Regression‐based covariance functions for nonstationary spatial modeling

Abstract

In many environmental applications involving spatially-referenced data, limitations on the number and locations of observations motivate the need for practical and efficient models for spatial interpolation, or kriging. A key component of models for continuously-indexed spatial data is the covariance function, which is traditionally assumed to belong to a parametric class of stationary models. However, stationarity is rarely a realistic assumption. Alternative methods which more appropriately model the nonstationarity present in environmental processes often involve high-dimensional parameter spaces, which lead to difficulties in model fitting and interpretability. To overcome this issue, we build on the growing literature of covariate-driven nonstationary spatial modeling. Using process convolution techniques, we propose a Bayesian model for continuously-indexed spatial data based on a flexible parametric covariance regression structure for a convolution-kernel covariance matrix. The resulting model is a parsimonious representation of the kernel process, and we explore properties of the implied model, including a description of the resulting nonstationary covariance function and the interpretational benefits in the kernel parameters. Furthermore, we demonstrate that our model provides a practical compromise between stationary and highly parameterized nonstationary spatial covariance functions that do not perform well in practice. We illustrate our approach through an analysis of annual precipitation data

Modeling Tangential Vector Fields on a Sphere

“…Nonetheless, we can extend the model using an anisotropic underlying potential field, such as non-stationary bivariate Matérn models with spatially varying parameters (Kleiber and Nychka, 2012;Jun, 2014). The spatially varying parameters can be modeled by parametric functions of spherical harmonics (Bolin and Lindgren, 2011) or covariates observed together with the vector field (Risser and Calder, 2015). This remains a topic of future research.…”

A generalized convolution model and estimation for non-stationary random functions

“…This can be achieved by setting the variance and anisotropy functions for covariates. This can be done by following the work of Neto et al [23] and Risser [24]. To date, the non-stationary modelling convolution approach does not provide closed-form non-stationary covariance functions with compact support, this remains an open problem.…”

Second-order non-stationary modeling approaches for univariate geostatistical data

“…(21), (22), and (23). Calder (2008), Neto et al (2014), and Risser and Calder (2015) show how covariates can be incorporated to the convolution modeling approach. The approach is applied to ambient particulate matter concentration data, ozone data, and precipitation data.…”