Local polynomial modeling of time-varying autoregressive models with application to time-frequency analysis of event-related EEG

This paper proposes a new local polynomial modeling (LPM) method for identification of time-varying autoregressive (TVAR) models and applies it to time-frequency analysis (TFA) of event-related electroencephalogram (ER-EEG). The LPM method models the TVAR coefficients locally by polynomials and estimates the polynomial coefficients using weighted least-squares with a window having a certain bandwidth. A data-driven variable bandwidth selection method is developed to determine the optimal bandwidth that minimizes the mean squared error. The resultant time-varying power spectral density estimation of the signal is capable of achieving both high time resolution and high frequency resolution in the time-frequency domain, making it a powerful TFA technique for nonstationary biomedical signals like ER-EEG. Experimental results on synthesized signals and real EEG data show that the LPM method can achieve a more accurate and complete time-frequency representation of the signal.     

Estimation of event-related synchronization changes by a new TVARmethod

The modeling of nonstationary electroencephalogram (EEG) with time-varying autoregressive (TVAR) models is discussed. The classical least squares TVAR approach is modified so that prior assumptions about the signal can be taken into account in an optimal way. The method is then applied to the estimation of event-related synchronization changes in the EEG. The results show that the new approach enables effective estimation of the parameter evolution of the time-varying EEG with better time resolution compared to previous methods. The new method also allows single-trial analysis of the event-related synchronization     

Stabilization of Subba Rao-Liporace models

The stability of time-varying autoregressive (TVAR) models is an important issue in many applications such as time-varying spectral estimation, EEG simulation and analysis, and time-varying linear prediction coding (TVLPC). For stationary AR models there are methods that guarantee stability, but the for nonadaptive time-varying approaches there are no such methods. On the other hand, in some situations, such as in EEG analysis, the models that temporarily exhibit roots with almost unit moduli are difficult to use. Thus we may need a tighter stability condition such as stability with margin 1–. In this paper we propose a method for the estimation of TVAR models that guarantees stability with margin 1–, that is, the moduli of the roots of the time-varying characteristic polynomial are less than or equal to some arbitrary positive number for every time instant. The model class is the Subba Rao-Liporace class, in which the time-varying coefficients are constrained to a subspace of the coefficient time evolutions. The method is based on sequential linearization of the associated nonlinear constraints and the subsequent use of a Gauss-Newton-type algorithm. The method is also applied to a simulated autoregressive process.     

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     

Deterministic regression methods for unbiased estimation of time-varying autoregressive parameters from noisy observations

A great deal of interest has been paid to autoregressive parameter estimation in the noise-free case or when the observation data are disturbed by random noise. Tracking time-varying autoregressive (TVAR) parameters has been also discussed, but few papers deal with this issue when there is an additive zero-mean white Gaussian measurement noise. In this paper, one considers deterministic regression methods (or evolutive methods) where the TVAR parameters are assumed to be weighted combinations of basis functions. However, the additive white measurement noise leads to a weight-estimation bias when standard least squares methods are used. Therefore, we propose two alternative blind off-line methods that allow both the variance of the additive noise and the weights to be estimated. The first one is based on the errors-in-variable issue whereas the second consists in viewing the estimation issue as a generalized eigenvalue problem. A comparative study with other existing methods confirms the effectiveness of the proposed methods.     

粒子滤波在目标跟踪算法中的应用研究

针对非高斯、强噪声背景下的高机动目标实施跟踪时,卡尔曼滤波、扩展卡尔曼滤波等算法将出现滤波精度下降甚至发散现象。粒子滤波方法作为一种基于贝叶斯估计的非线性滤波算法,在处理非高斯非线性时变系统的参数估计和状态滤波问题方面有独到的优势。以目标跟踪问题为背景,将粒子滤波与卡尔曼滤波算法进行了对比研究。     

Estimation of nonstationary EEG with Kalman smoother approach: an application to event-related synchronization (ERS)

An adaptive spectrum estimation method for nonstationary electroencephalogram by means of time-varying autoregressive moving average modeling is presented. The time-varying parameter estimation problem is solved by Kalman filtering along with a fixed-interval smoothing procedure. Kalman filter is an optimal filter in the mean square sense and it is a generalization of other adaptive filters such as recursive least squares or least mean square. Furthermore, by using the smoother the unavoidable tracking lag of adaptive filters can be avoided. Due to the properties of Kalman filter and benefits of the smoothing the time-frequency resolution of the presented Kalman smoother spectra is extremely high. The presented approach is applied to estimation of event-related synchronization/desynchronization (ERS/ERD) dynamics of occipital alpha rhythm measured from three healthy subjects. With the Kalman smoother approach detailed spectral information can be extracted from single ERS/ERD samples.     

