutions of the equation to the critical points of a suitable functional. Furthermore, the nonlinearity is general and does not satisfy the well-known AmbrosettiRabinowitz condition, which makes the study of the compactness associated with the problem and the boundedness of Palais-Smale sequences more sophisticated. In this context, the main tools used to achieve our main results were the use of the mountain pass theorem and the Lions’ compactness principle, in addition to the common basis which consists of basic results of measure theory and Lebesgue integration, functional analysis and nonlinear functional analysis.Master Thesis Modelagem matemática e computacional do processo de reinjeção de água produzida em reservatórios de petróleo(Universidade Federal do Rio Grande do Norte, 2025-02-26) Assunção, Brunna Karla de Morais Souza; Lima, Sidarta Araújo de; http://lattes.cnpq.br/7285935215651168; http://lattes.cnpq.br/6826569527056669; Santos, Adriano dos; Fernando, Honório Joaquim; Radtke, Luiz CarlosProduced water reinjection is a process used in oil reservoirs, consisting of reinjecting the water produced during oil production. However, this technique can be a!ected by reactive phenomena such as adsorption and mechanical retention. These phenomena may cause formation damage, leading to injectivity loss and, consequently, negatively impacting oil production curves. Given this, it is essential to develop mathematical and computational models that accurately describe the processes governing particle injection in porous media. In this context, the mathematical modeling developed in this dissertation aims to derive a system of partial di!erential equations that govern the hydrodynamics and particle transport in an oil reservoir. The hydrodynamics are governed by the total mass balance of the fluid phases, along with the total Darcy’s law, while particle transport is described by the advective-dispersive-reactive transport equation. The reactive nature of particle transport is modeled using kinetic laws expressed through ordinary di!erential equations. Regarding computational modeling, the hydrodynamics are discretized using the dual mixed finite element method, while the movement of the fluid phases is approximated using the explicit Central-Upwind finite volume method. For the discretization of the particle transport equation, we employ a predictor-corrector algorithm, which, due to the nature of the di!usive term, allows the equation to be numerically solved using the two aforementioned methods. The implemented computational simulator was validated against analytical solutions to assess its accuracy. The obtained numerical results demonstrate that the simulator can accurately and stably quantify the produced water reinjection process in oil reservoirs. Finally, the computational model is applied to a more realistic scenario, considering a quarter of a five-spot domain.Master Thesis Involuções da álgebra de Grassmann: um estudo sobre *- polinômios centrais, *-identidades e *-isomorfismos(Universidade Federal do Rio Grande do Norte, 2025-01-24) Santos, Lígia Danielly Rocha dos; Guimarães, Alan de Araújo; https://orcid.org/0000-0002-4492-8818; http://lattes.cnpq.br/6286902372305694; http://lattes.cnpq.br/8469906596468208; Kuzmin, Alexey; http://lattes.cnpq.br/6586935974677811; Tsurkov, Arkady; http://lattes.cnpq.br/0825556032686787; Macêdo, David Levi da Silva; https://orcid.org/0000-0001-8151-0843; http://lattes.cnpq.br/3432177281707815; Silva, Diogo Diniz Pereira da Silva e; https://orcid.org/0000-0002-7475-7538; http://lattes.cnpq.br/5154042218439017; Teixeira, José Victor Gomes; https://orcid.org/0000-0002-4831-0310; http://lattes.cnpq.br/9368785489596803; Araújo, Laise Dias Alves; http://lattes.cnpq.br/4617200292068975; Morais, Pedro Henrique Martins de; http://lattes.cnpq.br/8110339213590421Let K be a field of characteristic distinct from two, E be the Grassmann algebra of a K-vector space with infinite and countable base L and let φ be any involution. Based on the article (CENTRONE; GONCALVES; SILVA, 2020), we present generating sets for the ∗-polynomial identities and ∗-central polynomials of the Grassmann algebra, with a strong distinction between the cases char(K) = 0 and char(K) > 2. According to (DINIZ; GUIMARÃES; ROCHA, 2024), when φ satisfies certain conditions, we show that there is homogeneous involution φl (i.e., φl(L) = L) such that (E, φ) and (E, φl) are ∗-isomorphic. Additionally, we present certain necessary and sufficient condition for two homogeneous involutions to produce ∗-isomorphic structures. As a consequence, we obtain examples of algebras that are ∗-PI-equivalent and that are not ∗-isomorphic algebras.Master Thesis Modelos espaciais e espaço-temporais para contagem de chuvas extremas na costa leste do Nordeste do Brasil(Universidade Federal do Rio Grande do Norte, 2025-01-10) Luz, Josiel Oliveira da; Morales, Fidel Ernesto Castro; https://orcid.org/0000-0002-7227-8023; http://lattes.cnpq.br/8552159154343151; https://orcid.org/0000-0001-6128-9432; http://lattes.cnpq.br/6106700517329582; Costa, Eliardo Guimarães da; https://orcid.org/0000-0003-4528-0379; http://lattes.cnpq.br/3160805152538713; Cabral Júnior, Jório Bezerra; https://orcid.org/0000-0002-4207-2155; http://lattes.cnpq.br/7439808091974845The study of climatic extremes is essential for understanding the occurrence of natural disasters. It becomes even more important when the tools available for this analysis do not perform their functions accurately. This is the case with the extreme rainfall events that occur on the east coast of north-eastern Brazil, where estimates from the Tropical Rainfall Measuring Mission satellite (TRMM) and the Tropical Rainfall Measuring Mission satellite (TRMM) are not accurate. (TRMM) satellite and the Global Precipitation Measurement (GPM) satellite tend to underestimate extreme precipitation. This study therefore set out to analyse the performance of spatial and spatio-temporal models, available in the literature and implemented in the R language, when applied to extreme rainfall count data from the east coast of the Northeast, as well as comparing the performance of the models with the performance of the satellite for