The growing complexity and data in modern engineering and physical systems require robust frameworks for real-time decision-making. Data-driven models trained on observational data enable faster predictions but face key challenges—data corruption, bias, limited interpretability, and uncertainty misrepresentation—which can compromise their reliability. Propagating uncertainties from sources like model parameters and input features is crucial in data-driven models to ensure trustworthy predictions and informed decisions. Uncertainty quantification (UQ) methods are broadly categorized into surrogate-based models, which approximate simulators for speed and efficiency, and probabilistic approaches, such as Bayesian models and Gaussian processes, that inherently capture uncertainty into predictions. For real-time UQ, leveraging recent data instead of historical records enables more accurate and efficient uncertainty characterization, making it inherently data-driven. In dynamical analysis, the Koopman operator represents nonlinear system dynamics as linear systems by lifting state functions, enabling data-driven estimation through its applied form. By analyzing its spectral properties—eigenvalues, eigenfunctions, and modes—the Koopman operator reveals key insights into system dynamics and simplifies control design. However, inherent measurement uncertainty poses challenges for efficient estimation with dynamic mode and extended dynamic mode decomposition algorithms. This dissertation develops a statistical framework to propagate measurement uncertainties in the elements of the Koopman operator. This dissertation also develops robust estimation of model parameters, considering observational data, which is often corrupted, in Gaussian process settings. The proposed approaches adapt to evolving data and process agnostic— in which reliance on predefined source distributions is avoided. / Doctor of Philosophy / Modern engineering and scientific systems are increasingly complex and interconnected— operating in environments with significant uncertainties and dynamic changes. Traditional mathematical models and simulations often fall short in capturing the complexity of largescale real-world, ever-evolving systems—struggling to adapt to dynamic changes and fully utilize today's data-rich environments. This is especially critical in fields like renewable integrated power systems, robotics, etc., where real-time decisions must account for uncertainties in the environment, measurements, and operations. The growing availability of observational data—enabled by advanced sensors and computational tools—has driven a shift toward data-driven approaches. Unlike traditional simulators, these models are faster and learn directly from data. However, their reliability depends on robust methods to quantify and manage uncertainties, as corrupted data, biases, and measurement noise challenge their accuracy. This dissertation focuses on characterizing uncertainties at the source using recent data, instead of relying on assumed distributions or historical data, as is common in the literature. Given that observational data is often corrupted by outliers, this dissertation also develops robust parameter estimation within the Gaussian process setting. A central focus is the Koopman operator theory—a transformative framework that converts complex, nonlinear systems into simpler, linear representations. This research integrates measurement uncertainty quantification into Koopman-based models, providing a metric to assess the reliability of the Koopman operator under measurement noise.