Solved – Kalman filter parameter estimation

Question

From what I've known about Kalman filter, it requires all the parameters of the underlying state space model. Say the state space model is:
$$\xi_{t+1} = F\xi_t + v_{t+1}$$
$$y_t = H\xi_t + w_{t}$$
where $v$ and $w$ are disturbances.
Kalman filter needs the $F$, $H$, $Q$ (the covariance matrix of $v$) and $R$ (the covariance matrix of $w$) as well as $\xi_1$ as the initial state and the corresponding $P_1$ (the mean squared error of $\xi_1$) to start the recursion.
However, these parameters generally have to be estimated by numerical methods. Assume $y_t$ is Gaussian, how can the $H$ be estimated if we don't have any external knowledge about $\xi_t$? Alternatively, if we have some external knowledge of $H$, the estimation of all the parameters can be proceeded as well. What is the general rule to determine the initial parameter values or the state vector if we don't have any external knowledge?

Answer 1

Everything you will ever need regarding estimation of parameters in a state space model is in this document:

https://cran.r-project.org/web/packages/MARSS/vignettes/EMDerivation.pdf

Kalman Filter/Smoother assumes that the parameters are known in advance so that the unobserved state can be estimated. The initial values for the state can be user supplied or a diffuse initialization approach can be used.

For this the standard reference is:

https://www.amazon.com/Time-Analysis-State-Space-Methods/dp/019964117X

For the matrix parameters and noise co-variance estimation the standard procedure is Expectation Maximization which is described in detail in the first reference, and also in

https://www.stat.pitt.edu/stoffer/tsa4/tsa4.pdf

chapter 6.

Archived version of tsa4.pdf (Time Series Analysis and Its Applications): https://web.archive.org/web/20210401070804/https://www.stat.pitt.edu/stoffer/tsa4/tsa4.pdf