arXiv subjects

Fredrik Gustafsson

Publications and source records attributed to Fredrik Gustafsson.

34 records · Page 2Linked to original sources

arXiv2019

Exploring Positive Noise in Estimation Theory

Estimation of a deterministic quantity observed in non-Gaussian additive noise is explored via order statistics approach. More specifically, we study the estimation problem when measurement noises either have positive supports or follow a mixture of normal and uniform distribution. This is a problem of great interest specially in cellular positioning systems where the wireless signal is prone to multiple sources of noises which generally have a positive support. Multiple noise distributions are investigated and, if possible, minimum variance unbiased (MVU) estimators are derived. In case of uniform, exponential and Rayleigh noise distributions, unbiased estimators without any knowledge of the hyper parameters of the noise distributions are also given. For each noise distribution, the proposed order statistic-based estimator's performance, in terms of mean squared error, is compared to the best linear unbiased estimator (BLUE), as a function of sample size, in a simulation study.

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arXiv2019

Asynchronous Averaging of Gait Cycles for Classification of Gait and Device Modes

An approach for computing unique gait signature using measurements collected from body-worn inertial measurement units (IMUs) is proposed. The gait signature represents one full cycle of the human gait, and is suitable for off-line or on-line classification of the gait mode. The signature can also be used to jointly classify the gait mode and the device mode. The device mode identifies how the IMU-equipped device is being carried by the user. The method is based on precise segmentation and resampling of the measured IMU signal, as an initial step, further tuned by minimizing the variability of the obtained signature within each gait cycle. Finally, a Fourier series expansion of the gait signature is introduced which provides a low-dimensional feature vector well suited for classification purposes. The proposed method is evaluated on a large dataset involving several subjects, each one containing two different gait modes and four different device modes. The gait signatures enable a high classification rate for each step cycle.

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arXiv2019

Zero-Velocity Detection - A Bayesian Approach to Adaptive Thresholding

A Bayesian zero-velocity detector for foot-mounted inertial navigation systems is presented. The detector extends existing zero-velocity detectors based on the likelihood-ratio test, and allows, possibly time-dependent, prior information about the two hypotheses - the sensors being stationary or in motion - to be incorporated into the test. It is also possible to incorporate information about the cost of a missed detection or a false alarm. Specifically, we consider an hypothesis prior based on the velocity estimates provided by the navigation system and an exponential model for how the cost of a missed detection increases with the time since the last zero-velocity update. Thereby, we obtain a detection threshold that adapts to the motion characteristics of the user. Thus, the proposed detection framework efficiently solves one of the key challenges in current zero-velocity-aided inertial navigation systems: the tuning of the zero-velocity detection threshold. A performance evaluation on data with normal and fast gait demonstrates that the proposed detection framework outperforms any detector that chooses two separate fixed thresholds for the two gait speeds.

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arXiv2019

Bayesian Cooperative Localization Using Received Signal Strength With Unknown Path Loss Exponent: Message Passing Approaches

We propose a Bayesian framework for the received-signal-strength-based cooperative localization problem with unknown path loss exponent. Our purpose is to infer the marginal posterior of each unknown parameter: the position or the path loss exponent. This probabilistic inference problem is solved using message passing algorithms that update messages and beliefs iteratively. To enable the numerical tractability, we combine the variable discretization and Monte-Carlo-based numerical approximation schemes. To further improve computational efficiency, we develop an auxiliary importance sampler that updates the beliefs with the help of an auxiliary variable. To sample from a normalized likelihood function, which is an important ingredient of the proposed auxiliary importance sampler, we develop a stochastic sampling strategy that mathematically interprets and corrects an existing heuristic strategy. The proposed message passing algorithms are analyzed systematically in terms of computational complexity, demonstrating the computational efficiency of the proposed auxiliary importance sampler. Various simulations are conducted to validate the overall good performance of the proposed algorithms.

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arXiv2018

Skew-t Filter and Smoother with Improved Covariance Matrix Approximation

Filtering and smoothing algorithms for linear discrete-time state-space models with skew-t-distributed measurement noise are proposed. The algorithms use a variational Bayes based posterior approximation with coupled location and skewness variables to reduce the error caused by the variational approximation. Although the variational update is done suboptimally using an expectation propagation algorithm, our simulations show that the proposed method gives a more accurate approximation of the posterior covariance matrix than an earlier proposed variational algorithm. Consequently, the novel filter and smoother outperform the earlier proposed robust filter and smoother and other existing low-complexity alternatives in accuracy and speed. We present both simulations and tests based on real-world navigation data, in particular GPS data in an urban area, to demonstrate the performance of the novel methods. Moreover, the extension of the proposed algorithms to cover the case where the distribution of the measurement noise is multivariate skew-$t$ is outlined. Finally, the paper presents a study of theoretical performance bounds for the proposed algorithms.

