Robust Kalman tracking and smoothing with propagating and non-propagating outliers

A common situation in filtering where classical Kalman filtering does not perform particularly well is tracking in the presence of propagating outliers. This calls for robustness understood in a distributional sense, i.e.; we enlarge the distribution assumptions made in the ideal model by suitable neighborhoods. Based on optimality results for distributional-robust Kalman filtering from Ruckdeschel[01,10], we propose new robust recursive filters and smoothers designed for this purpose as well as specialized versions for non-propagating outliers. We apply these procedures in the context of a GPS problem arising in the car industry. To better understand these filters, we study their behavior at stylized outlier patterns (for which they are not designed) and compare them to other approaches for the tracking problem. Finally, in a simulation study we discuss efficiency of our procedures in comparison to competitors.Comment: 27 pages, 12 figures, 2 table

Multi-Target Tracking in Distributed Sensor Networks using Particle PHD Filters

Multi-target tracking is an important problem in civilian and military applications. This paper investigates multi-target tracking in distributed sensor networks. Data association, which arises particularly in multi-object scenarios, can be tackled by various solutions. We consider sequential Monte Carlo implementations of the Probability Hypothesis Density (PHD) filter based on random finite sets. This approach circumvents the data association issue by jointly estimating all targets in the region of interest. To this end, we develop the Diffusion Particle PHD Filter (D-PPHDF) as well as a centralized version, called the Multi-Sensor Particle PHD Filter (MS-PPHDF). Their performance is evaluated in terms of the Optimal Subpattern Assignment (OSPA) metric, benchmarked against a distributed extension of the Posterior Cram\'er-Rao Lower Bound (PCRLB), and compared to the performance of an existing distributed PHD Particle Filter. Furthermore, the robustness of the proposed tracking algorithms against outliers and their performance with respect to different amounts of clutter is investigated.Comment: 27 pages, 6 figure

Does median filtering truly preserve edges better than linear filtering?

Image processing researchers commonly assert that "median filtering is better than linear filtering for removing noise in the presence of edges." Using a straightforward large-$n$ decision-theory framework, this folk-theorem is seen to be false in general. We show that median filtering and linear filtering have similar asymptotic worst-case mean-squared error (MSE) when the signal-to-noise ratio (SNR) is of order 1, which corresponds to the case of constant per-pixel noise level in a digital signal. To see dramatic benefits of median smoothing in an asymptotic setting, the per-pixel noise level should tend to zero (i.e., SNR should grow very large). We show that a two-stage median filtering using two very different window widths can dramatically outperform traditional linear and median filtering in settings where the underlying object has edges. In this two-stage procedure, the first pass, at a fine scale, aims at increasing the SNR. The second pass, at a coarser scale, correctly exploits the nonlinearity of the median. Image processing methods based on nonlinear partial differential equations (PDEs) are often said to improve on linear filtering in the presence of edges. Such methods seem difficult to analyze rigorously in a decision-theoretic framework. A popular example is mean curvature motion (MCM), which is formally a kind of iterated median filtering. Our results on iterated median filtering suggest that some PDE-based methods are candidates to rigorously outperform linear filtering in an asymptotic framework.Comment: Published in at http://dx.doi.org/10.1214/08-AOS604 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Threshold Regularization Method for Inverse Problems

A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. However, these so-called filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. In this paper, we point out that in several cases, non-monotonic sequences of filters are more efficient. We study a regularization method that naturally extends the spectral cut-off procedure to non-monotonic sequences and provide several oracle inequalities, showing the method to be nearly optimal under mild assumptions. Then, we extend the method to inverse problems with noisy operator and provide efficiency results in a newly introduced conditional framework

Enhancing Produce Safety: State Estimation-based Robust Adaptive Control of a Produce Wash System

The rapid introduction of fresh-cut produce into a produce wash system can dramatically decrease the free chlorine (FC) concentration level in the wash water, resulting in potential widespread cross-contamination throughout the entire wash system. To minimize such contamination, a sufficient level of FC must be maintained in the wash water. This paper presents a state estimation-based robust adaptive sliding mode (RASM) control strategy for the wash system to stabilize the FC concentration level during fresh-cut iceberg lettuce washing. This feedback control law for FC dosing is suggested to provide a sufficient FC injection rate (FCIR) to the wash system in order to compensate for the fall in the FC level and in turn to minimize the Escherichia coli (E. coli) O157:H7 levels on washed lettuce and in the wash water. The proposed controller uses the estimated chemical oxygen demand (COD) and FC concentration as feedback signals while system states are estimated by a hybrid extended Kalman filter (HEKF) and the unknown noise statistics are identified by a noise identification (NI) algorithm. Uniformly ultimately boundedness (UUB) of the FC concentration tracking error in the presence of unmodelled dynamics is proven using the Lyapunov framework and Barbalat\u27s lemma. The E. coli O157:H7 contamination levels are predicted from the joint estimator and controller properties. Simulation results show that the proposed NI-based HEKF/RASM control methodology achieves FC tracking while the pathogens converge to their predicted levels. The E. coli O157:H7 levels decrease as FC concentration increases and in particular, no E. coli O157:H7 is detected when FC concentration is regulated at 15 mg/L. Two robustness tests are performed to show the performance of the proposed controller in the presence of chlorine actuator failure and system parameter uncertainties. Finally, cross-contamination management is examined in terms of the prevalence and mean pathogen levels of incoming pre-wash lettuce in the context of FC regulation at 15 mg/L

Sever: A Robust Meta-Algorithm for Stochastic Optimization

In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers. To address this, we introduce a new meta-algorithm that can take in a base learner such as least squares or stochastic gradient descent, and harden the learner to be resistant to outliers. Our method, Sever, possesses strong theoretical guarantees yet is also highly scalable -- beyond running the base learner itself, it only requires computing the top singular vector of a certain $n \times d$ matrix. We apply Sever on a drug design dataset and a spam classification dataset, and find that in both cases it has substantially greater robustness than several baselines. On the spam dataset, with $1\%$ corruptions, we achieved $7.4\%$ test error, compared to $13.4\%-20.5\%$ for the baselines, and $3\%$ error on the uncorrupted dataset. Similarly, on the drug design dataset, with $10\%$ corruptions, we achieved $1.42$ mean-squared error test error, compared to $1.51$-$2.33$ for the baselines, and $1.23$ error on the uncorrupted dataset.Comment: To appear in ICML 201