Dans cette thèse, nous étudions l’estimation de modèles autorégressif-moyenne mobile<p>vectoriels ou VARMA, `a coefficients dépendant du temps, et avec une matrice de covariance<p>des innovations dépendant du temps. Ces modèles sont appel´es tdVARMA. Les éléments<p>des matrices des coefficients et de la matrice de covariance sont des fonctions déterministes<p>du temps dépendant d’un petit nombre de paramètres. Une première partie de la thèse<p>est consacrée à l’étude des propriétés asymptotiques de l’estimateur du quasi-maximum<p>de vraisemblance gaussienne. La convergence presque sûre et la normalité asymptotique<p>de cet estimateur sont démontrées sous certaine hypothèses vérifiables, dans le cas o`u les<p>coefficients dépendent du temps t mais pas de la taille des séries n. Avant cela nous considérons les propriétés asymptotiques des estimateurs de modèles non-stationnaires assez<p>généraux, pour une fonction de pénalité générale. Nous passons ensuite à l’application de<p>ces théorèmes en considérant que la fonction de pénalité est la fonction de vraisemblance<p>gaussienne (Chapitre 2). L’étude du comportement asymptotique de l’estimateur lorsque<p>les coefficients du modèle dépendent du temps t et aussi de n fait l’objet du Chapitre 3.<p>Dans ce cas, nous utilisons une loi faible des grands nombres et un théorème central limite<p>pour des tableaux de différences de martingales. Ensuite, nous présentons des conditions<p>qui assurent la consistance faible et la normalité asymptotique. Les principaux<p>résultats asymptotiques sont illustrés par des expériences de simulation et des exemples<p>dans la littérature. La deuxième partie de cette thèse est consacrée à un algorithme qui nous<p>permet d’évaluer la fonction de vraisemblance exacte d’un processus tdVARMA d’ordre (p, q) gaussien. Notre algorithme est basé sur la factorisation de Cholesky d’une matrice<p>bande partitionnée. Le point de départ est une généralisation au cas multivarié de Mélard<p>(1982) pour évaluer la fonction de vraisemblance exacte d’un modèle ARMA(p, q) univarié. Aussi, nous utilisons quelques résultats de Jonasson et Ferrando (2008) ainsi que les programmes Matlab de Jonasson (2008) dans le cadre d’une fonction de vraisemblance<p>gaussienne de modèles VARMA à coefficients constants. Par ailleurs, nous déduisons que<p>le nombre d’opérations requis pour l’évaluation de la fonction de vraisemblance en fonction de p, q et n est approximativement le double par rapport à un modèle VARMA à coefficients<p>constants. L’implémentation de cet algorithme a été testée en comparant ses résultats avec<p>d’autres programmes et logiciels très connus. L’utilisation des modèles VARMA à coefficients<p>dépendant du temps apparaît particulièrement adaptée pour la dynamique de quelques<p>séries financières en mettant en évidence l’existence de la dépendance des paramètres en<p>fonction du temps.<p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Statistics / M. Sc. (Statistics)

The idea of time series forecasting techniques is that the past has certain information about future. So, the question of how the information is encoded in the past can be interpreted and later used to extrapolate events of future constitute the crux of time series analysis and forecasting. Several methods such as qualitative techniques (e.g., Delphi method), causal techniques (e.g., least squares regression), quantitative techniques (e.g., smoothing method, time series models) have been developed in the past in which the concept lies in establishing a model either theoretically or mathematically from past observations and estimate future from it. Of all the models, time series methods such as autoregressive moving average (ARMA) process have gained popularity because of their simplicity in implementation and accuracy in obtaining forecasts. But, these models were formulated based on certain properties that a time series is assumed to possess. Classical decomposition techniques were developed to supplement the requirements of time series models. These methods try to define a time series in terms of simple patterns called trend, cyclical and seasonal patterns along with noise. So, the idea of decomposing a time series into component patterns, later modeling each component using forecasting processes and finally combining the component forecasts to obtain actual time series predictions yielded superior performance over standard forecasting techniques. All these methods involve basic principle of moving average computation. But, the developed classical decomposition methods are disadvantageous in terms of containing fixed number of components for any time series, data independent decompositions. During moving average computation, edges of time series might not get modeled properly which affects long range forecasting. So, these issues are to be addressed by more efficient and advanced decomposition techniques such as Wavelets and Empirical Mode Decomposition (EMD). Wavelets and EMD are some of the most innovative concepts considered in time series analysis and are focused on processing nonlinear and nonstationary time series. Hence, this research has been undertaken to ascertain the predictability of nonstationary