1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Statistical Interpretation of Least Squares ASEN 5070 LECTURE 15 9/28/09

2. Colorado Center for Astrodynamics Research The University of Colorado 2 Statistical Interpretation of Least Squares Show that is an unbiased estimator of, i.e., where Assume that 1 st two moments of the probability density function for are known, i.e.,, From the least squares solution we have that But, i.e., is the true value Hence, and is an unbiased estimate of.

3. Colorado Center for Astrodynamics Research The University of Colorado 3 Statistical Interpretation of Least Squares Determine the mean and variance-covariance matrix of the estimation error,, for the weighted least squares estimate,. Because is unbiased With variance-covariance matrix (usually simply referred to as the “covariance matrix”), P From the normal equation

4. Colorado Center for Astrodynamics Research The University of Colorado 4 Statistical Interpretation of Least Squares (Cont.) Substitute into Eq (1) or If Then,

5. Colorado Center for Astrodynamics Research The University of Colorado 5 Statistical Interpretation of Least Squares If apriori information and are given, where True value Apriori value Error in Assume Then We have shown that the normal equation in this case is Recall that the estimation error is given by

6. Colorado Center for Astrodynamics Research The University of Colorado 6 Statistical Interpretation of Least Squares It can be shown that and, hence, and

7. Colorado Center for Astrodynamics Research The University of Colorado 7 Statistical Interpretation of Least Squares Define Then Assume that and are uncorrelated, then and

8. Colorado Center for Astrodynamics Research The University of Colorado 8 Statistical Interpretation of Least Squares Hence, the observation error covariance matrix in given by The quantity is called the information matrix,, i.e. For example, ifthen If are x and y unobservable?

9. Colorado Center for Astrodynamics Research The University of Colorado 9 Statistical Interpretation of Least Squares Recall that Generally will be a diagonal matrix If we do not wish to estimate a parameter (say ) we may leave it out of the state deviation vector or set its apriori variance to a very small number i.e., (say ) This will result in

10. Colorado Center for Astrodynamics Research The University of Colorado 10 Statistical Interpretation of Least Squares Demonstrate this for a 2-D case. Let, assume Where and so and

11. Colorado Center for Astrodynamics Research The University of Colorado 11 Statistical Interpretation of Least Squares let, then but,,, Thus

12. Colorado Center for Astrodynamics Research The University of Colorado 12 Statistical Interpretation of Least Squares with Hence Note that and is effected by

13. Colorado Center for Astrodynamics Research The University of Colorado 13 Statistical Interpretation of Least Squares Example Problem I. The state vector is given by where and are constants. An a priori value is given with its relative accuracy described by the matrix,. Three observations of the state are given by: The relative accuracy of the three observations is given by the weighting matrix, 1. Set up a step by step algorithm describing how you would solve for the weighted least squares estimate,, including use of the a priori information. Define the, and matrices (i.e., what is each element of the matrices).

14. Colorado Center for Astrodynamics Research The University of Colorado 14 Statistical Interpretation of Least Squares Answer the following questions: 2. Is the observation – state relationship linear or nonlinear? 3. Was it necessary to use both a state and observation deviation vector? 4. Do the and matrices differ for this problem? Why or why not? 5. Was it necessary to generated a computed observation for this problem? Why or why not?

15. Colorado Center for Astrodynamics Research The University of Colorado 15 Statistical Interpretation of Least Squares II. Given the system with the state vector defined by a.Write the linearized equations in state space form, b.How would you determine the state transition matrix for this system? Is additional information needed to generate the state transition matrix?

16. Colorado Center for Astrodynamics Research The University of Colorado Statistical Interpretation of Least Squares 16