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.

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.