time series using wavelet and Empirical Mode Decomposition (EMD) based ARMA models. The development of wavelets has been made based on concepts of Fourier analysis and Window Fourier Transform. In accordance with this, initially, the necessity of involving the advent of wavelets has been presented. This is followed by the discussion regarding the advantages that are provided by wavelets. Primarily, the wavelets were defined in the sense of continuous time series. Later, in order to match the real world requirements, wavelets analysis has been defined in discrete scenario which is called as Discrete Wavelet Transform (DWT). The current thesis utilized DWT for performing time series decomposition. The detailed discussion regarding the theory behind time series decomposition is presented in the thesis. This is followed by description regarding mathematical viewpoint of time series decomposition using DWT, which involves decomposition algorithm. EMD also comes under same class as wavelets in the consequence of time series decomposition. EMD is developed out of the fact that most of the time series in nature contain multiple frequencies leading to existence of different scales simultaneously. This method, when compared to standard Fourier analysis and wavelet algorithms, has greater scope of adaptation in processing various nonstationary time series. The method involves decomposing any complicated time series into a very small number of finite empirical modes (IMFs-Intrinsic Mode Functions), where each mode contains information of the original time series. The algorithm of time series decomposition using EMD is presented post conceptual elucidation in the current thesis. Later, the proposed time series forecasting algorithm that couples EMD and ARMA model is presented that even considers the number of time steps ahead of which forecasting needs to be performed. In order to test the methodologies of wavelet and EMD based algorithms for prediction of time series with non stationarity, series of streamflow data from USA and rainfall data from India are used in the study. Four non-stationary streamflow sites (USGS data resources) of monthly total volumes and two non-stationary gridded rainfall sites (IMD) of monthly total rainfall are considered for the study. The predictability by the proposed algorithm is checked in two scenarios, first being six months ahead forecast and the second being twelve months ahead forecast. Normalized Root Mean Square Error (NRMSE) and Nash Sutcliffe Efficiency Index (Ef) are considered to evaluate the performance of the proposed techniques. Based on the performance measures, the results indicate that wavelet based analyses generate good variations in the case of six months ahead forecast maintaining harmony with the observed values at most of the sites. Although the methods are observed to capture the minima of the time series effectively both in the case of six and twelve months ahead predictions, better forecasts are obtained with wavelet based method over EMD based method in the case of twelve months ahead predictions. It is therefore inferred that wavelet based method has better prediction capabilities over EMD based method despite some of the limitations of time series methods and the manner in which decomposition takes place. Finally, the study concludes that the wavelet based time series algorithm could be used to model events such as droughts with reasonable accuracy. Also, some modifications that could be made in the model have been suggested which can extend the scope of applicability to other areas in the field of hydrology.

金融市場是個變化莫測的環境，看似隨機，在隨機中卻隱藏著某些特性與關係。不論是自然現象中的氣象預測或是金融領域中對下一時刻價格的預測, 都有相似的複雜性。 時間序列的預測一直都是許多領域中重要的項目之一, 金融時間序列的預測也不例外。在本論文中我們針對金融時間序列的非線性與非穩態關係引入類神經網路(ANNs) 與集合經驗模態分解法(EEMD), 藉由ANNs處理非線性問題的能力與EEMD處理時間序列信號的優點，並進一步與傳統上使用於金融時間序列分析的自回歸滑動平均模型(ARMA)進行複合式的模型建構，引入燭型圖概念嘗試進行高頻下的台指期貨TAIEX交易。在不計交易成本的績效測試下本研究的高頻交易模型有突出的績效，證明以ANNs、EEMD方法與ARMA組成的混合式模型在高頻時間尺度交易下有相當的發展潛力，具有進一步發展的價值。在處理高頻時間尺度下所產生的大型數據方面，引入平行運算架構SPMD(single program, multiple data)以增進其處理大型資料下的運算效率。本研究亦透過分析高頻時間尺度的本質模態函數(IMFs)探討在高頻尺度下影響台指期貨價格的因素。 / Financial market is complex, unstable and non-linear system, it looks like have some principle but the principle usually have exception. The forecasting of time series always an issue in several field include finance. In this thesis we propose several version of hybrid models, they combine Ensemble Empirical Mode Decomposition (EEMD), Back-Propagation Neural Networks(BPNN) and ARMA model, try to improve the forecast performance of financial time series forecast. We also found the physical means or impact factors of IMFs under high-frequency time-scale. For processing the massive data generated by high-frequency time-scale, we pull in the concept of big data processing, adopt parallel computing method ”single program, multiple data (SPMD)” to construct the model improve the computing performance. As the result of backtesting, we prove the enhanced hybrid models we proposed outperform the standard EEMD-BPNN model and obtain a good performance. It shows adopt ANN, EEMD and ARMA in the hybrid model configure for high-frequency trading modeling is effective and it have the potential of development.