Este trabalho apresenta um novo modelo de redes neurais para análise e previsão de séries temporais: o modelo ARMA-CIGMN (do inglês, Autoregressive Moving Average Classical Incremental Gaussian Mixture Network) além dos resultados obtidos pelo mesmo. Este modelo se baseia em modificações realizadas em uma versão reformulada da IGMN. A IGMN Clássica, CIGMN, é similar à versão original da IGMN, porém baseada em uma abordagem estatística clássica, a qual também é apresentada neste trabalho. As modificações do algoritmo da IGMN foram feitas para melhor adpatação a séries temporais. O modelo ARMA-CIGMN demonstra boa capacidade preditiva e a modelagem ainda pode ser auxiliada por conhecidas ferramentas estatísticas como a função de autorrelação (acf, do original em inglês autocorrelation function) e a de autocorrelação parcial (pacf, do original em inglês partial autocorrelation function), já utilizadas em modelagem de séries temporais e nos modelos da IGMN original. As comparações foram feitas utilizando-se séries conhecidas e dados simulados. Foram selecionados para comparação os modelos estatísticos clássicos ARIMA (do inglês, Autoregressive Integrated Moving Average), a IGMN original e duas modificações feitas ainda na IGMN original:(i) um modelo similar ao modelo ARMA (do inglês, Autoregressive Moving Average) clássico e (ii) um modelo similar ao modelo NOE (do inglês, Nonlinear Output Error). Também é apresentada um versão reformulada da IGMN, usando a abordagem clássica da estatística, necessária para o desenvolvimento do modelo ARMA-CIGMN. / This work presents a new model of neural network for time series analysis and forecasting: the ARMA-CIGMN (Autoregressive Moving Average Classical Incremental Gaussian Mixture Network) model and its analysis. This model is based on modifications made to a reformulated IGMN, the Classical IGMN (CIGMN). The CIGMN is similar to the original IGMN, but based on a classical statistical approach. The modifications to the IGMN algorithm were made to better fit it to time series. The proposed ARMA-CIGMN model demonstrates good forecasts and the modeling procedure can also be aided by known statistical tools as the autocorrelation (acf) and partial autocorrelation functions (pacf), already used in classical statistical time series modeling and also with the original IGMN algorithm models. The ARMA-CIGMN model was evaluated using known series and simulated data. The models used for comparisons were the classical statistical ARIMA model and its variants, the original IGMN and two modifications over the original IGMN: (i) a modification similar to a classical ARMA (Autoregressive Moving Average) model and (ii) a similar NOE (Nonlinear Output Error) model. It is also presented a reformulated IGMN version with a classical statistical approach, which is needed for the ARMA-CIGMN model.

Estudos recentes em séries temporais direcionam-se àquelas que apresentam característica de longa dependência, ou seja, séries temporais nas quais a dependência entre observações distantes não é desprezível. Neste trabalho, analisamos o modelo ARFIN!A(p, d,q ), para dE (0,0;0,5), que apresenta a. característica de longa dependência. Como estimativas para o grau de diferenciação d consideramos os estimadores obtidos através da função periodograma, da função periodograma suavizado e da função de máxima verossimilhança sugerida por Whittle, comparando a variância e o erro quadrático médio destes estimadores através de diversas simulações. / Recent work on time series analysis is concerned with the property of long mcmory, that is, time series in which the dependence between distant observations is not negligible. In this work we analyzc the ARF I .NI A(p, d, q) model, for d E (0.0; 0.5), that has the property of long memory. We consider estimators for the degree of differencing d based on the perioclogram function, on the smoothed periodogram function , anel on the maximum likelihood function suggested by Whittle. Through several simulations we compare the variance anel the mean squared error for these estimators.

