A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination

1992 ◽  
Vol 41 (5) ◽  
pp. 646-652 ◽  
Author(s):  
J. Baker-Jarvis ◽  
R.G. Geyer ◽  
P.D. Domich
Keyword(s):  
Transmission Line ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Least Squares Solution ◽  
Permittivity And Permeability

Unbiased Nonlinear Least Squares Estimations of Short-distance Equations

2017 ◽  
Vol 70 (4) ◽  
pp. 810-828 ◽  
Author(s):  
Shuqiang Xue ◽  
Yuanxi Yang
Keyword(s):  
Least Squares ◽  
Short Distance ◽  
Iterative Procedure ◽  
Nonlinear Least Squares ◽  
Long Distance ◽  
Geometrical Condition ◽  
Least Squares Solution ◽  
The Monte Carlo Method ◽  
Linearization Error ◽  
Geometric Dilution Of Precision

Nonlinear least squares estimations have been widely applied in positioning. However, nonlinear least squares estimations are generally biased. As the Gauss-Newton method has been widely applied to obtain a nonlinear least squares solution, we propose an iterative procedure for obtaining unbiased estimations with this method. The characteristics of the linearization error are discussed and a systematic error source of the linearization error needs to be removed to guarantee the unbiasedness. Both the geometrical condition and the statistical condition for unbiased nonlinear least squares estimations are revealed. It is shown that for long-distance observations of high precision, or for a positioning configuration with the lowest Geometric Dilution Of Precision (GDOP), the nonlinear least squares estimations tend to be unbiased; but for short-distance cases, the bias in the nonlinear least squares solution should be estimated to obtain unbiased values by removing the bias from the nonlinear least squares solution. The proposed results are verified by the Monte Carlo method and this shows that the bias in nonlinear least squares solution of short-distance distances cannot be ignored.

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