基于高斯和近似的扩展切片高斯混合滤波器及其在多径估计中的应用

全球卫星导航系统(Global navigation satellite system, GNSS)信号的多径估计问题实际上是条件线性状态空间模型下的状态估计问题. 根据高斯和理论提出了适用于非高斯噪声环境的扩展切片高斯混合滤波(Extension of sliced Gaussian mixture filter, ESGMF)算法. 该算法将非高斯噪声的状态概率密度函数(Probability density function, PDF)表示为高斯和的形式,将ESGMF通过一组并行的切片高斯混合滤波器(Sliced Gaussian mixture filter, SGMF)来实现.同时, 在ESGMF算法中利用粒子滤波(Particle filter, PF)中重采样的思想对成指数增加的状态预测PDF的高斯混合个体进行约简, 以提高贝叶斯推理的效率.该算法可以获得非高斯噪声下状态PDF的迭代解析表达式. 最后, 将ESGMF应用于GPS多径参数估计, 仿真结果表明, ESGMF算法的估计精度优于基于PF和扩展卡尔曼滤波(Extended Kalman filter, EKF)的算法.     

Maximum correntropy unscented filter

The unscented transformation (UT) is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilising a derivative-free higher-order approximation by approximating a Gaussian distribution rather than approximating a non-linear function. Applying the UT to a Kalman filter type estimator leads to the well-known unscented Kalman filter (UKF). Although the UKF works very well in Gaussian noises, its performance may deteriorate significantly when the noises are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises. To improve the robustness of the UKF against impulsive noises, a new filter for non-linear systems is proposed in this work, namely the maximum correntropy unscented filter (MCUF). In MCUF, the UT is applied to obtain the prior estimates of the state and covariance matrix, and a robust statistical linearisation regression based on the maximum correntropy criterion is then used to obtain the posterior estimates of the state and covariance matrix. The satisfying performance of the new algorithm is confirmed by two illustrative examples.     

Bayesian spatial modeling and interpolation using copulas

Classical Bayesian spatial interpolation methods are based on the Gaussian assumption and therefore lead to unreliable results when applied to extreme valued data. Specifically, they give wrong estimates of the prediction uncertainty. Copulas have recently attracted much attention in spatial statistics and are used as a flexible alternative to traditional methods for non-Gaussian spatial modeling and interpolation. We adopt this methodology and show how it can be incorporated in a Bayesian framework by assigning priors to all model parameters. In the absence of simple analytical expressions for the joint posterior distribution we propose a Metropolis-Hastings algorithm to obtain posterior samples. The posterior predictive density is approximated by averaging the plug-in predictive densities. Furthermore, we discuss the deficiencies of the existing spatial copula models with regard to modeling extreme events. It is shown that the non-Gaussian χ²-copula model suffers from the same lack of tail dependence as the Gaussian copula and thus offers no advantage over the latter with respect to modeling extremes. We illustrate the proposed methodology by analyzing a dataset here referred to as the Helicopter dataset, which includes strongly skewed radioactivity measurements in the city of Oranienburg, Germany.     

高斯渐进贝叶斯滤波器

渐进贝叶斯方法将先验分布到后验分布的演化描述为一阶动态系统,通过在伪时间上连续地引入观测信息实现后验状态估计.该方法的一般形式解,即动态系统的时间导数,是难以得到的.本文提出一种高斯型渐进贝叶斯滤波器.首先在线性高斯条件下推导了时间导数的解析解;然后证明了在该条件下,由该解析解确定的一阶动态系统与常量状态估计的Kalman-Bucy滤波器是一致的,且由此导出的高斯渐进贝叶斯滤波器与卡尔曼滤波器是一致的.最后利用一阶Taylor展开推导了滤波器在非线性高斯条件下的近似解表达式,并采用Monte Carlo方法给出了具体实现方法.通过若干仿真算例表明,新滤波器具有较高的精度,且在一定精度条件下的时间复杂度低于一般粒子滤波器.     

Elementary quantum gates with Gaussian states

We study the question of converting initially Gaussian states into non-Gaussian ones by two- and three-photon subtraction to improve non-classical properties of the conditional optical fields. We show the photon subtraction may effectively generate non-Gaussian states only in case of small values of the mean values of the position and momentum operators. In particular, the photon-subtracted state can be made arbitrary close to Gaussian state in limiting case of large initial amplitude of displacement. Use of initial displacement in input Gaussian states opens wider prospects to manipulate them. In particular, realization of probabilistic Hadamard gate with input Gaussian states is discussed where photon subtraction is motive force able unevenly to increase measure of non-classicality of the output state. Subtraction of larger number of photons enables to increase fidelity and non-classical measure of the conditional states.     

