Combining a linear Kalman Filter with additional linear constraints?

Question

This question contains a relatively long prelude, since I want to explain as clearly as possible the motivation for the question. It may well be the case that I am asking the wrong question (i.e. there is a better way of solving the problem than what I am attempting) and I would welcome any good answers along those lines.

I have implemented a real time map for a physical quantity over a large region (a continent) that combines many hundreds of data points and uses a Kalman Filter to jointly estimate the coefficients of a linear combination of basis functions (the map) plus a large number of instrumental biases and other nuisance parameters.

A problem with this approach is that the data points are distributed in a very uneven way; some regions have a lot of closely spaced data points and other regions are much more sparse. Since the map is represented by a global set of basis functions, there is no real happy medium in choosing the best order of expansion for the generating function. I either have a too high order for the sparse regions leading to spurious artefacts, or too low order for the data rich regions, smoothing out real structure that is supported by the data.

In order to be more adaptive to local data density, I am thinking of using some kind of radial basis function (RBF) approach utilising unsupervised clustering (e.g. K-means) to distribute a set number of basis function origins to match the local data density. I still need to use some kind of filtering to estimate the nuisance parameters along with the map, but this RBF approach maintains the linearity of the system so the Kalman Filter can be used. I'd 'just' need to experiment with different forms of the basis functions and their parameters, the number of basis functions to use etc.

However, it would be ideal to also incorporate a smoothness constraint, as in the case of a thin plate spline (TPS). This can be expressed as an additional set of linear constraints on the map values at each basis origin and the standard TPS formulation has a closed form solution. What I would like to do is incorporate these linear constraints into the filtering mechanism, as is required due to the noisiness of some parts of the system, rather than the one off minimisation of the TPS. I can vaguely conceive of how this might be done, but my linear algebra skills are not up to the task of fully deriving this (or even demonstrating that it can be done).

So my question is whether this approach is feasible and if so what the form of the resulting filter looks like. Is there a specific piece of terminology that succinctly describes what I am trying to do? Is there a standard result out there for this situation?

Answer 1

Among other spatio-temporal techniques, the Kriged Kalman filter can be a solution. The spatial smoothing captures the spatial low frequencies thanks to the Principal Kriging Functions and seems comparable to that of thin plate splines. RBF are used if the covariance kernel is square exponential.

See

Sujit K. Sahu and Kanti V. Mardia A Bayesian kriged Kalman model for short-term forecasting of air pollution levels Applied Statistics 2005, vol. 54 pp. 223–244.

as well as later papers by the authors on spatio-temporal modeling.

Answer 2

Kalman filtering with constraints on the elements of the state vector has been dealt with in the literature and is easy to implement. A few references follow:

Chia, T.L. and Simon, D. and Chizeck, H.J (2006) Kalman Filtering With
Statistical State Constraints, Control and Intelligent Systems, vol. 34(1), doi 10.2316/Journal.201.2006.1.201-1556.

Simon, D. (2002) Kalman filtering with state equality constraints, IEEE Transactions on Aerospace and Electronic Systems, vol. 38(1), p. 128--136. doi 10.1109/7.993234.

Simon, D. (2010) Kalman filtering with state constraints: a survey of linear and nonlinear algorithms, IET Control Theory & Applications, vol. 4(8), p. 1303. doi 10.1049/iet-cta.2009.0032

Also the book by Simon, D. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, Wiley, 2006 contains a summary on the topic.