Gaussian State Space Models Research Articles

Gaussian Process-Integrated State Space Model for Continuous Joint Angle Prediction from EMG and Interactive Force in a Human-Exoskeleton System

As one of the most direct indicators of the transparency between a human and an exoskeleton, interactive force has rarely been fused with electromyography (EMG) in the control of human-exoskeleton systems, the performances of which are largely determined by the accuracy of the continuous joint angle prediction. To achieve intuitive and naturalistic human intent learning, a state space model (SSM) for continuous angle prediction of knee joint is developed. When the influence of the interactive force is often ignored in the existing models of human-exoskeleton systems, interactive force is applied as the measurement model output of the proposed SSM, and the EMG signal is used as the state model input signal to indicate muscle activation. The forward dynamics of joint motion and the human-machine interaction mechanism, i.e., the biomechanical interpretations of the interactive force generation mechanism, are derived as the bases for the state model and measurement model based on Hill’s muscle model and semiphenomenological (SP) muscular model, respectively. Gaussian process (GP)-based nonlinear autoregressive with the exogenous inputs (NARX) model and back-propagation neural network (BPNN) are applied to provide better adaptivity for the SSM in practical applications. Corresponding experimental results demonstrate the validity and superiority of the method.

Computing exact score vectors for linear Gaussian state space models

A recursive formula for computing the exact value of score vectors is proposed for a general form of the linear Gaussian state space model, which is more desirable than approximate values in some statistical analyses. Unlike most extant methods, our formula calculates all components of the score vector simultaneously. This approach significantly simplifies its programing, in particular, with some matrix-oriented programing languages, such as MATLAB. We also consider a way of handling initial conditions that depend on unknown parameters. This issue has not yet been explicitly addressed in the existing literature in the context of exact score computing for a general case, such as the one that we consider in this paper. It is also shown that our formula is especially useful for calculating score tests with an outer product of gradient asymptotic covariance matrix estimator.

Marginalized Particle Filtering and Related Filtering Techniques as Message Passing

In this paper, a factor graph approach is employed to investigate the recursive filtering problem for conditionally linear Gaussian state-space models. First, we derive a new factor graph for the considered filtering problem; then, we show that applying the sum-product rule to our graphical model results in both known and novel filtering techniques. In particular, we prove that: 1) marginalized particle filtering can be interpreted as a form of forward only message passing over the devised graph; 2) novel filtering methods can be easily developed by exploiting the graph structure and/or simplifying probabilistic messages.

A New Adaptive Kalman Filter with Inaccurate Noise Statistics

In this paper, a new adaptive Kalman filter is proposed for a linear Gaussian state-space model with inaccurate noise statistics based on the variational Bayesian (VB) approach. Both the prior joint probability density function (PDF) of the one-step prediction and corresponding prediction error covariance matrix and the joint PDF of the mean vector and covariance matrix of measurement noise are selected as Normal-inverse-Wishart (NIW), from which a new NIW-based hierarchical Gaussian state-space model is constructed. The state vector, the one-step prediction and corresponding prediction error covariance matrix, and the mean vector and covariance matrix of measurement noise are jointly estimated based on the constructed hierarchical Gaussian state-space model using the VB approach. Simulation results show that the proposed filter has better estimation accuracy than existing state-of-the-art adaptive Kalman filters.

Rao–Blackwellized Gaussian Smoothing

In this paper, we consider Rao–Blackwellization of linear substructures in sigma-point-based Gaussian assumed density smoothers. We derive marginalized prediction, smoothing, and update steps for the mixed linear/nonlinear Gaussian state-space model as well as for a hierarchical model for both conventional and iterated posterior linearization Gaussian smoothers. The proposed method is evaluated in a numerical example and it is shown that the computational complexity is reduced considerably compared to non-Rao–Blackwellized Gaussian smoothers for systems with high-dimensional linear subspaces.

Uncertainty-based Human Motion Tracking with Stable Gaussian Process State Space Models

Abstract Data-driven approaches are well suited to represent human motion because arbitrary complex trajectories can be captured. Gaussian process state space models allow to encode human motion while quantifying uncertainty due to missing data. Such human motion models are relevant for many application domains such as learning by demonstration and motion prediction in human-robot collaboration. For goal-directed tasks it is essential to impose stability constraints on the model representing the human motion. Motivated by learning by demonstration applications, this paper proposes an uncertainty-based control Lyapunov function approach for goal-directed path tracking. We exploit the model fidelity which is related to the location of the training and test data: Our approach actively strives into regions with more demonstration data and thus higher model certainty. This achieves accurate reproduction of the human motion independent of the initial condition and we show that generated trajectories are uniformly globally asymptotically stable. The approach is validated in a nonlinear learning by demonstration task where human-demonstrated motions are reproduced by the learned dynamical system, and higher precision than competitive state of the art methods is achieved.

