Transcript Lecture 17
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 17: Minimum Variance Estimator 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 University of Colorado Boulder 2 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! University of Colorado Boulder 3 Minimum Variance Estimator University of Colorado Boulder 4 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? University of Colorado Boulder 5 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 University of Colorado Boulder 6 To be linear, the estimated state is a linear combination of the observations: What is the matrix M? This ambiguous M matrix gives us the solution to the minimum variance estimator University of Colorado Boulder 7 To be unbiased, then Solution Constraint! University of Colorado Boulder 8 Must satisfy previous requirements: University of Colorado Boulder 9 Put into the context of scalars: University of Colorado Boulder 10 University of Colorado Boulder 11 We seek to minimize: Subject to the equality constraint: Using the method of Lagrange Multipliers, we seek to minimize: University of Colorado Boulder 12 Using calculus of variations, we need the first variation to vanish to achieve a minimum: University of Colorado Boulder 13 In order for the above to be satisfied: We will focus on the first University of Colorado Boulder 14 We now have two constraints, which will give us a solution: University of Colorado Boulder 15 University of Colorado Boulder 16 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 University of Colorado Boulder 17 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 University of Colorado Boulder 18 Propagation of Estimate and Covariance Matrix University of Colorado Boulder 19 Well, we’ve kind of covered this one before: Note: Yesterdays estimate can become today’s a priori… ◦ Not used much for the batch, but will be used for sequential processing University of Colorado Boulder 20 How do we map our uncertainty forward in time? X* University of Colorado Boulder 21 University of Colorado Boulder 22