Optimal smoothing of non-linear dynamic systems via Monte Carlo Markov chains

We consider the smoothing problem of estimating a sequence of state vectors given a nonlinear state space model with additive white Gaussian noise, and measurements of the system output. The system output may also be nonlinearly related to the system state. Often, obtaining the minimum variance state estimates conditioned on output data is not analytically intractable. To tackle this difficulty, a Markov chain Monte Carlo technique is presented. The proposal density for this method efficiently draws samples from the Laplace approximation of the posterior distribution of the state sequence given the measurement sequence. This proposal density is combined with the Metropolis-Hastings algorithm to generate realizations of the state sequence that converges to the proper posterior distribution. The minimum variance estimate and confidence intervals are approximated using these realizations. Simulations of a fed-batch bioreactor model are used to demonstrate that the proposed method can obtain significantly better estimates than the iterated Kalman-Bucy smoother.     

Distributed adaptive Kalman filter based on variational Bayesian technique

In this paper, distributed Kalman filter design is studied for linear dynamics with unknown measurement noise variance, which modeled by Wishart distribution. To solve the problem in a multi-agent network, a distributed adaptive Kalman filter is proposed with the help of variational Bayesian, where the posterior distribution of joint state and noise variance is approximated by a free-form distribution. The convergence of the proposed algorithm is proved in two main steps: noise statistics is estimated, where each agent only use its local information in variational Bayesian expectation (VB-E) step, and state is estimated by a consensus algorithm in variational Bayesian maximum (VB-M) step. Finally, a distributed target tracking problem is investigated with simulations for illustration.     

Information-driven receptor placement for contaminant source determination

Isolating the source of an unknown contaminant emission is a challenging problem when provided with only a limited and noisy set of mean concentration measurements. A Bayesian approach to this inverse problem yields estimates for the source parameters (location and strength) which depend directly on the quality of the information obtained from an array of detectors. If the data are of poor quality, uncertainties associated with the source parameter estimates may be large, necessitating further exploration (e.g., using mobile detection) in order to better isolate the putative source. We employ Bayesian experimental design with the goal of strategically placing an additional detector in order to maximize the ‘expected information’ contained in subsequent posterior distributions for the source parameters. The methodology is demonstrated using synthetic data from detectors lying in a horizontally homogeneous, neutrally-stratified atmospheric surface layer. Markov chain Monte Carlo and a posterior sampling technique are used to calculate the expected information over a grid of potential detector locations, and an auxiliary detector measuring both mean concentration and turbulent scalar fluxes is added where the expected information reaches a maximum. The updated posterior distribution (calculated based on the additional measurements) yields significantly improved estimates for the source location and strength.     

Particle filtering-based fault detection in non-linear stochastic systems

Much of the development in model-based fault detection techniques for dynamic stochastic systems has relied on the system model being linear and the noise and disturbances being Gaussian. Linearized approximations have been used in the non-linear systems case. However, linearization techniques, being approximate, tend to suffer from poor detection or high false alarm rates. A novel particle filtering based approach to fault detection in non-linear stochastic systems is developed here. One of the appealing advantages of the new approach is that the complete probability distribution information of the state estimates from particle filter is utilized for fault detection, whereas, only the mean and covariance of an approximate Gaussian distribution are used in a coventional extended Kalman filter-based approach. Another advantage of the new approach is its applicability to general non-linear system with non-Gaussian noise and disturbances. The effectiveness of this new method is demonstrated through Monte Carlo simulations and the detection performance is compared with that using the extended Kalman filter on a non-linear system.     

适用处理非高斯观测噪声的强跟踪卡尔曼滤波器

在高斯噪声条件下,卡尔曼滤波器(KF)能够获得系统状态的一致最小方差线性无偏估计.但当噪声非高斯,KF性能将严重下降.观测噪声非高斯现象在深空探测自主导航中经常遇到,然而现有模型可能存在着精度不高、稳定性不强或者计算复杂度较高的缺点.针对这种现状,本文在传统强跟踪卡尔曼滤波器(STKF)中新息正交原则的基础上,推导了适用处理非高斯观测噪声的强跟踪卡尔曼滤波器(STKFNO),并将其嵌入到无迹卡尔曼滤波(UKF)框架下形成适用处理非线性系统非高斯观测噪声的强跟踪无迹卡尔曼滤波器(STUKFNO).所提出的算法被应用到深空光学自主导航系统中,仿真结果表明所提出的算法能够较好地应对观测噪声的非高斯性.     

