Reparameterization for statistical state estimation applied to differential equations

The ensemble Kalman filter is a widely applied data assimilation technique useful for improving the forecast of computational models. The main computational cost of the ensemble Kalman filter comes from the numerical integration of each ensemble member forward in time. When the computational model involves a partial differential equation, the degrees of freedom of the solution in the discretization of the spatial domain are oftentimes used for the representation of the state of the system, and the filter is applied to this state vector. We propose a method of approximating the state of a partial differential equation in a representation space developed separately from the numerical method. This representation space represents a reparameterization of the state vector and can be chosen to retain desirable physical features of the solutions. We apply the ensemble Kalman filter to this representation of the state, and numerically demonstrate that acceptable results are obtained with substantially smaller ensemble sizes.     

集合卡尔曼滤波在时变声速剖面追踪中的性能分析

对集合卡尔曼滤波在时变海洋环境下的声速剖面追踪性能进行了分析。将南海实验背景下普林斯顿海洋模型预报的声速剖面正交分解为3阶系数组成的状态-空间形式,其状态转移方程建模为3阶自回归过程;基于卡尔曼反馈理论,利用适合于水平非均匀模型RAM仿真的观测声压场对系统状态进行校正,实现声速剖面的动态追踪。在水平均匀、水平非均匀和海底参数失配环境下的仿真结果均能较好地实现对声速剖面的追踪,验证了算法的可行性。同时对不同信噪比、粒子数、阵元数和海底参数失配等情况下的分析表明,观测信息量的增加可以有效抑制观测误差和模型误差的影响,相关结论得到了实验数据的验证。      

强非线性时间演化声速剖面的序贯反演

受海面波浪起伏、降雨和内波等海洋动力学过程的影响,浅水声速剖面的时间演化具有高度非线性,针对该问题提出使用改进的粒子滤波方法进行声速剖面序贯反演.该方法通过建立声速剖面的经验正交模型(EOF)以及描述声速剖面时间演化特征的状态空间模型,将声速剖面反演问题建模为状态跟踪问题,利用不敏粒子滤波(UPF:Uncented Particle Filter)算法进行声速剖面序贯反演。仿真试验通过实测声速剖面数据和先验地声参数信息产生接收声场数据,再利用模拟声场数据估计声速剖面的时间变化.结果表明,相比于集合卡尔曼滤波(EnKF:Ensemble Kalman Filter),在计算效率等同的情形下,该方法可以在状态参数的时间跳变点保持良好的跟踪性能,一定程度上克服了现有反演算法在跳变点发散的问题,可以有效提高声速剖面反演精度,尤其在声速剖面时变性较强时具有显著优势.      

Hybrid three-dimensional variation and particle filtering for nonlinear systems

This work addresses the problem of estimating the states of nonlinear dynamic systems with sparse observations.We present a hybrid three-dimensional variation(3DVar) and particle piltering(PF) method,which combines the advantages of 3DVar and particle-based filters.By minimizing the cost function,this approach will produce a better proposal distribution of the state.Afterwards the stochastic resampling step in standard PF can be avoided through a deterministic scheme.The simulation results show that the performance of the new method is superior to the traditional ensemble Kalman filtering(EnKF) and the standard PF,especially in highly nonlinear systems.     

State and parameter estimation in stochastic dynamical models

This paper derives generalized maximum likelihood estimates of state and model parameters of a stochastic dynamical model. In contrast to previous studies, the change in background distribution due to changes in model parameters is taken into account. An ensemble approach to solving the maximum likelihood estimates is proposed. An exact solution for the ensemble update based on a square root Kalman Filter is derived. This solution involves a two step procedure in which an ensemble is first produced by a standard ensemble Kalman Filter, and then “corrected” to account for parameter estimation, thereby allowing a user to take advantage of an existing ensemble filter. The solution is illustrated with simple, low-dimensional stochastic dynamical models and shown to work well and outperform augmentation methods for estimating stochastic parameters.     

