Maximum likelihood mixture modeling for three-dimensional non-Gaussian measurements

In the case of a non-linear system, the dynamic state of the targets (position, velocity, and acceleration) is estimated by an extended Kalman filter (EKF). The theory of EKF is established on the assumption that measurements follow Gaussian distribution. However, in practice, this assumption falls short and limits the application of EKF. In literature, to deal with the non-Gaussianity, the maximum correntropy criterion (MCC)-based EKF (EKF-MCC) has been studied well. The MCC, an information-theoretic criterion, claims to effectively deal with the system's non-Gaussianity. Nevertheless, like EKF, EKF-MCC also approximates the known system non-linearity with a Jacobian. The Jacobian provides the first-order approximation of the non-linearity and hinders the estimation accuracy achieved by EKF-MCC, particularly for complex target motion models. Therefore, in this work, firstly, we propose to use EKF-MCC for estimating the dynamic state of the target from non-Gaussian measurement. After that, utilizing MCC, we propose reproducing kernel Hilbert space (RKHS) based non-linear estimation of system non-linearity and using it with EKF-MCC. Amid non-linear estimation utilizing MCC, the proposed filter is named EKF-MCC-RKHS. The simulation performed to estimate the dynamic states of the complex constant acceleration (CA) target motion model validates the superiority of EKF-MCC-RKHS over recently introduced EKF-MCC and traditional EKF.

The stochastic nature of noise signals affects the vehicle’s internal states and the outputs, resulting in the difficulty in estimation. The unknown or time-varying nature of noise signals if not taken into account for estimation, the results will diverge and be highly deteriorated. In this paper, a maximum likelihood principle (MLP) based adaptive robust extended kalman filter for estimating the states of the adopted non-linear vehicle model is proposed. An observability test is done for the purpose of estimation. A covariance matching (CM) based robust adaptive high forgetting factor based fault tolerant technique is also employed on the robust adaptive extended and unscented kalman filters for comparison purpose. The Robustness of the filter is analyzed by varying the parameter of the vehicle through a local sensitivity analysis. The results show that the MLP based approach to the extended kalman filter performs well in three simulations for sinusoidal steering, Double Lance Change, J-Turn, Fishhook, Slalom maneuver in comparison to robust adaptive unscented kalman filter. Friction coefficient of 0.8 (dry road) and 0.4 (wet road) is chosen for the simulation. The sideslip angle RMSE value for MLP based estimation is obtained as 2.62e-05, 4.545e-06 for Sine and DLC maneuver.

Three-dimensional ground reaction forces (3D GRFs) in various gait patterns (e.g., walking and turning) provide essential information for clinical assessment. Previous scholars proposed to estimate GRFs from insoles with few pressure sensors using machine learning techniques. However, estimating GRFs during turning using a model learned from walking proved difficult because relationships between plantar pressure and GRFs vary according to the gait pattern. In this study, GRFs were estimated based on gait pattern recognition. GRF estimation models were learned from various gait patterns in advance. These models represented relationships between the insole measurements and GRFs using a Gaussian process regression (GPR). The GRFs were estimated from the insole measurements using a maximum likelihood (ML) model recognized as the current gait pattern. The ML model had the largest time frames, in which its standard deviation (SD) of the probabilistically estimated vertical GRF was the smallest among all the models (i.e., its training data were closest to the inputs), in the initial and terminal stance phases (subphases). In addition, the time frames of the contralateral leg were considered to enhance the recognition accuracy based on interlimb coordination. Experiment results showed that the proposed system, which used a common model learned from all the gait patterns for the vertical GRF and the ML model in the subphases of both legs for the horizontal GRFs, estimated 3D GRFs during walking and turning with different curvatures and directions with higher accuracy. The system can be widely applied for clinical walking tests and monitoring in daily life.

