Reduced Order Nonlinear Filters for Multi-Scale Systems with Correlated Sensor Noise

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

“…Finally, the first term in (14b) is built after the Gubinelli derivative from (12). Crucially, the scheme (14) takes the lifted component ∆Y k (representing iterated integrals of the path (Yt) t≥0 ) as an input. This dependence allows our methodology to appropriately take into account multiscale structure and other information encoded in the signature, but also necessitates appropriate ways to estimate Y k from data.…”

“…Itô, Stratonovich and rough integrals. Before concluding this introduction, let us comment on the difference between the RP-EnKF scheme (14) and the more conventional EnKF updates…”

“…• We suggest the RP-EnKF scheme (14) and in particular devise a subsampling based method to estimate the lift components Y k from data. The robustness of the RP-EnKF (and the nonrobustness of the EnKF) is demonstrated using numerical examples in the context of combined state-parameter estimation.…”

“…We would also like to mention the works [38,57] that allow treating the Zakai SPDE (governing the unnormalised filtering distribution [10, Section 3.5]) in a rough paths framework, however noticing that the numerical treatment of SPDEs is faced with enormous challenges, in particular in high-dimensional settings. Some other works addressing issues in robust or multiscale filtering include [3,4] (assuming uncertainty in the coefficients) as well as a sequence of works by N. Perkowski and coworkers in the context of averaging and homogenisation [14,12,13,59,60,73,72,101,102].…”

“…As mentioned in Section 2.3, ideas related to subsampling have been considered extensively in the context of multiscale parameter estimation (without observational noise), see, for instance, [1,5,6,51,52,61,63,64,83,88]. The method proposed in this section is different in that we use the subsampled paths in order to estimate the Lévy area, but otherwise input the original data path into the RP-EnKF dynamics (14). Moreover, the motivations are distinct: While subsampling in the aforementioned works is used in order to eliminate small-scale fluctuations, our method is specifically designed to estimate Lévy area terms.…”