Gaussian Sum Filter Research Articles

Carrier Phase-based RAIM Algorithm Using a Gaussian Sum Filter

Carrier phase measurements are used to provide high-accuracy estimates of position. For safety-of-life navigation applications such as precision approach and landing, integrity plays a critical role. Carrier phase-based Receiver Autonomous Integrity Monitoring (CRAIM) has been investigated for many years (Pervan et al, 1998; Feng et al, 2007). Assuming that the carrier phase error has a Gaussian distribution, conventional CRAIM algorithms were directly derived from the Pseudorange-based RAIM (PRAIM). However, the actual carrier phase error does not exactly follow the Gaussian distribution, hence the performance of the conventional CRAIM algorithm is not optimal.To approach this problem, this paper proposes a new CRAIM algorithm that uses Gaussian sum filters. A Gaussian sum filter can deal with any non-Gaussian error distribution and accurately present the posterior distributions of states. In this paper, a new method of making a Gaussian mixture model, which follows the true error distribution, is introduced. Additionally an integrity monitoring algorithm, using a Gaussian sum filter, is described in detail. The simulation results show that the proposed algorithm can have about 18% smaller Minimum Detectable Bias (MDB) and generates about 20% lower protection levels than those of the conventional CRAIM algorithm. In other words, by considering a non-Gaussian carrier phase error distribution, the new algorithm can improve the accuracy and the availability.

MMSE-Based Filtering in Presence of Non-Gaussian System and Measurement Noise

The problem of sequential Bayesian estimation in linear non-Gaussian problems is addressed. In the Gaussian sum filter (GSF), the non-Gaussian system noise, the measurement noise, and the posterior state densities are modeled by the Gaussian mixture model (GMM). The GSF is optimal under the minimum-mean-square error (MMSE) criterion, however it is impractical due to the exponential model order growth of the system probability density function (pdf). The proposed recursive estimator, named the Gaussian mixture Kalman filter (GMKF), combines the GSF and the model order reduction procedure. The posterior state density at each iteration is approximated by a lower order density. This model order reduction procedure minimizes the estimated Kullback-Leibler divergence (KLD) of the reduced order density from the original density at each step. The estimation performance of the proposed GMKF is compared with the interactive multiple modeling (IMM), particle filter (PF), Gaussian sum PF (GSPF), and the GSF with mixture reduction (MR) method via simulations. It is shown in several examples that the proposed GMKF outperforms the other tested algorithms in terms of estimation accuracy. The superior estimation performance of the GMKF is obtained at the expense of its computational complexity, which is higher than the IMM and the MR algorithms.

A Gaussian Sum Filter Based on Stochastic Equivalent Linearization

Tn this paper, we propose a new Gaussian sum filter [1] based on the stochastic equivalent linearization technique [2] for suboptimal state estimation of discrete time nonlinear systems. The derived nonlinear filter will be expected to be a low computational cost and high quality for recursive state estimation.

Scaled unscented transform Gaussian sum filter: Theory and application

In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. We introduce a framework, called the scaled unscented transform Gaussian sum filter (SUT-GSF), which combines two ideas: the scaled unscented Kalman filter (SUKF) based on the concept of scaled unscented transform (SUT) (Julier and Uhlmann (2004) [16]), and the Gaussian mixture model (GMM). The SUT is used to approximate the mean and covariance of a Gaussian random variable which is transformed by a nonlinear function, while the GMM is adopted to approximate the probability density function (pdf) of a random variable through a set of Gaussian distributions. With these two tools, a framework can be set up to assimilate nonlinear systems in a recursive way. Within this framework, one can treat a nonlinear stochastic system as a mixture model of a set of sub-systems, each of which takes the form of a nonlinear system driven by a known Gaussian random process. Then, for each sub-system, one applies the SUKF to estimate the mean and covariance of the underlying Gaussian random variable transformed by the nonlinear governing equations of the sub-system. Incorporating the estimations of the sub-systems into the GMM gives an explicit (approximate) form of the pdf, which can be regarded as a “complete” solution to the state estimation problem, as all of the statistical information of interest can be obtained from the explicit form of the pdf (Arulampalam et al. (2002) [7]). In applications, a potential problem of a Gaussian sum filter is that the number of Gaussian distributions may increase very rapidly. To this end, we also propose an auxiliary algorithm to conduct pdf re-approximation so that the number of Gaussian distributions can be reduced. With the auxiliary algorithm, in principle the SUT-GSF can achieve almost the same computational speed as the SUKF if the SUT-GSF is implemented in parallel. As an example, we will use the SUT-GSF to assimilate a 40-dimensional system due to Lorenz and Emanuel (1998) [27]. We will present the details of implementing the SUT-GSF and examine the effects of filter parameters on the performance of the SUT-GSF.