A New Robust Kalman Filter for SINS/DVL Integrated Navigation System

A new robust strap-down inertial navigation system (SINS) and Doppler velocity log (DVL) integrated navigation algorithm are proposed in this paper with a focus on suppressing the process uncertainty and measurement outliers induced by severe manoeuvering. In the proposed algorithm, the one-step predicted probability density function is modeled as Student's t-distribution to deal with the heavy-tailed process noise, and hierarchical Gaussian state-space model for SINS/DVL integrated navigation algorithm is constructed. To detect and eliminate the measurement outliers, each measurement is marked by a binary indicator variable modeled as a beta-Bernoulli distribution. The variational Bayesian approach is used to jointly estimate state vector, auxiliary random variable, scale matrix, Bernoulli variable, and beta variable. The experimental results illustrate that the proposed algorithm has better robustness and navigation accuracy to deal with process uncertainty and measurement outliers than existing state-of-the-art algorithms.

Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. We employ a step-by-step approach to develop an implementation of the PMH algorithm (and the particle filter within) together with the reader. This final implementation is also available as the package pmhtutorial from the Comprehensive R Archive Network (CRAN) repository. Throughout the tutorial, we provide some intuition as to how the algorithm operates and discuss some solutions to problems that might occur in practice. To illustrate the use of PMH, we consider parameter inference in a linear Gaussian state-space model with synthetic data and a nonlinear stochastic volatility model with real-world data.

Latent Parameter Estimation in Fusion Networks Using Separable Likelihoods

Multisensor state-space models underpin fusion applications in networks of sensors. Estimation of latent parameters in these models has the potential to provide highly desirable capabilities such as network self-calibration. Conventional solutions to the problem pose difficulties in scaling with the number of sensors due to the joint multisensor filtering involved when evaluating the parameter likelihood. In this paper, we propose a separable pseudo-likelihood which is a more accurate approximation compared to a previously proposed alternative under typical operating conditions. In addition, we consider using separable likelihoods in the presence of many objects and ambiguity in associating measurements with objects that originated them. To this end, we use a state-space model with a hypothesis-based parameterization, and develop an empirical Bayesian perspective in order to evaluate separable likelihoods on this model using local filtering. Bayesian inference with this likelihood is carried out using belief propagation on the associated pairwise Markov random field. We specify a particle algorithm for latent parameter estimation in a linear Gaussian state-space model and demonstrate its efficacy for network self-calibration using measurements from noncooperative targets in comparison with alternatives.

An Application of a Linear Gaussian State Space Model and the Kalman Filter In Estimating Expectations of the Natural Logarithmic Monthly Market Returns of the US 1838 Bond Debenture Closed-End Fund

In this article, we have tested a linear Gaussian state space model and the kalman filter in testing ARMA(2,3) models of the natural logarithmic monthly market returns of the US 1838 bond debenture closed-end fund. The aim is to estimate expectations that arises from the interaction of arbitrageurs and noise traders. The state space model is composed from the observation and the state equation. It is analyzed by using the Kalman recursive algorithm filter to calculate finite sample forecasts for Gaussian ARMA models. We have found that The AR coefficients as shown in terms of C(6) and C(7) are statistically significant at 5% significance level. The MA coefficients as displayed by C(2), C(3), C(4) are not statistically significant at 5% significance level. The variance of the error is denoted by C(5) and it is statistically significant as the p-value is 0.0000. The Final one – step ahead values of the state vector, SV1, SV2, SV3 and SV4 are not statistically significant at the 5% significance level, as all the p-values are above the 5% level. We did not found unanticipated changes in the share prices of the closed-end fund. The total dataset includes monthly data starting from 31/05/1981 to 31/10/2004 and total to 282 observations. The total data for the natural logarithmic returns of the market prices are accounted to 281 observations. The data was obtained from Thomson Financial Investment View database.

Particle Smoothing for Conditionally Linear Gaussian Models as Message Passing Over Factor Graphs

In this paper, the fixed-lag smoothing problem for conditionally linear Gaussian state-space models is investigated from a factor graph perspective. More specifically, after formulating Bayesian smoothing for an arbitrary state-space model as a forward–backward message passing over a factor graph, we focus on the above-mentioned class of models and derive two novel particle smoothers for it. Both the proposed techniques are based on the well-known two-filter smoothing approach and employ marginalized particle filtering in their forward pass. However, on the one hand, the first smoothing technique can only be employed to improve the accuracy of state estimates with respect to that achieved by forward filtering. On the other hand, the second method, that belongs to the class of Rao–Blackwellized particle smoothers, also provides a point mass approximation of the so-called joint smoothing distribution. Finally, our smoothing algorithms are compared, in terms of estimation accuracy and computational requirements, with a Rao–Blackwellized particle smoother recently proposed by Lindsten et al. (“Rao–Blackwellized particle smoothers for conditionally linear Gaussian models,” IEEE J. Sel. Topics Signal Process. , vol. 10, no. 2, pp. 353–365, 2016).