Asymptotic analysis of an algorithm for parameter estimation andidentification of 1-b quantized AR time series

Krishnamurthy and Mareels (1995) presented a parameter estimation algorithm called the binary series estimation algorithm (BSEA) for Gaussian autoregressive (AR) time series given 1-b quantized noisy measurements. Of particular interest were the rather surprising computer simulation results that showed that for certain AR series in multiplicative noise, the BSEA based on 1-b quantized measurements yielded significantly better parameter estimates than Yule-Walker methods that are based on the unquantized measurements. The present paper carries out an asymptotic analysis of the BSEA for Gaussian AR models. In particular, from a central limit theorem, the authors obtain expressions for the asymptotic covariances of the parameter estimates. From this they (1) present an algorithm for estimating the order of an AR series from 1-b quantized measurements and (2) theoretically justify why BSEA can yield better estimates than the Yule-Walker methods in some cases. Computer simulations show that the theoretically predicted parameter estimate covariances are extremely accurate. In addition, the authors present examples of their order estimation algorithm     

Recursive estimation of images using non-Gaussian autoregressivemodels

We consider recursive estimation of images modeled by non-Gaussian autoregressive (AR) models and corrupted by spatially white Gaussian noise. The goal is to find a recursive algorithm to compute a near minimum mean square error (MMSE) estimate of each pixel of the scene using a fixed lookahead of D rows and D columns of the observations. Our method is based on a simple approximation that makes possible the development of a useful suboptimal nonlinear estimator. The algorithm is first developed for a non-Gaussian AR time-series and then generalized to two dimensions. In the process, we draw on the well-known reduced update Kalman filter (KF) technique of Woods and Radewan (1977) to circumvent computational load problems. Several examples demonstrate the non-Gaussian nature of residuals for AR image models and that our algorithm compares favorably with the Kalman filtering techniques in such cases.     

基于HFM和感知后滤波器的语音增强

研究了只能获得带噪信号的情况下的语音增强问题。将语音信号看作由高斯噪声激励的自回归(AR)过程,观测噪声为加性高斯白噪声,把信号转化为状态空间模型。首先用隐马尔可夫模型(HMM)估计AR参数和噪声的方差作为卡尔曼滤波器初值,估计信号作为参数估计的中间值给出,然后将估计信号通过一个感知滤波器平滑以消除残余噪声。仿真结果表明该算法有良好的性能。     

High resolution two-dimensional minimum free energy AR spectral estimation

We extend the minimum free energy (MFE) parameter estimation method to 2-D fields. This 2-D MFE method may be used to determine autoregressive (AR) model parameters for spectral estimation of 2-D fields. It may also be used to provide AR models for texture synthesis. The performance of the technique for closely spaced sinusoids in white noise is demonstrated by numerical example. Better results can be achieved than with the multidimensional Levinson algorithm.     

Two-dimensional minimum free energy spectral estimation

The author proposes a 2-D extension of the minimum free energy (MFE) parameter estimation method which may be used to determine autoregressive (AR) model parameters for 2-D spectral estimation. The performance of the technique for spectral estimation of 2-D sinusoids in white noise is demonstrated by numerical example. It is seen that MFE can provide superior spectral estimation over that which can be achieved with the multidimensional Levinson algorithm with equivalent computational burden. The performance of the technique in terms of computational expense and accuracy of spectral estimation over a number of simulation trials is compared with a modified covariance technique     

Modeling non-Gaussian time-varying vector autoregressive processes by particle filtering

We present a novel and general methodology for modeling time-varying vector autoregressive processes which are widely used in many areas such as modeling of chemical processes, mobile communication channels and biomedical signals. In the literature, most work utilize multivariate Gaussian models for the mentioned applications, mainly due to the lack of efficient analytical tools for modeling with non-Gaussian distributions. In this paper, we propose a particle filtering approach which can model non-Gaussian autoregressive processes having cross-correlations among them. Moreover, time-varying parameters of the process can be modeled as the most general case by using this sequential Bayesian estimation method. Simulation results justify the performance of the proposed technique, which potentially can model also Gaussian processes as a sub-case.     

Root modulus constraints in autoregressive model estimation

The stability of autoregressive (AR) models is an important issue in many applications such as spectral estimation, simulation of EEG, and synthesis of speech. There are methods for AR parameter estimation that guarantee the stability of the model, that is, all roots of the characteristic polynomial of the model have moduli less than unity. However, in some situations, such as EEG simulation, the models that exhibit roots with almost unit moduli are difficult to use. In this paper we propose a method for estimating AR models that guarantees hyperstability, that is, the moduli of the roots are less than or equal to some arbitrary positive number. The method is based on an iterative minimization scheme in which the associated nonlinear constraints are linearized sequentially.     

