Background Information for Continuous-time Filtering and Estimation on Manifolds
The preprint A Primer on Stochastic Differential Geometry in Signal Processing discusses, among other things, the following in simple but rigorous terms:
- How Brownian motion can be generated on Riemannian manifolds;
- How “coloured” (technically, left-invariant) Brownian motion can be generated on Lie groups;
- Ito and Stratonovich integrals, and the transfer principle of Stratonovich integrals making them convenient to use for stochastic differential equations on manifolds;
- The special orthogonal groups SO(n);
- How a “Gaussian random variable” can be generated on a Riemannian manifold;
- How state-space models extend to manifolds;
- How stochastic development provides a convenient framework for understanding stochastic processes on manifolds;
- Whether or not stochastic integrals are “pathwise” computable.
The last section of the paper includes the following:
Several concepts normally taken for granted, such as unbiasedness of an estimator, are not geometric concepts and hence raise the question of their correct generalisations to manifolds. The answer is that the difficulty lies not with manifolds, but with the absence of meaning to ask for an estimate of a parameter. The author believes firmly that asking for an estimate of a parameter is, a priori, a meaningless question. It has been given meaning by force of habit. An estimate only becomes useful once it is used to make a decision, serving as a proxy for the unknown true parameter value. Decisions include: the action taken by a pilot in response to estimates from the flight computer; an automated control action in response to feedback; and, what someone decides they hear over a mobile phone (with the pertinent question being whether the estimate produced by the phone of the transmitted message is intelligible). Without knowing the decision to be made, whether an estimator is good or bad is unanswerable. One could hope for an estimator that works well for a large class of decisions, and the author sees this as the context of estimation theory.