Gaussian Covariance Matrix Research Articles

Completely Operator Semi-Selfdecomposable Distributions

The class $L_\infty(b, Q)$ of completely operator semi-selfdecomposable distributions on $\mathbf{R}^d$ for $b$ and $Q$ is studied. Here $0<b<1$ and $Q$ is a $d\times d$ matrix whose eigenvalues have positive real parts. This is the limiting class of the decreasing sequence of classes $L_m(b, Q)$, $m=-1,0,1,\ldots$, where $L_{-1}(b, Q)$ is the class of all infinitely divisible distributions on $\mathbf{R}^d$ and $L_m(b,Q)$ is defined inductively as the class of distributions $\mu$ with characteristic function $\hat{\mu}(z)$ satisfying $\hat{\mu}(z)=\hat{\mu}(b^{Q'}z)\hat{\rho}(z)$ for some $\rho\in L_{m-1}(b,Q)$. $Q'$ is the transpose of $Q$. Distributions in $L_\infty(b,Q)$ are characterized in terms of Gaussian covariance matrices and Lévy measures. The connection with the class $OSS(b,Q)$ of operator semi-stable distributions on $\mathbf{R}^{d}$ for $b$ and $Q$ is established.

The Asteroid Identification Problem

A large fraction of the asteroids observed so far are to be considered lost, that is they cannot be recovered by pointing the telescope at the predicted position only. The catalogues of asteroid orbits are therefore polluted by large numbers of low accuracy orbits, which cannot be easily improved by observation. Two of these inaccurate orbits can belong to the same physical object, and indeed identifications are regularly found when the orbital elements are close; an unknown number of real identifications have not been found in the existing catalogues because the orbital elements as computed are far apart. It is impossible to systematically check for possible identification of all the couples of orbital element sets; an algorithm is required to single out a comparatively small number of proposed identifications, and to find a suitable first guess within the region of convergence of the differential corrections procedure to be used to confirm the identification.In the linear approximation the target function to be minimized is a quadratic form; thus a fully analytic algorithm can be given to compute both the first guess for an identification and the cost, that is the increase in the target function resulting from joining the two sets of observations. This allows to define a number of distances in the elements space, which are functions, not only of the two orbital element sets, but also of their uncertainties, as measured by the Gaussian covariance matrices. Tests on the values of these distances are used as preliminary filters, to allow more extensive computations to be performed on a fraction of the total number of possible couples.

A note on the iterative application of Bayes' rule

Alterations in probability densities produced by iterative application of Bayes' rule are analyzed. Computational requirements for finding a sequence of a posteriori densities remain reasonable despite a growing sample size if and only if a sufficient statistic expressible as a vector of fixed dimension exists. The existence of such a sufficient statistic also insures existence of reproducing a priori probability densities (a priori probability densities insuring that the a posteriori densities are in the same family). The theory is applied to find a class of sequential Bayes estimators for a Gaussian covariance matrix, and to treat a variety of adaptive "Bayesian learning" schemes in a unified manner.