Square-root Filter Research Articles

Scale-dependent inflation algorithms for ensemble Kalman filters

Abstract Ensemble-based data assimilation methods often suffer sampling errors due to limited ensemble sizes and model errors, which can result in filter divergence. One method to avoid filter divergence is inflation, which inflates ensemble perturbations to increase ensemble spread and account for model errors. The commonly applied inflation methods, including the multiplicative inflation, relax to prior spread (RTPS), and additive inflation, often use a constant inflation parameter. To capture different error growths at different scales, a scale-dependent inflation is proposed here, which applies different inflation magnitudes for variables associated with different scales. Results from the two-scale Lorenz05 model III show that for the ensemble square-root filter (EnSRF) and integrated hybrid ensemble Kalman filter with ensemble mean updated by hybrid background error covariances (IHEnKF-Mean), scale-dependent inflation is superior to constant inflation. Constant inflation overinflates small-scale variables and results in increased small-scale errors, which then propagate to large-scale variables through the coupling between large- and small-scale variables and lead to increased large-scale errors. Scale-dependent inflation applies larger inflation for large-scale variables and imposes no inflation for small-scale variables, since large-scale errors have larger magnitudes than small-scale ones, and small-scale errors grow faster than large-scale ones. But IHEnKF-Ensemble that updates both the ensemble mean and perturbations with hybrid background error covariances is much less sensitive to scale-dependent inflation, compared to EnSRF and IHEnKF-Mean, because updating ensemble perturbations with hybrid background error covariances can play a role similar to the scale-dependent inflation.

Square-root filtering via covariance SVD factors in the accurate continuous-discrete extended-cubature Kalman filter

This paper continues our research devoted to an accurate nonlinear Bayesian filters' design. Our solution implies numerical methods for solving ordinary differential equations (ODE) when propagating the mean and error covariance of the dynamic state. The key idea is that an accurate implementation strategy implies the methods with a discretization error control involved. This means that the filters' moment differential equations are to be solved accurately, i.e. with negligible error. In this paper, we explore the continuous-discrete extended-cubature Kalman filter that is a hybrid method between Extended and Cubature Kalman filters (CKF). Motivated by recent results obtained for the continuous-discrete CKF in Bayesian filtering realm, we propose the numerically stable (to roundoff) square-root approach within a singular value decomposition (SVD) for the hybrid filter. The new method is extensively tested on a few application examples including stiff systems.

Validation of Prognostic Radiomic Features From Resectable Pancreatic Ductal Adenocarcinoma in Patients With Advanced Disease Undergoing Chemotherapy.

Radiomic features in pancreatic ductal adenocarcinoma (PDAC) often lack validation in independent test sets or are limited to early or late stage disease. Given the lethal nature of PDAC it is possible that there are similarities in radiomic features of both early and advanced disease reflective of aggressive biology. To assess the performance of prognostic radiomic features previously published in patients with resectable PDAC in a test set of patients with unresectable PDAC undergoing chemotherapy. The pre-treatment CT of 108 patients enrolled in a prospective chemotherapy trial were used as a test cohort for 2 previously published prognostic radiomic features in resectable PDAC (Sum Entropy and Cluster Tendency with square-root filter[Sqrt]). We assessed the performance of these 2 radiomic features for the prediction of overall survival (OS) and time to progression (TTP) using Cox proportional-hazard models. Sqrt Cluster Tendency was significantly associated with outcome with a hazard ratio (HR) of 1.27(for primary pancreatic tumor plus local nodes), (Confidence Interval(CI):1.01 -1.6, P-value = 0.039) for OS and a HR of 1.25(CI:1.00 -1.55, P-value = 0.047) for TTP. Sum entropy was not associated with outcomes. Sqrt Cluster Tendency remained significant in multivariate analysis. The CT radiomic feature Sqrt Cluster Tendency, previously demonstrated to be prognostic in resectable PDAC, remained a significant prognostic factor for OS and TTP in a test set of unresectable PDAC patients. This radiomic feature warrants further investigation to understand its biologic correlates and CT applicability in PDAC patients.

