1) 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.
2) 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.