Derivative-free estimation methods: New results and performance analysis

The derivative-free nonlinear estimation methods exploiting the Stirling’s interpolation and the unscented transformation for discrete-time nonlinear stochastic systems are treated. The divided difference and unscented filters, smoothers, and predictors based on the methods are introduced in the unified framework. The new relations among the first order Stirling’s interpolation, the second order Stirling’s interpolation, and the unscented transformation are derived and their impact on the covariance matrices of the state estimates of the corresponding filters is analysed. The theoretical results are illustrated and used for the explanation of the unexpected behaviour of the sigma point Gaussian sum filters given as a mixture of the derivative-free filters.

Constrained State Estimation for Nonlinear Systems with Non-Gaussain Noise

This paper addresses a state-estimation problem for nonlinear systems with non-Gaussian noise and an interval constraint on the state vector. We propose new efficient algorithms, which are based on Unscented Kalman filter(UKF) and Ensemble Kalman filter(EnKF). We use Truncated UKF in Gaussian sum filter(GSF) framework, which is named Constrained Unscented GSF. And we proposed an Efficient constrained EnKF, which does not require to solve complex optimization problem like a conventional method. The validity of the proposed methods is illustrated in numerical examples.

Spacecraft Attitude Estimation Using Adaptive Gaussian Sum Filter

This paper is concerned with improving the attitude estimation accuracy by implementing an adaptive Gaussian sum filter where the a posteriori density function is approximated by a sum of Gaussian density functions. Compared to the traditional Gaussian sum filter, this adaptive approach utilizes the Fokker-Planck-Kolmogorov residual minimization to update the weights associated with different components of the Gaussian mixture model. Updating the weights provides an accurate approximation of the a posteriori density function and thus superior estimates. Simulation results show that updating the weights during the propagation stage not only provides better estimates between the observations but also provides superior estimator performance where the measurements are ambiguous.

A pre-identification for electron reconstruction in the CMS particle-flow algorithm

In the CMS software, a dedicated electron track reconstruction algorithm, based on a Gaussian Sum Filter (GSF), is used. This algorithm is able to follow an electron along its complete path up to the electromagnetic calorimeter, even in the case of a large amount of Bremsstrahlung emission. Because of the significant CPU consumption of this algorithm, it can, however, only be run on a limited number of candidates. The standard GSF electron track reconstruction is triggered by the presence of high energy isolated electromagnetic clusters, but it is not suited for electrons in jets (usually soft and not isolated). A pre-identification algorithm based on both the tracker and the calorimeter, was therefore recently developed. It allows electron tracks within jets to be efficiently reconstructed even for small electron transverse momentum. This algorithm as well as its performance in terms of efficiency, mis-identification probability are presented.

A Gaussian Sum Filter Approach for DGNSS Integrity Monitoring

With conventional snapshot RAIM algorithms, it is difficult to detect small errors and simultaneous multiple faults. Assuming that we know the system dynamics, filtering algorithms, such as the Kalman filter, can provide better integrity-monitoring performance than the snapshot algorithms because the filter reduces the noise level of measurements. However, because the Kalman filter presumes that measurement noise and disturbance follow the Gaussian distribution, its performance might degrade if the assumption is wrong. To address this problem, we propose a fault detection and exclusion algorithm using Gaussian sum filters. Because GNSS measurement noise does not follow the Gaussian distribution perfectly, the Gaussian sum filter can estimate the posterior distribution more accurately; therefore it has better integrity-monitoring performance. This paper describes the detailed algorithms and shows simulation results to evaluate the integrity-monitoring performance of the algorithms. The proposed algorithms detect about 30% smaller faults and generate 35% lower protection levels than the conventional methods. The results show that the proposed algorithms can provide better accuracy and availability performance.

