Given an unconditional diffusion model targeting a joint model $\pi(x, y)$, using it to perform conditional simulation $\pi(x \mid y)$ is still largely an open question and is typically achieved by learning conditional drifts to the denoising SDE after the fact. In this work, we express \emph{exact} conditional simulation within the \emph{approximate} diffusion model as an inference problem on an augmented space corresponding to a partial SDE bridge. This perspective allows us to implement efficient and principled particle Gibbs and pseudo-marginal samplers marginally targeting the conditional distribution $\pi(x \mid y)$. Contrary to existing methodology, our methods do not introduce any additional approximation to the unconditional diffusion model aside from the Monte Carlo error. We showcase the benefits and drawbacks of our approach on a series of synthetic and real data examples.

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.

Motion capture systems can provide ground truth in millimeter accuracy for evaluating positioning algorithms but the high cost and specialized hardware requirements of current commercial solutions limit accessibility for many researchers. This paper presents OPTICS, an open-source, low-cost marker tracking system built using readily available hardware (smart-phones and consumer-grade measuring tools). The methodological contribution of this paper is related to the implementation of the camera calibration. Particularly, we present a novel approach for estimating reference points for extrinsic calibration using pairwise distances between ground reference markers without precisely manufactured reference objects. While the system shows some accuracy limitations compared to professional setups, OPTICS offers a viable alternative for preliminary evaluations and studies without access to a high-end motion capture system.

Optimal decision-making under partial observability requires agents to balance reducing uncertainty (exploration) against pursuing immediate objectives (exploitation). In this paper, we introduce a novel policy optimization framework for continuous partially observable Markov decision processes (POMDPs) that explicitly addresses this challenge. Our method casts policy learning as probabilistic inference in a non-Markovian Feynman--Kac model that inherently captures the value of information gathering by anticipating future observations, without requiring suboptimal approximations or handcrafted heuristics. To optimize policies under this model, we develop a nested sequential Monte Carlo (SMC) algorithm that efficiently estimates a history-dependent policy gradient under samples from the optimal trajectory distribution induced by the POMDP. We demonstrate the effectiveness of our algorithm across standard continuous POMDP benchmarks, where existing methods struggle to act under uncertainty.

This paper presents parallel-in-time state estimation methods for systems with Slow-Rate inTegrated Measurements (SRTM). Integrated measurements are common in various applications, and they appear in analysis of data resulting from processes that require material collection or integration over the sampling period. Current state estimation methods for SRTM are inherently sequential, preventing temporal parallelization in their standard form. This paper proposes parallel Bayesian filters and smoothers for linear Gaussian SRTM models. For that purpose, we develop a novel smoother for SRTM models and develop parallel-in-time filters and smoother for them using an associative scan-based parallel formulation. Empirical experiments ran on a GPU demonstrate the superior time complexity of the proposed methods over traditional sequential approaches.

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.

A common assumption in state space models is that the state and observation noise is Gaussian. However, there are cases where this assumption is violated and is chosen for computational convenience. In this article, we present a state space model whose noise processes are modeled via highly flexible density functions based on Bayesian nonparametric priors with decreasing weights. We are focusing on a system identification problem were the aim is to estimate the parameters and the states of the (possibly) nonlinear dynamical system along with its noise processes using Gibbs sampling. Experiments in simulated data show that the nonparametric model outperforms parametric models especially when the distributions of the noise processes depart from Gaussianity.

In this paper, we propose a novel approach to Bayesian experimental design for non-exchangeable data that formulates it as risk-sensitive policy optimization. We develop the Inside-Out SMC$^2$ algorithm, a nested sequential Monte Carlo technique to infer optimal designs, and embed it into a particle Markov chain Monte Carlo framework to perform gradient-based policy amortization. Our approach is distinct from other amortized experimental design techniques, as it does not rely on contrastive estimators. Numerical validation on a set of dynamical systems showcases the efficacy of our method in comparison to other state-of-the-art strategies.

Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the computational complexity from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.

