# Multi-Scale Process Modelling and Distributed Computation for Spatial Data

## TL;DR

This paper introduces a scalable, multi-scale spatial modeling approach using Gaussian Markov random fields and parallel MCMC, enabling efficient analysis of large, complex, nonstationary spatial datasets like sea-surface temperature.

## Contribution

It presents a novel multi-scale process model that combines nonstationarity and distributed computation for large spatial data, advancing beyond existing global models.

## Key findings

- Successfully modeled sea-surface temperature with ~1 million observations.
- Demonstrated improved scalability and flexibility over state-of-the-art methods.
- Enabled distributed computing and data sharing for complex spatial models.

## Abstract

Recent years have seen a huge development in spatial modelling and prediction methodology, driven by the increased availability of remote-sensing data and the reduced cost of distributed-processing technology. It is well known that modelling and prediction using infinite-dimensional process models is not possible with large data sets, and that both approximate models and, often, approximate-inference methods, are needed. The problem of fitting simple global spatial models to large data sets has been solved through the likes of multi-resolution approximations and nearest-neighbour techniques. Here we tackle the next challenge, that of fitting complex, nonstationary, multi-scale models to large data sets. We propose doing this through the use of superpositions of spatial processes with increasing spatial scale and increasing degrees of nonstationarity. Computation is facilitated through the use of Gaussian Markov random fields and parallel Markov chain Monte Carlo based on graph colouring. The resulting model allows for both distributed computing and distributed data. Importantly, it provides opportunities for genuine model and data scaleability and yet is still able to borrow strength across large spatial scales. We illustrate a two-scale version on a data set of sea-surface temperature containing on the order of one million observations, and compare our approach to state-of-the-art spatial modelling and prediction methods.