With the growing practical interest in vision-based tasks for autonomous systems, the need for efficient and complex methods becomes increasingly larger. In the rush to develop new methods with the aim to outperform the current state of the art, an analysis of the underlying theory is often neglected and simply replaced with empirical evaluations in simulated or real-world experiments. While such methods might yield favorable performance in practice, they are often less well understood, which prevents them from being applied in safety-critical systems. The goal of this work is to design an algorithm for the Next Best View (NBV) problem in the context of active object reconstruction, for which we can provide qualitative performance guarantees with respect to true optimality. To the best of our knowledge, no previous work in this field addresses such an analysis for their proposed methods. Based on existing work on Gaussian process optimization, we rigorously derive sublinear bounds for the cumulative regret of our algorithm, which guarantees near-optimality. Complementing this, we evaluate the performance of our algorithm empirically within our simulation framework. We further provide additional insights through an extensive study of potential objective functions and analyze the differences to the results of related work.

Gaussian process state-space models (GPSSMs) have emerged as a powerful framework for modeling dynamical systems, offering interpretable uncertainty quantification and inherent regularization. However, existing GPSSMs face significant challenges in handling high-dimensional, non-stationary systems due to computational inefficiencies, limited scalability, and restrictive stationarity assumptions. In this paper, we propose an efficient transformed Gaussian process state-space model (ETGPSSM) to address these limitations. Our approach leverages a single shared Gaussian process (GP) combined with normalizing flows and Bayesian neural networks, enabling efficient modeling of complex, high-dimensional state transitions while preserving scalability. To address the lack of closed-form expressions for the implicit process in the transformed GP, we follow its generative process and introduce an efficient variational inference algorithm, aided by the ensemble Kalman filter (EnKF), to enable computationally tractable learning and inference. Extensive empirical evaluations on synthetic and real-world datasets demonstrate the superior performance of our ETGPSSM in system dynamics learning, high-dimensional state estimation, and time-series forecasting, outperforming existing GPSSMs and neural network-based methods in both accuracy and computational efficiency.

Container terminals, pivotal nodes in the network of empty container movement, hold significant potential for enhancing operational efficiency within terminal depots through effective collaboration between transporters and terminal operators. This collaboration is crucial for achieving optimization, leading to streamlined operations and reduced congestion, thereby benefiting both parties. Consequently, there is a pressing need to develop the most suitable forecasting approaches to address this challenge. This study focuses on developing and evaluating a data-driven approach for forecasting empty container availability at container terminal depots within a Vehicle Booking System (VBS) framework. It addresses the gap in research concerning optimizing empty container dwell time and aims to enhance operational efficiencies in container terminal operations. Four forecasting models-Naive, ARIMA, Prophet, and LSTM-are comprehensively analyzed for their predictive capabilities, with LSTM emerging as the top performer due to its ability to capture complex time series patterns. The research underscores the significance of selecting appropriate forecasting techniques tailored to the specific requirements of container terminal operations, contributing to improved operational planning and management in maritime logistics.

Transformers have increasingly become the de facto method to model sequential data with state-of-the-art performance. Due to its widespread use, being able to estimate and calibrate its modeling uncertainty is important to understand and design robust transformer models. To achieve this, previous works have used Gaussian processes (GPs) to perform uncertainty calibration for the attention units of transformers and attained notable successes. However, such approaches have to confine the transformers to the space of symmetric attention to ensure the necessary symmetric requirement of their GP's kernel specification, which reduces the representation capacity of the model. To mitigate this restriction, we propose the Correlated Gaussian Process Transformer (CGPT), a new class of transformers whose self-attention units are modeled as cross-covariance between two correlated GPs (CGPs). This allows asymmetries in attention and can enhance the representation capacity of GP-based transformers. We also derive a sparse approximation for CGP to make it scale better. Our empirical studies show that both CGP-based and sparse CGP-based transformers achieve better performance than state-of-the-art GP-based transformers on a variety of benchmark tasks. The code for our experiments is available at https://github.com/MinhLong210/CGP-Transformers.

