Singular value decomposition (SVD) – a tool for VLBI simulations Markus Vennebusch

1. Singular value decomposition (SVD) –a tool for VLBI simulations Markus Vennebusch VLBI – group of the Geodetic Institute of the University of Bonn IVS-General Meeting 2006 M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 1

2. Idea • Idea: • - Use of an advanced (not very popular) method for parameter estimation (without normal equations) • So far: Investigations of parameters (cofactors, EVD, correlations, ....) • Now: Investigation of parameters AND observations (entire schedule) • Goals: • - Improvement of Schedules by (direct) investigations of design matrix • - Gain insight into adjustment process: „Which observation affects which parameter? Is a particular observation more important than another observation?“ • This talk: No explanations of theoretical background, just applications M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 2

3. Least-squares adjustment Least-squares adjustment: Overdetermined system of linear equations: Normal equation method: direct methods (design matrix used): Solution of normal equations: QR-decomposition Singular valuedecomposition + numerically more stable(even with bad condition) + no inversion (S = diagonal)+ yields geometrical insight into adjustment process - complex, slowly + simple + fast + yields accuracy of parameters- inversion necessary - numerical problems (condition becomes worse) M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 3

4. Least-squares adjustment • Linear Algebra: • provides different methods for solution of (overdetermined) systems of linear equations: Matrix decompositions: Cholesky, LU (Gauß-Algorithm), QR, EVD, SVD, ... • Singular value decomposition (Extended EVD for rectangular matrices): = Jacobi-Matrix / Designmatrix Right singular vectors(= eigenvectors of NEQ) Left singular vectors Singular values M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 4

5. Singular value decomposition = + + + ... + r = Rank elements = scaling factors Least-Squares-solution by SVD: • Left and right singular vectors (ui and vi): • ui shows impact (or weight) of observations - vi shows parameters affected • si shows whether vi can be determined or not • SVD yields geometrical insight into LSM-adjustment: • - Vector spaces, bases of vector spaces, projections - column space = Data space, row space = Model space => further investigations M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 5

6. Singular value decomposition • Data resolution matrix (= Ur• UrT, Ur = U(:,1:r)): • shows „importance“ of observations in general (information of observation already contained?) • shows observations which might be neglected • system / solution is very sensitive to errors in observations with large importance (!) • Model resolution matrix (= Vr• VrT, Vr = V(:,1:r)): • shows dependencies (correlations, separability) between parameters (even in case of rank deficiency) • Qxl-Matrix (= Vr• Sr-1• UrT): • shows „correlations“/dependencies between observations and unknowns (more precise: common stochastic behaviour) • - shows impact of variations in observations on parameters (measure of sensitivity to errors) M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 6

7. Example Example I04267 (04SEP23XU): 1 hour duration: 16 observations, 6 pseudo-observations(not necessary) • SVD of (22 x 5)-design matrix (from OCCAM) yields: • Matrix U: Dimension 22 x 22 • Matrix S: Dimension 22 x 5 (Singular values on main diagonal) • Matrix V: Dimension 5 x 5 • => Rank r = 5 M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 7

8. Example Singular vectors 1 – 3 (out of 5): v1 u1 u2 v2 u3 v3 Number of observation DUT1 CL0 KOKEE XPOL ZD KOKEE ZD WETTZELL M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 8

9. Data resolution matrix Source statistics: Largest importance M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 9

10. Outlook • Problems: • Parameters must have the same unit • (=> parameter rates are difficult to investigate) • SVD heavily depends on parametrisation • Further investigations: • Piecewise linear parameters: useful parametrisation? • Comparison of different schedules • Identification of important observations • Identification of undeterminable parameters • Impact of datum constraints • ... • Conclusion: • SVD is a very useful tool to understand adjustment processes • Contribution to improve VLBI schedules and data analysis • => Suitable method for VLBI simulations M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 10

11. Singular value decomposition –a tool for VLBI simulations Markus Vennebusch VLBI – group of the Geodetic Institute of the University of Bonn IVS-General Meeting 2006 M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 11

12. Qxl-Matrix Qxl-Matrix: - impact of variations in observations on parameters (sensitivity on errors) M. Vennebusch – singular value decomposition – a tool for VLBI simulations January 9, 2006 slide 12