the study context. The aim is to build up the knowledge needed to support the development of mitigation measures for natural disasters. To this end, the study variable was the number of times the daily rainfall accumulated exceeded or was equal to the threshold of 30 millimetres over the course of a year, so that the counts were made per rainfall station. This variable is one of the 27 indices adopted by the World Meteorological Organisation (WMO) to study climate variations. The time series studied has records of daily rainfall accumulations from 36 stations during the period from 1991 to 2022. In addition to precipitation, it contains the latitude, longitude and altitude of the stations. The data was taken from the National Meteorological Institute (INMET) and the National Water and Sanitation Agency (ANA). With regard to the methodology adopted for the main parts of the study, the research began with a literature search for R language packages that worked with count data from a spatial and spatio-temporal perspective. After the search and before using the models, a descriptive analysis of the data was carried out, in which the means, medians, standard deviations, coefficients of variation and amplitudes by year and season were calculated. The investigation continued by analysing the study variable and the covariates latitude, longitude and altitude using scatter plots. Spatial and temporal dependencies were also analysed using variograms. Once the descriptive analysis had been completed, the cross-validation methodology was used to assess the performance of the models. The comparison between the models and the satellite data was made using mean square error, bias and correlation coefficient statistics. The results achieved in this research are encouraging. Extremes occur more frequently in regions close to the coast, with spatial dependence and temporal dependence for only a few stations. Furthermore, in a purely observational analysis, there is an indication of patterns of extremes repeating every 10 years from 1991 onwards. It was also possible to see that satellite data underestimates the extremes in the region studied, especially in the south of the area. Four packages implemented in the R language capable of working with the study data were identified in the literature, and six models from these packages were adjusted. The models obtained excellent results when compared to the satellite estimates. They proved to be better in most of the locations tested. They also show that they have great potential for generating more reliable data for analysing precipitation extremes.Master Thesis Avaliação do desempenho do gráfico de controle de Shewhart para o processo OIB-INAR(1) e comparação com o desempenho do gráfico de Shewhart do processo Borel INAR(1)(Universidade Federal do Rio Grande do Norte, 2024-09-13) Ferreira, Camiliane Azevedo; Pinho, André Luís Santos de; https://orcid.org/0000-0002-2975-4637; http://lattes.cnpq.br/7753762932186347; https://orcid.org/0000-0002-3268-991X; http://lattes.cnpq.br/0542984976258690; Souza, Isaac Jales Costa; Sales, Lucas de Oliveira Ferreira de; Fernandez, Luz Milena ZeaIn the face of technological advances and the large amount of information generated, count data is becoming increasingly autocorrelated. This creates a growing need for new models and monitoring tools that take into account the specific characteristics of this data. In this paper, we propose a Shewhart control chart for autocorrelated count data that can be modeled by an INAR(1) process with innovations from the inflated Borel distribution of ones (OIB-INAR(1)). This model is well-suited for modeling data with an inflation of ones, as well as underdispesion, equidispersion, or overdispersion. Addicionally, we will compare the performance of this control chart with that of a Borel INAR(1) process, which models series of counts truncated at zero that show underdispersion, equidispersion or overdispersion. The Borel INAR(1) process can be seen as a particular case of the OIB-INAR(1) process. The performance evaluation of the proposed approach is based on the average number of samples required to detect an alarm (NMAF and NMA) in different scenarios. The determination of the upper control limit (UCL) and the evaluation of the performance of the graphs are carried out by means of computational studies using Monte Carlo simulations. To illustrate the applicability of the proposed method, we present an example with real data.Master Thesis A distribuição Borel inflacionada de uns e um processo autoregressivo de primeira ordem para valores inteiros com inovações Borel inflacionada de uns(Universidade Federal do Rio Grande do Norte, 2024-02-02) Ciriaco, Beatriz Ariadna da Silva; Pinho, André Luís Santos de; Fernandez, Luz Milena Zea; https://orcid.org/0000-0001-8335-9446; http://lattes.cnpq.br/0576675498537949; https://orcid.org/0000-0002-2975-4637; http://lattes.cnpq.br/7753762932186347; http://lattes.cnpq.br/3864348097536219; Nascimento, Antônio Marcos Batista do; Souza, Isaac Jales CostaAs time series of integer values are based on counts, such as the autoregressive processes of integer values (INAR), they may contain an excessive number of repeated values that could harm the inferential analysis if this behavior is not known. Then it becomes necessary to understand time series inflated models for integer values. Some models have been developed to study zero-inflated data, however, this work focuses on the one-inflated model (OI) and the autoregressive process with one-inflated innovations (OI-INAR(1)). Thus, the Oneinflated Borel (OIB) model and the INAR(1) process with one-inflated Borel innovations (OIB-INAR(1)) are proposed. In this work the properties, such as expectation, variance, dispersion index and probability generating function, of these models are developed, as well as the methods for estimating the parameters of the OIB model, such as the method of moments and maximum likelihood. Likewise, for the OIB-INAR(1) process, the conditional least squares and conditional maximum likelihood methods, in addition to one-step-ahead prediction methodologies are studied. Finally, an application is made adjusting the OIBINAR(1) model.