Estudos recentes em séries temporais direcionam-se àquelas que apresentam característica de longa dependência, ou seja, séries temporais nas quais a dependência entre observações distantes não é desprezível. Neste trabalho, analisamos o modelo ARFIN!A(p, d,q ), para dE (0,0;0,5), que apresenta a. característica de longa dependência. Como estimativas para o grau de diferenciação d consideramos os estimadores obtidos através da função periodograma, da função periodograma suavizado e da função de máxima verossimilhança sugerida por Whittle, comparando a variância e o erro quadrático médio destes estimadores através de diversas simulações. / Recent work on time series analysis is concerned with the property of long mcmory, that is, time series in which the dependence between distant observations is not negligible. In this work we analyzc the ARF I .NI A(p, d, q) model, for d E (0.0; 0.5), that has the property of long memory. We consider estimators for the degree of differencing d based on the perioclogram function, on the smoothed periodogram function , anel on the maximum likelihood function suggested by Whittle. Through several simulations we compare the variance anel the mean squared error for these estimators.

Este trabalho apresenta um novo modelo de redes neurais para análise e previsão de séries temporais: o modelo ARMA-CIGMN (do inglês, Autoregressive Moving Average Classical Incremental Gaussian Mixture Network) além dos resultados obtidos pelo mesmo. Este modelo se baseia em modificações realizadas em uma versão reformulada da IGMN. A IGMN Clássica, CIGMN, é similar à versão original da IGMN, porém baseada em uma abordagem estatística clássica, a qual também é apresentada neste trabalho. As modificações do algoritmo da IGMN foram feitas para melhor adpatação a séries temporais. O modelo ARMA-CIGMN demonstra boa capacidade preditiva e a modelagem ainda pode ser auxiliada por conhecidas ferramentas estatísticas como a função de autorrelação (acf, do original em inglês autocorrelation function) e a de autocorrelação parcial (pacf, do original em inglês partial autocorrelation function), já utilizadas em modelagem de séries temporais e nos modelos da IGMN original. As comparações foram feitas utilizando-se séries conhecidas e dados simulados. Foram selecionados para comparação os modelos estatísticos clássicos ARIMA (do inglês, Autoregressive Integrated Moving Average), a IGMN original e duas modificações feitas ainda na IGMN original:(i) um modelo similar ao modelo ARMA (do inglês, Autoregressive Moving Average) clássico e (ii) um modelo similar ao modelo NOE (do inglês, Nonlinear Output Error). Também é apresentada um versão reformulada da IGMN, usando a abordagem clássica da estatística, necessária para o desenvolvimento do modelo ARMA-CIGMN. / This work presents a new model of neural network for time series analysis and forecasting: the ARMA-CIGMN (Autoregressive Moving Average Classical Incremental Gaussian Mixture Network) model and its analysis. This model is based on modifications made to a reformulated IGMN, the Classical IGMN (CIGMN). The CIGMN is similar to the original IGMN, but based on a classical statistical approach. The modifications to the IGMN algorithm were made to better fit it to time series. The proposed ARMA-CIGMN model demonstrates good forecasts and the modeling procedure can also be aided by known statistical tools as the autocorrelation (acf) and partial autocorrelation functions (pacf), already used in classical statistical time series modeling and also with the original IGMN algorithm models. The ARMA-CIGMN model was evaluated using known series and simulated data. The models used for comparisons were the classical statistical ARIMA model and its variants, the original IGMN and two modifications over the original IGMN: (i) a modification similar to a classical ARMA (Autoregressive Moving Average) model and (ii) a similar NOE (Nonlinear Output Error) model. It is also presented a reformulated IGMN version with a classical statistical approach, which is needed for the ARMA-CIGMN model.

The primary purpose of this thesis was to investigate if the modern Phillips Curve is valid for ASEAN five (Indonesia, Malaysia, Thailand, Singapore and Philippines) countries using a time-varying approach in the form of an ARMA-GARCH model. The method enables us to investigate how the inflation volatility reacts to economic shocks and if its history can predict the conditional variance of inflation. This study also aimed to investigate whether financial liberalisation affects the conditional variance of inflation. Moreover, we introduce a new parameter into the Phillips Curve. We propose the inclusion of a globally decomposed financial spillover index to see how it affects the inflation dynamics. Examining the period between 1996-2020, using monthly data. We find weak results, and the Phillips Curve was only valid for Singapore. Our findings also suggest that the inflation volatility is highly time-varying, indicating the suitability of the ARMA-GARCH framework. Significant coefficients in the model allow forecasting the conditional variance of inflation. The results support the idea that financial liberalisation to be volatility augmenting in some countries, suggesting a negative relationship between the degree of financial integration and received spillover effects. The globally decomposed spillover indices demonstrated weak results. For further investigations, we, therefore, propose the usage of regionally decomposed spillover indices.

