Softly Unbiased Prediction Part 2: The Random Effects Model

  • Burkhard Schaffrin
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 122)


At the First Hotine-Marussi Symposion, this author compared varieous methods to pretict the sate vector” within a Random Effects Model ineluding inhomBLIP (”lesst-squar” collocation”), homI3LIP and horn- BLUP (”robust collocation”); see (1986). The term ”robust” refers to the specific restance of homBLUP against scale errore In the prior infonnation. The case of resistance against multiple scale errors was later discussed in tvo popers by 8. Sehaf-frin/B, Middd tl989; 1991) related to terrestrial gravity field studies.

In the meantime, the question was asked whether there ia a continuous transition between homBLUP (which is weakly unbiased) and homBLIP (which is ”more efficient” than homBLUP). Such an intermediate predictor would allow us to give up some rigor in the un-biasednes condition in order to gain more ”efficiency”, i. e., a smaller Mean-Square-Error matrix. First results have been presented for univariate spatial processes in the context of Ordinary Kriging by (1997).

In the following we shall introduce the notion of ”Softly Unbiased Prediction” and present the corresponding formulas for the homBLISUP (Best homogeneously Linear Softly Unbiased Predictor) within a Random Effects Model.

A similar approach within the GaussMarkov Model led to the BLUSUE (Best Linear Uniformly Softly Unbiased Eatimator) of the parameters; see (1998).


Varieous Method Random Effect Model Gravity Field Unbiased Estimation Continuous Transition 
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  1. Schaffrin, B. (1986). On robust collocation. In Proc. of the First Hotine-Marussi Symp. on Math. Geodesy, pages 343–361.Google Scholar
  2. Schaffrin, B. (1997). Kriging with soft unbiasedness. In Baafi, E. and Schofield, N., editors, Geostatistics Wollonong’96, volume 1, pages 69–79. Kluwer: Dordrecht/Boston/London.Google Scholar
  3. Schaffrin, B. (1998). Softly unbiased estimation. part1: Gauss-markov model.Google Scholar
  4. Schaffrin, B. and Middel, B. (1989). Stabilized determination of the geopotential coemcientsby the mixed-homblup approach. In R. Rapp, editor, Progress in the Determination of the Earth’s Gravity Field, volume 397 of Report of Dep. of Geodetic Sei and Surv., pages 27–30. Ohio State University, Columbus/HO.Google Scholar
  5. Schaffrin, B. and Middel, B. (1991). Experiences with the mixed-blup approach in the earth’s gravity field determination. IUGG General Assembly, IAG Sei. Meeting GM3, Vienna, Austria Aug. 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Burkhard Schaffrin
    • 1
  1. 1.Dept. of Civil and Environmental Engineerig and Geodetic ScienceThe Ohio State UniversityColumbusUSA

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