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.     

A minimum misadjustment adaptive FIR filter

The performance of an adaptive filter is limited by the misadjustment resulting from the variance of adapting parameters. This paper develops a method to reduce the misadjustment when the additive noise in the desired signal is correlated. Given a static linear model, the linear estimator that can achieve the minimum parameter variance estimate is known as the best linear unbiased estimator (BLUE). Starting from classical estimation theory and a Gaussian autoregressive (AR) noise model, a maximum likelihood (ML) estimator that jointly estimates the filter parameters and the noise statistics is established. The estimator is shown to approach the best linear unbiased estimator asymptotically. The proposed adaptive filtering method follows by modifying the commonly used mean-square error (MSE) criterion in accordance with the ML cost function. The new configuration consists of two adaptive components: a modeling filter and a noise whitening filter. Convergence study reveals that there is only one minimum in the error surface, and global convergence is guaranteed. Analysis of the adaptive system when optimized by LMS or RLS is made, together with the tracking capability investigation. The proposed adaptive method performs significantly better than a usual adaptive filter with correlated additive noise and tracks a time-varying system more effectively     

A novel robust filtering strategy for systems with Non-Gaussian noises

For estimating the states of moving targets in the nonlinear system with non-Gaussian noise, the combination of Gaussian Sum Filter (GSF) and other nonlinear filters has been chosen as the filtering algorithm conventionally. The Smooth Variable Structure Filter (SVSF) is a new predictor-corrector method used for state and parameter estimation, which has good stability and robustness. In this paper we propose a new strategy called the modified GS-EKF-SVSF, which inherits good robustness of Gaussian Sum and Smooth Variable Structure Filter (GS-SVSF) and high accuracy of Gaussian Sum and Extended Kalman Filter (GS-EKF). A nonlinear system with non-Gaussian noise for target tracking is used to test the proposed new strategy. The simulation results demonstrate that our proposed strategy has higher accuracy and better robustness when there are modelling uncertainties existing in the system.     

有色噪声下基于Unscented粒子滤波的语音增强方法

针对含有色噪声的语音,提出了一种基于Unscented粒子滤波的单通道语音增强方法.采用时变自回归模型(TVAR)对干净语音建模,通过Unscented粒子滤波器估计AR模型的参数并滤除有色噪声.与大多数常用的粒子滤波选择的建议分布不同,Unscented粒子滤波器采用Unscented卡尔曼滤波器生成粒子滤波的建议分布.由于在粒子的更新过程中考虑了最近的观测值,Unscented粒子滤波器能够在粒子数少于传统粒子滤波算法所需粒子数目的基础上改善估计的性能.仿真实验结果表明,在有色噪声背景下该算法具有良好的语音增强效果.     

On-line spectral estimation of nonstationary time series based onAR model parameter estimation and order selection with a forgettingfactor

A new method for on-line spectral estimation of nonstationary time series via autoregressive (AR) model construction is proposed. The method consists of on-line parameter estimation based on the recursive least squares ladder estimation algorithm with a forgetting factor and on-line order determination based on AIC with some modifications. The effectiveness of the proposed method is demonstrated by computer simulation study and applying to the actual data of electroencephalogram (EEG)     

Analysis of Acoustic Signatures from Moving Vehicles Using Time-Varying Autoregressive Models

Time-varying autoregressive (TVAR) modeling approach for the analysis of acoustic signatures from moving vehicles is presented in this paper. Acoustic signatures from moving vehicles are nonstationary, and features extracted under the stationary assumption often result unsatisfactory performance. In TVAR modeling approach, the time-varying parameters are expanded as a linear combination of deterministic time functions. In this paper, the TVAR parameters are expanded by a low-order discrete cosine transform (DCT), since DCT is known to be close to the optimal Kahrunen-Loève transform when the signal is Markov. The maximum likelihood estimation and order selection in TVAR models are also discussed. Many attributes of vehicle activities, such as vehicle type, engine speed, loading, road condition, etc., may be inferred from the estimated model parameters. The performance of the TVAR modeling approach is tested with both synthetic and real acoustic signatures. A synthetic signal containing multiple time-varying sinusoids are used to compare the performances in the estimation of time-frequency distribution with other approaches. In the experiment with acoustic signatures from moving vehicles, it is shown that the TVAR models can be effectively used to determine vehicle activities and types at close range and cruising speed.