Two real time series, one discussing the area of energy, other discussing the area of economy. By the energetic area we will be dealing with the electric power consumption in the USA, by the economic area we will be dealing with the progress of index PX50. We will try to approve the validity of hypothesis that with some test functions we will be able to set down the accidental unit distribution in these two time series.

This thesis compares the results of the performance of the statistical Autoregressive integrated moving average (ARIMA) model and the neural network Long short-term model (LSTM) on a data set, which represents a market index. Both models are used to predict monthly, daily, and minute close prices of the OMX Stockholm 30 Index. Chosen data were preprocessed, models were fitted to data and their prediction was evaluated and compared. To evaluate forecast accuracy as well as to compare two models fitted to a financial time series, we have used the two performance measures: mean square error (MSE) and mean absolute percentage error (MAPE). In addition, the computation time of fitting models was measured in this thesis to evaluate and compare the computational workload associated with the two models. Also, other factors were discussed, such as the number of parameters and explainability. The analysis revealed that the minute and the daily data of the OMX 30 Stockholm index closely resembled white noise, indicating random fluctuations. However, for the monthly data, the LSTM model outperformed the ARIMA model in terms of MSE, with values of 15,230 and 14,380, respectively. Additionally, the LSTM model demonstrated superior capability in capturing the dynamics of price movement compared to ARIMA. Regarding MAPE, both models exhibited similar values, with ARIMA at 4.8 and LSTM at 4.9. In addition, the ARIMA model had significantly fewer parameters compared to the LSTM model and offered the advantages of being more transparent and easier to interpret.

In this work, the random vibration of higher-order nonlinear beams and composite plates subjected to stochastic loading is studied. The fourth-order nonlinear beam equation is examined to study the effect of rotary inertia and shear deformation on the root mean square values of displacement response. A new linearly coupled equivalent linearization method is proposed and compared with the widely used traditional equivalent linearization method. The new method is proven to yield closer predictions to the numerical simulation results of the nonlinear beam vibration. A systematical investigation of the nonlinear random vibration of composite plates is conducted in which effects of nonlinearity, choices of different plate theories (the first order shear deformation plate theory and the classical plate theory), and temperature gradient on the plate statistical transverse response are addressed. Attention is paid to calculate the R.M.S. values of stress components since they directly affect the fatigue life of the structure. A statistical data reconstruction technique named ARMA modeling and its applications in random vibration data analysis are discussed. The model is applied to the simulation data of nonlinear beams. It is shown that good estimations of both the nonlinear frequencies and the power spectral densities are given by the technique. / Ph. D.

