Spectral estimation of nonstationary EEG using particle filtering with application to event-related desynchronization (ERD)

This paper proposes non-Gaussian models for parametric spectral estimation with application to event-related desynchronization (ERD) estimation of nonstationary EEG. Existing approaches for time-varying spectral estimation use time-varying autoregressive (TVAR) state-space models with Gaussian state noise. The parameter estimation is solved by a conventional Kalman filtering. This study uses non-Gaussian state noise to model autoregressive (AR) parameter variation with estimation by a Monte Carlo particle filter (PF). Use of non-Gaussian noise such as heavy-tailed distribution is motivated by its ability to track abrupt and smooth AR parameter changes, which are inadequately modeled by Gaussian models. Thus, more accurate spectral estimates and better ERD tracking can be obtained. This study further proposes a non-Gaussian state space formulation of time-varying autoregressive moving average (TVARMA) models to improve the spectral estimation. Simulation on TVAR process with abrupt parameter variation shows superior tracking performance of non-Gaussian models. Evaluation on motor-imagery EEG data shows that the non-Gaussian models provide more accurate detection of abrupt changes in alpha rhythm ERD. Among the proposed non-Gaussian models, TVARMA shows better spectral representations while maintaining reasonable good ERD tracking performance.     

基于高斯和均方根容积卡尔曼滤波的姿态角辅助目标跟踪算法

根据目标2维运动速度与姿态角的关系,该文提出一种姿态角辅助目标跟踪算法。在目标运动学基础上建立状态向量中包含姿态角的跟踪模型,实现姿态角对目标跟踪的辅助;针对基于模板匹配姿态角量测的噪声为非高斯情况,将均方根容积卡尔曼滤波引入到高斯和滤波框架下,提出新的高斯和均方根容积卡尔曼滤波算法,提高非线性非高斯处理能力,同时结合目标运动中姿态角的变化规律,建立姿态角分量不同的跟踪模型,通过模型切换实现机动姿态角的滤波。算法对姿态角量测进行滤波,同时实现了姿态角信息与位置信息的有效融合。仿真结果验证了该算法的有效性和正确性。     

On the recursive solution of the normal equations of bilateralmultivariate autoregressive models

A multivariate version of the bilateral autoregressive (AR) model is proposed, and a recursive algorithm is presented to solve the normal equations of the bilateral multivariate AR models. The recursive algorithm is computationally efficient and easy to implement as a computer program. The recursive algorithm is useful for identifying and smoothing not only bilateral multivariate AR processes but multidimensional multivariate AR processes and multivariate spatio-temporal processes as well     

一种新的基于粒子滤波的多模型跟踪算法

粒子滤波技术通过非参数化的蒙特卡罗模拟方法实现递推贝叶斯滤波,适用于非线性目标运动模型、非线性传感器测量模型和非高斯噪声的目标跟踪。但需已知目标和量测模型,而实际情况往往难以满足此条件。交互多模型算法(IMM)依据各模型对目标前一时刻状态估计的方差,确定各模型在当前时刻状态下存在的概率,利用各模型对目标状态估计的加权和,确定目标的状态。本文采用粒子滤波代替IMM算法中各模型的Kalman滤波,将粒子滤波与IMM的优点相结合。同时,采用UKF(UnscentedKalmanFilter)产生粒子,由于考虑了当前量测,使得粒子的分布更加接近后验概率分布,用较少的粒子就可以逼近目标的真实状态。仿真实验结果表明,本算法可用于标准IMM算法无法实现跟踪的复杂情形,而且使用的粒子数目仅是同类算法的二十分之一。     

Recursive blind identification of non-Gaussian time-varying ARmodel and application to blind equalisation of time-varying channel

A novel method for the blind identification of a non-Gaussian time-varying autoregressive model is presented. By approximating the non-Gaussian probability density function of the model driving noise sequence with a Gaussian-mixture density, a pseudo maximum-likelihood estimation algorithm is proposed for model parameter estimation. The real model identification is then converted to a recursive least squares estimation of the model time-varying parameters and an inference of the Gaussian-mixture parameters, so that the entire identification algorithm can be recursively performed. As an important application, the proposed algorithm is applied to the problem of blind equalisation of a time-varying AR communication channel online. Simulation results show that the new blind equalisation algorithm can achieve accurate channel estimation and input symbol recovery     

Particle filtering based channel estimation in OFDM power line communication

Particle filtering （PF） algorithm has the powerful potential for coping with difficult non-linear and non-Gaussian problems. Aiming at non-linear, non-Gaussian and time-varying characteristics of power line channel, a time-varying channel estimation scheme combined PF algorithm with decision feedback method is proposed. In the proposed scheme, firstly the indoor power line channel is measured using the pseudo-noise （PN） correlation method, and a first-order dynamic autoregressive （AR） model is set up to describe the measured channel, then, the channel states are estimated dynamically from the received signals by exploiting the proposed scheme. Meanwhile, due to the complex noise distribution of power line channel, the performance of channel estimation based on the proposed scheme under the Middleton class A impulsive noise environment is analyzed. Comparisons are made with the channel estimation scheme respectively based on least square （LS）, Kalman filtering （KF） and the proposed algorithm. Simulation indicates that PF algorithm dealing with this power line channel estimation difficult non-linear and non-Gaussian problems performance is superior to those of LS and KF respectively, so the proposed scheme achieves higher estimation accuracy. Therefore, it is confirmed that PF algorithm has its own unique advantage for power line channel estimation.     

