Made available in DSpace on 2016-06-02T20:06:02Z (GMT). No. of bitstreams: 1 2524.pdf: 3173626 bytes, checksum: 444d75f97bd088459e470db31df717a5 (MD5) Previous issue date: 2009-02-26 / Financiadora de Estudos e Projetos / In this work they are presented extensions of Auto Regressive and Auto Regressive Conditional Heteroscedasticity models with coefficients varying in time. These coefficients have been used as models for non stationary real time series, specially for financial series. The objective of this work is to present the models and the techniques involved in estimating time-varying coefficients, moreover, it is made an introduction to financial modeling and some suggestions in order to facilitate implementation and use of models with time-varying coefficients. The simulation studies and the application on real data showed that the models have great potential to be exploited in the analysis of non-stationary series. The suggestions in confidence band and forecasting for the Auto regressive models with time-varying coefficients enable the use of models in financial data and other series that show a non-stationary characteristic. The modified algorithm for estimation of ARCH models varying in time was to increase the rate of convergence. The creation of a method for forecasting for ARCH models require a deeper study, although the algorithm has shown promising results in simulation study, giving some evidences that it can be applied in real situation. Finally, the contributions in the creation of functions for a free software that facilitate the use and the analysis of the models studied and the use of the proposed methods. / No presente trabalho são apresentadas extensões dos modelos Auto Regressivo e Auto Regressivo Condicionalmente Heteroscedasticos com coeficientes variando ao longo do tempo. Estes têm sido utilizados como modelos para séries temporais reais não estacionárias, em especial as séries financeiras. O objetivo desse trabalho é apresentar os modelos e as técnicas envolvidas para estimar esses coeficientes que variam no tempo, além disso, é feito uma introdução a modelagem financeira e algumas sugestões para facilitar a aplicação e utilização dos modelos com coeficientes variando no tempo. Os estudos de simulação e a aplicação em dados reais mostraram que os modelos têm um grande potencial a ser explorados na análise de séries não estacionárias. As sugestões de banda de confiança e previsão para os modelos Auto Regressivos com coeficientes variando no tempo viabilizam a utilização dos modelos em dados financeiros e outras séries que apresentam uma característica de não estacionariedade. As modificações no algoritmo de estimação dos modelos ARCH variando no tempo foram para aumentar a taxa de convergência. A criação de um método para previsão dos modelos ARCH necessitam de um estudo mais profundo, porém o algoritmo mostrou resultados promissores no estudo de simulação, dando alguns indícios de que pode ser aplicada na prática. Por fim, as contribuições na criação de funções para um software livre que facilitam a utilização e a análise dos modelos estudados bem como a utilização dos métodos propostos.

The main goal of this study is to propose a methodology that measures directed relations between locally stationary processes. Unlike stationary processes, locally stationary processes may present sudden pattern changes and have local characteristics in specific intervals. This behavior causes instability in measures based on Fourier transforms. The relevance of this study relies on considering these processes and propose robust methodologies that are not affected by outliers, sudden pattern changes or local behavior. We start reviewing the Partial Directed Coherence (PDC) and the Wavelet Coherence. PDC measures the directed relation between components of a multivariate stationary Vector Autoregressive (VAR) model in the frequency domain, while Wavelet Coherence is based on complex wavelets decomposition. We then propose a causal wavelet decomposition of the covariance structure for bivariate locally stationary processes: the Directed Wavelet Covariance (DWC). Compared to Fourier-based quantities, wavelet-based estimators are more appropriate for non-stationary processes and processes with local patterns, outliers and rapid regime changes like in EEG experiments with the introduction of stimuli. We then propose its estimators and calculate its expectation and analyze its variance. Next we propose a decomposition for the variance of multivariate processes with more than two components: the Partial Directed Wavelet Covariance (pDWC). Considering a N-variate locally stationary process, the pDWC calculates the Directed Wavelet Covariance of X_1(t) with X_2(t) eliminating the effect of the other components X_3(t), ... ,X_N(t). We propose two approaches to this situation. First we filter the multivariate process to remove all the exogenous influences and then we calculate the directed relation between the components. In the second case, as in Partial Directed Coherence, we consider the multivariate process as a time-varying Vector Autoregressive Model (tv-VAR) and use its coefficients in the decomposition of the covariance function to isolate the effects of the other components. We also compare results of the PDC, Wavelet Coherence and Directed Wavelet Covariance with simulated data. Finally, we present an application of the proposed Directed Wavelet Covariance and Partial Directed Wavelet Covariance on EEG data. Simulation results show that the proposed measures capture the simulated relations. The pDWC with linear filter has shown more stable estimations than the proposed pDWC considering the tv-VAR. Future studies will discuss the DWC\'s and pDWC\'s