La plupart des modèles en statistique classique repose sur une hypothèse sur la distribution des données ou sur une distribution sous-jacente aux données. La validité de cette hypothèse permet de faire de l’inférence, de construire des intervalles de confiance ou encore de tester la fiabilité du modèle. La problématique des tests d’ajustement vise à s’assurer de la conformité ou de la cohérence de l’hypothèse avec les données disponibles. Dans la présente thèse, nous proposons des tests d’ajustement à la loi normale dans le cadre des séries chronologiques univariées et vectorielles. Nous nous sommes limités à une classe de séries chronologiques linéaires, à savoir les modèles autorégressifs à moyenne mobile (ARMA ou VARMA dans le cas vectoriel). Dans un premier temps, au cas univarié, nous proposons une généralisation du travail de Ducharme et Lafaye de Micheaux (2004) dans le cas où la moyenne est inconnue et estimée. Nous avons estimé les paramètres par une méthode rarement utilisée dans la littérature et pourtant asymptotiquement efficace. En effet, nous avons rigoureusement montré que l’estimateur proposé par Brockwell et Davis (1991, section 10.8) converge presque sûrement vers la vraie valeur inconnue du paramètre. De plus, nous fournissons une preuve rigoureuse de l’inversibilité de la matrice des variances et des covariances de la statistique de test à partir de certaines propriétés d’algèbre linéaire. Le résultat s’applique aussi au cas où la moyenne est supposée connue et égale à zéro. Enfin, nous proposons une méthode de sélection de la dimension de la famille d’alternatives de type AIC, et nous étudions les propriétés asymptotiques de cette méthode. L’outil proposé ici est basé sur une famille spécifique de polynômes orthogonaux, à savoir les polynômes de Legendre. Dans un second temps, dans le cas vectoriel, nous proposons un test d’ajustement pour les modèles autorégressifs à moyenne mobile avec une paramétrisation structurée. La paramétrisation structurée permet de réduire le nombre élevé de paramètres dans ces modèles ou encore de tenir compte de certaines contraintes particulières. Ce projet inclut le cas standard d’absence de paramétrisation. Le test que nous proposons s’applique à une famille quelconque de fonctions orthogonales. Nous illustrons cela dans le cas particulier des polynômes de Legendre et d’Hermite. Dans le cas particulier des polynômes d’Hermite, nous montrons que le test obtenu est invariant aux transformations affines et qu’il est en fait une généralisation de nombreux tests existants dans la littérature. Ce projet peut être vu comme une généralisation du premier dans trois directions, notamment le passage de l’univarié au multivarié ; le choix d’une famille quelconque de fonctions orthogonales ; et enfin la possibilité de spécifier des relations ou des contraintes dans la formulation VARMA. Nous avons procédé dans chacun des projets à une étude de simulation afin d’évaluer le niveau et la puissance des tests proposés ainsi que de les comparer aux tests existants. De plus des applications aux données réelles sont fournies. Nous avons appliqué les tests à la prévision de la température moyenne annuelle du globe terrestre (univarié), ainsi qu’aux données relatives au marché du travail canadien (bivarié). Ces travaux ont été exposés à plusieurs congrès (voir par exemple Tagne, Duchesne et Lafaye de Micheaux (2013a, 2013b, 2014) pour plus de détails). Un article basé sur le premier projet est également soumis dans une revue avec comité de lecture (Voir Duchesne, Lafaye de Micheaux et Tagne (2016)). / Several phenomena from natural and social sciences rely on distribution’s assumption among which the normal distribution is the most popular. The validity of that assumption is useful to setting up forecast intervals or for checking model adequacy of the underlying model. The goodness-of-fit procedures are tools to assess the adequacy of the data’s underlying assumptions. Autoregressive and moving average time series models are often used to find the mathematical behavior of these phenomena from natural and social sciences, and especially in the finance area. These models are based on some assumptions including normality distribution for the innovations. Normality assumption may be helpful for some testing procedures. Furthermore, stronger conclusions can be drawn from the adjusted model if the white noise can be assumed Gaussian. In this work, goodness-of-fit tests for checking normality for the innovations from autoregressive moving average time series models are proposed for both univariate and multivariate cases (ARMA and VARMA models). In our first project, a smooth test of normality for ARMA time series models with unknown mean based on a least square type estimator is proposed. We derive the asymptotic null distribution of the test statistic. The result here is an extension of the paper of Ducharme et Lafaye de Micheaux (2004), where they supposed the mean known and equal to zero. We use the least square type estimator proposed by Brockwell et Davis (1991, section 10.8) and we provide a rigorous proof that it is almost surely convergent. We show that the covariance matrix of the test is nonsingular regardless if the mean is known. We have also studied a data driven approach for the choice of the dimension of the family and we gave a finite sample approximation of the null distribution. Finally, the finite and asymptotic sample properties of the proposed test statistic are studied via a small simulation study. In the second project, goodness-of-fit tests for checking multivariate normality for the innovations from vector autoregressive moving average time series models are proposed. Since these time series models may rely on a large number of parameters, structured parameterization of the functional form is allowed. The methodology also relies on the smooth test paradigm and on families of orthonormal functions with respect to the multivariate normal density. It is shown that the smooth tests converge to convenient chi-square distributions asymptotically. An important special case makes use of Hermite polynomials, and in that situation we demonstrate that the tests are invariant under linear transformations. We observed that the test is not invariant under linear transformations with Legendre polynomials. A consistent data driven method is discussed to choose the family order from the data. In a simulation study, exact levels are studied and the empirical powers of the smooth tests are compared to those of other methods. Finally, an application to real data is provided, specifically on Canadian labour market data and annual global temperature. These works were exposed at several meeting (see for example Tagne, Duchesne and Lafaye de Micheaux (2013a, 2013b, 2014) for more details). A paper based on the first project is submitted in a refereed journal (see Duchesne, Lafaye de Micheaux et Tagne (2016)).