The zero velocity update (ZUPT) offers an effective correction method for the sensor drift in indoor positioning systems using foot-mounted inertial measurement units (IMUs). However, the heading drift is still problematic in positioning systems using a single IMU. This paper studies methods for positioning using two foot-mounted IMUs. We propose two methods for this purpose, which are based on the use of a distance constraint and a spacing-vector constraint, respectively. Our methods are compared against other distance-constraint-based methods. The results reveal that our methods are able to achieve a better distribution concentration than the other methods, and they better control the separation between the trajectories of the feet.

We present a deterministic algorithm for solving a wide range of dynamic programming problems in trees in O(log D) rounds in the massively parallel computation model (MPC), with O(nδ) words of local memory per machine, for any given constant 0 < δ < 1. Here D is the diameter of the tree and n is the number of nodes---we emphasize that our running time is independent of n.

We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this 'Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite (PSD) covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models.

Particle smoothers are SMC (Sequential Monte Carlo) algorithms designed to approximate the joint distribution of the states given observations from a state-space model. We propose dSMC (de-Sequentialized Monte Carlo), a new particle smoother that is able to process $T$ observations in $\mathcal{O}(\log_2 T)$ time on parallel architectures. This compares favorably with standard particle smoothers, the complexity of which is linear in $T$. We derive $\mathcal{L}_p$ convergence results for dSMC, with an explicit upper bound, polynomial in $T$. We then discuss how to reduce the variance of the smoothing estimates computed by dSMC by (i) designing good proposal distributions for sampling the particles at the initialization of the algorithm, as well as by (ii) using lazy resampling to increase the number of particles used in dSMC. Finally, we design a particle Gibbs sampler based on dSMC, which is able to perform parameter inference in a state-space model at a $\mathcal{O}(\log_2 T)$ cost on parallel hardware.

This letter is concerned with solving continuous-discrete Gaussian smoothing problems by using the Taylor moment expansion (TME) scheme. In the proposed smoothing method, we apply the TME method to approximate the transition density of the stochastic differential equation in the dynamic model. Furthermore, we derive a theoretical error bound (in the mean square sense) of the TME smoothing estimates showing that the smoother is stable under weak assumptions. Numerical experiments are presented in order to illustrate practical use of the method.

We address the problem of autonomous tracking and state estimation for marine vessels, autonomous vehicles, and other dynamic signals under a (structured) sparsity assumption. The aim is to improve the tracking and estimation accuracy with respect to the classical Bayesian filters and smoothers. We formulate the estimation problem as a dynamic generalized group Lasso problem and develop a class of smoothing-and-splitting methods to solve it. The Levenberg-Marquardt iterated extended Kalman smoother-based multiblock alternating direction method of multipliers (LM-IEKS-mADMMs) algorithms are based on the alternating direction method of multipliers (ADMMs) framework. This leads to minimization subproblems with an inherent structure to which three new augmented recursive smoothers are applied. Our methods can deal with large-scale problems without preprocessing for dimensionality reduction. Moreover, the methods allow one to solve nonsmooth nonconvex optimization problems. We then prove that under mild conditions, the proposed methods converge to a stationary point of the optimization problem. By simulated and real-data experiments, including multisensor range measurement problems, marine vessel tracking, autonomous vehicle tracking, and audio signal restoration, we show the practical effectiveness of the proposed methods.

Autonomous ships are expected to improve the level of safety and efficiency in future maritime navigation. Such vessels need perception for two purposes: to perform autonomous situational awareness and to monitor the integrity of the sensor system itself. In order to meet these needs, the perception system must fuse data from novel and traditional perception sensors using Artificial Intelligence (AI) techniques. This article overviews the recognized operational requirements that are imposed on regular and autonomous seafaring vessels, and then proceeds to consider suitable sensors and relevant AI techniques for an operational sensor system. The integration of four sensors families is considered: sensors for precise absolute positioning (Global Navigation Satellite System (GNSS) receivers and Inertial Measurement Unit (IMU)), visual sensors (monocular and stereo cameras), audio sensors (microphones), and sensors for remote-sensing (RADAR and LiDAR). Additionally, sources of auxiliary data, such as Automatic Identification System (AIS) and external data archives are discussed. The perception tasks are related to well-defined problems, such as situational abnormality detection, vessel classification, and localization, that are solvable using AI techniques. Machine learning methods, such as deep learning and Gaussian processes, are identified to be especially relevant for these problems. The different sensors and AI techniques are characterized keeping in view the operational requirements, and some example state-of-the-art options are compared based on accuracy, complexity, required resources, compatibility and adaptability to maritime environment, and especially towards practical realization of autonomous systems.