Gaussian mixture density modeling of non-Gaussian source forautoregressive process

A new approach is taken to model non-Gaussian sources of AR processes using Gaussian mixture densities that are known to be effective for approximating wide varieties of probability distributions. A maximum likelihood estimation algorithm is derived for estimating the AR parameters by solving a generalized normal equation, and a clustering algorithm is used for estimating the parameters of Gaussian mixture density of the source signals. The correlation matrix of the generalized normal equation is not Toeplitz but is symmetric and in general positive definite. Higher order statistics of skewness and kurtosis are used for identifying the source distribution as being Gaussian or non-Gaussian and, consequently, determining the parameter estimation technique between the conventional method and the proposed method. Experiments on non-Gaussian source AR processes demonstrate that under high SNR conditions (SNR⩾20 dB), the proposed algorithm outperforms the conventional AR estimation algorithm and the cumulant-based algorithm by an order-of-magnitude reduction of average estimation errors. The proposed algorithm also has very low estimation errors with short data records. Finally, a maximum likelihood prediction method is formulated for non-Gaussian source AR processes that has shown potential in achieving higher efficiency signal coding than linear predictive coding     

A Least-Squares-Based Algorithm for Identification of Non-Gaussian ARMA Models

A new recursive method for estimating the parameters of autoregressive moving average (ARMA) models is presented in this paper. The recursive linear identification method is developed using higher-order statistics of the observed output data and is based on a least-squares solution. Namely, a matrix consisting of third-order statistics (or cumulants) of the observed output data is constructed so that it almost possesses a full rank structure. The signal is embedded in a Gaussian noise that may be colored. The system is driven by a zero-mean independent and identically distributed non-Gaussian process. The excitation signal is unobserved. Simulation results are given to illustrate the performance of the proposed algorithm with respect to existing well-known methods.     

EM-based recursive estimation of channel parameters

Recursive (online) expectation-maximization (EM) algorithm along with stochastic approximation is employed in this paper to estimate unknown time-invariant/variant parameters. The impulse response of a linear system (channel) is modeled as an unknown deterministic vector/process and as a Gaussian vector/process with unknown stochastic characteristics. Using these models which are embedded in white or colored Gaussian noise, different types of recursive least squares (RLS), Kalman filtering and smoothing and combined RLS and Kalman-type algorithms are derived directly from the recursive EM algorithm. The estimation of unknown parameters also generates new recursive algorithms for situations, such as additive colored noise modeled by an autoregressive process. The recursive EM algorithm is shown as a powerful tool which unifies the derivations of many adaptive estimation methods     

On 2-D recursive LMS algorithms using ARMA prediction for ADPCMencoding of images

A two-dimensional (2D) linear predictor which has an autoregressive moving average (ARMA) representation well as a bias term is adapted for adaptive differential pulse code modulation (ADPCM) encoding of nonnegative images. The predictor coefficients are updated by using a 2D recursive LMS (TRLMS) algorithm. A constraint on optimum values for the convergence factors and an updating algorithm based on the constraint are developed. The coefficient updating algorithm can be modified with a stability control factor. This realization can operate in real time and in the spatial domain. A comparison of three different types of predictors is made for real images. ARMA predictors show improved performance relative to an AR algorithm.     

Recursive filtering with non-Gaussian noises

The Kalman filter is an optimal recursive filter, although its optimality can only be claimed under the Gaussian noise environment. In this paper, we consider the problem of recursive filtering with non-Gaussian noises. One of the most promising schemes, which was proposed by Masreliez (1972, 1975), uses the nonlinear score function as the correction term in the state estimate. Unfortunately, the score function cannot be easily implemented except for simple cases. In this paper, a new method for efficient evaluation of the score function is developed. The method employs an adaptive normal expansion to expand the score function followed by truncation of the higher order terms. Consequently, the score function can be approximated by a few central moments. The normal expansion is made adaptive by using the concept of conjugate recentering and the saddle point method. It is shown that the approximation is satisfactory, and the method is simple and practically feasible. Experimental results are reported to demonstrate the effectiveness of the new algorithm     

Estimation and identification for 2-D block Kalman filtering

The development of a recursive identification and estimation procedure for two-dimensional block Kalman filtering is discussed. The recursive identification scheme can be used online to update the image model parameters at each iteration based on the local statistics within a block of the observed noisy image. The covariance matrix of the driving noise can also be estimated at each iteration of this algorithm. A recursive procedure for computing the parameters of the higher order models is given. Simulation results are also provided     

