169 research outputs found
Perturbed Datasets Methods for Hypothesis Testing and Structure of Corresponding Confidence Sets
Hypothesis testing methods that do not rely on exact distribution assumptions
have been emerging lately. The method of sign-perturbed sums (SPS) is capable
of characterizing confidence regions with exact confidence levels for linear
regression and linear dynamical systems parameter estimation problems if the
noise distribution is symmetric. This paper describes a general family of
hypothesis testing methods that have an exact user chosen confidence level
based on finite sample count and without relying on an assumed noise
distribution. It is shown that the SPS method belongs to this family and we
provide another hypothesis test for the case where the symmetry assumption is
replaced with exchangeability. In the case of linear regression problems it is
shown that the confidence regions are connected, bounded and possibly
non-convex sets in both cases. To highlight the importance of understanding the
structure of confidence regions corresponding to such hypothesis tests it is
shown that confidence sets for linear dynamical systems parameter estimates
generated using the SPS method can have non-connected parts, which have far
reaching consequences
Decoupling Multivariate Polynomials Using First-Order Information
We present a method to decompose a set of multivariate real polynomials into
linear combinations of univariate polynomials in linear forms of the input
variables. The method proceeds by collecting the first-order information of the
polynomials in a set of operating points, which is captured by the Jacobian
matrix evaluated at the operating points. The polyadic canonical decomposition
of the three-way tensor of Jacobian matrices directly returns the unknown
linear relations, as well as the necessary information to reconstruct the
univariate polynomials. The conditions under which this decoupling procedure
works are discussed, and the method is illustrated on several numerical
examples
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