The Rauch-Tung-Striebel (RTS) smoother is an algorithm for computing state estimates in time series using noisy measurements from all time steps. The RTS smoother works by first filtering the state when measurements arrive and then using a backward pass to obtain the smoothed state estimates for all time steps that use measurement information from all time steps as well. The backward pass goes through all time steps for which a filtering estimate were obtained in backward order. We propose a smoother that does the backward pass using only a fraction of the time steps and provides the same results as the conventional RTS smoother for these time steps. This reduces computational complexity and required memory significantly as only data for these interesting time steps need to be stored. We also propose the extension of the reduced sampling-rate smoother for non-linear systems. We show an example application involving position estimation using a state space model that uses a high filtering rate, but where it is suitable to present the final smoothed route with a considerably lower rate. In a second example, we show how the reduced rate smoother works in a nonlinear case.

This article introduces Gaussian approximation-based smoothing algorithms for nonlinear stochastic state space models using the polynomial chaos expansion (PCE). Initially, we present a smoothing algorithm, where the nonlinear functions of the state space model are approximated using a PCE that is formed using a set of collocation points generated from the filtering distribution. Subsequently, an iterative variant of the proposed smoothing algorithm is also presented. It iteratively forms a PCE approximation to the nonlinear functions by using collocation points generated from the current posterior approximation. The performance of the algorithms is evaluated on pendulum and aircraft tracking problems.

The Kalman Filter (KF) is a classical algorithm that was developed for estimating a state that evolves in time based on noisy measurements by assuming linear state transition and measurements models. There exist various KF extensions for non-linear situations, but they are not exact and provide different linearization errors. The Iterated Posterior Linearization Filter (IPLF) does the linearizations iteratively to achieve better linearizations. However, it is possible that some measurements cannot be well linearized using the current knowledge, but their linearization may be better after more measurements are available. Thus, we propose an algorithm that can store the older state elements and measurements when their linearization error is high. The resulting algorithm, the Stacked Iterated Posterior Linearization Filter (S-IPLF), is based on linear dynamic models and uses information from multiple time instances to make the linearization of the measurement function. Results show that the proposed algorithm outperforms traditional KF extensions when some of the measurements cannot be well linearized with the current knowledge, but can be when future information is available.

Electrocardiography (ECG) using lightweight and inexpensive ambulatory ECG devices makes it possible to monitor patients during their daily activities and can give important insight in arrhythmias and other cardiac diseases. However, everyday activities cause several kinds of motion artifacts which deteriorate the ECG quality and thus complicate both automated and manual ECG analysis. In this paper, we discuss some of the challenges associated with long-term ambulatory ECG and propose a baseline wander compensation algorithm based on inertial measurement units (IMUs) attached to each ECG electrode. The IMUs are used for estimating the local electrode motion which in turn is used as the reference signal for baseline wander reduction. We evaluate the proposed algorithm on data gathered in clinical trials and show that the baseline wander is successfully removed, without compromising the ECG's morphology.

This paper is concerned with tracking of reference vectors in the continuous-discrete-time setting. For this end, an Itô stochastic differential equation, using the gyroscope as input, is formulated that explicitly accounts for the geometry of the problem. The filtering problem is solved by restricting the prediction and filtering distributions to the von Mises-Fisher class, resulting in ordinary differential equations for the parameters. A strategy for approximating Bayesian updates and marginal likelihoods is developed for the class of conditionally spherical measurement distributions' which is realistic for sensors such as accelerometers and magnetometers, and includes robust likelihoods. Furthermore, computationally efficient and numerically robust implementations are presented. The method is compared to other state-of-the-art filters in simulation experiments involving tracking of the local gravity vector. Additionally, the methodology is demonstrated in the calibration of a smartphone's accelerometer and magnetometer. Lastly, the method is compared to state-of-the-art in gravity vector tracking for smartphones in two use cases, where it is shown to be more robust to unmodeled accelerations.

This paper is concerned with inferring the state of a Itô stochastic differential equation (SDE) from noisy discrete-time measurements. The problem is approached by considering basis function expansions of Brownian motion, that as a consequence give approximations to the underlying stochastic differential equation in terms of an ordinary differential equation with random coefficients. This allows for representing the latent process at the measurement points as a discrete time system with a non-linear transformation of the previous state and a noise term. The smoothing problem can then be solved by sigma-point or Taylor series approximations of this non-linear function, implementations of which are detailed. Furthermore, a method for interpolating the smoothing solution between measurement instances is developed. The developed methods are compared to the Type III smoother in simulation examples involving (i) hyperbolic tangent drift and (ii) the Lorenz 63 system where the present method is found to be better at reconstructing the smoothing solution at the measurement points, while the interpolation scheme between measurement instances appear to suffer from edge effects, serving as an invitation to future research.

The aim of this article is to design a moment transformation for Student-t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable whose variance provides information about the incurred integration error. Advantage of the Student-t process quadrature over the traditional Gaussian process quadrature, is that the integral variance depends also on the function values, allowing for a more robust modelling of the integration error. The moment transform is applied in nonlinear sigma-point filtering and evaluated on two numerical examples, where it is shown to outperform the state-of-the-art moment transforms.