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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 14: Probability Wrap-Up and Statistical Least Squares. Announcements. Lecture Quiz Due by 5pm Wednesday Exam 1 – Friday, October 11

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 14: Probability Wrap-Up and Statistical Least Squares

  2. Announcements • Lecture Quiz Due by 5pm Wednesday • Exam 1 – Friday, October 11 • Open book, open notes, calculator (no phone calculator) • Material presented up to, and including, this week may be on the exam • Homework Assignments 1-5 • Sample exams from previous years on the public website • Exam Review on Monday • E-mail Marco if there is anything you would like him to discuss

  3. Outlines of Topics to Date • Flat Earth Problem • Linearization Procedure • State Transition Matrix • A(t), H(t), etc. • Least Squares (weighted and w/ and w/o a priori) • Minimum Norm • Probability and Statistics • Statistical Least Squares

  4. Today’s Topics • Central Limit Theorem • Bayes’ Theorem • Last Lecture Quiz • Statistical Least Squares

  5. Central Limit Theorem

  6. Central Limit Theorem

  7. CLT Illustration

  8. CLT and Orbit Determination • When we have many observations, the CLT implies that we can treat the mean observation errors as a Gaussian random variables • What about when we don’t have many observations?

  9. Bayes’ Theorem

  10. A Quick Derivation

  11. Bayes’ Theorem • Allows for updating a hypothesis’ probability when given additional information • Known as Bayesian Inference • Modern estimation research is rooted in Bayesian Inference!

  12. Bayes’ Theorem w/ Conditional Densities

  13. Importance in Estimation

  14. Lecture Quiz

  15. Question 1

  16. Question 2

  17. Question 3

  18. Question 4 Only if the weight is zero will an observation be ignored! What if all weights are small? If the weight is small, it may have little impact on the solution, but it is not ignored

  19. Question 5

  20. Statistical Interpretation of Least Squares

  21. Weighted Least Squares

  22. Observation Errors

  23. State Estimation Error Description

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