Ensemble Kalman filter with the unscented transform

A modification scheme to the ensemble Kalman filter (EnKF) is introduced based on the concept of the unscented transform [S. Julier, J. Uhlmann, H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Automat. Control. 45 (2000) 477-482; S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation, Proc. IEEE 92 (2004) 401-422], which therefore will be called the ensemble unscented Kalman filter (EnUKF) in this work. When the error distribution of the analysis is symmetric (not necessarily Gaussian), it can be shown that, compared with the ordinary EnKF, the EnUKF has more accurate estimations of the ensemble mean and covariance of the background by examining the multidimensional Taylor series expansion term by term. This implies that, the EnUKF may have better performance in state estimation than the ordinary EnKF in the sense that the deviations from the true states are smaller. For verification, some numerical experiments are conducted on a 40-dimensional system due to Lorenz and Emanuel [E.N. Lorenz, K.A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55 (1998) 399-414]. Simulation results support our argument.     

基于滚动时域估计的飞行器姿态估计及三轴磁强计在线校正

针对飞行器姿态估计以及三轴磁强计在线校正问题, 提出了一种实时滚动时域估计算法. 首先, 为了解决在卡尔曼滤波框架下系统约束不能显式求解的问题, 设计了滚动时域估计滤波算法. 该算法将飞行器姿态估计问题转化为优化问题, 显式求解四元数归一化性质, 缩小搜索空间的同时提高了搜索效率和精度. 其次, 滤波时域窗内应用高斯-牛顿迭代法求解最优状态估计值, 满足了实时性要求. 最后, 在没有增加系统状态维数的情况下, 在线求解了三轴磁强计校正参数, 保证了磁强计量测值以矢量形式输入系统. 仿真结果表明, 由于合理地利用了历史信息, 该方法精度较高, 且对初始误差、系统误差均不敏感, 具有一定鲁棒性.     

基于时频差的正交容积卡尔曼滤波跟踪算法

针对基于时频差测量的无源跟踪中面临的非线性估计问题, 提出一种正交容积卡尔曼滤波跟踪算法. 该算法在容积卡尔曼滤波算法的基础上, 通过引入特定正交矩阵改进容积采样方法, 在高维状态估计下减小因采样产生的误差, 在没有增加计算量的前提下, 有效提高收敛速度及跟踪精度. 仿真结果表明, 在基于到达时差和到达频差的联合无源跟踪问题中, 与扩展卡尔曼滤波及容积卡尔曼滤波算法相比, 本文所提算法在跟踪性能上有明显提升.     

Tracking-positioning of sound speed profiles and moving acoustic source in shallow water

An ensemble Kalman filter(EnKF) approach is proposed to perform sequential tracking of water column sound speed profile(SSP) using a moving acoustic source. First,the SSPs are discretized in depth and range, and are expressed by the empirical orthogonal functions(EOFs). Second, the acoustic source state information and the first three orders of EOF coefficients are expressed as the state variable, and the acoustic field information received by the vertical line array are the measured values. Successively, the state variables and measured values are used to establish the state-measure model. Last, the EnKF is utilized to track the state variables. The simulation results show that the root mean square error of SSP and the absolute error of source are all small, and thus the acoustic source tracking-positioning has high accuracy. Moreover, increasing the number of sample collection, the signal-to-noise ratio and the number of receiving elements can improve the tracking-positioning results. The method is verified using the experimental data of the East China Sea.     

植被双向反射分布函数的模型及其发展

介绍了双向反射分布函数(BRDF)的概念和反映植被特性的归一化植被指数(NDVI)、增强植被指数(EVI)、差值植被指数(DVI)等一些重要指数参数。评述了近年发展起来的植被BRDF模型的构建方法,例如统计反演方法,MODIS植被指数合成法和集合卡曼滤波法(EnKF)。分析了目标表面BRDF数据的获取方法和改进的驱动核算法。对BRDF模型的发展趋势进行了展望。     

遥感反演时间序列叶面积指数的集合卡尔曼平滑算法

基于MODIS LAI产品数据集(MOD15A2)构建经验性的LAI动态模型,以LAI作为连接参数,将LAI动态模型与植被辐射传输模型MCRM2相耦合,提出了将耦合模型与时间序列MODIS反射率观测数据集(MOD09A1)同化进行LAI反演的方案.将集合卡尔曼平滑(EnKS)方法引入到LAI同化反演中,为更好地评价该算...     