Recalibrating probabilistic classifiers is vital for enhancing the reliability and accuracy of predictive models. Despite the development of numerous recalibration algorithms, there is still a lack of a comprehensive theory that integrates calibration and sharpness (which is essential for maintaining predictive power). In this paper, we introduce the concept of minimum-risk recalibration within the framework of mean-squared-error (MSE) decomposition, offering a principled approach for evaluating and recalibrating probabilistic classifiers.

This thesis contains a series of independent contributions to statistics, unified by a model-free perspective. The first chapter elaborates on how a model-free perspective can be used to formulate flexible methods that leverage prediction techniques from machine learning. Mathematical insights are obtained from concrete examples, and these insights are generalized to principles that permeate the rest of the thesis. The second chapter studies the concept of local independence, which describes whether the evolution of one stochastic process is directly influenced by another. To test local independence, we define a model-free parameter called the Local Covariance Measure (LCM). We formulate an estimator for the LCM, from which a test of local independence is proposed. We discuss how the size and power of the proposed test can be controlled uniformly and investigate the test in a simulation study. The third chapter focuses on covariate adjustment, a method used to estimate the effect of a treatment by accounting for observed confounding. We formulate a general framework that facilitates adjustment for any subset of covariate information. We identify the optimal covariate information for adjustment and, based on this, introduce the Debiased Outcome-adapted Propensity Estimator (DOPE) for efficient estimation of treatment effects. An instance of DOPE is implemented using neural networks, and we demonstrate its performance on simulated and real data. The fourth and final chapter introduces a model-free measure of the conditional association between an exposure and a time-to-event, which we call the Aalen Covariance Measure (ACM). We develop a model-free estimation method and show that it is doubly robust, ensuring $\sqrt{n}$-consistency provided that the nuisance functions can be estimated with modest rates. A simulation study demonstrates the use of our estimator in several settings.

The theory of homogenization provides a systematic approach to the derivation of macroscale constitutive laws, obviating the need to repeatedly resolve complex microstructure. However, the unit cell problem that defines the constitutive model is typically not amenable to explicit evaluation. It is therefore of interest to learn constitutive models from data generated by the unit cell problem. Many viscoelastic and elastoviscoplastic materials are characterized by memory-dependent constitutive laws. In order to amortize the computational investment in finding such memory-dependent constitutive laws, it is desirable to learn their dependence on the material microstructure. While prior work has addressed learning memory dependence and material dependence separately, their joint learning has not been considered. This paper focuses on the joint learning problem and proposes a novel neural operator framework to address it. In order to provide firm foundations, the homogenization problem for linear Kelvin-Voigt viscoelastic materials is studied. The theoretical properties of the cell problem in this Kelvin-Voigt setting are used to motivate the proposed general neural operator framework; these theoretical properties are also used to prove a universal approximation theorem for the learned macroscale constitutive model. This formulation of learnable constitutive models is then deployed beyond the Kelvin-Voigt setting. Numerical experiments are presented showing that the resulting data-driven methodology accurately learns history- and microstructure-dependent linear viscoelastic and nonlinear elastoviscoplastic constitutive models, and numerical results also demonstrate that the resulting constitutive models can be deployed in macroscale simulation of material deformation.

The integration of renewable energy into electricity markets poses significant challenges to price stability and increases the complexity of market operations. Accurate and reliable electricity price forecasting is crucial for effective market participation, where price dynamics can be significantly more challenging to predict. Probabilistic forecasting, through prediction intervals, efficiently quantifies the inherent uncertainties in electricity prices, supporting better decision-making for market participants. This study explores the enhancement of probabilistic price prediction using Conformal Prediction (CP) techniques, specifically Ensemble Batch Prediction Intervals and Sequential Predictive Conformal Inference. These methods provide precise and reliable prediction intervals, outperforming traditional models in validity metrics. We propose an ensemble approach that combines the efficiency of quantile regression models with the robust coverage properties of time series adapted CP techniques. This ensemble delivers both narrow prediction intervals and high coverage, leading to more reliable and accurate forecasts. We further evaluate the practical implications of CP techniques through a simulated trading algorithm applied to a battery storage system. The ensemble approach demonstrates improved financial returns in energy trading in both the Day-Ahead and Balancing Markets, highlighting its practical benefits for market participants.