asymptotic distributions and significance tests. The proposed Directed Wavelet Covariance decomposition is a different approach to deal with non-stationary processes in the context of causality. The use of wavelets is a gain and adds to the number of studies that can be addressed when Fourier transform does not apply. The pDWC is an alternative for multivariate processes and it removes linear influences from observed external components. / O objetivo deste trabalho é propor uma metodologia para mensurar o impacto direcionado entre processos localmente estacionários. Diferente de processos estacionários, processos localmente estacionários podem apresentar mudanças bruscas e características específicas em determinados intervalos. Tal comportamento pode causar instabilidade em medidas baseadas na transformada de Fourier. A importância deste estudo se dá em englobar processos com tais características, propondo metodologias robustas que não são afetadas pela existência de mudanças bruscas, pontos discrepantes e comportamentos locais. Inicialmente apresentamos conceitos já existentes na literatura, como a Coerência Parcial Direcionada (PDC) e a Coerência de Ondaletas. A PDC mede o impacto direcionado entre componentes de um modelo vetorial autoregressivo (VAR) no domínio da frequência. A coerência de ondaletas é baseada em transformadas complexas de ondaletas. Propomos então uma decomposição no domínio de ondaletas para a estrutura de covariância de processos bivariados localmente estacionários: a Covariância Direcionada de Ondaletas (DWC). Em comparação com as quantidades baseadas na tranformada Fourier, os estimadores baseados em ondaletas são mais apropriados para processos não estacionários com padrões locais, pontos discrepantes ou mudanças rápidas de regime, como em experimentos de eletroencefalograma (EEG) com a introdução de estímulo. Ainda, propomos um estimador para a DWC, calculamos a esperança deste estimador e avaliamos sua variância. Em seguida, propomos uma quantidade análoga à DWC para processos multivariados com mais de duas componentes: a Covariância Parcial Direcionada de Ondaletas (pDWC). Considerando um processo N-variado localmente estacionário, a pDWC calcula a Covariância Direcionada de Ondaletas entre X_1(t) e X_2(t) eliminando o efeito das outras componentes X_3(t), ... , X_N(t). Propomos duas abordagens para a pDWC: na primeira, a pDWC é calculada após a aplicação de um filtro linear que remove o efeito das variáveis exógenas. No segundo caso, a exemplo da Coerência Parcial Direcionada, consideramos o processo multivariado como um Modelo Autoregressivo de Vetorial variante no tempo (tv-VAR) e usamos seus coeficientes na decomposição da função de covariância para isolar os efeitos das demais componentes. Também comparamos os resultados da PDC, Coerência de Ondaletas e Covariância Direcionada de Ondaletas com dados simulados. Por fim, apresentamos uma aplicação da DWC e da pDWC em dados de EEG. Identificamos nas simulações que tanto as medidas já existentes na literatura quanto as quantidades propostas identificaram as relações simuladas. A pDWC proposta com filtros lineares apresentou estimações mais estáveis do que a pDWC considerando os modelos tv-VAR. Estudos futuros discutirão as propriedades assintóticas e testes de significância da DWC e pDWC.

L'objectif principal de cette recherche est d'analyser les effets de l'agrégation sur la prévision de la demande. Cet effet est examiné par l'analyse mathématique et l’étude de simulation. L'analyse est complétée en examinant les résultats sur un ensemble de données réelles. Dans la première partie de cette étude, l'impact de l'agrégation temporelle sur la prévision de la demande a été évalué. En suite, Dans la deuxième partie de cette recherche, l'efficacité des approches BU(Bottom-Up) et TD (Top-Down) est analytiquement évaluée pour prévoir la demande au niveau agrégé et désagrégé. Nous supposons que la série désagrégée suit soit un processus moyenne mobile intégrée d’ordre un, ARIMA (0,1,1), soit un processus autoregressif moyenne mobile d’ordre un, ARIMA (1,0,1) avec leur cas spéciales. / Demand forecasting performance is subject to the uncertainty underlying the time series an organisation is dealing with. There are many approaches that may be used to reduce demand uncertainty and consequently improve the forecasting (and inventory control) performance. An intuitively appealing such approach that is known to be effective is demand aggregation. One approach is to aggregate demand in lower-frequency ‘time buckets’. Such an approach is often referred to, in the academic literature, as temporal aggregation. Another approach discussed in the literature is that associated with cross-sectional aggregation, which involves aggregating different time series to obtain higher level forecasts.This research discusses whether it is appropriate to use the original (not aggregated) data to generate a forecast or one should rather aggregate data first and then generate a forecast. This Ph.D. thesis reveals the conditions under which each approach leads to a superior performance as judged based on forecast accuracy. Throughout this work, it is assumed that the underlying structure of the demand time series follows an AutoRegressive Integrated Moving Average (ARIMA) process.In the first part of our1 research, the effect of temporal aggregation on demand forecasting is analysed. It is assumed that the non-aggregate demand follows an autoregressive moving average process of order one, ARMA(1,1). Additionally, the associated special cases of a first-order autoregressive process, AR(1) and a moving average process of order one, MA(1) are also considered, and a Single Exponential Smoothing (SES) procedure is used to forecast