Is it possible to generate multi-step forecasts with the Kalman Filter (KF)?

Question

I wanted to double check with the community but I think I already have the answer to this question. Can the KF (or its variants such as the ensemble KF) produce meaningful multi-step forecasts?

My intuition is obviously no because the KF essentially provides a posterior gaussian distribution with mean and variance parameters and any future forecasts will just be the posterior mean for all future time steps.

Would immensely appreciate if someone could confirm this for me please.

Answer 1

It depends. The multi-step forward prediction of KF would be as imprecise as the current knowledge of it's state plus the error that would be accumulated from its evolution starting from now till the point in the future.

• If the hidden signal is highly correlated with itself, and the estimated variance of the current knowledge of the hidden state is small, the prediction can be very precise.
• If the hidden signal is loosely correlated with itself or if the estimated variance of the current knowledge of the hidden state is too high, the prediction would have too much variance and therefore often useless.

In both cases, you may compute the estimated variance of the prediction and compare with the unconditional variance of the hidden state at the future point to decide.