Tag-aware dynamic music recommendation

We present a tag-aware dynamic music recommendation framework that achieves personalized and accurate music recommendations to users. The proposed framework leverages the available semantic labels (in terms of tags) of music tracks to complement a highly sparse user-item interaction matrix, which effectively addresses the data sparsity issue faced by most music recommendation systems. Music tracks are more accurately represented by aggregating the latent factors derived from both the tag space and the user interaction information. The proposed framework further employs a Gaussian state-space model to capture the evolving nature of users’ preferences over time, which helps achieve time-sensitive recommendation of music. A variational approximation is developed to achieve fast inference and learning of model parameters. Experiments conducted using actual music data and comparison with state-of-the-art competitive recommendation algorithms help demonstrate the effectiveness of the proposed framework.

An efficient Bayesian experimental calibration of dynamic thermal models

Experimental calibration of dynamic thermal models is required for model predictive control and characterization of building energy performance. In these applications, the uncertainty assessment of the parameter estimates is decisive; this is why a Bayesian calibration procedure (selection, calibration and validation) is presented. The calibration is based on an improved Metropolis-Hastings algorithm suitable for linear and Gaussian state-space models. The procedure, illustrated on a real house experiment, shows that the algorithm is more robust to initial conditions than a maximum likelihood optimization with a quasi-Newton algorithm. Furthermore, when the data are not informative enough, the use of prior distributions helps to regularize the problem.

A Novel Adaptive Kalman Filter With Inaccurate Process and Measurement Noise Covariance Matrices

In this paper, a novel variational Bayesian (VB)-based adaptive Kalman filter (VBAKF) for linear Gaussian state-space models with inaccurate process and measurement noise covariance matrices is proposed. By choosing inverse Wishart priors, the state together with the predicted error and measurement noise covariance matrices are inferred based on the VB approach. Simulation results for a target tracking example illustrate that the proposed VBAKF has better robustness to resist the uncertainties of process and measurement noise covariance matrices than existing state-of-the-art filters.

Learning Robot Dynamics from Raw Image Time Series using Kalman Variational Auto-Encoders

Learning dynamical models from high dimensional time series data is an crucial research topic for a variety of applications. Kalman Variational Auto-Encoders (KVAE) is one of the recent methods, which is composed of VAE and Linear Gaussian State Space Models (LGSSM). In this paper, the predictive performance of KVAE is investigated with control inputs through a 2DOF robot arm reaching task in simulations.

Application of a Linear Gaussian State Space Model and the Kalman Filter in Estimating Expectations of the Natural Logarithmic Monthly Nav Returns of the Open – End Funds of Major Investment Banks.

In this article, we have tested a linear Gaussian state space model and the Kalman filter in testing ARMA(2,4) model of the natural logarithmic monthly market returns of the of the open – end funds of major investment banks such as: Merrill Lynch small cap value A, Fidelity equity income I, Goldman Sachs capital growth A, JPMorgan capital growth A, Invesco value equity investment, and Credit Suisse blue chip A. The purpose of this article is to estimate expectations that arises from the interaction of arbitrageurs and noise traders. The software that we have used is EViews 6. The state space model is composed from the observation and the state equation. It is analyzed by using the Kalman recursive algorithm filter to calculate finite sample forecasts for Gaussian ARMA models. We have found that the natural logarithmic monthly NAV returns of all open-end funds are a stationary series. The natural logarithmic monthly NAV returns for all open-end funds have no serial correlation and are not normally distributed. We have found that the AR coefficients as shown in terms of C(7) and C(8) are not statistically significant at the 5% significance level for Merrill Lynch small cap value A, Fidelity equity income I. The AR coefficient, C(7) is statistically significant and the C(8) coefficient is not statistically significant for Invesco value equity investment, Credit Suisse blue chip A, JPMorgan capital growth A, and Goldman Sachs capital growth A. The MA coefficient C(2) of the Goldman Sachs capital growth A, and Credit Suisse blue chip A, is statistically significant and C(3), C(4), and C(5) are not statistically significant at the 5% significance level. The MA coefficients as displayed by C(2), C(3), C(4), and C(5) are not statistically significant at the 5% significance level for the remaining open-end funds, namely, Merrill Lynch small cap value A, Fidelity equity income I, JPMorgan capital growth A, and Invesco value equity investment. The variance of the error is denoted by C(6) and it is statistically significant for all open-end funds as the p-value is 0.0000. The Final one – step ahead values of the state vector, SV1, SV2, SV3,SV4, SV5 are not statistically significant for all open-end funds except Merril Lynch at the 5% significance level. We did not found unanticipated changes in the NAV prices of the open-end funds. The whole dataset is from 31/01/1990 to 31/12/2001. The total observations are 144 and the logarithmic monthly NAV returns observations are 143. The data was obtained from Thomson Financial Investment View database.

