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Weighted total least squares: necessary and sufficient conditions, fixed and random parameters

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Abstract

A standard errors-in-variables (EIV) model refers to a Gauss–Markov model with an uncertain model matrix from a geodetic perspective. Least squares within the EIV model is usually called the total least squares (TLS) technique because of its symmetrical adjustment. However, the solutions and computational advantages of the weighted TLS problem with a general weight matrix (WTLS) are mostly unknown. In this study, the WTLS problem was solved using three different approaches: iterative methods based on the normal equation, the iteratively linearized Gauss–Helmert model with algebraic Jacobian matrices, and numerical analysis. Furthermore, sufficient conditions for WTLS optimization were investigated systematically as proposed solutions yield only necessary conditions for optimality. A WTLS solution was considered to treat random parameters within the EIV model. Last, applications to test these novel algorithms are presented.

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Acknowledgments

The research work presented in this paper was conducted during my doctoral studies at the Leibniz University Hanover in Germany. I am grateful to Professor Hansjörg Kutterer for his guidance and helpful comments. Particularly, I would like to acknowledge Prof. Burkhard Schaffrin from the Ohio State University for his valuable research on TLS and critical comments related to my previous work.

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  1. School of Geodesy and Geomatics, Wuhan University, Wuhan, China

    Xing Fang

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Appendix: Second derivative of the WTLS optimality

Appendix: Second derivative of the WTLS optimality

In order to obtain sufficient conditions of the WTLS optimality, the second derivative of the unconstrained WTLS objective function is required, analytically. Furthermore, based on the second derivative, one can design the Newton method for the WTLS problem.

According to the calculated first derivative, the second derivative is organized in two parts as follows:

$$\begin{aligned}&\frac{\partial ^{2}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_k \partial \varvec{\upxi }_j}\\&\quad =-2\frac{\partial \mathbf{a}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_{j}}\ldots \\&\qquad -2\frac{\partial \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_{k}^{T} \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_j} \end{aligned}$$

The first part of the equation above can be analytically formulated by

$$\begin{aligned}&-2\frac{\partial \mathbf{a}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_j }\\&\quad =-2\mathbf{a}_k^T \frac{\partial \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}}{\partial \varvec{\upxi }_j }\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&\qquad -2\mathbf{a}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\frac{\partial \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_j }\\&\quad =2\mathbf{a}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_j \mathbf{B}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \\&\qquad {\times } \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) +2\mathbf{a}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\\&\qquad {\times } \mathbf{BQ}_j^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&\qquad +2\mathbf{a}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{a}_j , \end{aligned}$$

and second part is presented as

$$\begin{aligned}&-2\frac{\partial \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_{k}^{T} \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_{j}} \\&=-2\frac{\partial \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}}{\partial \varvec{\upxi }_j }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \mathbf{BQ}_k^T \\&\quad {\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&\quad -2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \mathbf{BQ}_k^T\\&\quad {\times } \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\frac{\partial \left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }_j } \\&\quad -2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\frac{\partial \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}}{\partial \varvec{\upxi }_j } \mathbf{BQ}_k^T \\&\quad {\times } \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&\quad -2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T\\&\quad {\times }\frac{\partial \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}}{\partial \varvec{\upxi }_j }\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&\quad -2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\frac{\partial \mathbf{BQ}_k^T }{\partial \varvec{\upxi }_j }\\&\quad {\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&= 2\mathbf{a}_{j}^{T} \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_{k}^{T} \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{a}_j\ldots \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_j \mathbf{B}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T\\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_j^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T \\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_{k}^{T}\\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_j \mathbf{B}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_j^T \\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&-2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_{kj}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&= 2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_k \mathbf{B}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{a}_j\\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{a}_{j}\ldots \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_k \mathbf{B}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_j^T \\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_k \mathbf{B}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_j \mathbf{B}^{T}\\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_j \mathbf{B}^{T}\\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&+2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_k^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{BQ}_j^T\\&{\times }\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \\&-2\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{Q}_{kj}^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) \end{aligned}$$

In the last step the property introduced in Eq. (37) and the property of the transposed scalar are used. Thus, combining the solved equations of first and second part, the Hessian matrix is given as follows:

$$\begin{aligned} \mathbf{H}&= \frac{\partial ^{2}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) }{\partial \varvec{\upxi }\partial \varvec{\upxi }^{T}} \\&= 2\mathbf{A}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{A}^{*}+2\mathbf{A}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \mathbf{A}^{**}\\&+2\mathbf{A}^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{A} +2\left( {\mathbf{A}^{**}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \mathbf{A}\\&+2\left( {\mathbf{A}^{*}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \mathbf{A}\\&+2\left( {\mathbf{A}^{**}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{A}^{**} \\&+2\left( {\mathbf{A}^{**}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{A}^{*} \\&+2\left( {\mathbf{A}^{*}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1} \mathbf{A}^{*}\\&+2\left( {\mathbf{A}^{*}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\mathbf{A}^{**}-2\left[ {\varpi _{kj} } \right] \\&= 2\left( {\mathbf{A}+\mathbf{A}^{*}+\mathbf{A}^{**}} \right) ^{T}\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\\&\left( {\mathbf{A}+\mathbf{A}^{*}+\mathbf{A}^{**}} \right) -2\left[ {\varpi _{kj} } \right] \end{aligned}$$

with \(\varpi _{kj} =(\mathbf{y}-{\mathbf{A}\varvec{\upxi }} )^{T}({\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}})^{-1}\mathbf{Q}_{kj} ({\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}})^{-1}(\mathbf{y}-{\mathbf{A}\varvec{\upxi }})\) and

$$\begin{aligned} {\mathbf{A}}^{**}&= \left[ \mathbf{BQ}_1^T \left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) ,\ldots ,\mathbf{BQ}_u^{T} \right. \\&\left. {\left( {\mathbf{BQ}_{\mathbf{ll}} \mathbf{B}^{T}} \right) ^{-1}\left( \mathbf{y}-{\mathbf{A}\varvec{\upxi }} \right) } \right] . \end{aligned}$$

Note that \(\mathbf{A}^{**}=\mathbf{A}^{*}\) unless \(\mathbf{Q}_{kj}^{T} =\mathbf{Q}_{kj}\) for any k and j.

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Fang, X. Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J Geod 87, 733–749 (2013). https://doi.org/10.1007/s00190-013-0643-2

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