Emerging Challenges in Spatial Statistics

Organizer: Hao Zhang, Professor of Statistics, Professor of Forestry and Natural Resources, Purdue University

Speakers

Aritra Halder, Assistant Professor of Biostatistics, Drexel University
Juan Du, Associate Professor and Graduate Certificate Coordinator, Department of Statistics, Kansas State University
Yulia Gel, Professor, Deparment of Mathematical Sciences, University of Texas at Dallas
Mikyoung Jun, Professor, Department of Mathematics, University of Houston

Speaker	Title
Aritra Halder	Bayesian Modeling with Spatial Curvature Processes
Abstract: Spatial process models are widely used for modeling point-referenced variables arising from diverse scientific domains. Analyzing the resulting random surface provides deeper insights into the nature of latent dependence within the studied response. We develop Bayesian modeling and inference for rapid changes on the response surface to assess directional curvature along a given trajectory. Such trajectories or curves of rapid change, often referred to as wombling boundaries, occur in geographic space in the form of rivers in a flood plain, roads, mountains or plateaus or other topographic features leading to high gradients on the response surface. We demonstrate fully model based Bayesian inference on directional curvature processes to analyze differential behavior in responses along wombling boundaries. We illustrate our methodology with a number of simulated experiments followed by multiple applications featuring the Boston Housing data; Meuse river data; and temperature data from the Northeastern United States.
Juan Du	Covariance Modeling of Some Data Sets In Continuous Space and Discrete Time
Abstract: Space- time data are often multivariate and collected at monitored discrete time lags, which are usually viewed as a component of time series. Valid and practical covariance models are needed to characterize the complex dependence structure of these types of data sets in various disciplines, such as environmental science, climatology, and agriculture. We propose several classes of univariate and multivariate spatio-temporal functions whose discrete temporal margins are some celebrated autoregressive and moving average (ARMA) models, and obtain necessary and/or sufficient conditions for them to be valid covariance (matrix) functions. This model specification enables us to take advantage of well-established time series and spatial statistics tools, which make the model identification relatively straightforward in practice. The applications of proposed univariate and multivariate covariance (matrix) models are illustrated using Kansas weather data in terms of (co-)kriging, compared with some traditional space-time models for prediction. In addition, a simple approach for simulating multivariate spatio-temporal data is explored based on the proposed models.
Yulia Gel	Coupling Time-Aware Multipersistence Knowledge Representation with Graph Convolutional Networks for Time Series Forecasting
Abstract: Graph Neural Networks (GNNs) are proven to be a powerful machinery for learning complex dependencies in multivariate spatio-temporal processes. However, most existing GNNs have inherently static architectures, and as a result, do not explicitly account for time dependencies of the encoded knowledge and are limited in their ability to simultaneously infer latent time-conditioned relations among entities. We postulate that such hidden time-conditioned properties may be captured by the tools of multipersistence, i.e., an emerging machinery in topological data analysis which allows us to quantify dynamics of the data shape along multiple geometric dimensions. We propose to summarize inherent time-conditioned topological properties of the data as time-aware multipersistence Euler-Poincar\'e surface and prove its stability. We then construct a supragraph convolution module which simultaneously accounts for the extracted intra- and inter-dependencies in the data. We illustrate the utility of the proposed approach in application to forecasting highway traffic flow, blockchain Ethereum token prices, and COVID-19 hospitalizations.
Mikyoung Jun	3D Bivariate Spatial Modelling of Argo Ocean Temperature and Salinity Profiles
Abstract: Variables contained within the global oceans can detect and reveal the effects of the warming climate as the oceans absorb huge amounts of solar energy. Hence, information regarding the joint spatial distribution of ocean variables is critical for climate monitoring. In this paper, we investigate the spatial correlation structure between ocean temperature and salinity using data harvested from the Argo program and construct a model to capture their bivariate spatial dependence from the surface to the ocean's interior. We develop a flexible class of multivariate nonstationary covariance models defined in 3-dimensional (3D) space (longitude x latitude x depth) that allows for the variances and correlation to change along the vertical pressure dimension. These models are able to describe the joint spatial distribution of the two variables while incorporating the underlying vertical structure of the ocean. We demonstrate that proposed cross-covariance models describe the complex vertical cross-covariance structure well, while existing cross-covariance models including bivariate Matérn models poorly fit empirical cross-covariance structure. Furthermore, the results show that using one more variable significantly enhances the prediction of the other variable and that the estimated spatial dependence structures are consistent with the ocean stratification. This is joint work with Mary Lai Salvaña.