Browsing by Subject "diagonal covariance matrix"

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Access status: Open Access ,
Application of advanced statistical procedures for adjustment of measurement results in engineering surveying
(Wydawnictwa AGH, 2018) Czaja, Józef; Dąbrowski, Janusz
Measurements in engineering surveying are aimed at determining the coordinates of the points of a geodetic control, spatially setting out a technical design of an engineering structure, determining the spatial coordinates of points (or their displacement) that represent an engineering structure, and identifying the displacement and deformation of a studied engineering structure. Provided that the aforementioned measurements are to represent the same engineering structure, such observation results should be settled (adjusted) in one calculation process. The application of the Gauss-Markov theorem for this adjustment using covariance matrix Cov(L) for observed values L is the classical approach for adjusting the results of surveying observations of various accuracy (taking into account accuracy weights). Determining the displacements of points in the process of adjusting the results of periodic measurements, applying different methods of tying geodetic controls to national networks, and using various instruments and measurement methods result in the individual displacement components or coordinates of the observed points being determined with different accuracies. This circumstance forms the basis for the assumption that the estimated parameters (unknown values) should be random. This paper will formulate the principles of estimation of Gauss-Markov models in which the estimated parameters (X) are random. For this purpose, methods for the prior definition of covariance matrix CX for the estimated parameters will be provided, which will be used to determine the conditional covariance matrix of observation vector L and then to estimate the most probable values of the Xˆ parameters. Covariance matrix Cov(Xˆ) obtained as a result of this estimation will be used to define the limit values of the variances of these parameters.
Access status: Open Access ,
Application of advanced statistical procedures for adjustment of results in measurements of displacements
(Wydawnictwa AGH, 2019) Czaja, Józef; Dąbrowski, Janusz
In this paper, the authors verified the formulated principles of the estimation of Gauss-Markov models in which estimated parameters X were random. For this purpose, methods for the prior definition of covariance matrix Cx for the estimated parameters were provided, which were used to determine the conditional covariance matrix of observation vector L and then estimate the most probable values of parameters Xˆ. Covariance matrix Cov(Xˆ) obtained as a result of this estimation was used to define the limit values of the variance of these parameters. Practical application of the proposed method for the Gauss-Markov model estimation for random parameters was illustrated on a fragment of a leveling network of points to determine the vertical displacements of a landslide surface.