Dans cette thèse nous proposons de nouveaux modèles paramétriques en traitement du signal et de l'image, fondés sur la décomposition de Wold des processus stochastiques. Les approches de modélisation font appel à l'analyse fonctionnelle et harmonique, l'analyse par ondelettes, ainsi qu'à la théorie des champs stochastiques. Le premier chapitre a un caractère introductif théorique et précise les éléments de base concernant le contexte de la prédiction linéaire des processus stochastiques stationnaires et la décomposition Wold, dans le cas 1-D et multi-D. On montre comment les différentes parties de la décomposition sont obtenues à partir de l'hypothèse de stationnarité, via la représentation du processus comme l'orbite d'un certain opérateur unitaire, l'isomorphisme canonique de Kolmogorov et les conséquences sur la prédiction linéaire du théorème de Szégö et de ses extensions multidimensionnelles. Le deuxième chapitre traite une approche de factorisation spectrale de la densité spectrale de puissance qu'on utilisera pour l'identification des modèles de type Moyenne Ajustée (MA), Autorégressif (AR) et ARMA. On utilise la représentation par le noyau reproduisant de Poisson d'une fonction extérieure pour construire un algorithme d'estimation d'un modèle MA avec une densité spectrale de puissance donnée. Cette méthode d'estimation est présentée dans le cadre de deux applications: - Dans la simulation de canaux sans fil de type Rayleigh (cas 1-D). - Dans le cadre d'une approche de décomposition de Wold des images texturées (cas 2-D). Dans le troisième chapitre nous abordons la représentation et la compression hybride d'images. Nous proposons une approche de compression d'images qui utilise conjointement : - les modèles issus de la décomposition de Wold pour la représentation des régions dites texturées de l'image; - une approche fondée sur les ondelettes pour le codage de la partie "cartoon" (ou non-texturée) de l' image. Dans ce cadre, nous proposons une nouvelle approche pour la décomposition d'une image dans une partie texturée et une partie non-texturée fondée sur la régularité locale. Chaque partie est ensuite codée à l'aide de sa représentation particulière. / In this thesis we propose new parametric models in signal and image processing based on the Wold decomposition of stationary stochastic processes. These models rely upon several theoretical results from functional and harmonic analysis, wavelet analysis and the theory of stochastic fields, The first chapter presents the theoretical background of the linear prediction for stationary processes and of the Wold decomposition theorems in 1-D and n-D. It is shown how the different parts of the decomposition are obtained and represented, by the means of the unitary orbit representation of stationary processes, the Kolmogorov canonical model and Szego-type extensions. The second chapter deals with a spectral factorisation approach of the power spectral density used for the parameter estimation of Moving Avergage (MA), AutoRegressif (AR) and ARMA models. The method uses the Poisson integral representation in Hardy spaces in order to estimate an outer transfer function from its power spectral density. - Simulators for Rayleigh fading channels (1-D). - A scheme for the Wold decomposition for texture images (2-D). In the third chapter we deal with hybrid models for image representation and compression. We propose a compression scheme which jointly uses, on one hand, Wold models for textured regions of the image, and on the other hand a wavelet-based approach for coding the 'cartoon' (or non-textured) part of the image. In this context, we propose a new algorithm for the decomposing images in a textured part and a non-textured part. The separate parts are then coded with the appropriate representation.