demand. These demand processes are often encountered in practice and SES is one of the standard estimators used in industry. Theoretical Mean Squared Error expressions are derived for the aggregate and the non-aggregate demand in order to contrast the relevant forecasting performances. The theoretical analysis is validated by an extensive numerical investigation and experimentation with an empirical dataset. The results indicate that performance improvements achieved through the aggregation approach are a function of the aggregation level, the smoothing constant value used for SES and the process parameters.In the second part of our research, the effect of cross-sectional aggregation on demand forecasting is evaluated. More specifically, the relative effectiveness of top-down (TD) and bottom-up (BU) approaches are compared for forecasting the aggregate and sub-aggregate demands. It is assumed that that the sub-aggregate demand follows either a ARMA(1,1) or a non-stationary Integrated Moving Average process of order one, IMA(1,1) and a SES procedure is used to extrapolate future requirements. Such demand processes are often encountered in practice and, as discussed above, SES is one of the standard estimators used in industry (in addition to being the optimal estimator for an IMA(1) process). Theoretical Mean Squared Errors are derived for the BU and TD approach in order to contrast the relevant forecasting performances. The theoretical analysis is supported by an extensive numerical investigation at both the aggregate and sub-aggregate levels in addition to empirically validating our findings on a real dataset from a European superstore. The results show that the superiority of each approach is a function of the series autocorrelation, the cross-correlation between series and the comparison level.Finally, for both parts of the research, valuable insights are offered to practitioners and an agenda for further research in this area is provided.

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés. / Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés. / Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

Demand forecasting performance is subject to the uncertainty underlying the time series an organisation is dealing with. There are many approaches that may be used to reduce demand uncertainty and consequently improve the forecasting (and inventory control) performance. An intuitively appealing such approach that is known to be effective is demand aggregation. One approach is to aggregate demand in lower-frequency 'time buckets'. Such an approach is often referred to, in the academic literature, as temporal aggregation. Another approach discussed in the literature is that associated with cross-sectional aggregation, which involves aggregating different time series to obtain higher level forecasts.This research discusses whether it is appropriate to use the original (not aggregated) data to generate a forecast or one should rather aggregate data first and then generate a forecast. This Ph.D. thesis reveals the conditions under which each approach leads to a superior performance as judged based on forecast accuracy. Throughout this work, it is assumed that the underlying structure of the demand time series follows an AutoRegressive Integrated Moving Average (ARIMA) process.In the first part of our1 research, the effect of temporal aggregation on demand forecasting is analysed. It is assumed that the non-aggregate demand follows an autoregressive moving average process of order one, ARMA(1,1). Additionally, the associated special cases of a first-order autoregressive process, AR(1) and a moving average process of order one, MA(1) are also considered, and a Single Exponential Smoothing (SES) procedure is used to forecast demand. These demand processes are often encountered in practice and SES is one of the standard estimators used in industry. Theoretical Mean Squared Error expressions are derived for the aggregate and the non-aggregate demand in order to contrast the relevant forecasting performances. The theoretical analysis is validated by an extensive numerical investigation and experimentation with an empirical dataset. The results indicate that performance improvements achieved through the aggregation approach are a function of the aggregation level, the smoothing constant value used for SES and the process parameters.In the second part of our research, the effect of cross-sectional aggregation on demand forecasting is evaluated. More specifically, the relative effectiveness of top-down (TD) and bottom-up (BU) approaches are compared for forecasting the aggregate and sub-aggregate demands. It is assumed that that the sub-aggregate demand follows either a ARMA(1,1) or a non-stationary Integrated Moving Average process of order one, IMA(1,1) and a SES procedure is used to extrapolate future requirements. Such demand processes are often encountered in practice and, as discussed above, SES is one of the standard estimators used in industry (in addition to being the optimal estimator for an IMA(1) process). Theoretical Mean Squared Errors are derived for the BU and TD approach in order to contrast the relevant forecasting performances. The theoretical analysis is supported by an extensive numerical investigation at both the aggregate and sub-aggregate levels in addition to empirically validating our findings on a real dataset from a European superstore. The results show that the superiority of each approach is a function of the series autocorrelation, the cross-correlation between series and the comparison level.Finally, for both parts of the research, valuable insights are offered to practitioners and an agenda for further research in this area is provided.

