Stochastic partial differential equation based modelling of large space-time data sets

Prediction and model evaluation for space-time data

Evaluation metrics for prediction error, model selection and model averaging on space-time data are understudied and poorly understood. The absence of independent replication makes prediction ambiguous as a concept and renders evaluation procedures developed for independent data inappropriate for most space-time prediction problems. Motivated by air pollution data collected during California wildfires in 2008, this manuscript attempts a formalization of the true prediction error associated with spatial interpolation. We investigate a variety of cross-validation (CV) procedures employing both simulations and case studies to provide insight into the nature of the estimand targeted by alternative data partition strategies. Consistent with recent best practice, we find that location-based cross-validation is appropriate for estimating spatial interpolation error as in our analysis of the California wildfire data. Interestingly, commonly held notions of bias-variance trade-off of CV fold size do not trivially apply to dependent data, and we recommend leave-one-location-out (LOLO) CV as the preferred prediction error metric for spatial interpolation.

Spatiotemporal covariance functions for Laplacian ARMA fields in higher dimensions

This paper presents clear formulae of the covariance functions of Laplacian ARMA fields in terms of coefficients and Bessel functions in higher spatial dimensions. Spectral methods are used for the study of spatiotemporal Laplacian ARMA fields in Euclidean spaces and spheres therein with dimension d ≥ 2 d\geq 2 .

A Mechanistic Model of Annual Sulfate Concentrations in the United States

Understanding how individual pollution sources contribute to ambient sulfate pollution is critical for assessing past and future air quality regulations. Since attribution to specific sources is typically not encoded in spatial air pollution data, we develop a mechanistic model which we use to estimate, with uncertainty, the contribution of ambient sulfate concentrations attributable specifically to sulfur dioxide (SO₂) emissions from individual coal-fired power plants in the central United States. We propose a multivariate Ornstein-Uhlenbeck (OU) process approximation to the dynamics of the underlying space-time chemical transport process, and its distributional properties are leveraged to specify novel probability models for spatial data that are viewed as either a snapshot or time-averaged observation of the OU process. Using US EPA SO₂ emissions data from 193 power plants and state-of-the-art estimates of ground-level annual mean sulfate concentrations, we estimate that in 2011 - a time of active power plant regulatory action - existing flue-gas desulfurization (FGD) technologies at 66 power plants reduced population-weighted exposure to ambient sulfate by 1.97 μg/m³ (95% CI: 1.80 - 2.15). Furthermore, we anticipate future regulatory benefits by estimating that installing FGD technologies at the five largest SO₂-emitting facilities would reduce human exposure to ambient sulfate by an additional 0.45 μg/m³ (95% CI: 0.33 - 0.54).

Smoothing spatio-temporal data with complex missing data patterns

We consider spatio-temporal data and functional data with spatial dependence, characterized by complicated missing data patterns. We propose a new method capable to efficiently handle these data structures, including the case where data are missing over large portions of the spatio-temporal domain. The method is based on regression with partial differential equation regularization. The proposed model can accurately deal with data scattered over domains with irregular shapes and can accurately estimate fields exhibiting complicated local features. We demonstrate the consistency and asymptotic normality of the estimators. Moreover, we illustrate the good performances of the method in simulations studies, considering different missing data scenarios, from sparse data to more challenging scenarios where the data are missing over large portions of the spatial and temporal domains and the missing data are clustered in space and/or in time. The proposed method is compared to competing techniques, considering predictive accuracy and uncertainty quantification measures. Finally, we show an application to the analysis of lake surface water temperature data, that further illustrates the ability of the method to handle data featuring complicated patterns of missingness and highlights its potentiality for environmental studies.

Gradient Boosted Trees for Spatial Data and Its Application to Medical Imaging Data

Boosting Trees are one of the most successful statistical learning approaches that involve sequentially growing an ensemble of simple regression trees ("weak learners"). This paper proposes a gradient Boosted Trees algorithm for Spatial Data (Boost-S) with covariate information. Boost-S integrates the spatial correlation into the classical framework of eXtreme Gradient Boosting. Each tree is constructed by solving a regularized optimization problem, where the objective function takes into account the underlying spatial correlation and involves two penalty terms on tree complexity. A computationally-efficient greedy heuristic algorithm is proposed to obtain an ensemble of trees. The proposed Boost-S is applied to the spatially-correlated FDG-PET (fluorodeoxyglucose-positron emission tomography) imaging data collected from clinical trials of cancer chemoradiotherapy. Our numerical investigations successfully demonstrate the advantages of the proposed Boost-S over existing approaches for this particular application.

Progression of COVID-19 From Urban to Rural Areas in the United States: A Spatiotemporal Analysis of Prevalence Rates

Purpose

There are growing signs that the COVID‐19 virus has started to spread to rural areas and can impact the rural health care system that is already stretched and lacks resources. To aid in the legislative decision process and proper channelizing of resources, we estimated and compared the county‐level change in prevalence rates of COVID‐19 by rural‐urban status over 3 weeks. Additionally, we identified hotspots based on estimated prevalence rates.

Methods

We used crowdsourced data on COVID‐19 and linked them to county‐level demographics, smoking rates, and chronic diseases. We fitted a Bayesian hierarchical spatiotemporal model using the Markov Chain Monte Carlo algorithm in R‐studio. We mapped the estimated prevalence rates using ArcGIS 10.8, and identified hotspots using Gettis‐Ord local statistics.

Findings

In the rural counties, the mean prevalence of COVID‐19 increased from 3.6 per 100,000 population to 43.6 per 100,000 within 3 weeks from April 3 to April 22, 2020. In the urban counties, the median prevalence of COVID‐19 increased from 10.1 per 100,000 population to 107.6 per 100,000 within the same period. The COVID‐19 adjusted prevalence rates in rural counties were substantially elevated in counties with higher black populations, smoking rates, and obesity rates. Counties with high rates of people aged 25‐49 years had increased COVID‐19 prevalence rates.

Conclusions

Our findings show a rapid spread of COVID‐19 across urban and rural areas in 21 days. Studies based on quality data are needed to explain further the role of social determinants of health on COVID‐19 prevalence.