ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 26: Cholesky and Singular Value
Decomposition
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Homework due Friday
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Lecture quiz due Friday
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Exam 2 – Friday, November 6
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Cholesky-Based Least Squares
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Recall the weighted least squares:
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Instead, we will write:
M is the information
matrix
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Usually, we solve via matrix inversion
If the number of estimated parameters is
large, then this is expensive and possibly
inaccurate
◦ Estimate gravity field of degree 360
◦ n ≈ 129,600
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Instead, let’s write the equations in terms of
the Cholesky decomposition
R here is not the obs.
error covariance matrix!
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Eq. 5.2.7 in the Book
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Eq. 5.2.8 in the Book
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We may also solve for the covariance matrix
using the Cholesky decomposition
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Using this directly still requires an n×n
matrix inversion!
Eq. 5.2.9 provides a simple algorithm to get S
by leveraging
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Eq. 5.2.9:
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SVD-Based Least Squares (not in book)
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The SVD of any real m×n matrix H is
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It turns out that we can solve the linear
system
using the pseudoinverse given by the SVD
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For the linear system
the solution
minimizes the least squares cost function
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Recall that for the normal solution,
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This squares the condition number of H !
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Instead, SVD operates on H, thereby
improving solution accuracy
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The covariance matrix P with R the identity
matrix is:
Home Practice Exercise: Derive the equation for P above
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Solving the LS problem via SVD provides one of
(if not the most) numerically stable solutions
Also a square-root method (does not square
the condition number of H )
Generating the SVD is more computationally
intensive than most methods
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Bias Estimation
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As shown in the homework,
i.e., biased observations, yields a biased
estimator.
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To compensate, we can estimate the bias:
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What are some example sources of bias in an
observation?
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GPS receiver solutions for Jason-2
Antenna is offset ~1.4 meters from COM
What could be causing the bias change after
80 hours?
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