The Solution of Linear Equation Systems

1965 ◽  
Vol 19 (1) ◽  
pp. 78-83
Author(s):  
Peter Wilson
Keyword(s):  
Linear Equation ◽  
Least Squares ◽  
Covariance Matrix ◽  
Error Propagation ◽  
Normal Equations ◽  
The Law ◽  
Linear Equation Systems ◽  
Variance Covariance Matrix

Several methods for solving normal equations in least squares solutions are explained and the variance-covariance matrix is developed from the law of error propagation.

Progress towards a rigorous error propagation for total least-squares estimates

2020 ◽  
Vol 14 (2) ◽  
pp. 159-166 ◽  
Author(s):  
Burkhard Schaffrin ◽  
Kyle Snow
Keyword(s):  
Least Squares ◽  
Error Propagation ◽  
Total Least Squares ◽  
Iteration Step ◽  
Dispersion Matrix ◽  
Normal Equations ◽  
Estimated Parameters ◽  
Nonlinear Normal ◽  
Rigorous Error ◽  
Least Squares Estimates

AbstractAfter several attempts at a formal derivation of the dispersion matrix for Total Least-Squares (TLS) estimates within an Errors-In-Variables (EIV) Model, here a refined approach is presented that makes rigorous use of the nonlinear normal equations, though assuming a Kronecker product structure for both observational dispersion matrices at this point. In this way, iterative linearization of a model (that can be established as being equivalent to the original EIV-Model) is avoided, which might be preferred since such techniques are based on the last iteration step only and, therefore, produce dispersion matrices for the estimated parameters that are generally too optimistic. Here, the error propagation is based on the (linearized total differential of the) exact nonlinear normal equations, which should lead to more trustworthy measures of precision.

scholarly journals Robust Estimations of Current Velocities with Four-Beam Broadband ADCPs

2009 ◽  
Vol 26 (12) ◽  
pp. 2642-2654 ◽  
Author(s):  
M. Gilcoto ◽  
Emlyn Jones ◽  
Luis Fariña-Busto
Keyword(s):  
Least Squares ◽  
Covariance Matrix ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Robust Solution ◽  
Least Squares Solution ◽  
Weighted Least Squares Method ◽  
Radial Beam ◽  
Least Squares Approach ◽  
Variance Covariance Matrix

Abstract An extended explanation of the hypothesis and equations traditionally used to transform between four-beam ADCP radial beam velocities and current velocity components is presented. This explanation includes a dissertation about the meaning of the RD Instrument error velocity and a description of the standard beam-to-current components transformation as a least squares solution. Afterward, the variance–covariance matrix associated with the least squares solution is found. Then, a robust solution for transforming radial beam velocities into current components is derived under the formality of a weighted least squares approach. The associated variance–covariance matrix is also formulated and theoretically proves that the modulus of its elements will be generally lower than the corresponding modulus of the variance–covariance matrix associated with the standard least squares solution. Finally, a comparison between the results obtained using the standard least squares solution and the results of the weighted least squares method, using a high-resolution ADCP dataset, is presented. The results show that, in this case, the weighted least squares solution provides variance estimations that are 4% lower over the entire record period (8 days) and 7% lower during a shorter, more energetic period (12 h).

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