Inference life-models of electrical insulating materials by using aKalman filter

A Kalman filter is applied to life tests for characterizing electrical or thermal endurance of electrical insulating materials. This recursive estimator provides updated life model parameter values after each life test. The life models are: (1) inverse power law and the exponential law, used for electrical or multi-stress ageing; and (2) Arrhenius model, used for thermal ageing. The state, prediction, and updating equations of the Kalman filter algorithm are specified for insulation endurance inference. Insight into the definition of the state variables, which are directly related to the model parameters, and determination of system and observation errors are developed. A recursive breakdown test detects important changes in the prevailing ageing process. The range of validity of the life model, as well as information on electrical and thermal threshold are considered. A flow chart of the filtering algorithm is presented. Example experimental results relevant to insulating materials and systems subjected to electrical and thermal life tests are processed according to the algorithm     

James-Stein state filtering algorithms

In 1961, James and Stein discovered a remarkable estimator that dominates the maximum-likelihood estimate of the mean of a p-variate normal distribution, provided the dimension p is greater than two. This paper extends the James-Stein estimator and highlights the benefits of applying these extensions to adaptive signal processing problems. The main contribution of this paper is the derivation of the James-Stein state filter (JSSF), which is a robust version of the Kalman filter. The JSSF is designed for situations where the parameters of the state-space evolution model are not known with any certainty. In deriving the JSSF, we derive several other results. We first derive a James-Stein estimator for estimating the regression parameter in a linear regression. A recursive implementation, which we call the James-Stein recursive least squares (JS-RLS) algorithm, is derived. The resulting estimate, although biased, has a smaller mean-square error than the traditional RLS algorithm. Finally, several heuristic algorithms are presented, including a James-Stein version of the Yule-Walker equations for AR parameter estimation     

Image motion estimation algorithms using cumulants

A class of algorithms is presented that estimates the displacement vector from two successive image frames consisting of signal plus noise. In the model, the signals are assumed to be either non-Gaussian or (quasistationary) deterministic; and, via a consistency result for cumulant estimators, the authors unify the stochastic and deterministic signal viewpoints. The noise sources are assumed to be Gaussian (perhaps spatially and temporally correlated) and of unknown covariance. Viewing image motion estimation as a 2D time delay estimation problem, the displacement vector of a moving object is estimated by solving linear equations involving third-order auto-cumulants and cross-cumulants. Additionally, a block-matching algorithm is developed that follows from a cumulant-error optimality criterion. Finally, the displacement vector for each pel is estimated using a recursive algorithm that minimizes a mean 2D fourth-order cumulant criterion. Simulation results are presented and discussed.     

Finite dimensional smoothers for MAP state estimation of bilinearsystems

In this paper, we present two finite-dimensional iterative algorithms for maximum a posteriori (MAP) state sequence estimation of bilinear systems. Bilinear models are appealing in their ability to represent or approximate a broad class of nonlinear systems. Our iterative algorithms for state estimation are based on the expectation-maximization (EM) algorithm and outperform the widely used extended Kalman smoother (EKS). Unlike the EKS, these EM algorithms are optimal (in the MAP sense) finite-dimensional solutions to the state sequence estimation problem for bilinear models. We also present recursive (on-line) versions of the two algorithms and show that they outperform the extended Kalman filter (EKF). Our main conclusion is that the EM-based algorithms presented in this paper are novel nonlinear filtering methods that perform better than traditional methods such as the EKF     

Recursive parameter estimation for noisy autoregressive signals (Corresp.)

The problem of recursively estimating the unknown parameters of a scalar autoregressive (AR) signal observed in additive white noise, including signal power and noise variance, is considered. A state-space model in a canonical but noninnovations form is used to represent the noisy AR signal. An algorithm based on a system identification/parameter estimation technique known as the recursive prediction error method is presented for recursive parameter estimation. Two simulation examples illustrate the effectiveness of the proposed algorithm.     

非高斯噪声情况下多平台机动目标被动跟踪新方法

针对仅有角度测量信息条件下,被动传感器融合目标跟踪问题,提出了扩维UKF滤波算法;并对经典IMM进行改进提出变维IMM算法,利用不同维数模型之间的交互式融合解决对机动目标的跟踪问题;进一步考虑实际情况中往往存在的测量噪声为非高斯情况,引入自适应滤波方法。最终提出变维交互式多模型自适应抗差扩维无迹滤波方法（VDIMM-AAUKF）,成功实现了被动多传感器在高斯和非高斯噪声情况下对机动目标跟踪。仿真实验结果表明该算法跟踪精度高、稳定性好,具有较好的实际应用价值。     

二维分数阶卡尔曼滤波及其在图像处理中的应用

该文研究了二维分数阶卡尔曼滤波及其在图像增强与滤波中的应用问题。首先基于分数微积分的定义,建立了二维线性离散系统的分数阶差分状态空间模型。然后,提出了一种可应用于图像信息处理的二维分数阶卡尔曼滤波算法,并通过实验验证了该文提出算法的有效性。仿真结果证明,该算法增强了图像中的细节特征,同时消弱了图像中的背景噪声。