# Prediction in regression models with continuous observations

## TL;DR

This paper derives optimal linear predictors and their mean squared errors for continuous observation regression models with correlated errors, extending kriging to continuous data and derivatives, with applications to Matérn kernels.

## Contribution

It provides explicit formulas for best linear unbiased predictors in continuous observation settings, including derivatives, expanding kriging theory.

## Key findings

- Derived explicit predictor formulas for continuous observations.
- Extended kriging to include derivatives of the process.
- Illustrated results with Matérn 3/2 kernel examples.

## Abstract

We consider the problem of predicting values of a random process or field satisfying a linear model $y(x)=\theta^\top f(x) + \varepsilon(x)$, where errors $\varepsilon(x)$ are correlated. This is a common problem in kriging, where the case of discrete observations is standard. By focussing on the case of continuous observations, we derive expressions for the best linear unbiased predictors and their mean squared error. Our results are also applicable in the case where the derivatives of the process $y$ are available, and either a response or one of its derivatives need to be predicted. The theoretical results are illustrated by several examples in particular for the popular Mat\'{e}rn $3/2$ kernel.