A new filtering method is proposed to accurately estimate target state via decreasing the nonlinearity between radar polar measurements (or spherical measurements in three-dimensional (3D) radar) and target position in Cartesian coordinate. The degree of linearity is quantified here by utilizing correlation coefficient and Taylor series expansion. With the proposed method, the original measurements are converted from polar or spherical coordinate to a carefully chosen Cartesian coordinate system that is obtained by coordinate rotation transformation to maximize the linearity degree of the conversion function from polar/spherical to Cartesian coordinate. Then the target state is filtered along each axis of the chosen Cartesian coordinate. This method is compared with extended Kalman filter (EKF), Converted Measurement Kalman filter (CMKF), unscented Kalman filter (UKF) as well as Decoupled Converted Measurement Kalman filter (DECMKF). This new method provides highly accurate position and velocity with consistent estimation.

Tracking a ballistic target in its reentry mode by considering the radar measurements is a highly complex problem in nonlinear filtering. Kalman Filter (KF) is used to estimate the position of target when the measurements are corrupted with noise. If the measurements are nonlinear (radar measurements) then Extended kalman filter (EKF) is used. For obtaining reliable estimate of the target state, filter has to be tuned before the operation which is offline. Tuning an EKF is the process of estimating the process noise covariance matrix (Q) and measurement noise covariance matrix (R). This paper presents a new method of tuning the EKF using different evolutionary algorithms.

Within the general framework of chemical risk assessment, a difficult step in dietary exposure assessment is the handling of concentration data reported to be below the limit of detection (LOD). These data are known as non-detects and the resulting distribution of occurrence values is left-censored. Handling left-censored data represents a challenge for EFSA?s collection and statistical analysis of chemical occurrence data. EFSA has so far treated left-censored data with widely used substitution methods recommended by international organisations. The appropriateness of this approach has a natural limitation in the computation of percentiles and in the application of statistical techniques. An EFSA working group was established to estimate the accuracy of methods currently used and to propose recommendations for more advanced alternative statistical approaches. Based on a simulation study and on analyses of real data, an ad hoc evaluation was carried out to assess the performance of different statistical methods to handle non-detects, i.e. parametric Maximum likelihood (ML) models, the log-probit regression method and the non-parametric Kaplan-Meier (KM) method. Results showed that the number of samples had a relatively limited impact on the accuracy and precision of estimates, but the degree of censoring had a large effect. When analysing a complex set of data, it was also shown that it is essential to identify possible sources of heterogeneity in a dataset, such as country of sample collection/origin, food group, laboratory, etc. Statistical analyses should either be conducted separately from these factors, or, to explicitly account for this heterogeneity, fixed/random effect ML models could be used. Based on a minimum number of available samples and to different values of censoring percentages, the working group outlined recommendations, including the use of appropriate statistical tests, to handle left-censored distributions of chemical contaminant data in the context of exposure assessment.

In the work presented here, we consider state and parameter estimation of a semi-nonlinear Markov jump system in a non-Gaussian noise environment. The non-Gaussian measurement noise is approximated by a finite Gaussian mixture model (GMM). We propose a maximum likelihood (ML) solution to this state estimation problem which leads to two expectation-maximization (EM) algorithms. The first is a batch EM method which takes all the available data in the conditional expectation of the state in the E-step. An interacting multiple model (IMM) smoother is employed to evaluate the conditional expectation of the state by which a suboptimal estimate of system state is directly obtained. The Gaussian mixture parameters are then updated in the M-step. The second is a recursive EM algorithm which results from a stochastic approximation procedure and uses a standard IMM filter. For performance evaluation, posterior Cramer-Rao bound (PCRB) on the state estimation is adopted. Simulation results verify the effectiveness of the proposed algorithms.

