A Robust Kalman Framework with Resampling and Optimal Smoothing (original) (raw)
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We introduce a novel approach for processing sequential data in the presence of outliers. The outlier-robust Kalman filter we propose is a discrete-time model for sequential data corrupted with non-Gaussian and heavy-tailed noise. We present efficient filtering and smoothing algorithms which are straightforward modifications of the standard Kalman filter Rauch-Tung-Striebel recursions and yet are much more robust to outliers and anomalous observations. Additionally, we present an algorithm for learning all of the parameters of our outlier-robust Kalman filter in a completely unsupervised manner. The potential of our approach is borne out in experiments with synthetic and real data.
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A new robust Kalman filter is proposed that detects and bounds the influence of outliers in a discrete linear system, including those generated by thick-tailed noise distributions such as impulsive noise. Besides outliers induced in the process and observation noises, we consider in this paper a new type called structural outliers. For a filter to be able to counter the effect of these outliers, observation redundancy in the system is necessary. We have therefore developed a robust filter in a batch-mode regression form to process the observations and predictions together, making it very effective in suppressing multiple outliers. A key step in this filter is a new prewhitening method that incorporates a robust multivariate estimator of location and covariance. The other main step is the use of a generalized maximum likelihood-type (GM) estimator based on Schweppe's proposal and the Huber function, which has a high statistical efficiency at the Gaussian distribution and a positive breakdown point in regression. The latter is defined as the largest fraction of contamination for which the estimator yields a finite maximum bias under contamination. This GM-estimator enables our filter to bound the influence of residual and position, where the former measures the effects of observation and innovation outliers and the latter assesses that of structural outliers. The estimator is solved via the iteratively reweighted least squares (IRLS) algorithm, in which the residuals are standardized utilizing robust weights and scale estimates. Finally, the state estimation error covariance matrix of the proposed GM-Kalman filter is derived from its influence function. Simulation results revealed that our filter compares favorably with the H -filter in the presence of outliers.
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In this paper, we propose a robust Kalman filter and smoother for the errors-in-variables (EIV) state space model subject to observation noise with outliers. We introduce the EIV problem with outliers and then we present the Least-Trimmed-Squares (LTS) estimator which is highly robust estimator to detect outliers. As a result, a new statistical test to check the existence of outliers which is based on the Kalman filter and smoother has been formulated. Since the LTS is combinatorial optimization problem the randomized algorithm has been proposed in order to achieve the optimal estimate. However, the uniform sampling method has a high computational cost and may lead to biased estimate, therefore we apply the subsampling method.
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A Statistical and Computational Theory for Robust and Sparse Kalman Smoothing
16th IFAC Symposium on System Identification, 2012
Kalman smoothers reconstruct the state of a dynamical system starting from noisy output samples. While the classical estimator relies on quadratic penalization of process deviations and measurement errors, extensions that exploit Piecewise Linear Quadratic (PLQ) penalties have been recently proposed in the literature. These new formulations include smoothers robust with respect to outliers in the data, and smoothers that keep better track of fast system dynamics, e.g. jumps in the state values. In addition to L 2 , well known examples of PLQ penalties include the L 1 , Huber and Vapnik losses. In this paper, we use a dual representation for PLQ penalties to build a statistical modeling framework and a computational theory for Kalman smoothing. We develop a statistical framework by establishing conditions required to interpret PLQ penalties as negative logs of true probability densities. Then, we present a computational framework, based on interior-point methods, that solves the Kalman smoothing problem with PLQ penalties and maintains the linear complexity in the size of the time series, just as in the L 2 case. The framework presented extends the computational efficiency of the Mayne-Fraser and Rauch-Tung-Striebel algorithms to a much broader non-smooth setting, and includes many known robust and sparse smoothers as special cases.
A Modified Kalman Filter for Non-gaussian Measurement Noise
A novel modification is proposed to the Kalman filter for the case of non-Gaussian measurement noise. We model the non-Gaussian data as outliers. Measurement data is robustly discriminated between Gaussian (valid data) and outliers by Robust Sequential Estimator (RSE). The measurement update is carried out for the valid data only. The modified algorithm proceeds as follows. Initially, the robust parameter and scale estimates of the measurement data are obtained for a sample of data using maximum likelihood estimates for a t-distribution error model through Iteratively Reweighted Least Squares (IRLS). The sample is dynamically updated with each new observation. Sequential classification of each new measurement is decided through a weighting scheme determined by RSE. State updates are carried out for the valid data only. Simulations provide satisfactory results and a significant improvement in mean square error with the proposed scheme.
Asian Journal of Control, 2011
In this paper, we propose a robust Kalman filter and smoother for the errors-in-variables (EIV) state space model subject to observation noise with outliers. We introduce the EIV problem with outliers and then we present the Least-Trimmed-Squares (LTS) estimator which is highly robust estimator to detect outliers. As a result, a new statistical test to check the existence of outliers which is based on the Kalman filter and smoother has been formulated. Since the LTS is combinatorial optimization problem the randomized algorithm has been proposed in order to achieve the optimal estimate. However, the uniform sampling method has a high computational cost and may lead to biased estimate, therefore we apply the subsampling method.
Robust kriged Kalman filtering
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Although the kriged Kalman filter (KKF) has welldocumented merits for prediction of spatial-temporal processes, its performance degrades in the presence of outliers due to anomalous events, or measurement equipment failures. This paper proposes a robust KKF model that explicitly accounts for presence of measurement outliers. Exploiting outlier sparsity, a novel 1-regularized estimator that jointly predicts the spatialtemporal process at unmonitored locations, while identifying measurement outliers is put forth. Numerical tests are conducted on a synthetic Internet protocol (IP) network, and real transformer load data. Test results corroborate the effectiveness of the novel estimator in joint spatial prediction and outlier identification.