stranded

simulating determinantal processes

In the plane to Atlanta, I happened to read a paper called Efficient simulation of the Ginibre point process by Laurent Decreusefond, Ian Flint, and Anaïs Vergne (from Telecom Paristech). “Happened to” as it was a conjunction of getting tipped by my new Dauphine colleague (and fellow blogger!) Djalil Chaffaï about the paper, having downloaded it prior to departure, and being stuck in a plane (after watching the only Chinese [somewhat] fantasy movie onboard, Saving General Yang).

This is mostly a mathematics paper. While indeed a large chunk of it is concerned with the rigorous definition of this point process in an abstract space, the last part is about simulating such processes. They are called determinantal (and not detrimental as I was tempted to interpret on my first read!) because the density of an n-set (x₁, x₂,…,x_n) is given by a kind of generalised Vandermonde determinant

where T is defined in terms of an orthonormal family,

(The number n of points can be simulated via an a.s. finite Bernoulli process.) Because of this representation, the sequence of conditional densities for the x_i‘s (i.e. x₁, x₂ given x₁, etc.) can be found in closed form. In the special case of the Ginibre process, the ψ_i‘s are of the form

and the process cannot be simulated for it has infinite mass, hence an a.s. infinite number of points. Somehow surprisingly (as I thought this was the point of the paper), the authors then switch to a truncated version of the process that always has a fixed number N of points. And whose density has the closed form

It has an interestingly repulsive quality in that points cannot get close to one another. (It reminded me of the pinball sampler proposed by Kerrie Mengersen and myself at one of the Valencia meetings and not pursued since.) The conclusion (of this section) is anticlimactic, though,  in that it is known that this density also corresponds to the distribution of the eigenvalues of an Hermitian matrix with standardized complex Gaussian entries. The authors mentions that the fact that the support is the whole complex space Cⁿ is a difficulty, although I do not see why.

The following sections of the paper move to the Ginibre process restricted to a compact and then to the truncated Ginibre process restricted to a compact, for which the authors develop corresponding simulation algorithms. There is however a drag in that the sequence of conditionals, while available in closed-form, cannot be simulated efficiently but rely on a uniform accept-reject instead. While I am certainly missing most of the points in the paper, I wonder if a Gibbs sampler would not be an interesting alternative given that the full (last) conditional is a Gaussian density…

over Greenland [from the other side]

over Greenland

Winter workshop, Gainesville (day 1)

After a rather long flight, I arrived in Gainesville for this special Winter workshop. We indeed had to wait for hours in Paris to get defrosted and then the ride to Atlanta is terribly lengthy (esp. after I realised that all one’s files arXived for the trip were on my “other” computer… and that the book intended for the trip still stood under the X’ mas tree…!) I just managed to read a book for review, rewrite my slides and watch two movies, plus the last part of one I had started on my way back from India…

Anyway, here I am, back in Gainesville, a few years after my last visit, quite glad to meet again with old friends while terribly missing George Casella. The conference is actually dedicated to his memory. The schedule is well-done, once again giving speakers plenty of times and participants plenty of breaks, along with a superb conference room with tables and plugs. I listened to and enjoy all of them, but the one that did not overlap with the latest workshop at ICERM was Dawn Woodard’s, with a challenging data analysis problem about ambulance routes in Toronto. I clearly was not the only one finding this problem interesting and coming up with (mostly hair-brained!) alternatives. (Another apex of the day was to find a 2007 Beaune premier cru at a very reasonable price in a local store!)