It[formula omitted]-Taylor-based square-root unscented Kalman filtering methods for state estimation in nonlinear continuous-discrete stochastic systems

This paper addresses the problem of square-rooting in the Unscented Kalman Filtering (UKF) methods rooted in the Ito^-Taylor approximation of the strong order 1.5. Since its discovery the UKF has become one of the most powerful state estimation means because of its outstanding performance in numerous stochastic systems of practical value, including continuous-discrete ones. Besides, the main shortcoming of this technique is the need for the Cholesky decomposition of covariance matrices derived in its time and measurement update steps. Such a factorization is time-consuming and highly sensitive to round-off and other errors committed in the course of computation, which can result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually overcome by means of square-root filter implementations, which propagate not the covariance itself but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of the UKF to high-dimensional stochastic systems preclude from designing conventional square-root UKF methods. We resolve it with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root filters are justified theoretically and examined and compared numerically to the existing UKF in a flight control scenario.

The J-Orthogonal Square-Root Euler-Maruyama-Based Unscented Kalman Filter for Nonlinear Stochastic Systems

This paper addresses the issue of square-rooting in the Unscented Kalman Filtering (UKF) methods. Since their discovery the UKF is considered to be among the most valued state estimation algorithms because of its outstanding performance in numerous real-world applications. However, the main shortcoming of such a technique is the need for the Cholesky decomposition of predicted and filtering covariances derived in all time and measurement update steps. Such a factorization is time-consuming and highly sensitive to round-off and other errors committed in the course of calculation, which can result in losing the covariance’s positivity and, hence, in failing the Cholesky decomposition. The latter problem is usually overcome via square-root filtering implementations, which propagate not the covariance itself but only its square root (Cholesky factor). Unfortunately, negative weights arising in applications of the UKF schemes to large stochastic systems preclude from designing conventional square-root UKF methods. So, we resolve it with a hyperbolic QR factorization used for yielding J-orthogonal square roots. Our novel square-root filter is grounded in the Euler-Maruyama discretization of order 0.5. It is justified theoretically and examined and compared numerically to the conventional (non-square-root) UKF in an aircraft’s coordinated turn scenario with ill-conditioned measurements.

Advanced square‐root cubature Kalman filters based on singular value decomposition and sequential processing

Numerical accuracy of the cubature Kalman filter (CKF) is crucially degraded by the accumulated round-off errors. As a systematic solution to reduce the influence of round-off errors, the square-root filters are appealing. This study proposes a square-root CKF based on the singular value decomposition (SVD) approach to enhance the robustness against round-off errors. In addition, motivated by the impact of matrix inversion operation on the positive definiteness and symmetry of the error covariance matrix and numerical conditioning, the authors derive a sequential square-root CKF. The sequential CKF avoids matrix inversion operation involved in the original CKF and directly propagates square factor in each cycle step. The variation is a valid way to better preserve the positive definiteness and symmetry and numerical stability. To evaluate the proposed approaches, four numerical experiments are simulated. The results illustrate the numerical robustness and stability of the proposed approaches.

A numerical integration-based Kalman filter for moderately nonlinear systems

This paper introduces a computationally efficient data assimilation scheme based on Gaussian quadrature filtering that potentially outperforms current methods in data assimilation for moderately nonlinear systems. Moderately nonlinear systems, in this case, are systems with numerical models with small fourth and higher derivative terms. Gaussian quadrature filters are a family of filters that make simplifying Gaussian assumptions about filtering pdfs in order to numerically evaluate the integrals found in Bayesian data assimilation. These filters are differentiated by the varying quadrature rules to evaluate the arising integrals. The approach we present, denoted by Assumed Gaussian Reduced (AGR) filter, uses a reduced order version of the polynomial quadrature first proposed in Ito and Xiong [2000. Gaussian filters for nonlinear filtering problems. IEEE Trans. Automat. Control. 45, 910–927]. This quadrature uses the properties of Gaussian distributions to form an effectively higher order method increasing its efficiency. To construct the AGR filter, this quadrature is used to form a reduced order square-root filter, which will reduce computational costs and improve numerical robustness. For cases of sufficiently small fourth derivatives of the nonlinear model, we demonstrate that the AGR filter outperforms ensemble Kalman filters (EnKFs) for a Korteweg-de Vries model and a Boussinesq model.

