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When modelling time series, it is common to decompose observed variation intoa "signal" process, the process of interest, and "noise", representing nuisancefactors that obfuscate the signal. To separate signal from noise, assumptionsmust be made about both parts of the system. If the signal process isincorrectly specified, our predictions using this model may generalise poorly;similarly, if the noise process is incorrectly specified, we can attribute toomuch or too little observed variation to the signal. With little justification,independent Gaussian noise is typically chosen, which defines a statisticalmodel that is simple to implement but often misstates system uncertainty andmay underestimate error autocorrelation. There are a range of alternative noiseprocesses available but, in practice, none of these may be entirelyappropriate, as actual noise may be better characterised as a time-varyingmixture of these various types. Here, we consider systems where the signal ismodelled with ordinary differential equations and present classes of flexiblenoise processes that adapt to a system's characteristics. Our noise modelsinclude a multivariate normal kernel where Gaussian processes allow fornon-stationary persistence and variance, and nonparametric Bayesian models thatpartition time series into distinct blocks of separate noise structures. Acrossthe scenarios we consider, these noise processes faithfully reproduce truesystem uncertainty: that is, parameter estimate uncertainty when doinginference using the correct noise model. The models themselves and the methodsfor fitting them are scalable to large datasets and could help to ensure moreappropriate quantification of uncertainty in a host of time series models.

Using flexible noise models to avoid noise model misspecification in inference of differential equation time series models