MR fingerprinting reconstruction with Kalman filter

Magnetic resonance fingerprinting (MR fingerprinting or MRF) is a newly introduced quantitative magnetic resonance imaging technique, which enables simultaneous multi-parameter mapping in a single acquisition with improved time efficiency. The current MRF reconstruction method is based on dictionary matching, which may be limited by the discrete and finite nature of the dictionary and the computational cost associated with dictionary construction, storage and matching.In this paper, we describe a reconstruction method based on Kalman filter for MRF, which avoids the use of dictionary to obtain continuous MR parameter measurements. With this Kalman filter framework, the Bloch equation of inversion-recovery balanced steady state free-precession (IR-bSSFP) MRF sequence was derived to predict signal evolution, and acquired signal was entered to update the prediction. The algorithm can gradually estimate the accurate MR parameters during the recursive calculation. Single pixel and numeric brain phantom simulation were implemented with Kalman filter and the results were compared with those from dictionary matching reconstruction algorithm to demonstrate the feasibility and assess the performance of Kalman filter algorithm.The results demonstrated that Kalman filter algorithm is applicable for MRF reconstruction, eliminating the need for a pre-define dictionary and obtaining continuous MR parameter in contrast to the dictionary matching algorithm.     

Tracking noisy limit cycle oscillation with nonlinear filters

For engineering systems, the dynamic state estimates provide valuable information for the detection and prediction of failure due to noise and vibration. From this perspective, nonlinear filtering techniques are applied to the problem of state estimation of dynamical systems undergoing noisy limit cycle oscillation. Specifically, the extended Kalman filter, ensemble Kalman filter and particle filter are used to track the limit cycle oscillations of a Duffing oscillator using noisy observational data. The noisy limit cycle oscillations feature highly non-Gaussian trends. The efficiency and limitations of the extended Kalman filter, ensemble Kalman filter and particle filter in tracking limit cycle oscillations are examined with respect to the model and measurement noise and sparsity of measurement data. For the limit cycle oscillations considered here, it is demonstrated that the ensemble Kalman filter and particle filter outperform the extended Kalman filter in the presence of sparse observational data or strong measurement noise. For moderate measurement noise and frequent measurement data, the ensemble Kalman filter and particle filter perform equally well in comparison to the extended Kalman filter.     

Sampling strong tracking nonlinear unscented Kalman filter and its application in eye tracking

The unscented Kalman filter is a developed well-known method for nonlinear motion estimation and tracking. However, the standard unscented Kalman filter has the inherent drawbacks, such as numerical instability and much more time spent on calculation in practical applications. In this paper, we present a novel sampling strong tracking nonlinear unscented Kalman filter, aiming to overcome the difficulty in nonlinear eye tracking. In the above proposed filter, the simplified unscented transform sampling strategy with n+2 sigma points leads to the computational efficiency, and suboptimal fading factor of strong tracking filtering is introduced to improve robustness and accuracy of eye tracking. Compared with the related unscented Kalman filter for eye tracking, the proposed filter has potential advantages in robustness, convergence speed, and tracking accuracy. The final experimental results show the validity of our method for eye tracking under realistic conditions.     

Stochastic stability of the derivative unscented Kalman filter

This is the second of two consecutive papers focusing on the filtering algorithm for a nonlinear stochastic discretetime system with linear system state equation. The first paper established a derivative unscented Kalman filter(DUKF) to eliminate the redundant computational load of the unscented Kalman filter(UKF) due to the use of unscented transformation(UT) in the prediction process. The present paper studies the error behavior of the DUKF using the boundedness property of stochastic processes. It is proved that the estimation error of the DUKF remains bounded if the system satisfies certain conditions. Furthermore, it is shown that the design of the measurement noise covariance matrix plays an important role in improvement of the algorithm stability. The DUKF can be significantly stabilized by adding small quantities to the measurement noise covariance matrix in the presence of large initial error. Simulation results demonstrate the effectiveness of the proposed technique.     