In radar tracking the target motion is best modeled in Cartesian coordinates. Its position is however measured in polar coordinates (range and azimuth). Tracking in Cartesian coordinates with noisy polar measurements requires either converting the measurements to a Cartesian frame of reference and then applying the linear Kalman filter to the converted measurement or using the extended Kalman filter (EKF) in mixed coordinates. The first approach is accurate only for moderate cross-range errors; the second approach is consistent only for small errors. A new efficient tracking algorithm using the multidimensional Gauss-Hermite quadratures to propagate the mean and the covariance of the conditional probability density function is presented. This method is compared with the EKF and the converted measurement Kalman filter (CMKF) and it is shown to be more accurate.

Data fusion algorithms are inevitable in improving the accuracy and reliability of radar target tracking. Considering the shortcomings of target tracking in Cartesian coordinates, all the fusion process are carried out in spherical coordinate system. To fuse data from bidimensional radar and Infrared Search and Track (IRST), five data fusion methods, Bayesian fusion, Signal variance based fusion, Optimal fusion, Error variance based fusion and Extended Kalman Filter (EKF) based fusion are proposed and analyzed. The performances of the methods are analyzed in terms of Root Mean Square Error in Position (RMSPE) and Mean Absolute Error in Position (MAPE) for a wide range of measurement noise variances. It is concluded that EKF based fusion has the best error performance. Kalman Filter State-Vector Fusion (KF-SVF) which is a hierarchical fusion architecture, having first level of fusion with Kalman filter and second level with error variance based fusion method, is available in the literature. In this work it has been expanded to Extended Kalman Filter—State-Vector Fusion (EKF-SVF) with either error variance based fusion, signal variance based fusion or optimal information fusion as second level. All the three EKF-SVF methods have similar error characteristics, but EKF-SVF with signal variance based fusion, unlike the other two, provides less computational cost since it does not involve matrix inversion. While comparing EKF based fusion and EKF-SVF, EKF-SVF offers increased error minimization.

The Extended Kalman Filter (EKF) has been the state of the art in integrated navigation systems and especially in Pedestrian Dead-Reckoning PDR for foot-mounted Inertial Measurements Units. However in most related work with PDR, indirect filtering approach is used based on linear error Kalman Filter (KF). In this work, it is proposed to outperform this approach by the use of Direct filtering approach, which involves the non-linearity in the propagation of the orientation, velocity and in some models, in position coordinates, where the EKF can not achieve optimal estimation. We propose then, the usage of the most recent algorithms developed in the last decade; Sigma Point Kalman Filters (SPKF), especially based on Cubature rule, called Cubature Kalman Filter (CKF) as the integration algorithm for the inertial measurements. The CKF improves the mean and covariance propagation consequently comparing with EKF and previous SPKF (UKF, CDKF). Although the CKF provides a better estimate of the orientation, velocity and position with Zero velocity UPdaTes (ZUPT) measurements and Zero Angular Rate UpdaTes (ZARUT), additional sensors are necessary to measure other states such as yaw angle and to estimate properly the gyroscope bias. We studied then the possibility to integrate electronic compass as additional measure and also Map street data base or Map building data base depending on the type of navigation "Indoor", "Outdoor". In order to get much better estimation based on the Cubature rule, it is proposed to synthesize CKF in order to get robust estimation against nonlinearity of the process and the multiple measurement sensors.

A particle filter guided by iterated extended Kalman filter (IEGPF) is proposed to solve the problem of particle impoverishment. Firstly, particles are divided into two groups adaptively according to the maximum likelihood ratio, which is defined to measure how well the particles, drawn from transition prior density, match the likelihood model. Particles in one group are drawn from transition prior density, while those in the other group are drawn from the Gaussian approximate posterior density, obtained by running an iterated extended Kalman filter (IEKF). The results of numerical simulations, conducted at different observation noise level, show that IEGPF can produce more accurate estimate than extended Kalman particle filter (EPF) and unscented Kalman particle filter (UPF), whereas the computational cost is comparable to that of SIR.