Recursive Sparse Estimation using a Gaussian Sum Filter

We develop a recursive estimator that systematically arrives at sparse parameter estimates. The algorithm is computationally feasible for moderate parameter estimation problems and leverages the Gaussian sum filter to provide both sparse parameter estimates and credible Bayesian intervals for non-zero parameters in a recursive fashion. Simulations show extremely promising accuracy, as well as a robustness not enjoyed by other sparse estimators.

Active State Estimation for Nonlinear Systems: A Neural Approximation Approach

In this paper, we consider the problem of actively providing an estimate of the state of a stochastic dynamic system over a (possibly long) finite time horizon. The active estimation problem (AEP) is formulated as a stochastic optimal control one, in which the minimization of a suitable uncertainty measure is carried out. Toward this end, the use of the Renyi entropy as an information measure is proposed and motivated. A neural control scheme, based on the application of the extended Ritz method (ERIM) and on the use of a Gaussian sum filter (GSF), is then presented. Simulation results show the effectiveness of the proposed approach.

Modified Gaussian Sum Filtering Methods for INS/GPS Integration

In INS (Inertial Navigation System) /GPS (Global Positioning System) integration, nonlinear models should be properly handled. The most popular and commonly used method is the Extended Kalman Filter (EKF) which approximates the nonlinear state and measurement equations using the first order Taylor series expansion. On the other hand, recently, some nonlinear filtering methods such as Gaussian Sum filter, particle filter and unscented Kalman filter have been applied to the integrated systems. In this paper, we propose a modified Gaussian Sum filtering method and apply it to land-vehicle INS/GPS integrated navigation as well as the in-motion alignment systems. The modification of Gaussian Sum filter is based on a combination of Gaussian Sum filter and so-called unscented transformation which is utilized in the unscented Kalman filter in order to improve the treatment of the nonlinearity in Gaussian Sum filter. In this paper, the performance of modified Gaussian Sum filter based integrated systems is compared with other filters in numerical simulations. From simulation results, it was found that the proposed filter can improve transient responses of the filter under large initial estimation errors.

Electron reconstruction in CMS

The reconstruction of the energy and momentum of isolated electrons in CMS combining tracking and electromagnetic calorimetry information is described. The emphasis is put on primary electrons with transverse momentum below 50 GeV/c. The energy deposited in the electromagnetic calorimeter is measured in clusters of clusters (superclusters) which collect bremsstrahlung photons emitted along the electron trajectory in the tracker volume. The electron tracks are built from seeds in the pixel detector found via a cluster-driven pixel hit matching algorithm, followed by a reconstruction of trajectories in the silicon strip tracker with a Gaussian sum filter. Electrons are classified using observables sensitive to the pattern of bremsstrahlung emission and electromagnetic showering in the tracker material. Energy scale corrections depending on the electron class are applied to the supercluster and estimates of associated errors are obtained. The electron energy is deduced from a weighted combination of the corrected supercluster energy and tracker momentum measurements. The electron direction is that of the reconstructed electron track at interaction vertex. The pre-selection of isolated electron candidates for physics analysis is described. Class-dependent observables combining tracking and calorimetry information are discussed for electron identification.

