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University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 26: Cholesky and Singular Value Decomposition
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University of Colorado Boulder Homework due Friday Lecture quiz due Friday Exam 2 – Friday, November 6 2
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University of Colorado Boulder 3 Cholesky-Based Least Squares
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University of Colorado Boulder Recall the weighted least squares: 4 Instead, we will write: M is the information matrix
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University of Colorado Boulder Usually, we solve via matrix inversion 5 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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University of Colorado Boulder Instead, let’s write the equations in terms of the Cholesky decomposition 6 R here is not the obs. error covariance matrix!
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University of Colorado Boulder Eq. 5.2.7 in the Book 7
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University of Colorado Boulder Eq. 5.2.8 in the Book 8
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University of Colorado Boulder We may also solve for the covariance matrix using the Cholesky decomposition 9
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University of Colorado Boulder Using this directly still requires an n×n matrix inversion! Eq. 5.2.9 provides a simple algorithm to get S by leveraging 10
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University of Colorado Boulder Eq. 5.2.9: 11
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University of Colorado Boulder 12 SVD-Based Least Squares (not in book)
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University of Colorado Boulder The SVD of any real m×n matrix H is 13
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University of Colorado Boulder 14
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University of Colorado Boulder It turns out that we can solve the linear system 15 using the pseudoinverse given by the SVD
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University of Colorado Boulder For the linear system 16 the solution minimizes the least squares cost function
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University of Colorado Boulder Recall that for the normal solution, 17 This squares the condition number of H ! Instead, SVD operates on H, thereby improving solution accuracy
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University of Colorado Boulder The covariance matrix P with R the identity matrix is: 18 Home Practice Exercise: Derive the equation for P above
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University of Colorado Boulder 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 19
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University of Colorado Boulder 20 Bias Estimation
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University of Colorado Boulder As shown in the homework, i.e., biased observations, yields a biased estimator. To compensate, we can estimate the bias: 21
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University of Colorado Boulder What are some example sources of bias in an observation? 22
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University of Colorado Boulder GPS receiver solutions for Jason-2 Antenna is offset ~1.4 meters from COM What could be causing the bias change after 80 hours? 23
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