T HIS Note describes a robust modification of the divided difference filtering technique. The robust technique relies on Huber’s generalized maximum likelihood approach to estimation [1]. Specifically,Huber’smethod is a combinedminimum ‘1and ‘2norm estimation technique, which exhibits robustnesswith respect to deviations from the commonly assumed Gaussian error probability density functions, for which the least–squares or minimum ‘2-norm technique exhibits a severe degradation in estimation accuracy [2]. TheHuber-based estimates are robust in the sense that theyminimize the maximum asymptotic estimation variance when applied to contaminated Gaussian densities. The Huber technique was originally developed as a generalization of maximum likelihood estimation, applied first to estimating the center of a probability distribution in [1] and generalized tomultiple linear regression in [2– 4]. The Kalman filter is a recursive minimum ‘2-norm technique and therefore exhibits sensitivity to deviations in the true underlying error probability distributions [5]. For this reason, theHuber technique has been further extended to dynamic estimation problems. Boncelet and Dickinson [6] first proposed to solve the robust filtering problem by means of the Huber technique at each measurement, by expressing the discrete-time Kalman filter as a sequence of linear regression problems. The authors do not provide any simulation results to validate the proposed technique. Kovacevic et al. [7] follow thework of [6] and develop a robust Kalman filter using the Huber technique applied to a linear regression problem at each measurement update. References [8–10] express the dynamic filtering problem as a sequential linear regression to be solved by the Huber technique and apply the filter to underwater vehicle tracking, power system state estimation, and spacecraft rendezvous navigation, respectively. The increase in computation due to the use of the Huber technique was found in [10] to be small. It should be noted that [6–10] apply the Huber methodology to linearized filters. The divided difference filter is one of several new estimation techniques that are collectively known as sigma-point Kalman filters (SPKF). The first-order (DD1) and second-order (DD2) divided difference filters [11,12] are generalizations of the filter introduced by Schei [13] and are two examples of SPKF-class estimators; other examples can be found in [14–16]. Like the basic Kalman filter, the SPKFs seek to determine a state estimate that minimizes the ‘2 norm of the residuals. The SPKF technique differs from the standard Kalmanfilter in the sense that the SPKFs do not linearize the dynamic system for the propagation, but instead propagate a cluster of points centered around the current estimate to form improved approximations of the conditional mean and covariance. Specifically, the divided difference filters make use of multidimensional interpolation formulas to approximate the nonlinear transformations. As a result of this approach, the filter does not require knowledge or existence of the partial derivatives of the system dynamics and measurement equations. SPKFs have the additional advantage over the basic Kalman filter in that they can easily be extended to determine second-order solutions to the minimum ‘2-norm filtering problem, which increases the estimation accuracywhen the system andmeasurement equations are nonlinear. It is important to note that the SPKFs use a minimum ‘2-norm measurement update and are therefore subject to the same sensitivity to non-Gaussian measurement errors as the Kalman filter. Therefore, the purpose of this Note is tomodify the DD1 andDD2measurement update equations by making use of the Huber technique to provide robustness against deviations from Gaussianity without a large increase in computation. This Note first provides a short review of the DD1 and DD2 filters and then shows how the measurement update can be expressed in terms of a standard regression problem, which can be solved using the robust Huber technique. The filtering techniques are then applied to a benchmark problem that involves estimating the trajectory of an entry body from discrete-time range data measured by a radar tracking station. The simulation is conducted using Monte Carlo techniques for both Gaussian and non-Gaussian cases. The computational cost associated with each filter is provided.

Particle filters can provide better results than Kalman filter based algorithms in the problems where the nonlinear elements are dominant. In this study the problem of tracking an airborne platform using radar measurements, which has nonlinearities both in measurement and system models is considered. Different models are used for different target maneuvers; extended Kalman filter (EKF), interacting multiple model (IMM) structure and multiple model particle filter are compared on a sample scenario.

Effect of Noises on the Performance of Nonlinear State Estimator for Semi Batch Autocatalytic Esterification Reactor

Kalman filter tracking in a Cherenkov neutrino telescope