Least-squares adjustment

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Least-squares adjustment is a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals. It is used extensively in the disciplines of surveying, geodesy, and photogrammetry—the field of geomatics, collectively.

Formulation

There are three forms of least squares adjustment: parametric, conditional, and combined:

  • In parametric adjustment, one can find an observation equation h(X) = Y relating observations Y explicitly in terms of parameters X (leading to the A-model below).
  • In conditional adjustment, there exists a condition equation which is g(Y) = 0 involving only observations Y (leading to the B-model below) — with no parameters X at all.
  • Finally, in a combined adjustment, both parameters X and observations Y are involved implicitly in a mixed-model equation f(X, Y) = 0.

Clearly, parametric and conditional adjustments correspond to the more general combined case when f(X,Y) = h(X) - Y and f(X, Y) = g(Y), respectively. Yet the special cases warrant simpler solutions, as detailed below. Often in the literature, Y may be denoted L.

Solution

The equalities above only hold for the estimated parameters [math]\displaystyle{ \hat{X} }[/math] and observations [math]\displaystyle{ \hat{Y} }[/math], thus [math]\displaystyle{ f\left(\hat{X},\hat{Y}\right) = 0 }[/math]. In contrast, measured observations [math]\displaystyle{ \tilde{Y} }[/math] and approximate parameters [math]\displaystyle{ \tilde{X} }[/math] produce a nonzero misclosure: [math]\displaystyle{ \tilde{w} = f\left(\tilde{X},\tilde{Y}\right). }[/math] One can proceed to Taylor series expansion of the equations, which results in the Jacobians or design matrices: the first one, [math]\displaystyle{ A = \partial{f}/\partial{X}; }[/math] and the second one, [math]\displaystyle{ B = \partial{f}/\partial{Y}. }[/math] The linearized model then reads: [math]\displaystyle{ \tilde{w} + A \hat{x} + B \hat{y} = 0, }[/math] where [math]\displaystyle{ \hat{x}=\hat{X}-\tilde{X} }[/math] are estimated parameter corrections to the a priori values, and [math]\displaystyle{ \hat{y} = \hat{Y} - \tilde{Y} }[/math] are post-fit observation residuals.

In the parametric adjustment, the second design matrix is an identity, B=-I, and the misclosure vector can be interpreted as the pre-fit residuals, [math]\displaystyle{ \tilde{y} = \tilde{w} = h(\tilde{X}) - \tilde{Y} }[/math], so the system simplifies to: [math]\displaystyle{ A \hat{x} = \hat{y} - \tilde{y}, }[/math] which is in the form of ordinary least squares. In the conditional adjustment, the first design matrix is null, A = 0. For the more general cases, Lagrange multipliers are introduced to relate the two Jacobian matrices, and transform the constrained least squares problem into an unconstrained one (albeit a larger one). In any case, their manipulation leads to the [math]\displaystyle{ \hat{X} }[/math] and [math]\displaystyle{ \hat{Y} }[/math] vectors as well as the respective parameters and observations a posteriori covariance matrices.

Computation

Given the matrices and vectors above, their solution is found via standard least-squares methods; e.g., forming the normal matrix and applying Cholesky decomposition, applying the QR factorization directly to the Jacobian matrix, iterative methods for very large systems, etc.

Worked-out examples

Applications

Related concepts

Extensions

If rank deficiency is encountered, it can often be rectified by the inclusion of additional equations imposing constraints on the parameters and/or observations, leading to constrained least squares.

References

  1. Kotz, Samuel; Read, Campbell B.; Balakrishnan, N.; Vidakovic, Brani; Johnson, Norman L. (2004-07-15). "Gauss-Helmert Model". Encyclopedia of Statistical Sciences. Hoboken, NJ, USA: John Wiley & Sons, Inc.. doi:10.1002/0471667196.ess0854.pub2. ISBN 978-0-471-66719-3. 
  2. Förstner, Wolfgang; Wrobel, Bernhard P. (2016). "Estimation". Photogrammetric Computer Vision. Geometry and Computing. 11. Cham: Springer International Publishing. pp. 75–190. doi:10.1007/978-3-319-11550-4_4. ISBN 978-3-319-11549-8. 
  3. Schaffrin, Burkhard; Snow, Kyle (2010). "Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth". Linear Algebra and Its Applications (Elsevier BV) 432 (8): 2061–2076. doi:10.1016/j.laa.2009.09.014. ISSN 0024-3795. 
  4. Neitzel, Frank (2010-09-17). "Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation". Journal of Geodesy (Springer Science and Business Media LLC) 84 (12): 751–762. doi:10.1007/s00190-010-0408-0. ISSN 0949-7714. Bibcode2010JGeod..84..751N. 