Discrimination Between Different Types of Material in Track Reconstruction With a Gaussian-Sum Filter

In the context of track reconstruction, the description of the material in a detector is frequently a simplified version of the detailed description used in the simulation. In particular, complex structures of different types of materials are often approximated by simple layers consisting of a material with average properties. Multiple scattering is then usually modeled by a Gaussian distribution, and energy loss by a narrow Gaussian or a delta function (Dirac distribution). More detailed information about the material can be taken into account by modeling multiple scattering and energy loss by mixtures of Gaussians and Dirac distributions, respectively. Each component of the mixture corresponds to a particular type of material, and the weight of the component is given by the probability to encounter this type of material. Track fitting using these mixtures can then be accomplished by the Gaussian-sum filter (GSF). Its performance is compared to a standard Kalman filter using the average material properties. The gain in precision of the estimated track parameters is studied in a series of simulation experiments, and the most promising scenarios are identified

Robust vertex fitters

While linear estimators are optimal when the model is linear and all random noise is Gaussian, they are very sensitive to outlying tracks. Non-linear vertex reconstruction algorithms offer a higher degree of robustness against such outliers. Two of the algorithms presented, the Adaptive filter and the Trimmed Kalman Filter are able to down-weight or discard these outlying tracks, while a third, the Gaussian-sum filter, offers a better treatment of non-Gaussian distributions of track parameter errors when these are modelled by Gaussian mixtures.

Nonlinear Filtering Methods for the INS/GPS In-Motion Alignment and Navigation

In this paper we present the algorithms of land-vehicle INS (Inertial Navigation System) / DGPS (Differential Global Positioning System) In-Motion Alignment based on nonlinear filtering techniques (Quasi-Linear Optimal filter [1], Gaussian Sum filter [2], the Extended Kalman filter [3] and Monte Carlo filter [4]). We also show results of comparative numerical experiments, and evaluate the nonlinear filtering performances under various error sources such as the errors of GPS, accelerometers and gyros.

A Gaussian-sum filter for vertex reconstruction

A vertex reconstruction algorithm that is based on the Gaussian-sum filter (GSF) was developed and implemented in the framework of the CMS reconstruction program. While linear least-square estimators are optimal in case all observation errors are Gaussian distributed, the GSF offers a better treatment of non-Gaussian distributions of track parameter errors when these are modeled by Gaussian mixtures. The algorithm has been verified and evaluated with simulated data. The results are compared to the Kalman filter and to an adaptive vertex estimator.

Application of adaptive filters to track finding

We study the application of adaptive methods such as the Gaussian-sum filter and the Deterministic Annealing Filter to track finding in various scenarios with large amounts of noise. It is shown in simulation studies that such adaptive methods are competitive alternatives to a combinatorial Kalman filter, and that in some cases there are appreciable gains in speed of the track finding procedure with respect to the Kalman filter.

Treatment of non-Gaussian tails of multiple Coulomb scattering in track fitting with a Gaussian-sum filter

If any of the probability densities involved in track fitting deviate from the Gaussian assumption, it is plausible that a non-linear estimator which better takes the actual shape of the distribution into account can do better. One such non-linear estimator is the Gaussian-sum filter, which is adequate if the distributions under consideration can be approximated by Gaussian mixtures. The main purpose of this paper is to present a Gaussian-sum filter for track fitting, based on a two-component approximation of the distribution of angular deflections due to multiple scattering. In a simulation study within a linear track model the Gaussian-sum filter is shown to be a competitive alternative to the Kalman filter. Scenarios at various momenta and with various maximum number of components in the Gaussian-sum filter are considered. Particularly at low momenta the Gaussian-sum filter yields a better estimate of the uncertainties than the Kalman filter, and it is also slightly more precise than the latter.

Reconstruction of electrons with the Gaussian-sum filter in the CMS tracker at the LHC

The bremsstrahlung energy loss distribution of electrons propagating in matter is highly non-Gaussian. Because the Kalman filter relies solely on Gaussian probability density functions, it is not necessarily the optimal reconstruction algorithm for electron tracks. A Gaussian-sum filter (GSF) algorithm for electron reconstruction in the CMS tracker has therefore been developed and implemented. The basic idea is to model the bremsstrahlung energy loss distribution by a Gaussian mixture rather than by a single Gaussian. It is shown that the GSF is able to improve the momentum resolution of electrons compared to the standard Kalman filter. The momentum resolution and the quality of the error estimate are studied both with a fast simulation, modelling the radiative energy loss in a simplified detector, and the full CMS tracker simulation.