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Mathematical Geodesy Kapitel lesen Erstes Kapitel lesen

2020 | OriginalPaper | Buchkapitel

9. Parameter Estimation, Variance Components and Statistical Analysis in Errors-in-Variables Models

verfasst von : Peiliang Xu

Erschienen in: Mathematische Geodäsie/Mathematical Geodesy

Verlag: Springer Berlin Heidelberg

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Abstract

This chapter discusses statistical and numerical aspects of constrained and unconstrained errors-in-variables (EIV) models. The parameters in an EIV model can often be estimated by using three categories of methods: the conventional weighted least squares (LS) method, normed orthogonal regression, and the weighted total least squares (TLS) method. The conventional weighted LS method is of significantly computational advantage but not rigorous statistically. We systematically investigate the effects of random errors in the design matrix on the weighted LS estimates of parameters and variance components, construct the N-calibrated almost unbiased weighted LS estimator of parameters and derive almost unbiased estimates for the variance of unit weight. Although orthogonal regression can be used to estimate the parameters in an EIV model, it is not statistically optimal either. The weighted TLS method is most rigorous and optimal to statistically estimate the parameters in an EIV model at the cost of substantially increasing computation. We reformulate an EIV model as a nonlinear adjustment model without constraints and investigate the statistical effects of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate. Closed form solutions to coordinate transformation have been presented as well. Finally, we prove that variance components in an EIV model with the simplest stochastic structure are not estimable.

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Vorheriges Kapitel Monte Carlo Methods
Nächstes Kapitel Fast Harmonic/Spherical Splines and Parameter Choice Methods
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Metadaten
Titel
Parameter Estimation, Variance Components and Statistical Analysis in Errors-in-Variables Models
verfasst von
Peiliang Xu
Copyright-Jahr
2020
Verlag
Springer Berlin Heidelberg
Buch
Mathematische Geodäsie/Mathematical Geodesy

Print ISBN: 978-3-662-55853-9

Electronic ISBN: 978-3-662-55854-6

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-3-662-55854-6_99