Bibliography

Lecture notes and technical reports
Books and chapters
  • Friedrich Robert Helmert. Die Ausgleichsrechnung nach der Methode der kleinsten Quadrate (Adjustment computation based on the method of least squares). Leipzig: Teubner, 1872. <http://eudml.org/doc/203764>.
  • Reino Antero Hirvonen, "Adjustments by least squares in geodesy and photogrammetry", Ungar, New York. 261 p., ISBN:0804443971, ISBN:978-0804443975, 1971.
  • Edward M. Mikhail, Friedrich E. Ackermann, "Observations and least squares", University Press of America, 1982
  • Wolf, Paul R. (1995). "Survey Measurement Adjustments by Least Squares". The Surveying Handbook. pp. 383–413. doi:10.1007/978-1-4615-2067-2_16. ISBN 978-1-4613-5858-9. 
  • Peter Vaníček and E.J. Krakiwsky, "Geodesy: The Concepts." Amsterdam: Elsevier. (third ed.): ISBN:0-444-87777-0, ISBN:978-0-444-87777-2; chap. 12, "Least-squares solution of overdetermined models", pp. 202–213, 1986.
  • Gilbert Strang and Kai Borre, "Linear Algebra, Geodesy, and GPS", SIAM, 624 pages, 1997.
  • Paul Wolf and Bon DeWitt, "Elements of Photogrammetry with Applications in GIS", McGraw-Hill, 2000
  • Karl-Rudolf Koch, "Parameter Estimation and Hypothesis Testing in Linear Models", 2a ed., Springer, 2000
  • P.J.G. Teunissen, "Adjustment theory, an introduction", Delft Academic Press, 2000
  • Edward M. Mikhail, James S. Bethel, J. Chris McGlone, "Introduction to Modern Photogrammetry", Wiley, 2001
  • Harvey, Bruce R., "Practical least squares and statistics for surveyors", Monograph 13, Third Edition, School of Surveying and Spatial Information Systems, University of New South Wales, 2006
  • Huaan Fan, "Theory of Errors and Least Squares Adjustment", Royal Institute of Technology (KTH), Division of Geodesy and Geoinformatics, Stockholm, Sweden, 2010, ISBN:91-7170-200-8.
  • Gielsdorf, F.; Hillmann, T. (2011). "Mathematics and Statistics". Springer Handbook of Geographic Information. pp. 7–10. doi:10.1007/978-3-540-72680-7_2. ISBN 978-3-540-72678-4. 
  • Charles D. Ghilani, "Adjustment Computations: Spatial Data Analysis", John Wiley & Sons, 2011
  • Charles D. Ghilani and Paul R. Wolf, "Elementary Surveying: An Introduction to Geomatics", 13th Edition, Prentice Hall, 2011
  • Erik Grafarend and Joseph Awange, "Applications of Linear and Nonlinear Models: Fixed Effects, Random Effects, and Total Least Squares", Springer, 2012
  • Alfred Leick, Lev Rapoport, and Dmitry Tatarnikov, "GPS Satellite Surveying", 4th Edition, John Wiley & Sons, ISBN:9781119018612; Chapter 2, "Least-Squares Adjustments", pp. 11–79, doi:10.1002/9781119018612.ch2
  • A. Fotiou (2018) "A Discussion on Least Squares Adjustment with Worked Examples" In: Fotiou A., D. Rossikopoulos, eds. (2018): “Quod erat demonstrandum. In quest for the ultimate geodetic insight.” Special issue for Professor Emeritus Athanasios Dermanis. Publication of the School of Rural and Surveying Engineering, Aristotle University of Thessaloniki, 405 pages. ISBN:978-960-89704-4-1 [1]
  • John Olusegun Ogundare (2018), "Understanding Least Squares Estimation and Geomatics Data Analysis", John Wiley & Sons, 720 pages, ISBN:9781119501404.
  • Shen, Yunzhong; Xu, Guochang (2012-07-31). "Regularization and Adjustment". Sciences of Geodesy - II. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 293–337. doi:10.1007/978-3-642-28000-9_6. ISBN 978-3-642-27999-7.