Feedback system identification, not necessarily linear

The order in which this is presented right now makes no sense.

If I have a system whose future evolution is important to predict, why not try to infer a plausible model instead of a convenient linear one?

To reconstruct the unobserved state, as opposed to the parameters of the process acting upon the state, we do state filtering. There can be interplay between these steps if we are doing simulation-based online parameter inference, as in recursive estimation (what is the division between this and that?) Or: we might decide the state is unimportant and attempt to estimate the evolution only of the observations. That is the Koopman operator trick.

A compact overview is inserted incidentally in Cosma’s review of Fan and Yao (2003) where he also recommends (Bosq and Blanke 2007; Bosq 1998; Taniguchi and Kakizawa 2000).

There are many methods. From an engineering/control perspective, we have (Brunton, Proctor, and Kutz 2016), which generalises the process for linear time series to a sparse regression version via Indirect inference, or recursive hierarchical generalised linear models, which is an obvious way to generalize linear systems in the same way GLM generalizes linear models. Kitagawa and Gersch (1996) is popular in a Bayes context.

Hefny, Downey, and Gordon (2015):

We address […] these problems with a new view of predictive state methods for dynamical system learning. In this view, a dynamical system learning problem is reduced to a sequence of supervised learning problems. So, we can directly apply the rich literature on supervised learning methods to incorporate many types of prior knowledge about problem structure. We give a general convergence rate analysis that allows a high degree of flexibility in designing estimators. And finally, implementing a new estimator becomes as simple as rearranging our data and calling the appropriate supervised learning subroutines.

[…] More specifically, our contribution is to show that we can use much-more-general supervised learning algorithms in place of linear regression, and still get a meaningful theoretical analysis. In more detail:

• we point out that we can equally well use any well-behaved supervised learning algorithm in place of linear regression in the first stage of instrumental-variable regression;

• for the second stage of instrumental-variable regression, we generalize ordinary linear regression to its RKHS counterpart;

• we analyze the resulting combination, and show that we get convergence to the correct answer, with a rate that depends on how quickly the individual supervised learners converge

State filters are cool for estimating time-varying hidden states given known fixed system parameters. How about learning those parameters of the model generating your states? Classic ways that you can do this in dynamical systems include basic linear system identification, and general system identification. But can you identify the fixed parameters (not just hidden states) with a state filter?

Yes. This is called recursive estimation.