Cramer-Rao bound and Ricci flow II

18/05/2009

The paper I presented about this matter (see here) has been accepted by EuRad 2009 Conference. This will result in a publication in IEEE Proceedings. IEEE is the most important engineering society. I cannot made public this paper until it will appear in the proceedings. After this date you can read it at IEEE Xplore where you can find another paper of mine about scattering of electromagnetic waves by a rough surface (see here). As you can see, the way publishing is operated by engineers is quite different from that of physicists.

Anyhow, the idea is quite simple and use the fact that for a two-dimensional Riemann manifold one has always a conformal metric. Then, Fischer information matrix can be expressed in a diagonal form with new estimators that are always optimal with respect to Cramer-Rao bound. So, due to the fact that there exists a vast set of probability distributions with two parameters, the application areas of this result are huge. In my paper I make the case of sea clutter for radar applications but what I prove is a theorem in statistics and you can realize by yourself the importance.

The n-parameter case can also be made but here there are two more demanding requests: the existence of a conformal metric and the existence of a potential for a vector field that satisfies Liouville equation. These cannot always be satisfied and so the two-dimensional case appears a rather lucky one.

Cramer-Rao bound and Ricci flow

04/02/2009

Two dimensional Ricci flow is really easy to manage. In this case the equation takes a very simple form and a wealth of results can be extracted. As you know from my preceding posts, I have been able to prove in a rigorous way that in this case the Ricci flow arises from Brownian motion (see here). So, the equation for  Einstein manifolds in this case takes the very simple form, $R=\Lambda$ being $\Lambda$ a constant, that is also the equation for a Ricci soliton. This equation is rather well-knwon to physicists as is the equation of 2d Einstein gravity. This equation is nothing else than Liouville equation

$\Delta_2\phi+\Lambda e^{\phi}=0$

that admits an exact solution notwithstanding being non-linear. There is an unexpected application of all this machinery of Riemann geometry to the case of statistics. Statistics has a wide body of application fields as radar tracking, digital communications and so on. Then, any new result about can be translated into a wealthy number of applications.

The problem one meets in this case is that of parameter estimation of a given probability distribution. For a sample of measured data the question is to determine the best probability distribution with respect to the spread of the data themselves with a proper choice of the parameters. A known result in this area is the so called Cramer-Rao bound. This inequality gives limit for the optimality of the chosen estimators of the data entering into the distribution. The result I have found is that, for a probability distribution with two parameters, an infinite class of optimal estimators exists that are all efficient. These estimators are given by the solution of Liouville equation! The result can be extended to the n-dimensional case granted the existence of isothermal coordinates that are the conformal ones.

This result arises from the deep link between differential geometry and statistics that was put forward by Calayampudi Radhakrishna Rao. My personal interest in this matter was arisen working in radar tracking but one can think on a large number of other areas. I should say, as a final consideration, that the work of Hamilton and Perelman can have a deep impact in a large body of our knowledge. We are just at the beginning.