<p>The Ensemble Kalman filter (EnKF) has achieved great successes in data assimilation in atmospheric and oceanic sciences, but its failure in convergence to the right filtering distribution precludes its use for uncertainty quantification. Other existing methods, such as particle filter or sequential importance sampler, do not scale well to the dimension of the system and the sample size of the datasets. In this dissertation, we address these difficulties in a coherent way.</p><p><br></p><p> </p><p>In the first part of the dissertation, we reformulate the EnKF under the framework of Langevin dynamics, which leads to a new particle filtering algorithm, the so-called Langevinized EnKF (LEnKF). The LEnKF algorithm inherits the forecast-analysis procedure from the EnKF and the use of mini-batch data from the stochastic gradient Langevin-type algorithms, which make it scalable with respect to both the dimension and sample size. We prove that the LEnKF converges to the right filtering distribution in Wasserstein distance under the big data scenario that the dynamic system consists of a large number of stages and has a large number of samples observed at each stage, and thus it can be used for uncertainty quantification. We reformulate the Bayesian inverse problem as a dynamic state estimation problem based on the techniques of subsampling and Langevin diffusion process. We illustrate the performance of the LEnKF using a variety of examples, including the Lorenz-96 model, high-dimensional variable selection, Bayesian deep learning, and Long Short-Term Memory (LSTM) network learning with dynamic data.</p><p><br></p><p> </p><p>In the second part of the dissertation, we focus on two extensions of the LEnKF algorithm. Like the EnKF, the LEnKF algorithm was developed for Gaussian dynamic systems containing no unknown parameters. We propose the so-called stochastic approximation- LEnKF (SA-LEnKF) for simultaneously estimating the states and parameters of dynamic systems, where the parameters are estimated on the fly based on the state variables simulated by the LEnKF under the framework of stochastic approximation. Under mild conditions, we prove the consistency of resulting parameter estimator and the ergodicity of the SA-LEnKF. For non-Gaussian dynamic systems, we extend the LEnKF algorithm (Extended LEnKF) by introducing a latent Gaussian measurement variable to dynamic systems. Those two extensions inherit the scalability of the LEnKF algorithm with respect to the dimension and sample size. The numerical results indicate that they outperform other existing methods in both states/parameters estimation and uncertainty quantification.</p>

L’objectif de cette recherche était d’évaluer l’efficacité du paramètre de classification pour prédire suivre les tendances boursières. Les méthodes traditionnelles basées sur la prévision, qui ciblent l’immédiat pas de temps suivant, rencontrent souvent des défis dus à des données non stationnaires, compromettant le modèle précision et stabilité. En revanche, notre approche de classification prédit une évolution plus large du cours des actions avec des mouvements sur plusieurs pas de temps, visant à réduire la non-stationnarité des données. Notre ensemble de données, dérivé de diverses actions du NASDAQ-100 et éclairé par plusieurs indicateurs techniques, a utilisé un mélange d'experts composé d'un mécanisme de déclenchement souple et d'une architecture basée sur les transformateurs. Bien que la méthode principale de cette expérience ne se soit pas révélée être aussi réussie que nous l'avions espéré et vu initialement, la méthodologie avait la capacité de dépasser toutes les lignes de base en termes de performance dans certains cas à quelques époques, en démontrant le niveau le plus bas taux de fausses découvertes tout en ayant un taux de rappel acceptable qui n'est pas zéro. Compte tenu de ces résultats, notre approche encourage non seulement la poursuite des recherches dans cette direction, dans lesquelles un ajustement plus précis du modèle peut être mis en œuvre, mais offre également aux personnes qui investissent avec l'aide de l'apprenstissage automatique un outil différent pour prédire les tendances boursières, en utilisant un cadre de classification et un problème défini différemment de la norme. Il est toutefois important de noter que notre étude est basée sur les données du NASDAQ-100, ce qui limite notre l’applicabilité immédiate du modèle à d’autres marchés boursiers ou à des conditions économiques variables. Les recherches futures pourraient améliorer la performance en intégrant les fondamentaux des entreprises et effectuer une analyse du sentiment sur l'actualité liée aux actions, car notre travail actuel considère uniquement indicateurs techniques et caractéristiques numériques spécifiques aux actions. / The objective of this research was to evaluate the classification setting's efficacy in predicting stock market trends. Traditional forecasting-based methods, which target the immediate next time step, often encounter challenges due to non-stationary data, compromising model accuracy and stability. In contrast, our classification approach predicts broader stock price movements over multiple time steps, aiming to reduce data non-stationarity. Our dataset, derived from various NASDAQ-100 stocks and informed by multiple technical indicators, utilized a Mixture of Experts composed of a soft gating mechanism and a transformer-based architecture. Although the main method of this experiment did not prove to be as successful as we had hoped and seen initially, the methodology had the capability in surpassing all baselines in certain instances at a few epochs, demonstrating the lowest false discovery rate while still having an acceptable recall rate. Given these results, our approach not only encourages further research in this direction, in which further fine-tuning of the model can be implemented, but also offers traders a different tool for predicting stock market trends, using a classification setting and a differently defined problem. It's important to note, however, that our study is based on NASDAQ-100 data, limiting our model's immediate applicability to other stock markets or varying economic conditions. Future research could enhance performance by integrating company fundamentals and conducting sentiment analysis on stock-related news, as our current work solely considers technical indicators and stock-specific numerical features.