Periodically correlated time series generally exist in several fields including hydrology, climatology, economics and finance, and are commonly modelled using periodic autoregressive (PAR) model. For a time series with stochastic periodic trend, for which a unit root is expected, a periodically integrated autoregressive PIAR model with periodic and/or seasonal unit root has been shown to be a satisfactory model. The existing theory used the multivariate methodology to study PIAR models. However, this theory is convoluted, majority of it only developed for quarterly time series and its generalisation to time series with larger number of periods is quite cumbersome. This thesis studies the existing theory and highlights its restrictions and flaws. It provides a coherent presentation of the steps for analysing PAR and PIAR models for different number of periods. It presents the different unit roots representations and compares the performance of different unit root tests available in literature. The restrictions of existing studies gave us the impetus to develop a unified theory that gives a clear understanding of the integration and unit roots in the periodic models. This theory is based on the spectral information of the multi-companion matrix of the periodic models. It is more general than the existing theory, since it can be applied to any number of periods whereas the existing methods are developed for quarterly time series. Using the multi-companion method, we specify and estimate the periodic models without the need to extract complicated restrictions on the model parameters corresponding to the unit roots, as required by NLS method. The multi-companion estimation method performed well and its performance is equivalent to the NLS estimation method that has been used in the literature. Analysing integrated multivariate models is a problematic issue in time series. The multi-companion theory provides a more general approach than the error correction method that is commonly used to analyse such time series. A modified state state representation for the seasonal periodically integrated autoregressive (SPIAR) model with periodic and seasonal unit roots is presented. Also an alternative state space representations from which the state space representations of PAR, PIAR and the seasonal periodic autoregressive (SPAR) models can be directly obtained is proposed. The seasons of the parameters in these representations have been clearly specified, which guarantees correct estimated parameters. Kalman filter have been used to estimate the parameters of these models and better estimation results are obtained when the initial values were estimated rather than when they were given.

No description available.

[pt] Os modelos autorregressivos são comumente encontrados dentro do contexto de séries hidrológicas, especificamente em séries de vazões e/ou ENA (Energia Natural Afluente). Muitos destes modelos são de ordem 1, possuem parâmetros constantes ou periódicos e necessitam do requisito de normalidade. Segundo a literatura, séries de vazões anuais podem ser aproximadas para distribuições normais, porém em períodos de tempo curtos como diário, semanal e mensal esta característica não é observada, especialmente pelo problema de assimetria. Devido a isto, uma nova classe de modelo de ordem 1 foi estudada na tentativa de suprir tal problema. O novo modelo mantém estrutura autorregressiva, pode ser aditivo, multiplicativo ou híbrido, onde incorpora propriedades aditivas e multiplicativas conjunta- mente, porém suas marginais assumirão distribuição gama. Além disso, a modelagem parte do pressuposto que os Métodos de Momentos são eficientes para estimação de seus parâmetros. Recentemente esta abordagem, sob a forma híbrida, não demonstrou sucesso para o contexto do despacho hidrotérmico brasileiro. O presente trabalho foca na análise completa do modelo híbrido para as séries do Setor Elétrico Brasileiro, trazendo como novidade a estimação via máxima verossimilhança além dos estudos isolados de modelos aditivos e multiplicativos. Os resultados revelaram uma linha de pesquisa promissora, abrindo um campo de possibilidades para que novas ordens superiores a primeira ou distribuições assimétricas possam ser estudadas partindo deste princípio. / [en] Autoregressive models are commonly found in the context of hydrological series, specifically in streamflow and/or ANE series (Affluent Natural Energy). Most of them are models of order 1, which have constant or periodic parameters and need the requirement of normality. According to the literature, annual streamflow series can be approximated for normal distributions, however, in short periods of time, like daily, weekly and monthly, this feature is not observed, especially because of the asymmetry issue. Due to this reason, a new class of model of order 1 was studied for attempting to solve such problem. The new model keeps autoregressive structure and can