Covariance Estimation

Diffusion models have proven to be a flexible and effective framework for modelling probability distributions on finite-dimensional spaces. However, many physical modelling problems such as time series are naturally described over function spaces. In this work we apply diffusion models to such stochastic processes. To do so we consider a spectral representation of the data, obtained using a kernel, thereby dissociating the stochastic part of the processes from their space-time structure. As a result, the stochasticity of the processes is entirely encoded in the spectral coefficients, which we truncate and model using standard finite-dimensional diffusion models. By truncating the representation in the spectral domain we ensure our resulting model defines valid stochastic processes, thereby naturally satisfying consistency and exchangeability criteria. Projecting our spectral diffusion models back to the original input space, we show that for any given marginals our approach corresponds to a diffusion model with correlated noise, with explicit covariance matrix given by the kernel. We demonstrate our method's effectiveness for modelling various multimodal datasets as well as conditional sampling by amortising our models with respect to a context set.

Inference for factor models is often hampered by the lack of tractable and accurate variance estimates, which can materially distort downstream analyses. In practice, uncertainty in the residual covariance matrix is frequently either ignored or addressed through computationally intensive resampling methods that tend to be unstable. This paper develops a unified analytical framework for inference in exploratory factor analysis under several widely used extraction rules, including least-squares, principal-factor, iterative principal-component, alpha, and image factoring. By treating these estimators as implicitly defined functions of the sample covariance matrix, we derive closed-form Jacobians that translate perturbations in the covariance matrix into changes in the resulting factor solutions. Combined with the delta method and consistent estimators of the sample covariance matrix, the proposed approach yields standard errors that are straightforward to compute and remain valid under non-Gaussianity, heteroskedasticity, and serial or cross-sectional dependence. Simulation evidence confirms that the analytical standard errors accurately capture finite-sample variability while avoiding both the instability of bootstrap procedures and the restrictive assumptions underlying Fisher information-based inference. An application to a factor-augmented structural vector autoregressive (SVAR) model further demonstrates how accounting for this source of uncertainty can substantially affect impulse-response inference. Taken together, the results provide a practical and general tool for propagating estimation uncertainty in settings where factor extraction serves as an intermediate step.

Advanced Air Mobility (AAM) operations are planned to utilize strategic flight planning services that predict temporal uncertainties to validate flight plans against hazards such as weather cells, restricted airspaces, and CNS disruption areas. This paper presents a Kalman Filter-based uncertainty propagation method that models Flight Management System (FMS) correction behavior through a sigmoid-blended measurement noise covariance. The sigmoid formulation generalizes existing discrete FMS activation thresholds into a continuous, tunable function that smoothly transitions the filter's measurement noise based on progress toward each waypoint. When the measurement noise is high, due to an inverse relationship, the Kalman gain is small and thus uncertainty grows; as the aircraft nears a waypoint, measurement noise decreases as a function of progress, the Kalman gain increases, and state covariance contracts which models the FMS progressively correcting toward the planned trajectory. The approach is computationally efficient (up to two orders of magnitude faster than Monte Carlo methods), scales with control inputs, and is parametrically tunable for different classes of aircraft. The measurement noise covariance is calibrated using real Automatic Dependent Surveillance-Broadcast (ADS-B) data from commercial Instrument Flight Rules (IFR) flights serving as surrogates for future AAM operations, achieving coverage probability conservative relative to theoretical Gaussian predictions at the 1-sigma confidence level on a hold out verification dataset (N = 36). Parameter sensitivity analysis across multiple flight routes demonstrates robust behavior, and comparative evaluation against Monte Carlo and Linear Propagation methods contextualizes the method's computational and accuracy trade-offs.