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arXiv2017

Robust Bayesian Filtering and Smoothing Using Student's t Distribution

State estimation in heavy-tailed process and measurement noise is an important challenge that must be addressed in, e.g., tracking scenarios with agile targets and outlier-corrupted measurements. The performance of the Kalman filter (KF) can deteriorate in such applications because of the close relation to the Gaussian distribution. Therefore, this paper describes the use of Student's t distribution to develop robust, scalable, and simple filtering and smoothing algorithms. After a discussion of Student's t distribution, exact filtering in linear state-space models with t noise is analyzed. Intermediate approximation steps are used to arrive at filtering and smoothing algorithms that closely resemble the KF and the Rauch-Tung-Striebel (RTS) smoother except for a nonlinear measurement-dependent matrix update. The required approximations are discussed and an undesirable behavior of moment matching for t densities is revealed. A favorable approximation based on minimization of the Kullback-Leibler divergence is presented. Because of its relation to the KF, some properties and algorithmic extensions are inherited by the t filter. Instructive simulation examples demonstrate the performance and robustness of the novel algorithms.

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arXiv2017

The Ensemble Kalman Filter: A Signal Processing Perspective

The ensemble Kalman filter (EnKF) is a Monte Carlo based implementation of the Kalman filter (KF) for extremely high-dimensional, possibly nonlinear and non-Gaussian state estimation problems. Its ability to handle state dimensions in the order of millions has made the EnKF a popular algorithm in different geoscientific disciplines. Despite a similarly vital need for scalable algorithms in signal processing, e.g., to make sense of the ever increasing amount of sensor data, the EnKF is hardly discussed in our field. This self-contained review paper is aimed at signal processing researchers and provides all the knowledge to get started with the EnKF. The algorithm is derived in a KF framework, without the often encountered geoscientific terminology. Algorithmic challenges and required extensions of the EnKF are provided, as well as relations to sigma-point KF and particle filters. The relevant EnKF literature is summarized in an extensive survey and unique simulation examples, including popular benchmark problems, complement the theory with practical insights. The signal processing perspective highlights new directions of research and facilitates the exchange of potentially beneficial ideas, both for the EnKF and high-dimensional nonlinear and non-Gaussian filtering in general.

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arXiv2016

State Estimation for Piecewise Affine State-Space Models

We propose a filter for piecewise affine state-space (PWASS) models. In each filtering recursion, the true filtering posterior distribution is a mixture of truncated normal distributions. The proposed filter approximates the mixture with a single normal distribution via moment matching. The proposed algorithm is compared with the extended Kalman filter (EKF) in a numerical simulation where the proposed method obtains, on average, better root mean square error (RMSE) than the EKF.

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arXiv2016

Distributed Localization of Tree-structured Scattered Sensor Networks

Many of the distributed localization algorithms are based on relaxed optimization formulations of the localization problem. These algorithms commonly rely on first-order optimization methods, and hence may require many iterations or communications among computational agents. Furthermore, some of these distributed algorithms put a considerable computational demand on the agents. In this paper, we show that for tree-structured scattered sensor networks, which are networks that their inter-sensor range measurement graphs have few edges (few range measurements among sensors) and can be represented using a tree, it is possible to devise an efficient distributed localization algorithm that solely relies on second-order methods. Particularly, we apply a state-of-the-art primal-dual interior-point method to a semidefinite relaxation of the maximum-likelihood formulation of the localization problem. We then show how it is possible to exploit the tree-structure in the network and use message-passing or dynamic programming over trees, to distribute computations among different computational agents. The resulting algorithm requires far fewer iterations and communications among agents to converge to an accurate estimate. Moreover, the number of required communications among agents, seems to be less sensitive and more robust to the number of sensors in the network, the number of available measurements and the quality of the measurements. This is in stark contrast to distributed algorithms that rely on first-order methods. We illustrate the performance of our algorithm using experiments based on simulated and real data.