The problem of Bayesian filtering and smoothing in nonlinear models with additive noise is an active area of research. Classical Taylor series as well as more recent sigma-point based methods are two well-known strategies to deal with this problem. However, these methods are inherently sequential and do not in their standard formulation allow for parallelization in the time domain. In this paper, we present a set of parallel formulas that replace the existing sequential ones in order to achieve lower time (span) complexity. Our experimental results done with a graphics processing unit (GPU) illustrate the efficiency of the proposed methods over their sequential counterparts.

The aim of this article is to present Levenberg-Marquardt and line-search extensions of the classical iterated extended Kalman smoother (IEKS) which has previously been shown to be equivalent to the Gauss-Newton method. The algorithms are derived by rewriting the algorithm's steps in forms that can be efficiently implemented using modified EKS iterations. The resulting algorithms are experimentally shown to have superior convergence properties over the classical IEKS.

This paper is concerned with the estimation of unknown drift functions of stochastic differential equations (SDEs) from observations of their sample paths. We propose to formulate this as a non-parametric Gaussian process regression problem and use an Ito-Taylor expansion for approximating the SDE. To address the computational complexity problem of Gaussian process regression, we cast the model in an equivalent state-space representation, such that (non-linear) Kalman filters and smoothers can be used. The benefit of these methods is that computational complexity scales linearly with respect to the number of measurements and hence the method remains tractable also with large amounts of data. The overall complexity of the proposed method is O(N log N), where N is the number of measurements, due to the requirement of sorting the input data. We evaluate the performance of the proposed method using simulated data as well as with real-data applications to sunspot activity and electromyography.

In this letter, we propose a class of efficient, accurate, and general methods for solving state-estimation problems with equality and inequality constraints. The methods are based on recent developments in variable splitting and partially observed Markov processes. We first present the generalized framework based on variable splitting, then develop efficient methods to solve the state-estimation subproblems arising in the framework. The solutions to these subproblems can be made efficient by leveraging the Markovian structure of the model as is classically done in so-called Bayesian filtering and smoothing methods. The numerical experiments demonstrate that our methods outperform conventional optimization methods in computation cost as well as the estimation performance.

In this letter, we consider Gaussian approximations of the optimal importance density in sequential importance sampling for nonlinear, non-Gaussian state-space models. The proposed method is based on generalized statistical linear regression and posterior linearization using conditional expectations. Simulation results show that the method outperforms the compared methods in terms of the effective sample size and provides a better local approximation of the optimal importance density.

Received signal strength based respiration rate monitoring is emerging as an alternative non-contact technology. These systems make use of the radio measurements of short-range commodity wireless devices, which vary due to the inhalation and exhalation motion of a person. The success of respiration rate estimation using such measurements depends on the signal-to-noise ratio, which alters with properties of the person and with the measurement system. To date, no model has been presented that allows evaluation of different deployments or system configurations for successful breathing rate estimation. In this paper, a received signal strength model for respiration rate monitoring is introduced. It is shown that measurements in linear and logarithmic scale have the same functional form, and the same estimation techniques can be used in both cases. The model is numerically and empirically evaluated, and its properties are discussed in depth. The most important model implications are validated under varying signal-to-noise ratio conditions using the performances of three estimators: batch frequency estimator, recursive Bayesian estimator, and model-based estimator. The results are in coherence with the findings, and they imply that different estimators are advantageous in different signal-to-noise ratio regimes.

In this paper, we develop a novel method for approximate continuous-discrete Bayesian filtering. The projection filtering framework is exploited to develop accurate approximations of posterior distributions within parametric classes of probability distributions. This is done by formulating an ordinary differential equation for the posterior distribution that has the prior as initial value and hits the exact posterior after a unit of time. Particular emphasis is put on exponential families, especially the Gaussian family of densities. Experimental results demonstrate the efficacy and flexibility of the method.

