Spatial Statistics for Complex Domains via Stochastic Network Dynamics (Leonardo Marchesin)
Spatial dependence can reflect both domain geometry and underlying physical dynamics, which can be difficult to capture using conventional covariance models. Physical barriers and directional transport can produce dependence patterns that differ substantially from those implied by geographical proximity alone. Environmental applications provide natural examples, including water currents in aquatic domains and wind-driven pollutant transport. This seminar presents a framework for flow-informed spatial statistics based on directed network representations of complex domains. Building on moving-average models for stream networks, the framework introduces Markovian particle trajectories into a stochastic convolution construction defined over a directed network. Covariances take the form of weighted sums over directed paths connecting two locations, providing an interpretable link between transport dynamics and spatial dependence.
Two environmental applications illustrate the framework. In the Mediterranean Sea, temperature exceedance analysis identifies thermal hotspots in projected sea surface temperatures while accounting for spatial uncertainty and current-driven connectivity. In Lake Garda, universal kriging combines spatial covariates with flow-informed dependence to fill gaps in satellite-derived turbidity observations.
Beyond environmental applications, the construction provides a basis for modelling other systems where network topology and transport shape spatial dependence.
Physics-informed spatial quantile regression for air quality analysis (Ilenia Di Battista)
Quantile regression is a key statistical tool for analyzing data that depart from the Gaussian assumption, for instance due to pronounced skewness or heteroscedasticity. These features are frequently observed in climate and environmental sciences, where data often exhibit heavy tails associated with extreme values that may pose greater risks to human health and are therefore more critical to capture. Quantile regression models address this issue, providing more insightful results than linear regression, which focuses only on the mean.
This talk focuses on a physics-informed spatial quantile regression model and its extensions in two key directions: simultaneous spatial quantile estimation and spatio-temporal analysis. The model incorporates prior physical knowledge of the phenomenon under study through a regularization term based on partial differential equations (PDEs). This approach is particularly relevant when the phenomenon is strongly influenced by physical processes, such as the effect of wind on air pollutant dispersion.
Estimating different quantile levels separately may lead to violations of monotonicity across quantiles, particularly for quantile levels in the tails of the distribution. To overcome this problem, we propose a multiple quantile regression model that simultaneously estimates a grid of quantile levels while ensuring their monotonicity. The estimation functional consists of three components: a data misfit term based on the pinball loss function, a roughness penalty encoding the PDE misfit, and a non-crossing penalty enforcing the monotonicity constraints.
We further extend the model to spatio-temporal analysis, incorporating information from both spatial and temporal domains through two distinct roughness penalty terms. This extension allows the model to account for spatio-temporal variability and broadens its applicability to a wide range of real-world studies