Square-root accurate continuous-discrete extended-unscented Kalman filtering methods with embedded orthogonal and J-orthogonal QR decompositions for estimation of nonlinear continuous-time stochastic models in radar tracking

This paper presents a number of new state estimation algorithms, which unify the best features of the accurate continuous-discrete extended and unscented Kalman filters in treating nonlinear continuous-time stochastic systems with discrete measurements. In particular, our mixed-type algorithms succeed in estimating continuous-discrete stochastic systems with nonlinear and/or nondifferentiable measurements. The main weakness of these methods is the need for the Cholesky decomposition of predicted covariance matrices. Such a factorization is highly sensitive to numerical integration and round-off errors committed, which may result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually solved in the form of square-root filtering implementations, which propagate not the covariance matrix but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of our mixed-type methods to high-dimensional stochastic systems preclude from designing conventional square-root filters. We address the mentioned issue with one-rank Cholesky factor updates or with hyperbolic QR transforms used for yielding J-orthogonal square-root filters. These novel algorithms are justified theoretically and examined and compared numerically to the non-square-root one in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, in the presence of Gaussian or glint noise.

Impact of ambiguity resolution with sequential constraints on real-time precise GPS satellite orbit determination

The real-time precise orbit is an essential prerequisite for a real-time precise point positioning service. Focusing on the impact of ambiguity resolution on real-time orbital precision, we propose an efficient approach for real-time ambiguity resolution, which consists of two modules. The first module resolves the double-differenced ambiguities according to their proximity to the nearest integer. The second module sequentially adds the resolved integer ambiguity constraints to the square-root information filter process, after confirming that the same constraints have not been imposed before the current epoch. To validate our method, GPS data collected from 100 globally distributed stations are used to simulate a real-time precise orbit determination. The convergence performance is analyzed, and the accuracy of the orbit is evaluated. The results show that: (1) Almost 90% of the double-differenced ambiguities are fixed correctly in real time for baselines shorter than 1000 km. (2) After application of the proposed approach, the root-mean-square errors of all the satellite orbits are reduced from (5.9, 3.4, 2.3) cm to (4.7, 2.6, 2.2) cm for the along-track, cross-track and radial directions, respectively, with improvements of about 22% for the along-track and cross-track directions, and 6% for the radial direction. (3) Simulated real-time orbits determined with this method can obtain almost the same accuracy as some ultra-rapid products and, particularly, better accuracy can be achieved for eclipsing satellites.

Considering the impact of observation error correlation in ensemble square-root Kalman filter

Data assimilation has been developed into an effective technology that can utilize a large number of multisource unconventional data. It cannot only provide the initial field for the ocean numerical prediction model, but also construct the ocean reanalysis datasets and provide the design basis for the ocean observation plan. In data assimilation, the estimation of the observation error is of paramount importance, because the quality of the analysis depends on it. In general, the observation error covariance matrix is diagonal or assumed to be diagonal, which means that the observation errors are independent from one another. However, there are indeed correlations in the observation errors. A diagnostic method has been developed, which can estimate a correlated and more accurate observation error covariance matrix. The proposed method combines an ensemble squareroot Kalman filter with the diagnostic method, providing an estimation of the observation error covariance matrix. In order to test the performance of the method, the numerical experiments are performed with the Lorenz 96 model and a Shallow water model. The more accurate observation error covariance matrix can be obtained to use in ensemble square-root Kalman filter by using the new method. We could find using the estimated correlated observation error in the data assimilation improves the analysis.

Adaptive Square-Root Unscented Particle Filtering Algorithm for Dynamic Navigation.

This paper presents a new adaptive square-root unscented particle filtering algorithm by combining the adaptive filtering and square-root filtering into the unscented particle filter to inhibit the disturbance of kinematic model noise and the instability of filtering data in the process of nonlinear filtering. To prevent particles from degeneracy, the proposed algorithm adaptively adjusts the adaptive factor, which is constructed from predicted residuals, to refrain from the disturbance of abnormal observation and the kinematic model noise. Cholesky factorization is also applied to suppress the negative definiteness of the covariance matrices of the predicted state vector and observation vector. Experiments and comparison analysis were conducted to comprehensively evaluate the performance of the proposed algorithm. The results demonstrate that the proposed algorithm exhibits a strong overall performance for integrated navigation systems.

