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Robust and Trend-following Kalman Smoothers using Student's t
Computation · Submitted 22 Jan 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers).
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The abstract, sentence by sentence
We propose two nonlinear Kalman smoothers that rely on Student's t distributions. The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers). The second estimator, which we call the T-Trend smoother, is able to follow sudden changes in the process model, and is derived as a MAP solver for a model with Student's t-process noise and Gaussian observation noise. We design specialized methods to solve both problems which exploit the special structure of the Student's t-distribution, and provide a convergence theory. Both smoothers can be implemented with only minor modifications to an existing L2 smoother implementation. Numerical results for linear and nonlinear models illustrating both robust and fast tracking applications are presented.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose two nonlinear Kalman smoothers that rely on Student's t distributions. | contribution 1.00 | 0.94 | 0.17 | asserted 1.00 |
| s1 | The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers). | contribution 0.48 | 0.93 | 0.27 | asserted 1.00 |
| s2 | The second estimator, which we call the T-Trend smoother, is able to follow sudden changes in the process model, and is derived as a MAP solver for a model with Student's t-process noise and Gaussian observation noise. | contribution 0.52 | 0.95 | 0.11 | asserted 1.00 |
| s3 | We design specialized methods to solve both problems which exploit the special structure of the Student's t-distribution, and provide a convergence theory. | method 0.70 | 0.95 | 0.09 | asserted 1.00 |
| s4 | Both smoothers can be implemented with only minor modifications to an existing L2 smoother implementation. | contribution 0.46 | 0.76 | 0.08 | asserted 0.93 |
| s5 | Numerical results for linear and nonlinear models illustrating both robust and fast tracking applications are presented. | result 0.71 | 0.92 | 0.15 | asserted 0.99 |
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The paper
Original abstract
We propose two nonlinear Kalman smoothers that rely on Student's t distributions. The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers). The second estimator, which we call the T-Trend smoother, is able to follow sudden changes in the process model, and is derived as a MAP solver for a model with Student's t-process noise and Gaussian observation noise. We design specialized methods to solve both problems which exploit the special structure of the Student's t-distribution, and provide a convergence theory. Both smoothers can be implemented with only minor modifications to an existing L2 smoother implementation. Numerical results for linear and nonlinear models illustrating both robust and fast tracking applications are presented.
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