We propose a deblurring method that incorporates gyroscope measurements into a convolutional neural network (CNN). With the help of such measurements, it can handle extremely strong and spatially-variant motion blur. At the same time, the image data is used to overcome the limitations of gyro-based blur estimation. To train our network, we also introduce a novel way of generating realistic training data using the gyroscope. The evaluation shows a clear improvement in visual quality over the state-of-the-art while achieving real-time performance. Furthermore, the method is shown to improve the performance of existing feature detectors and descriptors against the motion blur.

This letter proposes a new algorithm for Gaussian process classification based on posterior linearization (PL). In PL, a Gaussian approximation to the posterior density is obtained iteratively using the best possible linearization of the conditional mean of the labels and accounting for the linearization error. PL has some theoretical advantages over expectation propagation (EP): all calculated covariance matrices are positive definite and there is a local convergence theorem. In experimental data, PL has better performance than EP with the noisy threshold likelihood and the parallel implementation of the algorithms.

In this letter, we analyze certain student's t-filters for linear Gaussian systems with misspecified noise covariances. It is shown that under appropriate conditions, the filter both estimates the state and re-scales the noise covariance matrices in a Kullback-Leibler optimal fashion. If the noise covariances are misscaled by a common scalar, then the re-scaling is asymptotically exact. We also compare the student's t-filter scale estimates to the maximum-likelihood estimates. Simulations demonstrating the results on the Wiener velocity model are provided.

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.

Many computer vision and image processing applications rely on local features. It is well-known that motion blur decreases the performance of traditional feature detectors and descriptors. We propose an inertial-based deblurring method for improving the robustness of existing feature detectors and descriptors against the motion blur. Unlike most deblurring algorithms, the method can handle spatially-variant blur and rolling shutter distortion. Furthermore, it is capable of running in real-time contrary to state-of-the-art algorithms. The limitations of inertial-based blur estimation are taken into account by validating the blur estimates using image data. The evaluation shows that when the method is used with traditional feature detector and descriptor, it increases the number of detected keypoints, provides higher repeatability and improves the localization accuracy. We also demonstrate that such features will lead to more accurate and complete reconstructions when used in the application of 3D visual reconstruction.

This paper focusses on the formulation of numerical integration as an inferential task. To date, research effort has largely focussed on the development of Bayesian cubature, whose distributional output provides uncertainty quantification for the integral. However, the point estimators associated to Bayesian cubature can be inaccurate and acutely sensitive to the prior when the domain is high-dimensional. To address these drawbacks we introduce Bayes-Sard cubature, a probabilistic framework that combines the flexibility of Bayesian cubature with the robustness of classical cubatures which are well-established. This is achieved by considering a Gaussian process model for the integrand whose mean is a parametric regression model, with an improper prior on each regression coefficient. The features in the regression model consist of test functions which are guaranteed to be exactly integrated, with remaining degrees of freedom afforded to the non-parametric part. The asymptotic convergence of the Bayes-Sard cubature method is established and the theoretical results are numerically verified. In particular, we report two orders of magnitude reduction in error compared to Bayesian cubature in the context of a high-dimensional financial integral.

This letter presents the development of novel iterated filters and smoothers that only require specification of the conditional moments of the dynamic and measurement models. This leads to generalizations of the iterated extended Kalman filter, the iterated extended Kalman smoother, the iterated posterior linearization filter, and the iterated posterior linearization smoother. The connections to the previous algorithms are clarified and a convergence analysis is provided. Furthermore, the merits of the proposed algorithms are demonstrated in simulations of the stochastic Ricker map where they are shown to have a similar or superior performance to competing algorithms.

Anomalies in the ambient magnetic field can be used as features in indoor positioning and navigation. By using Maxwell's equations, we derive and present a Bayesian nonparametric probabilistic modeling approach for interpolation and extrapolation of the magnetic field. We model the magnetic field components jointly by imposing a Gaussian process (GP) prior to the latent scalar potential of the magnetic field. By rewriting the GP model in terms of a Hilbert space representation, we circumvent the computational pitfalls associated with GP modeling and provide a computationally efficient and physically justified modeling tool for the ambient magnetic field. The model allows for sequential updating of the estimate and time-dependent changes in the magnetic field. The model is shown to work well in practice in different applications. We demonstrate mapping of the magnetic field both with an inexpensive Raspberry Pi powered robot and on foot using a standard smartphone.

