Transformation model selection by multiple hypotheses testing
Tóm tắt
Transformations between different geodetic reference frames are often performed such that first the transformation parameters are determined from control points. If in the first place we do not know which of the numerous transformation models is appropriate then we can set up a multiple hypotheses test. The paper extends the common method of testing transformation parameters for significance, to the case that also constraints for such parameters are tested. This provides more flexibility when setting up such a test. One can formulate a general model with a maximum number of transformation parameters and specialize it by adding constraints to those parameters, which need to be tested. The proper test statistic in a multiple test is shown to be either the extreme normalized or the extreme studentized Lagrange multiplier. They are shown to perform superior to the more intuitive test statistics derived from misclosures. It is shown how model selection by multiple hypotheses testing relates to the use of information criteria like AICc and Mallows’
$${C}_{{p}}$$
, which are based on an information theoretic approach. Nevertheless, whenever comparable, the results of an exemplary computation almost coincide.
Tài liệu tham khảo
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723
Andrei CO (2006) 3D affine coordinate transformations. Masters of Science Thesis in Geodesy, No. 3091. School of Architecture and the Built Environment Royal Institute of Technology (KTH) Stockholm, Sweden
Burnham KP, Anderson DR (2002) Model selection and multimodel Inference: a practical-theoretic approach, 2nd edn. Springer, New York. ISBN:0-387-95364-7
Carosio A, Piras M, Salvini D (2006) The forward search method applied to geodetic transformations. Studies in classification, data analysis, and knowledge organization, part III. pp 181–188. doi:10.1007/3-540-35978-8_21
Deakin RE (1998) 3D coordinate transformations. In: Surveying and land information systems, vol 58, issue no 4. pp 223–234
Deakin RE (2007) Coordinate transformations for Cadstral Surveying. RMIT University, School of Mathematical and Geospatial Sciences, pp 1–33
Donato G, Belongie S (2002) Approximate thin plate spline mappings. In: Lecture notes in computer science, vol 2352. Springer, Berlin, pp 21–31. doi:10.1007/3-540-47977-5_2
Felus YA, Felus M (2009) On choosing the right coordinate transformation method. In: Proceedings of FIG working week 2009: surveyors key role in accelerated development. Eilat, Israel, 3–8 May 2009. http://www.fig.net/pub/fig2009/papers/ts04c/ts04c_felus_felus_3313.pdf
Ge Y, Yuan Y, Jia N (2013) More efficient methods among commonly used robust estimation methods for GPS coordinate transformation. Surv Rev 45(330):229–234. doi:10.1179/1752270612Y.0000000028
Goktepe A, Kocaman E (2010) Analysis of camera calibrations using direct linear transformation and bundle adjustment methods. Sci Res Essays 5(9):869–872
Kampmann G (1996) New adjustment techniques for the determination of transformation parameters for cadastral and engineering purposes. Geomatica 50(1):27–34
Kanani E (2000) Robust estimators for geodetic transformations and GIS. Ph.D. thesis, Institut für Geodäsie und Photogrammetrie, ETH Zürich
Kargoll B (2012) On the theory and application of model misspecification tests in geodesy. Deutsche Geodätsche Kommission Reihe C, Nr. 674, München
Koch KR (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, Berlin. ISBN:978-3-662-03976-2
Lehmann R (2010) Im Schwerpunkt der Anschlusspunkte - Zur Genauigkeit geodätischer Koordinatentransformationen (In the barycentre of the control points—on the accuracy of geodetic coordinate transformations). Allgemeine Vermessungsnachrichten 4/2010. VDE-Verlag Offenbach
Lehmann R (2012) Improved critical values for extreme normalized and studentized residuals in Gauss–Markov models. J Geod 86(12):1137–1146
Lehmann R, Neitzel F (2013) Testing the compatibility of constraints for parameters of a geodetic adjustment model. J Geod 87:555–566. doi:10.1007/s00190-013-0627-2
Lippus J (2004) Transformation of coordinates using piecewise conformal mapping. J Geod 78(1–2):40–46. doi:10.1007/s00190-003-0364-z
Mahboub V (2012) On weighted total least-squares for geodetic transformations. J Geod 86(5):359–367. doi:10.1007/s00190-011-0524-5
Mahboub V (2014) Variance component estimation in errors-in-variables models and a rigorous total least squares approach. Stud Geophys Geod 58(2014):17–40. doi:10.1007/s11200-013-1150-x
Mallows CL (1973) Some comments on CP. Technometrics 15(4):661–675. doi:10.2307/1267380
Miller RG (1981) Simultaneous statistical inference. Springer-Verlag New York Inc., Berlin. ISBN:0-387-90548-0.
Neyman J, Pearson ES (1933) On the problem of the most efficient tests of statistical hypotheses. Philos Trans R Soc A Math Phys Eng Sci 231(694–706):289–337. doi:10.1098/rsta.1933.0009
Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod 84(12):751–762. doi:10.1007/s00190-010-0408-0
Schaffrin B, Felus YA (2008) On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J Geod 82(6):373–383. doi:10.1007/s00190-007-0186-5
Tanizaki H (2004) Computational methods in statistics and econometrics. Marcel Dekker, New York. ISBN-13: 978–0824748043
Teunissen PJG (1985) Quality control in geodetic networks. In: Grafarend EW, Sanso F (eds) Optimization and design of geodetic networks. Springer, Berlin, pp 526–547
Teunissen PJG (2000) Testing theory; an introduction. In: Series on mathematical geodesy and positioning, 2nd edn. Delft University of Technology, The Netherlands. ISBN:90-407-1975-6
Zhang P, Xu C, Hu C, Chen Y (2012) Coordinate transformations in satellite navigation systems. In: Advances in electronic engineering, communication and management, vol. 2. Springer, Berlin, pp 249–257. doi:10.1007/978-3-642-27296-7_40
Ziggah YY, Youjian H, Amans ChO, Fan DL (2013) Determination of GPS coordinate transformation parameters of geodetic data between reference datums—a case study of ghana geodetic reference network. Int J Eng Sci Res Technol 2(4):956–971