1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 17: Minimum Variance Estimator

2. University of Colorado Boulder  Friday October 9 – Exam 1 ◦ Linearization (STM, A(t), H(t), etc.) ◦ Least Squares (weighted, with and without a priori, etc.) ◦ Probability and Statistics ◦ Linear Algebra ◦ Statistical Least Squares ◦ Minimum Variance Estimator 2

3. University of Colorado Boulder  Open book, open notes ◦ Bring a calculator! ◦ No internet enabled devices  Sample exams on website (under “Misc”) were created by a previous instructor ◦ They are not indicative of my exams, but are good practice  Very generous with partial credit ◦ The worst thing you can do is leave a problem blank! 3

4. University of Colorado Boulder 4 Minimum Variance Estimator

5. University of Colorado Boulder  With the least squares solution, we minimized the square of the residuals  Instead, what if we want the estimate that gives us the highest confidence in the solution: ◦ What is the linear, unbiased, minimum variance estimate of the state x? 5

6. University of Colorado Boulder  What is the linear, unbiased, minimum variance estimate of the state x ? ◦ This encompasses three elements  Linear  Unbiased, and  Minimum Variance  We consider each of these to formulate a solution 6

7. University of Colorado Boulder  To be linear, the estimated state is a linear combination of the observations: 7  What is the matrix M?  This ambiguous M matrix gives us the solution to the minimum variance estimator

8. University of Colorado Boulder  To be unbiased, then 8 Solution Constraint!

9. University of Colorado Boulder  Must satisfy previous requirements: 9

10. University of Colorado Boulder 10  Put into the context of scalars:

11. University of Colorado Boulder 11

12. University of Colorado Boulder  We seek to minimize: 12  Subject to the equality constraint:  Using the method of Lagrange Multipliers, we seek to minimize:

13. University of Colorado Boulder  Using calculus of variations, we need the first variation to vanish to achieve a minimum: 13

14. University of Colorado Boulder  In order for the above to be satisfied: 14  We will focus on the first

15. University of Colorado Boulder 15  We now have two constraints, which will give us a solution:

16. University of Colorado Boulder 16

17. University of Colorado Boulder  Showed that P satisfies the constraints, but do we have a “minimum” ◦ Must show that, for our solution, ◦ See book, p 186-187 for proof 17

18. University of Colorado Boulder  Turns out, we get the weighted, statistical least squares!  Hence, the linear least squares gives us the minimum variance solution ◦ Of course, this is predicated on all of our statistical/linearization assumptions 18

19. University of Colorado Boulder 19 Propagation of Estimate and Covariance Matrix

20. University of Colorado Boulder  Well, we’ve kind of covered this one before: 20  Note: Yesterdays estimate can become today’s a priori… ◦ Not used much for the batch, but will be used for sequential processing

21. University of Colorado Boulder  How do we map our uncertainty forward in time? 21 X*X*

22. University of Colorado Boulder 22