
Accession Number : ADA184198
Title : Resolution of a RankDeficient Adjustment Model Via an Isomorphic Geometrical Setup with Tensor Structure.
Descriptive Note : Final rept. 21 Feb 863 Mar 87,
Corporate Author : NOVA UNIV OCEANOGRAPHIC CENTER DANIA FL
Personal Author(s) : Blaha,Georges
PDF Url : ADA184198
Report Date : Mar 1987
Pagination or Media Count : 99
Abstract : This study develops the rank deficient adjustment theory in a geometrical manner. In accordance with most leastsquares (L.S.) applications, the adjustment model is considered linear or linearized. The fundamental building blocks consist of orthonormal vectors spanning the spaces and surfaces linked to the L.S. setup. From this setup to the desired results including the variancecovariance matrices, the standard adjustment quantities can be represented by first and second order tensors. It is thus possible to express them in terms of the components of the above vectors, allowing for an easy and clearcut geometrical interpretation of the L.S. process. By virtue of such a vectorization, the propagation of the contravariant and covariant metric tensors is shown to fit perfectly the variance covariance propagation law and even to establish a weight propagation law. The opportunity to obtain variancecovariances and weights as a coherent part of the geometrical development provides the motivation for using tensor structure in the analysis of various L.S. methods and their properties. An algorithm furnishing the pseudoinverse of a positive semidefinite matrix, which could be useful for its straightforward geometrical interpretation as well as for its computational efficiency, is developed as a byproduct of this analysis. The Choleski algorithm for positivedefinite as well as positive semidefinite matrices is interpreted in terms of orthonormal vector components.
Descriptors : *TENSOR ANALYSIS, *MATRIX THEORY, *ANALYSIS OF VARIANCE, COVARIANCE, LEAST SQUARES METHOD, WEIGHTING FUNCTIONS, ERROR ANALYSIS, ALGEBRAIC GEOMETRY
Subject Categories : Statistics and Probability
Theoretical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE