Gauss-Markov model and mixed models; collocation
technique; generalized matrices in geodetic science; estimable and projected
parameters; variance components; prior information; dynamic linear model and
Kalman filtering.
Objectives:
The course makes students aware of various special
adjustment techniques. Relations between Gauss-Markov model and traditional
least squares solutions are explored and compared to the Collocation technique.
Ranks of the matrices are discussed, and they are derived for matrices usually
encountered in adjustment computations. The introduction to generalized
matrices will give the possibility to solve rank-deficient systems. Estimable and
nonestimable quantities in adjustment are defined and discussed, as well as the
estimation of variance components. The role of prior information is clarified,
and it is shown how the least-squares adjustment in a Dynamic Linear Model
leads to Kalman filtering.
As a result, students should be able to make a
prudent choice of a proper model and the corresponding adjustment techniques
for a host of overdetermined problems in geodetic science, no matter how
complicated.
Syllabus:
1. Gauss-Markov
Model
1.1 Fundamental theory
1.2 The least squares adjustment
1.3 BLUUE for the parameters
1.4 BIQUUE for the variance component
1.5 BIQE for the variance component
1.6 The E-D correspondence and Kronecker-Zehfuss
products of matrices
List of
Acronyms and Notations.
Reference: Softly unbiased estimation, Part 1:
The Gauss-Markov model, Schaffrin, Burkhard
Linear Algebra and its Applications Vol: 289, Issue: 1-3, pp. 285-296, March 1, 1999
Kronecker
- Zehfuss Product formulas
2. Rank-deficient
Gauss-Markov model and generalized inverse matrices
2.1 The least-squares adjustment with multiple
solutions
2.2 g-inverse matrices and estimable parameters
2.3 Datum constraints
2.4 Restricted least-squares adjustment
Summary
of possible g-inverses
3. Variance Components
3.1
Mixed regression
3.2
The Variance Components Model
3.3
Repro-BIQUUE of the variance components
3.4 Unbiasedness of the “adaptive BLUUE”
4. Prior information and the Extended
Gauss-Morkov Model
4.1
Pseudo-observations and Wolf’s collocation solution
4.2
Sequential update of Wolf’s collocation
4.3
Singular dispersion matrix for pseudo-observations
4.4
Limit case of datum parameters
4.5 Dual form of
the sequential update
Models
of observation equations (GMM) with Constrains
5. Prior information and Mixed Model
5.1
Helmert’s knack and Moritz’s collocation solution
5.2
Bjerhammar’s modified least-squares principle
5.3 Duality via
Helmert’s knack
6. Introduction to the Dynamic Adjustment
6.1
The Dynamic Linear Model
6.2
Least-squares adjustment and best linear prediction
6.3
Kalman filtering, including the dual form
Required
Textbook:
References
(for further reading):
Brown, R. G.
and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman
Filtering, Wiley:
Lütkeohl, H., Handbook of
Matrices, Wiley,
Rao, C. R. and
H. Toutenburg, Linear Models. Least Squares and Alternatives,
Other related books:
Gilbert Strang and Kai
Borre, Linear Algebra, Geodesy, and
GPS Wellesley-Cambridge Press , 1997
WEB sources:
1. SIMPLE
STATISTICS & EFFECT STATISTICS