1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 26: Cholesky and Singular Value Decomposition

2. University of Colorado Boulder  Homework due Friday  Lecture quiz due Friday  Exam 2 – Friday, November 6 2

3. University of Colorado Boulder 3 Cholesky-Based Least Squares

4. University of Colorado Boulder  Recall the weighted least squares: 4  Instead, we will write: M is the information matrix

5. University of Colorado Boulder  Usually, we solve via matrix inversion 5  If the number of estimated parameters is large, then this is expensive and possibly inaccurate ◦ Estimate gravity field of degree 360 ◦ n ≈ 129,600

6. University of Colorado Boulder  Instead, let’s write the equations in terms of the Cholesky decomposition 6 R here is not the obs. error covariance matrix!

7. University of Colorado Boulder  Eq. 5.2.7 in the Book 7

8. University of Colorado Boulder  Eq. 5.2.8 in the Book 8

9. University of Colorado Boulder  We may also solve for the covariance matrix using the Cholesky decomposition 9

10. University of Colorado Boulder  Using this directly still requires an n×n matrix inversion!  Eq. 5.2.9 provides a simple algorithm to get S by leveraging 10

11. University of Colorado Boulder  Eq. 5.2.9: 11

12. University of Colorado Boulder 12 SVD-Based Least Squares (not in book)

13. University of Colorado Boulder  The SVD of any real m×n matrix H is 13

14. University of Colorado Boulder 14

15. University of Colorado Boulder  It turns out that we can solve the linear system 15 using the pseudoinverse given by the SVD

16. University of Colorado Boulder  For the linear system 16 the solution minimizes the least squares cost function

17. University of Colorado Boulder  Recall that for the normal solution, 17  This squares the condition number of H !  Instead, SVD operates on H, thereby improving solution accuracy

18. University of Colorado Boulder  The covariance matrix P with R the identity matrix is: 18 Home Practice Exercise: Derive the equation for P above

19. University of Colorado Boulder  Solving the LS problem via SVD provides one of (if not the most) numerically stable solutions  Also a square-root method (does not square the condition number of H )  Generating the SVD is more computationally intensive than most methods 19

20. University of Colorado Boulder 20 Bias Estimation

21. University of Colorado Boulder  As shown in the homework, i.e., biased observations, yields a biased estimator.  To compensate, we can estimate the bias: 21

22. University of Colorado Boulder  What are some example sources of bias in an observation? 22

23. University of Colorado Boulder  GPS receiver solutions for Jason-2  Antenna is offset ~1.4 meters from COM  What could be causing the bias change after 80 hours? 23