# Stochastic Local Interaction Model with Sparse Precision Matrix for Space-Time Interpolation

## TL;DR

This paper introduces a stochastic local interaction model for space-time data that uses sparse precision matrices to enable efficient computation and prediction without large covariance matrix inversion.

## Contribution

It develops a local dependence model with sparse precision matrices for space-time interpolation, reducing computational complexity compared to traditional Gaussian process methods.

## Key findings

- Sparse precision matrices enable efficient storage and computation.
- Model is equivalent to a Gaussian Markov Random Field on regular lattices.
- Explicit prediction and variance formulas are derived.

## Abstract

The application of geostatistical and machine learning methods based on Gaussian processes to big space-time data is beset by the requirement for storing and numerically inverting large and dense covariance matrices. Computationally efficient representations of space-time correlations can be constructed using local models of conditional dependence which can reduce the computational load. We formulate a stochastic local interaction model for regular and scattered space-time data that incorporates interactions within controlled space-time neighborhoods. The strength of the interaction and the size of the neighborhood are defined by means of kernel functions and adaptive local bandwidths. Compactly supported kernels lead to finite-size local neighborhoods and consequently to sparse precision matrices that admit explicit expression. Hence, the stochastic local interaction model's requirements for storage are modest and the costly covariance matrix inversion is not needed. We also derive a semi-explicit prediction equation and express the conditional variance of the prediction in terms of the diagonal of the precision matrix. For data on regular space-time lattices, the stochastic local interaction model is equivalent to a Gaussian Markov Random Field.