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A stochastic framework for inequality constrained estimation

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Abstract

Quality description is one of the key features of geodetic inference. This is even more true if additional information about the parameters is available that could improve the accuracy of the estimate. However, if such additional information is provided in the form of inequality constraints, most of the standard tools of quality description (variance propagation, confidence ellipses, etc.) cannot be applied, as there is no analytical relationship between parameters and observations. Some analytical methods have been developed for describing the quality of inequality constrained estimates. However, these methods either ignore the probability mass in the infeasible region or the influence of inactive constraints and therefore yield only approximate results. In this article, a frequentist framework for quality description of inequality constrained least-squares estimates is developed, based on the Monte Carlo method. The quality is described in terms of highest probability density regions. Beyond this accuracy estimate, the proposed method allows to determine the influence and contribution of each constraint on each parameter using Lagrange multipliers. Plausibility of the constraints is checked by hypothesis testing and estimating the probability mass in the infeasible region. As more probability mass concentrates in less space, applying the proposed method results in smaller confidence regions compared to the unconstrained ordinary least-squares solution. The method is applied to describe the quality of estimates in the problem of approximating a time series with positive definite functions.

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References

  • Albertella A, Cazzaniga N, Crespi M, Luzietti L, Sacerdote F, Sansò F (2006) Deformations detection by a Bayesian approach: prior information representation and testing criteria definition. In: Sansò F, Gill AJ (eds) IAG symposium on geodetic deformation monitoring: from geophysical to engineering roles. Springer, Jaén, Spain, pp 30–37

    Chapter  Google Scholar 

  • Alkhatib H, Schuh WD (2007) Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems. J Geodesy 81: 53–66. doi:10.1007/s00190-006-0034-z

    Article  Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Google Scholar 

  • Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8(1): 69–92

    Google Scholar 

  • Dantzig G (1998) Linear programming and extensions. Princeton University Press, New Jersey

    Google Scholar 

  • Fritsch D (1985) Some additional informations on the capacity of the linear complementarity algorithm. In: Grafarend E, Sansò F (eds) Optimization and design of geodetic networks. Springer, Berlin, pp 169–184

    Chapter  Google Scholar 

  • Geweke J (1986) Exact inference in the inequality constrained normal linear regression model. J Appl Econ 1(2): 127–141

    Article  Google Scholar 

  • Gill P, Murray W, Wright M (1981) Practical optimization. Academic Press, London

    Google Scholar 

  • Gill P, Murray W, Wright M (1990) Numerical linear algebra and optimization, vol 1. Addison-Wesley, Reading

    Google Scholar 

  • Joint Committee for Guides in Metrology (2008) Evaluation of measurement data—supplement 1 to the “Guide to the expression of uncertainty in measurement”—propagation of distributions using a Monte Carlo method. JCGM 101:2008

  • Koch A (2006) Semantische Integration von zweidimensionalen GIS-Daten und Digitalen Geländemodellen. PhD thesis, University of Hannover, DGK series C, no. 601

  • Koch KR (1981) Hypothesis testing with inequalities. Bollettino di geodesia e scienze affini 2: 136–144

    Google Scholar 

  • Koch KR (1982) Optimization of the configuration of geodetic networks. In: Proceedings of the international symposium on geodetic networks and computations, München, DGK series B, no. 3

  • Koch KR (1985) First order design: Optimization of the configuration of a network by introducing small position changes. In: Grafarend E, Sansò F (eds) Optimization and design of geodetic networks. Springer, Berlin, pp 56–73

    Chapter  Google Scholar 

  • Koch KR (2007) Parameter estimation and hypothesis testing in linear models. Springer, Berlin

    Google Scholar 

  • Koch KR (2007) Introduction to Bayesian statistics, 2nd edn. Springer, Berlin

    Google Scholar 

  • Liew CK (1976) Inequality constrained least-squares estimation. J Am Stat Assoc 71(355): 746–751

    Article  Google Scholar 

  • Peng J, Zhang H, Shong S, Guo C (2006) An aggregate constraint method for inequality-constrained least squares problems. J Geodesy 79: 705–713. doi:10.1007/s00190-006-0026-z

    Article  Google Scholar 

  • Roese-Koerner L, Devaraju B, Schuh WD, Sneeuw N (2011) Describing the quality of inequality constrained estimates. In: Proceedings of the 1st international workshop on the quality of geodetic observation and monitoring systems, Springer, Garching/München (submitted)

  • Schaffrin B (1981) Ausgleichung mit Bedingungs-Ungleichungen. Allgemeine Vermessungs-Nachrichten 88. Jg.: 227–238

    Google Scholar 

  • Schaffrin B, Krumm F, Fritsch D (1980) Positiv-diagonale Genauigkeitsoptimierung von Realnetzen über den Komplementaritäts-Algorithmus. In: Conzett R, Matthias H, Schmid H (eds) Ingenieurvermessung 80—Beiträge zum VIII. Internationalen Kurs für Ingenieurvermessung, Ferd. Dümmlers Verlag, ETH Zürich, vol 1, pp A15–A15/19

  • Song Y, Zhu J, Li Z (2010) The least-squares estimation of adjustment model constrained by some non-negative parameters. Surv Rev 42(315): 62–71. doi:10.1179/003962610X12572516251367

    Article  Google Scholar 

  • Wald A (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans Am Math Soc 54(3): 426–482

    Article  Google Scholar 

  • Zhu J, Santerre R, Chang XW (2005) A Bayesian method for linear, inequality-constrained adjustment and its application to GPS positioning. J Geodesy 78: 528–534. doi:10.1007/s00190-004-0425-y

    Article  Google Scholar 

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Correspondence to Lutz Roese-Koerner.

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Roese-Koerner, L., Devaraju, B., Sneeuw, N. et al. A stochastic framework for inequality constrained estimation. J Geod 86, 1005–1018 (2012). https://doi.org/10.1007/s00190-012-0560-9

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  • DOI: https://doi.org/10.1007/s00190-012-0560-9

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