TY - JOUR
N2 - The relationship between internal response-based reliability and conditionality is investigated for Gauss-Markov (GM) models with uncorrelated observations. The models with design matrices of full rank and of incomplete rank are taken into consideration. The formulas based on the Singular Value Decomposition (SVD) of the design matrix are derived which clearly indicate that the investigated concepts are independent of each other. The methods are presented of constructing for a given design matrix the matrices equivalent with respect to internal response-based reliability as well as the matrices equivalent with respect to conditionality. To analyze conditionality of GM models, in general being inconsistent systems, a substitute for condition number commonly used in numerical linear algebra is developed, called a pseudo-condition^number. Also on the basis of the SVD a formula for external reliability is proposed, being the 2-norm of a vector of parameter distortions induced by minimal detectable error in a particular observation. For systems with equal nonzero singular values of the design matrix, the formula can be expressed in terms of the index of internal response-based reliability and the pseudo-condition^number. With these measures appearing in explicit form, the formula shows, although only for the above specific systems, the character of the impact of internal response-based reliability and conditionality of the model upon its external reliability. Proofs for complementary properties concerning the pseudo-condition^number and the 2-norm of parameter distortions in systems with minimal constraints are given in the Appendices. Numerical examples are provided to illustrate the theory.
L1 - http://journals.pan.pl/Content/98342/PDF/art5_cor.pdf
L2 - http://journals.pan.pl/Content/98342
PY - 2013
IS - No 2
DO - 10.2478/geocart-2013-0010
KW - internal reliability
KW - external reliability
KW - system conditionality
KW - singular value decomposition
KW - uncorrelated observations
A1 - Prószyński, Witold
PB - Commitee on Geodesy PAS
VL - vol. 62
DA - 2013
T1 - Investigating the links of internal and external reliability with the system conditionality in Gauss-Markov models with uncorrelated observations
UR - http://journals.pan.pl/dlibra/publication/edition/98342
T2 - Geodesy and Cartography
ER -