Gaussian Sum Filter Research Articles

A Gaussian-sum filter for vertex reconstruction

A vertex reconstruction algorithm was developed based on the Gaussian-sum filter and implemented in the framework of the CMS reconstruction program. While linear least-squares estimators are optimal in case all observation errors are Gaussian-distributed, a GSF offers a better treatment of non-Gaussian distributions of track parameter errors when these are modeled by Gaussian mixtures. Results are compared to the Kalman filter.

Nonlinear continuous-discrete filtering using kernel density estimatesand functional integrals

We develop filter algorithms for nonlinear stochastic differential equations with discrete time measurements (continuous-discrete state space model). The apriori density (time update) is computed by Monte Carlo simulations of the Fokker-Planck equation using kernel density estimators and measurement updates are obtained by using the extended Kalman filter (EKF) updates. For small sampling intervals, a discretized continuous sampling approach (DCS) is used. A third algorithm utilizes a functional (path) integral representation of the transition density (functional integral filter FIF). The kernel density filter (KDF), DCS, and FIF are compared with the EKF and the Gaussian sum filter by using a Ginzburg-Landau-equation and a stochastic volatility model.

Gaussian filters for nonlinear filtering problems

We develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical tests demonstrate that new filters significantly improve the extended Kalman filter with no additional cost, and the new Gaussian sum filter has a nearly optimal performance.

An adaptive Gaussian sum algorithm for radar tracking

In this paper, we propose a new radar tracking algorithm based on the Gaussian sum filter. To alleviate the computational burden associated with the Gaussian sum filter, we have developed a new systematic and efficient way to approximate a non-Gaussian and measurement-dependent function by a weighted sum of Gaussian density functions and we have also suggested a way to alleviate the growing memory problem inherited in the Gaussian sum filter. Our method is compared with the extended Kalman filter (EKF) and the converted measurement Kalman filter (CMKF) and it is shown to be more accurate in term of position and velocity errors.

On the treatment of energy loss in track fitting

As the energy loss of electrons in matter is described by a strongly non-Gaussian distribution, the Kalman filter is not necessarily the optimal procedure for the fitting of electron tracks. We show that the momentum resolution can be improved by using nonlinear methods, such as the Gaussian-sum filter and the Metropolis—Hastings algorithm. We report results of a simple simulation experiment and comment on the respective merits of these methods.

Robust estimation iin time series: an approximation to the gaussian sum filter

This paper proposes a filter which can track the level of a time series robustly and adapt well to step jumps. The filter, called the approximate Gaussian sum filter (AGSF), is derived from the Gaussian sum filter by collapsing the terms in the normal mixtures. Besides producing one-step ahead forecasts robust towards additive and innovation outliers, the filter also estimates the scale parameters in the local level model separately from the variability caused by contamination. Simulation results show that the AGSF produces robust estimates of the hyperparameters regardless of the underlying distributions of the error terms. An example is included to illustrate the robust tracking of the series level by the AGSF as compared with the Kalman filter and an additive outlier filter.

Robust ε-contaminated gaussian filtering of discrete-time linear systems

A robust slate estimation of multi-input single-output discrete-time linear systems is considered, where both the system disturbance and observation noise sequences contain outliers. The robust estimation problem is mathematically formulated for a special case assuming that the samples of the system disturbance and observation noise are from a known e-contaminated gaussian density and a partially known e-contaminated gaussian density, respectively. Through Monte Carlo simulations, the performance of the proposed robust filter is compared with that of the gaussian sum filter, which is the best non-linear filter when the densities of the underlying uncertainties are completely known. Comparison is also made between the proposed filter and some other available candidates.