# Improving Bayesian Local Spatial Models in Large Data Sets

## TL;DR

This paper introduces a Bayesian three-step local modeling approach that enhances inference and prediction in large, non-stationary spatial data by effectively managing the bias-variance trade-off through smaller, more manageable regions.

## Contribution

A novel Bayesian three-step method enabling smaller regional models without increasing variance, improving inference and prediction in large spatial datasets.

## Key findings

- Improved inference accuracy demonstrated in simulated wind speed data.
- Effective propagation of uncertainty across modeling steps.
- Enhanced prediction performance over traditional methods.

## Abstract

Environmental processes resolved at a sufficiently small scale in space and time will inevitably display non-stationary behavior. Such processes are both challenging to model and computationally expensive when the data size is large. Instead of modeling the global non-stationarity explicitly, local models can be applied to disjoint regions of the domain. The choice of the size of these regions is dictated by a bias-variance trade-off; large regions will have smaller variance and larger bias, whereas small regions will have higher variance and smaller bias. From both the modeling and computational point of view, small regions are preferable to better accommodate the non-stationarity. However, in practice, large regions are necessary to control the variance. We propose a novel Bayesian three-step approach that allows for smaller regions without compromising the increase of the variance that would follow. We are able to propagate the uncertainty from one step to the next without issues caused by reusing the data. The improvement in inference also results in improved prediction, as our simulated example shows. We illustrate this new approach on a data set of simulated high-resolution wind speed data over Saudi Arabia.