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
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Boulder
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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!
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Minimum Variance Estimator
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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?
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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
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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
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To be unbiased, then
Solution Constraint!
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Must satisfy previous requirements:
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Put into the context of scalars:
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We seek to minimize:
Subject to the equality constraint:
Using the method of Lagrange Multipliers, we
seek to minimize:
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Using calculus of variations, we need the first
variation to vanish to achieve a minimum:
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In order for the above to be satisfied:
We will focus on the first
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We now have two constraints, which will give
us a solution:
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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
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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
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Propagation of Estimate and Covariance Matrix
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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
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How do we map our uncertainty forward in
time?
X*
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