# State filtering parameters and recursive estimation

### Tracking things that don’t move

a.k.a. recursive estimation, recursive estimation, state space model calibration, recursive identification. Possibly the same as, but differently framed to, online estimation.

State filters are cool for estimating time-varying hidden states. How about learning the non-time-varying 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.

According to LIFM12, here are some landmark papers:

Augmenting the unobserved state vector is a well known technique, used in the system identification community for decades, see e.g. Ljung (Ljun79); Söderström and Stoica (SöSt88); Lindström et al. (LSBW08). Similar ideas, using Sequential Monte Carlos methods, were suggested by Kitagawa (Kita98); Liu and West (LiWe01). Combined state and parameter estimation is also the standard technique for data assimilation in high-dimensional systems, see Moradkhani et al. (MSGH05); Evensen (Even09a, Even09b ).

However, introducing random walk dynamics to the parameters with fixed variance leads to a new dynamical stochastic system with properties that may be different from the properties of the original system. That implies that the variance of the random walk should be decreased, when the method is used for offline parameter estimation, cf. Hürzeler and Künsch (HüKü01).

TBD.

## Iterated filtering

Related: indirect inference. Precise relation will have to wait, since I currently do not care enough about indirect inference.

## Questions

• Ionides and King dominate my citations, at least for the frequentist stuff. Surely other people use this method too? But what are the keywords? This research is suspiciously concentrated in U Michigan, but the idea is not so esoteric. I think I am caught in a citation bubble.

Update: the oceanographic crew of Even03 etc seem to do this with Bayes a lot.

• a lot of the variational filtering literature turns out to be about attempting this with, effectively, neural nets.

• can I estimate regularisation this way, despite the lack of probabilistic interpretation? (leveraging Bayesian-prior parameter relations)

• How does this work with non-Markov systems? Do we need to bother, or can we just do the Hamiltonian trick and augment the state vector? Can we talk about mixing, or correlation decay? Should I then shoot for the new-wave mixing approaches of Kuznetsov and Mohri etc?

### Basic Construction

There are a few variations. We start with the basic continuous time state space model.

Here we have an unobserved Markov state process $$x(t)$$ on $$\mathcal{X}$$ and an observation process $$y(t)$$ on $$\mathcal{Y}$$. For now they will be assumed to be finite dimensional vectors over $$\mathbb{R}.$$ They will additionally depend upon a vector of parameters $$\theta$$ We observe the process at discrete times $$t(1:T)=(t_1, t_2,\dots, t_T),$$ and we will write the observations $$y(1:T)=(y(t_1), y(t_2),\dots, y(1_T)).$$

We presume our processes are completely specified by the following conditional densities (which might not have closed-form expression)

The transition density

$f(x(t_i)|x(t_{i-1}), \theta)$

The observation density…

TBC.

## Awaiting filing

Recently enjoyed: Sahani Pathiraja’s state filter does something cool, in attempting to identify process model noise – a conditional nonparametric density of process errors, that may be used to come up with some neat process models. I’m not convinced about her use of kernel density estimation, since these scale badly precisely when you need them most, in high dimension; but any nonparametric density estimator would, I assume, work, and that would be awesome.

## Implementations

pomp does state filtering inference in R.

For some example of doing this in Stan see Sinhrks’ stan-statespace.