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arXiv2016

Skew-t inference with improved covariance matrix approximation

Filtering and smoothing algorithms for linear discrete-time state-space models with skew-t distributed measurement noise are presented. The proposed algorithms improve upon our earlier proposed filter and smoother using the mean field variational Bayes approximation of the posterior distribution to a skew-t likelihood and normal prior. Our simulations show that the proposed variational Bayes approximation gives a more accurate approximation of the posterior covariance matrix than our earlier proposed method. Furthermore, the novel filter and smoother outperform our earlier proposed methods and conventional low complexity alternatives in accuracy and speed.

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arXiv2015

Bayesian Inference via Approximation of Log-likelihood for Priors in Exponential Family

In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case, where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is linearized with respect to the sufficient statistic of the prior distribution in exponential family such that the posterior obtains the same exponential family form as the prior. Similarities between the proposed method and the extended Kalman filter for nonlinear filtering are illustrated. Furthermore, an extended target measurement update for target models where the target extent is represented by a random matrix having an inverse Wishart distribution is derived. The approximate update covers the important case where the spread of measurement is due to the target extent as well as the measurement noise in the sensor.

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arXiv2015

Approximate Bayesian Smoothing with Unknown Process and Measurement Noise Covariances

We present an adaptive smoother for linear state-space models with unknown process and measurement noise covariances. The proposed method utilizes the variational Bayes technique to perform approximate inference. The resulting smoother is computationally efficient, easy to implement, and can be applied to high dimensional linear systems. The performance of the algorithm is illustrated on a target tracking example.

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arXiv2015

Robust Inference for State-Space Models with Skewed Measurement Noise

Filtering and smoothing algorithms for linear discrete-time state-space models with skewed and heavy-tailed measurement noise are presented. The algorithms use a variational Bayes approximation of the posterior distribution of models that have normal prior and skew-t-distributed measurement noise. The proposed filter and smoother are compared with conventional low-complexity alternatives in a simulated pseudorange positioning scenario. In the simulations the proposed methods achieve better accuracy than the alternative methods, the computational complexity of the filter being roughly 5 to 10 times that of the Kalman filter.

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arXiv2015

Data-Driven Robust Receding Horizon Fault Estimation

This paper presents a data-driven receding horizon fault estimation method for additive actuator and sensor faults in unknown linear time-invariant systems, with enhanced robustness to stochastic identification errors. State-of-the-art methods construct fault estimators with identified state-space models or Markov parameters, but they do not compensate for identification errors. Motivated by this limitation, we first propose a receding horizon fault estimator parameterized by predictor Markov parameters. This estimator provides (asymptotically) unbiased fault estimates as long as the subsystem from faults to outputs has no unstable transmission zeros. When the identified Markov parameters are used to construct the above fault estimator, zero-mean stochastic identification errors appear as model uncertainty multiplied with unknown fault signals and online system inputs/outputs (I/O). Based on this fault estimation error analysis, we formulate a mixed-norm problem for the offline robust design that regards online I/O data as unknown. An alternative online mixed-norm problem is also proposed that can further reduce estimation errors when the online I/O data have large amplitudes, at the cost of increased computational burden. Based on a geometrical interpretation of the two proposed mixed-norm problems, systematic methods to tune the user-defined parameters therein are given to achieve desired performance trade-offs. Simulation examples illustrate the benefits of our proposed methods compared to recent literature.

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arXiv2014

Discretizing stochastic dynamical systems using Lyapunov equations

Stochastic dynamical systems are fundamental in state estimation, system identification and control. System models are often provided in continuous time, while a major part of the applied theory is developed for discrete-time systems. Discretization of continuous-time models is hence fundamental. We present a novel algorithm using a combination of Lyapunov equations and analytical solutions, enabling efficient implementation in software. The proposed method circumvents numerical problems exhibited by standard algorithms in the literature. Both theoretical and simulation results are provided.

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arXiv2013

Recursive maximum likelihood identification of jump Markov nonlinear systems

In this contribution, we present an online method for joint state and parameter estimation in jump Markov non-linear systems (JMNLS). State inference is enabled via the use of particle filters which makes the method applicable to a wide range of non-linear models. To exploit the inherent structure of JMNLS, we design a Rao-Blackwellized particle filter (RBPF) where the discrete mode is marginalized out analytically. This results in an efficient implementation of the algorithm and reduces the estimation error variance. The proposed RBPF is then used to compute, recursively in time, smoothed estimates of complete data sufficient statistics. Together with the online expectation maximization algorithm, this enables recursive identification of unknown model parameters. The performance of the method is illustrated in simulations and on a localization problem in wireless networks using real data.

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