Likelihood Analysis of Power Spectra and Generalized Moment Problems

We develop an approach to spectral estimation that has been advocated by Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance extension problem, by Enqvist and Karlsson. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Georgiou and Lindquist and can also be viewed as a generalization of the classical work by Burg and Jaynes on the maximum entropy method. In the present paper we present a new fast algorithm in the general case (i.e., for general Gaussian priors) and show that for priors with a specific structure the solution can be given in closed form.Comment: 17 pages, 4 figure

Time and spectral domain relative entropy: A new approach to multivariate spectral estimation

The concept of spectral relative entropy rate is introduced for jointly stationary Gaussian processes. Using classical information-theoretic results, we establish a remarkable connection between time and spectral domain relative entropy rates. This naturally leads to a new spectral estimation technique where a multivariate version of the Itakura-Saito distance is employed}. It may be viewed as an extension of the approach, called THREE, introduced by Byrnes, Georgiou and Lindquist in 2000 which, in turn, followed in the footsteps of the Burg-Jaynes Maximum Entropy Method. Spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate. The corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework. Indeed, it is equal to the one featured by THREE in the scalar case. The solution is computed via a globally convergent matricial Newton-type algorithm. Simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks, especially in the case of short data records.Comment: 32 pages, submitted for publicatio

The status and programs of the New Relativity Theory

A review of the most recent results of the New Relativity Theory is presented. These include a straightforward derivation of the Black Hole Entropy-Area relation and its $logarithmic$ corrections; the derivation of the string uncertainty relations and generalizations ; ; the relation between the four dimensional gravitational conformal anomaly and the fine structure constant; the role of Noncommutative Geometry, Negative Probabilities and Cantorian-Fractal spacetime in the Young's two-slit experiment. We then generalize the recent construction of the Quenched-Minisuperspace bosonic $p$-brane propagator in $D$ dimensions ($AACS$ [18]) to the full multidimensional case involving all $p$-branes : the construction of the Multidimensional-Particle propagator in Clifford spaces ($C$-spaces) associated with a nested family of $p$-loop histories living in a target $D$-dim background spacetime . We show how the effective $C$-space geometry is related to $extrinsic$ curvature of ordinary spacetime. The motion of rigid particles/branes is studied to explain the natural $emergence$ of classical spin. The relation among $C$-space geometry and ${\cal W}$, Finsler Geometry and (Braided) Quantum Groups is discussed. Some final remarks about the Riemannian long distance limit of $C$-space geometry are made.Comment: Tex file, 21 page