Model aided airborne integrated navigation system based on an improved square-root unscented H∞ filter

An inertial navigation system (INS) aided with an aircraft dynamic model (ADM) is developed as a novel airborne integrated navigation system, coping with the absence of a global navigation satellite system. To overcome the shortcomings of the conventional linear integration of INS/ADM based on an extended Kalman filter, a nonlinear integration method is proposed. Fast-update ADM makes it possible to utilize a direct filtering method, which employs nonlinear INS mechanics as system equations and a nonlinear ADM as observation equations, substituting the indirect filtering based on linear error equations. The strong nonlinearity generally calls for an unscented Kalman filter to accomplish the fusion process. Dealing with the model uncertainty, the inaccurate statistical characteristics of the noise and the potential nonpositive definiteness of the covariance matrix, an improved square-root unscented H∞ filter (ISRUHF) is derived in the paper, in which the robust factor [Formula: see text] is further expanded into a diagonal matrix [Formula: see text], to improve the accuracy and robustness of the integrated navigation system. Corresponding simulations as well as real flight tests based on a small-scale fixed-wing aircraft are operated and ISRUHF shows superiority compared with the commonly used fusion algorithm.

Switched and Iterated Square-Root Gauss–Hermite Filter for Passive Target Tracking

In this paper, a novel algorithm referred to as the switched and iterated square-root Gauss–Hermite filter is developed for the nonlinear state estimation. The algorithm is derived by embedding the matrix triangularization, iteration update, and switching control into the Gauss–Hermite filter architecture. This new filtering approach is used and analyzed in passive localization and tracking system, which is troublesome to other filtering algorithms because of its intrinsic disadvantages such as large initial error and weak observability. Comparative simulation results are demonstrated that a more accurate state estimation can be obtained rapidly by the proposed algorithm with lower computational burden and stronger robustness.

Residual Echo Reduction for Multichannel Acoustic Echo Cancelers With a Complex-Valued Residual Echo Estimate

This paper proposes a new method to reduce residual echo for multichannel acoustic echo cancelers (AECs). This method estimates complex-valued residual echo and corrects its overestimated amplitude on the basis of a bias model of coherence estimation. Unlike conventional residual echo reduction methods for single-channel AECs, this method takes into account not only the magnitudes but also the phases of far-end signals and frequency responses. The inclusion of the phases gives the correct model of residual echo for multichannel AECs. The correction of the overestimated amplitude of residual echo decreases the distortion of near-end speech during double talk. The proposed method reduces the residual echo in the AEC output by 5–10 dB immediately after environmental changes such as a far-end talker change and an echo-path change. The proposed method provides less distorted near-end speech than the square-root Wiener filter during double talk.

Designing an Easy-to-Deploy Improved DVB-C Set-Top Boxes With a Backward Compatible Square-Root Nyquist Filter

The set-top boxes based on the digital video broadcasting-cable (DVB-C) standard have been deployed worldwide for decades, and the square-root raised cosine filter is specified as its receive filter. Recent years, plenty of the preferable square-root Nyquist filters have been developed. It is eagerly desired to upgrade the filter in DVB-C system. However, because a great number of the set-top boxes are still in operation, the upgraded new set-top boxes are required to coexist with the already deployed old ones. Theoretically, it requires the receive filters in both the old and new set-top boxes satisfy the Nyquist criterion with the same transmit filter. Up to now, there is no acceptable solution to this problem. This paper proposed a backward compatible square-root Nyquist filter design method, which enables the old set-top boxes to coexist with the upgraded new one. Theoretical analysis shows that the proposed filter enables both the old and new set-top boxes to work properly, in terms of satisfying the Nyquist criterion, achieving the required stop-band attenuation and satisfying other requirements defined by the DVB-C standard. The experimental tests and simulation results have also demonstrated that the proposed filter has increased the spectrum utilization by 14% compared with the DVB-C standard, and with much lower computational complexity.

An adaptive particle filter based on PSO and fuzzy inference system for nonlinear state systems

ABSTRACTParticle filters have been widely used in nonlinear/non-Gaussian Bayesian state estimation problems. However, the particle filter (PF) is inconsistent over time. The inconsistency of PF mainly results from the particle depletion in resampling step and an incorrect priori knowledge of process and measurement noise. To cope with this problem and enhance the accuracy and consistency of the state estimation, an adaptive particle filter(APF) is proposed in this paper. In APF, an adaptive fuzzy square-root unscented Kalman filter (AFSRUKF) is used to generate the proposal distribution. This causes that beside the merit of reducing the computational cost, APF has some other advantages such as increasing consistency that leads to more numerical stability and better performance. Moreover,APF can work in unknown statistical noise behaviour and is more robust. This is why the fuzzy inference system (FIS) supervises the performance of square-root unscented particle filter (SRUPF) using tuning statistical noises. In APF, to increase the diversity of particles, the resampling process is done based on the particle swarm optimization (PSO). With this resampling strategy, the small-weight particles are modified to the large-weight ones without duplication and elimination of particles. The effectiveness of APF is demonstrated by using two experiment examples through Monte Carlo simulations. The simulation results demonstrate the effectiveness of the proposed method.