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.

Structure from motion algorithms have an inherent limitation that the reconstruction can only be determined up to the unknown scale factor. Modern mobile devices are equipped with an inertial measurement unit (IMU), which can be used for estimating the scale of the reconstruction. We propose a method that recovers the metric scale given inertial measurements and camera poses. In the process, we also perform a temporal and spatial alignment of the camera and the IMU. Therefore, our solution can be easily combined with any existing visual reconstruction software. The method can cope with noisy camera pose estimates, typically caused by motion blur or rolling shutter artifacts, via utilizing a Rauch-Tung-Striebel (RTS) smoother. Furthermore, the scale estimation is performed in the frequency domain, which provides more robustness to inaccurate sensor time stamps and noisy IMU samples than the previously used time domain representation. In contrast to previous methods, our approach has no parameters that need to be tuned for achieving a good performance. In the experiments, we show that the algorithm outperforms the state-of-the-art in both accuracy and convergence speed of the scale estimate. The accuracy of the scale is around 1% from the ground truth depending on the recording. We also demonstrate that our method can improve the scale accuracy of the Project Tango's build-in motion tracking.

Gaussian processes allow for flexible specification of prior assumptions of unknown dynamics in state space models. We present a procedure for efficient Bayesian learning in Gaussian process state space models, where the representation is formed by projecting the problem onto a set of approximate eigenfunctions derived from the prior covariance structure. Learning under this family of models can be conducted using a carefully crafted particle MCMC algorithm. This scheme is computationally efficient and yet allows for a fully Bayesian treatment of the problem. Compared to conventional system identification tools or existing learning methods, we show competitive performance and reliable quantification of uncertainties in the model.

We analyze the Lp-convergence of a previously proposed Girsanov theorem based particle filter for discretely observed stochastic differential equation (SDE) models. We prove the convergence of the algorithm with the number of particles tending to infinity by requiring a moment condition and a step-wise initial condition boundedness for the stochastic exponential process giving the likelihood ratio of the SDEs. The practical implications of the condition are illustrated with an Ornstein-Uhlenbeck model and with a non-linear Benes model.

In this paper, IMU and magnetometer models that enable pedestrian dead-reckoning without step detection or zero velocity updates on hand-held devices such as smartphones are proposed. The models are suitable for usage with any standard Bayesian filtering or smoothing technique and thus, do not require customized estimators. The method is evaluated in a real scenario using a dataset of approximately three minutes with a trajectory length of 302m. It is found that the overall shape of the estimated trajectory matches well with the GPS reference trajectory with an estimated trajectory length of 293m. Some problems in the heading estimation are observed.

The added flexibility of Student-t processes (TPs) over Gaussian processes (GPs) robustifies inference in outlier-contaminated noisy data. The uncertainties are better accounted for than in GP regression, because the predictive covariances explicitly depend on the training observations. For an entangled noise model, the canonical-form TP regression problem can be solved analytically, but the naive TP and GP solutions share the same cubic computational cost in the number of training observations. We show how a large class of temporal TP regression models can be reformulated as state space models, and how a forward filtering and backward smoothing recursion can be derived for solving the inference analytically in linear time complexity. This is a novel finding that generalizes the previously known connection between Gaussian process regression and Kalman filtering to more general elliptical processes and non-Gaussian Bayesian filtering. We derive this connection, demonstrate the benefits of the approach with examples, and finally apply the method to empirical data.

Localization in global navigation satellite system denied environments using inertial sensors alone, or radio sensors alone or a combination of both are the currently active research topics. The current research works are primarily focused on static environments with earth fixed coordinate frames, having nonmoving maps. In this research work, we use micro electromechanical sensors based inertial sensors, band pass filtering, particle filtering, maps and map matching techniques for pedestrian localization with respect to on ground moving platforms such as train or bus. Since these platforms are moving, the maps of such platforms are moving maps with respect to earth centered, earth fixed coordinate frames. The techniques of this research work could further be extended and adapted to other moving platforms such as airplanes, boats and submarines.