be additive, multiplicative or hybrid, in which embodies additive and multiplicative properties together, but its marginals will assume gamma distribution. Moreover, this modeling departs from the presupposition that Methods of Moments are efficient to the estimation of its parameters. Recently, this approach, under the hybrid way, has not proved to be successful to the context of Brazilian hydrothermal dispatch. This work focuses on the complete analysis of hybrid model to the series of the Brazilian Electric Sector, bringing, as novelty, Maximum Likelihood Estimation, besides isolated studies of additive and multiplicative models. The results revealed a prosperous line of research, opening a field of possibilities for new orders or asymmetric distributions to be studied starting from this point.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-06-22T12:09:45Z No. of bitstreams: 1 cristinamarciabarrosdecastro.pdf: 9219339 bytes, checksum: 92fbbaf80500b5c629a4e62bcd9aa49d (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T15:29:14Z (GMT) No. of bitstreams: 1 cristinamarciabarrosdecastro.pdf: 9219339 bytes, checksum: 92fbbaf80500b5c629a4e62bcd9aa49d (MD5) / Made available in DSpace on 2016-07-13T15:29:14Z (GMT). No. of bitstreams: 1 cristinamarciabarrosdecastro.pdf: 9219339 bytes, checksum: 92fbbaf80500b5c629a4e62bcd9aa49d (MD5) Previous issue date: 2012-12-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Com o objetivo de atendimento à demanda de energia elétrica, buscando um baixo custo na geração de energia, é imprescindível o desenvolvimento do planejamento da operação do setor elétrico brasileiro. O planejamento da operação no horizonte de médio prazo leva em consideração a alta estocasticidade das afluências e é avaliado através da série histórica de Energia Natural Afluente (ENA). No modelo homologado pelo setor, o estudo da ENA tem sido feito por meio da metodologia Box e Jenkins, para determinar os modelos autorregressivos periódicos (PAR(p)), bem como sua ordem . Aos resíduos gerados na modelagem do PAR(p), são aplicados uma distribuição lognormal três parâmetros, como forma de gerar séries sintéticas hidrológicas semelhantes à série histórica original. Contudo, a transformação lognormal incorpora não linearidades que afetam o processo de convergência da Programação Dinâmica Dual Estocástica (PDDE). Este trabalho incorpora a técnica de bootstrap para a geração de cenários sintéticos que servirão de base para a aplicação da PDDE. A técnica estatística Bootstrap é um método alternativo a ser empregado ao problema de planejamento e que permite tanto determinar a ordem ( ) do modelo PAR(p), quanto gerar novas séries sintéticas hidrológicas. Assim, o objetivo do trabalho é analisar os impactos existentes com o uso do Bootstrap no planejamento da operação dos sistemas hidrotérmicos e, em seguida estabelecer uma comparação com a metodologia que tem sido aplicada no setor. Diante dos resultados foi possível concluir que a técnica bootstrap permite a obtenção de séries hidrológicas bem ajustadas e geram resultados confiáveis quanto ao planejamento da operação de sistemas hidrotérmicos, podendo ser usada como uma técnica alternativa ao problema em questão. / Aiming to match the long term load demand with a low cost in power generation, it is very important to improve more and more the operation planning of the Brazilian electric sector. The operation planning of medium/long term takes into account the water inflows, which are strongly stochastic, and it must be evaluated using the series of Natural Energy Inflows (NEI). In the current computational model applied to Brazilian operation planning of medium/long term, the study of ENA has been done by Box and Jenkins methodology, which determines the periodic autoregressive model (PAR (p)), as well as its order p. A lognormal distribution with three parameters is applied on the residues that are created by the PAR (p) model, as a way to generate synthetic hydrologic series similar to the original series. However, this lognormal transformation brings nonlinearities which can disturb the stability and convergence of Stochastic Dual Dynamic Programming (SDDP). This thesis incorporates the bootstrap technique to create synthetic scenarios which will be taken into account as a basis for the SDDP implementation. This statistical technique, called bootstrap, is an alternative method used to determine both the order (p) of the model PAR (p), and, after that, to produce synthetic hydrological series. Thus, the objective of this thesis is to analyze the impact of the Bootstrap technique compared to the current methodology. The results showed that the bootstrap technique is suitable to obtain adherent hydrological series. So, it was created reliable scenarios regarding the planning of the operation of hydrothermal systems. Finally, this new methodology can be used as an alternative technique to long term hydrothermal planning problems.