Square‐root Kalman‐like filters for estimation of stiff continuous‐time stochastic systems with ill‐conditioned measurements

This study addresses the performance of the extended and extended-cubature Kalman filters for state estimation of ill-conditioned stiff continuous-discrete stochastic systems. Stiff models arise naturally in practise as, for instance, the Van der Pol oscillator in electrical engineering and the Oregonator reaction in chemistry. Their accurate state estimation is a challenging task, which is made even more difficult by including ill-conditioned measurements, here. The authors' goal is to study accuracy of these methods when the Van der Pol oscillator and the Oregonator reaction become increasingly ill-conditioned. Thus, the authors determine efficient tools for estimation of stiff stochastic systems in the presence of round-off and other disturbances. They examine both conventional filters and their square-root forms. The square-root methods are obtained in two ways, namely, by solving the square-root moment differential equations and by square-rooting the filter itself. They prove that only the square-root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically stable and provide the highest state estimation accuracy in the ill-conditioned stiff stochastic examples under consideration.

Square-Root Sigma-Point Information Consensus Filters for Distributed Nonlinear Estimation.

This paper focuses on the convergence rate and numerical characteristics of the nonlinear information consensus filter for object tracking using a distributed sensor network. To avoid the Jacobian calculation, improve the numerical characteristic and achieve more accurate estimation results for nonlinear distributed estimation, we introduce square-root extensions of derivative-free information weighted consensus filters (IWCFs), which employ square-root versions of unscented transform, Stirling’s interpolation and cubature rules to linearize nonlinear models, respectively. In addition, to improve the convergence rate, we introduce the square-root dynamic hybrid consensus filters (DHCFs), which use an estimated factor to weight the information contributions and shows a faster convergence rate when the number of consensus iterations is limited. Finally, compared to the state of the art, the simulation shows that the proposed methods can improve the estimation results in the scenario of distributed camera networks.

Operational modal identification using variational Bayes

Operational modal analysis is the primary tool for modal parameter identification in civil engineering. Bayesian statistics offers an ideal framework for analyzing uncertainties associated with the identified modal parameters. However, the exact Bayesian formulation is usually intractable due to the high computational demand in obtaining the posterior distributions of modal parameters. In this paper, the variational Bayes method is employed to provide an approximate solution. Unlike the Laplace approximation and Monte Carlo sampling, the variational Bayes approach provides a gradient-free algorithm to analytically approximate the posterior distributions. Working with the state-space representation of a dynamical system, the variational Bayes approach for identification of modal parameters is derived by ignoring statistical correlation between latent variables and the model parameters. In this approach, the joint distribution of the state-transition and observation matrices as well as the joint distribution of the process noise and measurement error are firstly calculated analytically using conjugate priors. The distribution of modal parameters is extracted from these obtained joint distributions using a first-order Taylor series expansion. A robust implementation of the method is discussed by using square-root filtering and Cholesky decomposition. The proposed approach is illustrated by its application to an example mass-spring system and the One Rincon Hill Tower in San Francisco.

Robust square-root cubature Kalman filter based on Huber’s M-estimation methodology

In practical engineering applications, the performance of the standard cubature Kalman filter (CKF) and its square-root version can be severely degraded due to outliers in measurement or contaminated distribution. In order to address the problem, a robust version of CKF is presented using Huber’s M-estimation methodology and square-root filtering framework. By making use of the Huber technique to reformulate the measurement update of CKF in square-root filtering framework, the proposed filter can exhibit robustness and numerical stability against deviation from Gaussian distribution assumption. In simulation tests, four versions of CKF—the standard, the square-root, Huber-based, and the proposed are evaluated in terms of estimation accuracy, numerical stability, and robustness under Gaussian and non-Gaussian distribution. The results are concluded that the square-root version outperforms the others under Gaussian distribution, whereas the proposed filter has improved performance in maintaining the robustness and numerical stability under non-Gaussian distribution. The investigated robust framework can be extended to other Gaussian filtering algorithms and the study is expected to facilitate applications of CKF in practical engineering as well.