逼近积分点数下限的五阶容积卡尔曼滤波定轨算法

为了在保持滤波定轨精度不变的条件下提高定轨计算的实时性,提出一种新的逼近积分点个数下限的五阶容积卡尔曼滤波定轨算法.首先,采用一种数值容积准则对非线性函数的高斯加权积分进行近似,该准则所需的积分点个数仅比五阶代数精度容积准则积分点个数的理论下限多一个积分点,并在贝叶斯滤波算法框架下推导出本文算法的更新步骤.然后,给出实时定轨所需的状态方程和量测方程,在状态方程中考虑了J2项引力摄动和大气阻力摄动,在量测方程中利用坐标系转换推导了轨道状态与测量元素之间的非线性关系.仿真实验结果表明,本文所提算法在定轨精度方面与已有的五阶滤波算法相当,但所需的积分点个数最少,计算实时性最高,从而验证了本文算法的有效性.     

浅海声速剖面与移动声源的跟踪定位

在水平非均匀分布的浅海环境中,针对移动声源跟踪时,声速剖面的变化会对声场产生影响,提出了一种利用集合卡尔曼滤波算法的声速剖面跟踪反演和移动声源跟踪定位的方法。首先,将声速剖面进行距离和深度的参数化表示,从而将对声速剖面的跟踪转化为对声速剖面前3阶经验正交函数系数的跟踪;其次,通过将声源状态信息和声速剖面信息表示为状态变量,而将垂直线列阵接收到的声场信息作为测量值建立状态-测量模型,然后利用集合卡尔曼滤波方法对模型状态变量进行跟踪。仿真结果得出:声速剖面跟踪反演的均方根误差和移动声源跟踪定位的绝对误差都非常小,对声源的跟踪定位精度很高。并且通过增加集合样本数、增加接收信号信噪比以及增加接收阵元数目都可以提高跟踪定位结果精度。最后,利用东海实验数据对本方法进行了验证。      

A Girsanov particle filter in nonlinear engineering dynamics

In this Letter, we propose a novel variant of the particle filter (PF) for state and parameter estimations of nonlinear engineering dynamical systems, modelled through stochastic differential equations (SDEs). The aim is to address a possible loss of accuracy in the estimates due to the discretization errors, which are inevitable during numerical integration of the SDEs. In particular, we adopt an explicit local linearization of the governing nonlinear SDEs and the resulting linearization errors in the estimates are corrected using Girsanov transformation of measures. Indeed, the linearization scheme via transformation of measures provides a weak framework for computing moments and this fits in well with any stochastic filtering strategy wherein estimates are themselves statistical moments. We presently implement the strategy using a bootstrap PF and numerically illustrate its performance for state and parameter estimations of the Duffing oscillator with linear and nonlinear measurement equations.     

基于卡尔曼滤波的低复杂度去混响算法*

在电话会议、智能音箱等应用场景下,传声器往往处在声源的远场。混响信号的存在会掩蔽后续到达的直达声信号,降低传声器接收信号的语音质量,以及语音识别系统的准确识别率。多通道线性预测算法是一种经典的盲去混响算法,但该算法往往具有较高的计算复杂度。本文提出了一种简化的卡尔曼滤波更新算法,通过对角化卡尔曼滤波器状态向量误差协方差矩阵,降低了自适应多通道线性预测去混响算法的复杂度。通过与现有分块对角简化算法对比发现,本文提出的简化算法在保证语音质量的同时,进一步降低了原卡尔曼滤波算法的复杂度。     

自回归状态空间模型下时变声速剖面跟踪方法

讨论了一种适用于浅海的时变声速剖面跟踪方法。该方法将时变水体声速剖面的反演问题建模为由描述声速剖面时变特性的状态方程与包含声压场局部测量信息的测量方程组成的状态-空间模型,提出利用自回归分析拟合方法将声速场扰动建模为高阶自回归演化模型,并通过集合卡尔曼滤波序贯地估计时间演化的海洋声速场。利用2001年亚洲海实验环境与声速测量数据,仿真分析了基于高阶自回归演化模型的时变声速剖面集合卡尔曼滤波估计方法。结果表明,相比于利用传统随机游走状态演化模型的估计方法,该改进方法可有效降低声速的跟踪误差,并且在较低信噪比条件下仍具有较好的跟踪性能。