基于因子图和联合消息传递的无线网络协作定位算法

针对现有基于消息传递算法的无线网络节点定位算法复杂度和通信开销过高的问题,提出一种基于测距的、低复杂度低协作开销的联合消息传递节点定位算法。所提算法考虑参考节点位置的不确定性以减少误差累积,并将消息约束为高斯函数以降低通信开销。首先,根据系统的概率模型和因子分解设计因子图;然后,根据状态转移模型和测距模型的特点,分别使用置信传播和平均场方法计算预测消息和协作消息;最后,在每次迭代过程中,通过非线性项的泰勒展开将非高斯置信消息近似为高斯函数。仿真分析表明,所提算法的定位性能与基于粒子的SPAWN算法接近,但节点间传输的信息由大量粒子变为均值向量和协方差矩阵,同时计算复杂度也大幅降低。     

Hessian matrix distribution for Bayesian policy gradient reinforcement learning

Bayesian policy gradient algorithms have been recently proposed for modeling the policy gradient of the performance measure in reinforcement learning as a Gaussian process. These methods were known to reduce the variance and the number of samples needed to obtain accurate gradient estimates in comparison to the conventional Monte-Carlo policy gradient algorithms. In this paper, we propose an improvement over previous Bayesian frameworks for the policy gradient. We use the Hessian matrix distribution as a learning rate schedule to improve the performance of the Bayesian policy gradient algorithm in terms of the variance and the number of samples. As in computing the policy gradient distributions, the Bayesian quadrature method is used to estimate the Hessian matrix distributions. We prove that the posterior mean of the Hessian distribution estimate is symmetric, one of the important properties of the Hessian matrix. Moreover, we prove that with an appropriate choice of kernel, the computational complexity of Hessian distribution estimate is equal to that of the policy gradient distribution estimates. Using simulations, we show encouraging experimental results comparing the proposed algorithm to the Bayesian policy gradient and the Bayesian policy natural gradient algorithms described in Ghavamzadeh and Engel [10].     

A new fault diagnosis method of multimode processes using Bayesian inference based Gaussian mixture contribution decomposition

This paper presents a novel Bayesian inference based Gaussian mixture contribution (BIGMC) method to isolate and diagnose the faulty variables in chemical processes with multiple operating modes. The statistical confidence intervals of traditional principal component analysis (PCA) based T² and SPE diagnostics rely upon the assumption that the operating data follow a multivariate Gaussian distribution approximately and therefore may not be able to determine the faulty variables in multimode non-Gaussian processes accurately. As an alternative solution, the proposed BIGMC method first identifies the multiple Gaussian modes corresponding to different operating conditions and then integrates the Mahalanobis distance based variable contributions across all the Gaussian clusters through Bayesian inference strategy. The derived BIGMC index is of probabilistic feature and includes all operation scenarios with posterior probabilities as weighting factors. The Tennessee Eastman process (TEP) is used to demonstrate the utility of the proposed BIGMC method for fault diagnosis of multimode processes. The comparison of the single-PCA and multi-PCA based contribution approaches shows that the BIGMC method can effectively identify the leading faulty variables with superior diagnosis capability.     

基于局部熵双子空间的多模态过程故障检测

为了提高非高斯工业过程的检测性能, 提出局部熵双子空间(LEDS)的多模态过程故障检测方法. 运用局部 概率密度估计构建数据的局部熵矩阵, 消除数据的多模态特性. 用Kolmogorov-Smirnov (KS)检验局部熵数据中变 量的正态分布特性, 对高斯分布和非高斯分布的数据分别建立基于PCA的高斯子空间和ICA的非高斯子空间故障 检测模型. 利用Bayesian决策将检测结果转化成发生故障概率的形式, 将检测结果组合成最终的统计信息, 进行故 障检测. 将该方法应用于数值例子和田纳西–伊斯曼多模态过程, 仿真结果表明, 该方法在误报率较低的情况下, 故 障检测率最高, 优于PCA、局部熵PCA(LEPCA)和局部熵ICA(LEICA)方法.     

