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Papers by Richard Gibbs
We derive three new tests that can be applied to a Kalman filter to check for inconsistencies. Th... more We derive three new tests that can be applied to a Kalman filter to check for inconsistencies. The Filter Residual Test can detect observations that are outliers but would be missed by a basic residual test because the uncertainty of the expected observation is large relative to the uncertainty of the observation. The Smoother Residual Test uses the output from a Modified Bryson-Frazier (MBF) smoother to detect observations that are outliers. The Smoother State Test compares the state estimates from the filter and MBF smoother to detect model inconsistencies, in particular insufficient process noise.
We derive here an algorithm for a complete square root implementation of the modified Bryson-Fraz... more We derive here an algorithm for a complete square root implementation of the modified Bryson-Frazier (MBF) smoother. The MBF algorithm computes the smoothed covariance as the difference of two symmetric matrices. Numerical errors in this differencing can result in the covariance matrix not being positive semi-definite. Earlier algorithms implemented the computation of intermediate quantities in square root form but still computed the smoothed covariance as the difference of two matrices. We show how to compute the square root of the smoothed covariance by solving an equation in the form CCT=AAT-BBT using QR decomposition with hyperbolic Householder transformations.
Drafts by Richard Gibbs
The extended Kalman filter (EKF) is frequently tried for solving nonlinear estimation problems, b... more The extended Kalman filter (EKF) is frequently tried for solving nonlinear estimation problems, but often fails. There have been no published objective criteria that can be used to determine whether the EKF will work. The EKF is the most basic nonlinear Gaussian filter approximation (NGFA), where the estimate is correct to first order and the covariance correct to second order. For a filter observation with a nonlinear measurement function an NGFA uses a Taylor series expansion about the mean to compute the mean and covariance of the measurement. We show that a requirement for the EKF to work is that the part of the fourth order covariance term of the measurement covariance that involves the products of the second order derivatives of the measurement function must be negligible in comparison to the covariance of the observation error. We also show that when this condition is not met we can implement an NGFA using the standard Kalman filter update equations where instead of the observation error covariance we use the sum of the observation error covariance and the above fourth order covariance term. We also discuss the limitations of the various sigma point filters because they do not implement this fourth order term correctly.
We derive three new tests that can be applied to a Kalman filter to check for inconsistencies. Th... more We derive three new tests that can be applied to a Kalman filter to check for inconsistencies. The Filter Residual Test can detect observations that are outliers but would be missed by a basic residual test because the uncertainty of the expected observation is large relative to the uncertainty of the observation. The Smoother Residual Test uses the output from a Modified Bryson-Frazier (MBF) smoother to detect observations that are outliers. The Smoother State Test compares the state estimates from the filter and MBF smoother to detect model inconsistencies, in particular insufficient process noise.
We derive here an algorithm for a complete square root implementation of the modified Bryson-Fraz... more We derive here an algorithm for a complete square root implementation of the modified Bryson-Frazier (MBF) smoother. The MBF algorithm computes the smoothed covariance as the difference of two symmetric matrices. Numerical errors in this differencing can result in the covariance matrix not being positive semi-definite. Earlier algorithms implemented the computation of intermediate quantities in square root form but still computed the smoothed covariance as the difference of two matrices. We show how to compute the square root of the smoothed covariance by solving an equation in the form CCT=AAT-BBT using QR decomposition with hyperbolic Householder transformations.
The extended Kalman filter (EKF) is frequently tried for solving nonlinear estimation problems, b... more The extended Kalman filter (EKF) is frequently tried for solving nonlinear estimation problems, but often fails. There have been no published objective criteria that can be used to determine whether the EKF will work. The EKF is the most basic nonlinear Gaussian filter approximation (NGFA), where the estimate is correct to first order and the covariance correct to second order. For a filter observation with a nonlinear measurement function an NGFA uses a Taylor series expansion about the mean to compute the mean and covariance of the measurement. We show that a requirement for the EKF to work is that the part of the fourth order covariance term of the measurement covariance that involves the products of the second order derivatives of the measurement function must be negligible in comparison to the covariance of the observation error. We also show that when this condition is not met we can implement an NGFA using the standard Kalman filter update equations where instead of the observation error covariance we use the sum of the observation error covariance and the above fourth order covariance term. We also discuss the limitations of the various sigma point filters because they do not implement this fourth order term correctly.