Sensitivity And Out-Of-Sample Error in Continuous Time Data Assimilation

TL;DR

This paper investigates how the sensitivity of data assimilation methods affects out-of-sample error, proposing a criterion to optimize the trade-off between model adherence and observation fitting, with practical numerical demonstrations.

Contribution

It establishes a relation between sensitivity and out-of-sample error in continuous-time data assimilation, introducing a criterion for optimal sensitivity setting.

Findings

Sensitivity controls the trade-off in data assimilation.

A relation between sensitivity and out-of-sample error is derived.

Numerical examples validate the proposed approach.

Abstract

Data assimilation refers to the problem of finding trajectories of a prescribed dynamical model in such a way that the output of the model (usually some function of the model states) follows a given time series of observations. Typically though, these two requirements cannot both be met at the same time--tracking the observations is not possible without the trajectory deviating from the proposed model equations, while adherence to the model requires deviations from the observations. Thus, data assimilation faces a trade-off. In this contribution, the sensitivity of the data assimilation with respect to perturbations in the observations is identified as the parameter which controls the trade-off. A relation between the sensitivity and the out-of-sample error is established which allows to calculate the latter under operational conditions. A minimum out-of-sample error is proposed as a criterion to set an appropriate sensitivity and to settle the discussed trade-off. Two approaches to data assimilation are considered, namely variational data assimilation and Newtonian nudging, aka synchronisation. Numerical examples demonstrate the feasibility of the approach.