Development of grey box state estimators for systems subjected to time correlated unmeasured disturbances

Unmeasured disturbances, which arise from uncertainties in the physical input sources, are commonly encountered in a process operation. For the purpose of developing Bayesian state estimators, such disturbances have been traditionally treated as Gaussian white noise processes. In practice, however, such disturbances are often correlated in time and the simplistic white noise assumption may not hold. Thus, to generate accurate estimates of the states, it is essential to obtain a reasonably accurate characterisation of the dynamics associated with the unmeasured disturbances. In this work, a systematic approach has been developed for identifying discrete time stochastic disturbance models, which captures the dynamics associated with such unmeasured disturbances. Under certain simplifying assumptions, the discrete time unmeasured disturbance models are combined with a continuous time mechanistic model to derive a discrete nonlinear grey box model. The grey box model is further used to formulate a nonlinear Bayesian state estimator. A constrained optimisation problem, that maximizes the log likelihood function of the innovation sequence generated by the state estimator, is formulated and solved for estimation of the parameters of the unmeasured disturbance model and the measurement noise covariance from the input–output data. The efficacy of this approach is demonstrated by simulating a benchmark continuous fermenter system and using experimental data obtained from a heater-mixer setup. The simulation studies demonstrate that the proposed approach is able to identify correlated disturbance models that closely match the characteristics of the true unmeasured disturbance models.     

A dynamic programming approach to trajectory estimation

An iterative equation based on dynamic programming for finding the most likely trajectory of a dynamic system observed through a noisy measurement system is presented; the procedure can be applied to nonlinear systems with non-Gaussian noise. It differs from the recently developed Bayesian estimation procedure in that the most likely estimate of the entire trajectory up to the present time, rather than of the present state only, is generated. It is shown that the two procedures in general yield different estimates of the present state; however, in the case of linear systems with Gaussian noise, both procedures reduce to the Kalman-Bucy filter. Illustrative examples are presented, and the present procedure is compared with the Bayesian procedure and with other estimation techniques in terms of computational requirements and applicability.     

递推更新高斯粒子滤波器

传统高斯粒子滤波算法(Gaussian particle Filter,GPF)中,粒子的重要性密度函数是由高斯滤波器结合当前最新量测来构建的.由于传统高斯滤波器在量测更新阶段直接利用量测对状态进行线性更新,在某些条件下会导致所构建的重要性密度函数并不能很好地近似状态真实分布.为了解决这一问题,结合递推更新的思想,本文推导出了递推更新高斯滤波器(recursive update Gaussian filter,RUGF)的一般结构.并在此基础上,选用RUGF来构建粒子滤波的重要性密度函数,从而提出了基于递推更新的高斯粒子滤波算法(recursive update gaussian particle filter,RUGPF).仿真表明,在非线性系统状态估计问题中,递推更新可以很好的利用量测信息,相比于传统的GPF,本文所提出的RUGPF滤波算法可以提供更高精度的估计结果.     

A Test of the Gaussian-ness of a Data Set Using Clustering

The properties of the ``magnifying glass' method of clustering are discussed. These properties, which include unbiased and consistent estimation of the mean for Gaussian distributions and biased and inconsistent estimation of the mean for non-Gaussian distributions, lead to the development of a technique for testing data to determine whether or not it is Gaussian. An example of a non-Gaussian distribution is given to show the sensitivity of the proposed Gaussian test.     

The Rosenblatt Bayesian Algorithm Learning in a Nonstationary Environment

In this letter, we study online learning in neural networks (NNs) obtained by approximating Bayesian learning. The approach is applied to Gibbs learning with the Rosenblatt potential in a nonstationary environment. The online scheme is obtained by the minimization (maximization) of the Kullback-Leibler divergence (cross entropy) between the true posterior distribution and the parameterized one. The complexity of the learning algorithm is further decreased by projecting the posterior onto a Gaussian distribution and imposing a spherical covariance matrix. We study in detail the particular case of learning linearly separable rules. In the case of a fixed rule, we observe an asymptotic generalization error e_gpropalpha^-1 for both the spherical and the full covariance matrix approximations. However, in the case of drifting rule, only the full covariance matrix algorithm shows a good performance. This good performance is indeed a surprise since the algorithm is obtained by projecting without the benefit of the extra information on drifting     

Bayesian robust linear dynamic system approach for dynamic process monitoring

In this paper, a Bayesian robust linear dynamic system approach is proposed for process modeling. Traditional linear dynamic system (LDS) constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in non-Gaussian modeling situations, especially those with outliers. To deal with this issue, the conventional Gaussian-based Kalman filter is modified with heavy tailed Student's t-distribution so as to deal with the non-Gaussian noise and modeling outliers. Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning parameters of the robust linear dynamic system. For process monitoring, traditional monitoring scheme are discussed and the residual space monitoring mechanism has been improved. To explore the feasibility and effectiveness, the proposed method is applied for fault detection, with detailed comparative studies with several other methods through the Tennessee Eastman benchmark.