La qualité de communication dans un réseau sans fil est déterminée par la qualité du signal, et plus précisément par le rapport signal à interférence et bruit. Cela pousse chaque récepteur à se connecter à l'émetteur qui lui donne la meilleure qualité du signal. Nous utilisons la géométrie stochastique et la théorie des extrêmes pour obtenir la distribution de la meilleure qualité du signal, ainsi que celles de interférence et du maximum des puissances reçues. Nous mettons en évidence comment la singularité de la fonction d'affaiblissement modifie leurs comportements. Nous nous intéressons ensuite au comportement temporel des signaux radios en étudiant le franchissement de seuils par un processus stationnaire X (t). Nous démontrons que l'intervalle de temps que X (t) passe au-dessus d'un seuil γ → −∞ suit une distribution exponentielle, et obtenons 'egalement des r'esultats caract'erisant des franchissements par X (t) de plusieurs seuils adjacents. Ces r'esultats sont ensuite appliqu'es 'a la gestion de mobilit'e dans les r'eseaux cellulaires. Notre travail se concentre sur la fonction de 'handover measurement'. Nous identifions la meilleure cellule voisine lors d'un handover. Cette fonction joue un rôle central sur expérience perçue par l'utilisateur. Mais elle demande une coopération entre divers mécanismes de contrôle et reste une question difficile. Nous traitons ce problème en proposant des approches analytiques pour les réseaux émergents de types macro et pico cellulaires, ainsi qu'une approche d'auto- optimisation pour les listes de voisinage utilisées dans les réseaux cellulaires actuels.

Dans cette thèse, nous abordons l'estimation non paramétrique de la fonction de dépendance des distributions multivariées à valeurs extrêmes. Dans une première partie, on adopte l’hypothèse classique stipulant que les variables aléatoires sont indépendantes et identiquement distribuées (i.i.d). Plusieurs estimateurs non paramétriques sont comparés pour une fonction de dépendance trivariée de type logistique dans deux différents cas. Dans le premier cas, on suppose que les fonctions marginales sont des distributions généralisées à valeurs extrêmes. La distribution marginale est remplacée par la fonction de répartition empirique dans le deuxième cas. Les résultats des simulations Monte Carlo montrent que l'estimateur Gudendorf-Segers (Gudendorf et Segers, 2011) est plus efficient que les autres estimateurs pour différentes tailles de l’échantillon. Dans une deuxième partie, on ignore l’hypothèse i.i.d vue qu’elle n'est pas vérifiée dans l'analyse des séries temporelles. Dans le cadre univarié, on examine le comportement extrêmal d'un modèle autorégressif Gaussien stationnaire. Dans le cadre multivarié, on développe un nouveau théorème qui porte sur la convergence asymptotique de l'estimateur de Pickands vers la fonction de dépendance théorique. Ce fondement théorique est vérifié empiriquement dans les cas d’indépendance et de dépendance asymptotique. Dans la dernière partie de la thèse, l'estimateur Gudendorf-Segers est utilisé pour modéliser la structure de dépendance des concentrations extrêmes d’ozone observées dans les stations qui enregistrent des dépassements de la valeur guide et limite de la norme Tunisienne de la qualité d'air NT.106.04. / In this thesis, we investigate the nonparametric estimation of the dependence function for multivariate extreme value distributions. Firstly, we assume independent and identically distributed random variables (i.i.d). Several nonparametric estimators are compared for a trivariate dependence function of logistic type in two different cases. In a first analysis, we suppose that marginal functions are generalized extreme value distributions. In a second investigation, we substitute the marginal function by the empirical distribution function. Monte Carlo simulations show that the Gudendorf-Segers (Gudendorf and Segers, 2011) estimator outperforms the other estimators for different sample sizes. Secondly, we drop the i.i.d assumption as it’s not verified in time series analysis. Considering the univariate framework, we examine the extremal behavior of a stationary Gaussian autoregressive process. In the multivariate setting, we prove the asymptotic consistency of the Pickands dependence function estimator. This theoretical finding is confirmed by empirical investigations in the asymptotic independence case as well as the asymptotic dependence case. Finally, the Gudendorf-Segers estimator is used to model the dependence structure of extreme ozone concentrations in locations that record several exceedances for both guideline and limit values of the Tunisian air quality standard NT.106.04.