Coupling stochastic EM and approximate Bayesian computation for parameter inference in state-space models

We study the class of state-space models and perform maximum likelihood estimation for the model parameters. We consider a stochastic approximation expectation–maximization (SAEM) algorithm to maximize the likelihood function with the novelty of using approximate Bayesian computation (ABC) within SAEM. The task is to provide each iteration of SAEM with a filtered state of the system, and this is achieved using an ABC sampler for the hidden state, based on sequential Monte Carlo methodology. It is shown that the resulting SAEM-ABC algorithm can be calibrated to return accurate inference, and in some situations it can outperform a version of SAEM incorporating the bootstrap filter. Two simulation studies are presented, first a nonlinear Gaussian state-space model then a state-space model having dynamics expressed by a stochastic differential equation. Comparisons with iterated filtering for maximum likelihood inference, and Gibbs sampling and particle marginal methods for Bayesian inference are presented.

Forecasting Calendar Futures Spreads of Crude Oil Using Kalman Filter

The aim of this project is to forecast futures spreads of WTI Crude Oil. The motivation for this project springs from the fact that trading with calendar futures spreads is much more advantageous than trading with many other financial instruments. We make use of the fact that futures prices follow the mean-reverting process (Ornstein-Uhlenbeck process, OU). We develop a method, which was first proposed by Islyaev (2014) and the approach then extended by Sadik et al. (2020), that combines three linear Gaussian state space models, namely one factor model, one factor model with risk premium, and one factor model with seasonality. Thereafter, we directly model futures spreads. Kalman filter and the Maximum Likelihood Estimate (MLE) are used to estimate the model parameters. It is shown that this new approach, using the ratio between the nearest prices over spot prices as a latent variable and calendar futures spreads vector as the observed variable, is more accurate and robust than the indirect forecasting method which inputs both spot prices and futures prices as the latent variable and the observed variable respectively. Results on calibration and comparison for three models and two methods, as well as out-of-sample forecasting results are then presented and discussed.

A New Outlier-Robust Student's t Based Gaussian Approximate Filter for Cooperative Localization

In this paper, a new outlier-robust Student's t based Gaussian approximate filter is proposed to address the heavy-tailed process and measurement noises induced by the outlier measurements of velocity and range in cooperative localization of autonomous underwater vehicles (AUVs). The state vector, scale matrices, and degrees of freedom (DOF) parameters are jointly estimated based on the variational Bayesian approach by using the constructed Student's t based hierarchical Gaussian state-space model. The performances of the proposed filter and existing filters are tested in the cooperative localization of an AUV through a lake trial. Experimental results illustrate that the proposed filter has better localization accuracy and robustness than existing state-of-the-art outlier-robust filters.

Particle rejuvenation of Rao-Blackwellized sequential Monte Carlo smoothers for conditionally linear and Gaussian models

This paper focuses on sequential Monte Carlo approximations of smoothing distributions in conditionally linear and Gaussian state spaces. To reduce Monte Carlo variance of smoothers, it is typical in these models to use Rao-Blackwellization: particle approximation is used to sample sequences of hidden regimes while the Gaussian states are explicitly integrated conditional on the sequence of regimes and observations, using variants of the Kalman filter/smoother. The first successful attempt to use Rao-Blackwellization for smoothing extends the Bryson-Frazier smoother for Gaussian linear state space models using the generalized two-filter formula together with Kalman filters/smoothers. More recently, a forward-backward decomposition of smoothing distributions mimicking the Rauch-Tung-Striebel smoother for the regimes combined with backward Kalman updates has been introduced. This paper investigates the benefit of introducing additional rejuvenation steps in all these algorithms to sample at each time instant new regimes conditional on the forward and backward particles. This defines particle-based approximations of the smoothing distributions whose support is not restricted to the set of particles sampled in the forward or backward filter. These procedures are applied to commodity markets which are described using a two-factor model based on the spot price and a convenience yield for crude oil data.