Advanced Bayesian methodologies for parametric and non-parametric inference of
spatiotemporal phenomena
Advisor: Professor Paolo Gardoni
Abstract
Real-world physical processes are rarely independent across space and time. For example,
an above-average temperature in Urbana, IL, is often followed by a similar observation in
nearby Champaign, IL, due to geographic proximity. Similarly, an increase in COVID-19 cases
in one day may lead to higher cases the next day in the same city. These spatial and temporal
correlations are critical for systems operating over large geographic areas and long time
periods. Ignoring these correlations can result in significant underestimation of system
vulnerabilities to external stressors. As a result, spatiotemporal models have become
essential for capturing these dynamics, providing valuable tools for tasks like environmental
monitoring, resource planning, and risk management.
Spatiotemporal phenomena are observed across various domains, from natural hazards like
earthquakes and floods to structural deterioration due to environmental factors. These
phenomena exhibit considerable variability over diNerent spatial and temporal scales. For
example, a bridge may experience vehicle loads over short periods while also facing slow
corrosion over decades. These multi-scale processes, coupled with a lack of comprehensive
data and the computational expense of detailed physical models, make the modeling of
spatiotemporal phenomena particularly challenging.
One specific example of a complex spatiotemporal phenomenon is storm surge, which
refers to the abnormal rise in seawater caused by hurricanes or other intense storms. Storm
surge poses significant risks to coastal infrastructure and human populations, making it
crucial to develop accurate models for prediction and real-time response. However,
modeling storm surge involves integrating multiple interacting processes—such as oceanic
dynamics, atmospheric conditions, and geographical factors—all operating on diNerent
spatial and temporal scales. Additionally, the sparsity of direct surge measurements further
complicates this task.
To address these challenges, this dissertation develops advanced statistical models and
inference methods for spatiotemporal phenomena, with a particular focus on storm surge.
The central approach involves the use of random fields and Bayesian inference, which oNer
flexible and probabilistic frameworks for handling uncertainty in both data and model
predictions. Random fields provide smooth and adaptive representations of spatiotemporal
processes, while Bayesian inference allows for the integration of sparse data with prior
knowledge, producing comprehensive uncertainty estimates. Despite their advantages,
traditional Bayesian methods such as Markov Chain Monte Carlo (MCMC) can be
computationally prohibitive, particularly in large-scale spatiotemporal settings.
This dissertation introduces several key advancements to address these computational
challenges. For non-parametric random fields, the work adapts tools from Information Field
Theory, specifically the Diagrammatic Perturbation Technique (DPT). DPT organizes the
logarithm of the joint distribution into a Taylor series expansion, with each term represented
by a diagram, simplifying complex Bayesian calculations. This approach is demonstrated
with Gamma likelihoods and Gaussian priors, which are common in structural deterioration
problems. The dissertation shows that DPT significantly reduces computational costs
compared to MCMC, while maintaining accurate posterior estimates.
For parametric models, the dissertation extends DPT to oNer closed form estimates of
posterior moments, providing an eNicient alternative to MCMC, especially in large datasets
where the posterior deviates slightly from a Gaussian distribution. This method is applied to
spatiotemporal models, linear regression, and generalized linear models, demonstrating its
broad applicability.
Finally, the dissertation presents a novel modeling approach for storm surge estimation
within a Bayesian random field framework, leveraging DPT for eNicient posterior estimation.
This approach enables rapid, real-time predictions of storm surge, validated using historical
NOAA data from the Gulf Coast. The model’s ability to capture uncertainty while delivering
fast and accurate predictions oNers a powerful tool for disaster preparedness and risk
management in coastal regions.