This paper is concerned with inertial-sensor-based tracking of the gravitation direction in mobile devices such as smartphones. Although this tracking problem is a classical one, choosing a good state-space for this problem is not entirely trivial. Even though for many other orientation related tasks a quaternion-based representation tends to work well, for gravitation tracking their use is not always advisable. In this paper we present a convenient linear quaternion-free state-space model for gravitation tracking. We also discuss the efficient implementation of the Kalman filter and smoother for the model. Furthermore, we propose an adaption mechanism for the Kalman filter which is able to filter out shot-noises similarly as has been proposed in context of adaptive and robust Kalman filtering. We compare the proposed approach to other approaches using measurement data collected with a smartphone.

This paper shows how periodic covariance functions in Gaussian process regression can be reformulated as state space models, which can be solved with classical Kalman filtering theory. This reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The representation is based on expanding periodic covariance functions into a series of stochastic resonators. The explicit representation of the canonical periodic covariance function is written out and the expansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is generalized to quasi-periodic covariance functions by introducing damping terms in the system and applied to two sets of real data. The approach could be easily extended to non-stationary and spatio-temporal variants.

In this paper we extend the L4proof of Hu et al. (2008) from bootstrap type of particle filters to particle filters with general importance distributions. The result essentially shows that with general importance distributions the particle filter converges provided that the importance weights are bounded. By numerical simulations we also show that this condition is often also a practical requirement for a good performance of a particle filter.

Stochastic differential equations (SDE) are a natural tool for modelling systems that are inherently noisy or contain uncertainties that can be modelled as stochastic processes. Crucial to the process of using SDE to build mathematical models is the ability to estimate parameters of those models from observed data. Over the past few decades, significant progress has been made on this problem, but we are still far from having a definitive solution. We describe a novel method of approximating a diffusion process that we show to be useful in Markov chain Monte-Carlo (MCMC) inference algorithms. We take the ‘white’ noise that drives a diffusion process and decompose it into two terms. The first is a ‘coloured noise’ term that can be deterministically controlled by a set of auxilliary variables. The second term is small and enables us to form a linear Gaussian ‘small noise’ approximation. The decomposition allows us to take a diffusion process of interest and cast it in a form that is amenable to sampling by MCMC methods. We explain why many state-of-the-art inference methods fail on highly nonlinear inference problems. We demonstrate experimentally that our method performs well in such situations. Our results show that this method is a promising new tool for use in inference and parameter estimation problems.

Abstract—This article is concerned with accurate discretization of linear analog filters such that the frequency response of the discrete time filter accurately matches that of the continuous time filter. The approach is based on formal reconstruction of the continuous time signal using Shannon’s interpolation theorem and numerical solving of the differential equation corresponding to the analog filter. When the formal continuous time system is sampled, the resulting filter reduces to discrete linear filter, which can be realized either as a state space model or as an IIR filter. The proposed methodology is applied to design of filters for parametric equalizers. Index Terms—analog filter, discretization, Shannon’s interpolation, differential equation, parametric equalizer

This article presents a classical type of solution to the time series prediction competition, the CATS benchmark, which is organized as a special session of the IJCNN 2004 conference. The solution is based on sequential application of the Kalman smoother, which is a classical statistical tool for estimation and prediction of time series. The Kalman smoother belongs to the class of linear methods, because the underlying filtering model is linear and the distributions are assumed as Gaussian. Since the time series model of the Kalman smoother assumes that the densities of noise terms are known, these are determined by cross-validation.

This article presents a classical type of solution to the time series prediction competition, the CATS benchmark, which is organized as a special session of the IJCNN 2004 conference. The solution is based on sequential application of the Kalman smoother, which is a classical statistical tool for estimation and prediction of time series. The Kalman smoother belongs to the class of linear methods, because the underlying filtering model is linear and the distributions are assumed as Gaussian. Since the time series model of the Kalman smoother assumes that the densities of noise terms are known, these are determined by cross-validation.

Bayesian MLP neural networks are a flexible tool in complex nonlinear problems. The approach is complicated by need to evaluate integrals over high-dimensional probability distributions. The integrals are generally approximated with Markov chain Monte Carlo (MCMC) methods. There are several practical issues which arise when implementing MCMC. This article discusses the choice of starting values and the number of chains in Bayesian MLP models. We propose a new method for choosing the starting values based on early stopping and we demonstrate the benefits of using several independent chains.