Download presentation
Presentation is loading. Please wait.
Published byJared Warren Modified over 9 years ago
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.