Thinning out redundant empirical data

Given a set $X$ of "empirical" points, whose coordinates are perturbed by errors, we analyze whether it contains redundant information, that is whether some of its elements could be represented by a single equivalent point. If this is the case, the empirical information associated to $X$ could be described by fewer points, chosen in a suitable way. We present two different methods to reduce the cardinality of $X$ which compute a new set of points equivalent to the original one, that is representing the same empirical information. Though our algorithms use some basic notions of Cluster Analysis they are specifically designed for "thinning out" redundant data. We include some experimental results which illustrate the practical effectiveness of our methods.Comment: 14 pages; 3 figure

Stable Border Bases for Ideals of Points

Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $\widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $\widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $I(\widetilde X)$.Comment: This is an update version of "Notes on stable Border Bases" and it is submitted to JSC. 16 pages, 0 figure

A singular value decomposition based approach to handle ill-conditioning in optimization problems with applications to portfolio theory.

We identify a source of numerical instability of quadratic programming problems that is hidden in its linear equality constraints. We propose a new theoretical approach to rewrite the original optimization problem in an equivalent reformulation using the singular value decomposition and substituting the ill-conditioned original matrix of the restrictions with a suitable optimally conditioned one. The proposed novel approach is showed, both empirically and theoretically, to solve ill-conditioning related numerical issues, not only when they depend on bad scaling and are relative easy to handle, but also when they result from almost collinearity or when numerically rank-deficient matrices are involved. Furthermore, our strategy looks very promising even when additional inequality constraints are considered in the optimization problem, as it occurs in several practical applications. In this framework, even if no closed form solution is available, we show, through empirical evidence, how the equivalent reformulation of the original problem greatly improves the performances of MatLab®’s quadratic programming solver and Gurobi®. The experimental validation is provided through numerical examples performed on real financial data in the portfolio optimization context

Cubature rules and expected value of some complex functions

The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for nite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined

On the belonging of a perturbed vector to a subspace from a numerical point of view. Submitted

We propose a criterion in order to establish, dealing with perturbed data, when a vector v belongs to a subspace W of a vector space U from a numerical point of view. The criterion formalizes the intuitive idea that, due to the errors that affect the knowledge of the vector and the subspace, we consider the exact vector lying in the exact subspace if a “sufficiently small perturbation ” of v belongs to a “sufficiently small perturbation ” of W. The criterion, obviously chosen as independent of the norm of v and of the basis of W, is computationally simple to use, but it requires the choice of a threshold. Suitable values of the threshold such that the criterion adheres to the previous intuitive idea are found both when we know a basis of the exact subspace and when an orthonormal basis of the perturbed subspace is given.