Optimal Estimation of High-Dimensional Covariance Matrices with Missing and Noisy Data

The estimation of covariance matrices is very important in many fields, such as statistics. In real applications, data are frequently influenced by high dimensions and noise. However, most relevant studies are based on complete data. This paper studies the optimal estimation of high-dimensional covariance matrices based on missing and noisy sample under the norm. First, the model with sub-Gaussian additive noise is presented. The generalized sample covariance is then modified to define a hard thresholding estimator , and the minimax upper bound is derived. After that, the minimax lower bound is derived, and it is concluded that the estimator presented in this article is rate-optimal. Finally, numerical simulation analysis is performed. The result shows that for missing samples with sub-Gaussian noise, if the true covariance matrix is sparse, the hard thresholding estimator outperforms the traditional estimate method.

PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS

We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small a-stable noises, observed at n regularly spaced time points ti = i/n, i = 1,...,n on [0, 1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator （LSE） when a small dispersion parameter ε→0 and n →∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.

Application of local polynomial estimation in suppressing strong chaotic noise

In this paper, we propose a new method that combines chaotic series phase space reconstruction and local polynomial estimation to solve the problem of suppressing strong chaotic noise. First, chaotic noise time series are reconstructed to obtain multivariate time series according to Takens delay embedding theorem. Then the chaotic noise is estimated accurately using local polynomial estimation method. After chaotic noise is separated from observation signal, we can get the estimation of the useful signal. This local polynomial estimation method can combine the advantages of local and global law. Finally, it makes the estimation more exactly and we can calculate the formula of mean square error theoretically. The simulation results show that the method is effective for the suppression of strong chaotic noise when the signal to interference ratio is low.

Parameter estimation for chaotic systems with and without noise using differential evolution-based method

We present an approach in which the differential evolution （DE） algorithm is used to address identification problems in chaotic systems with or without delay terms. Unlike existing considerations, the scheme is able to simultaneously extract （i） the commonly considered parameters, （ii） the delay, and （iii） the initial state. The main goal is to present and verify the robustness against the common white Guassian noise of the DE-based method. Results of the time-delay logistic system, the Mackey Glass system and the Lorenz system are also presented.

A CROSS-REFERENCE METHOD OF PARAMETER ESTIMATION AND NOISE REDUCTION AND ITS APPLICATIONS

This paper proposes a cross-reference method of nonlinear time series analysis, combining the tasks of dynamical system parameter estimation and noise reduction which were fulfilled separately before. With the positive interaction between the two processing modules, the method is somewhat superior. Some prior works can be viewed as special cases of this general framework and effective new algorithms may be devised according to it. Two examples of chaotic time series analysis are also given to show the applicability of the proposed method.

A Singular Value Thresholding Based Matrix Completion Method for DOA Estimation in Nonuniform Noise

Usually,the problem of direction-of-arrival(DOA)estimation is performed based on the assumption of uniform noise.In many applications,however,the noise across the array may be nonuniform.In this situation,the performance of DOA estimators may be deteriorated greatly if the non-uniformity of noise is ignored.To tackle this problem,we consider the problem of DOA es-timation in the presence of nonuniform noise by leveraging a singular value thresholding(SVT)based matrix completion method.Different from that the traditional SVT method apply fixed threshold,to improve the performance,the proposed method can obtain a more suitable threshold based on careful estimation of the signal-to-noise ratio(SNR)levels.Specifically,we firstly employ an SVT-based matrix completion method to estimate the noise-free covariance matrix.On this basis,the signal and noise subspaces are obtained from the eigendecomposition of the noise-free cov-ariance matrix.Finally,traditional subspace-based DOA estimation approaches can be directly ap-plied to determine the DOAs.Numerical simulations are performed to demonstrate the effective-ness of the proposed method.

Noise level estimation method with application to EMD-based signal denoising

This paper proposes a new signal noise level estimation approach by local regions. The estimated noise variance is applied as the threshold for an improved empirical mode decomposition（EMD） based signal denoising method. The proposed estimation method can effectively extract the candidate regions for the noise level estimation by measuring the correlation coefficient between noisy signal and a Gaussian filtered signal. For the improved EMD based method, the situation of decomposed intrinsic mode function（IMFs） which contains noise and signal simultaneously are taken into account. Experimental results from two simulated signals and an X-ray pulsar signal demonstrate that the proposed method can achieve better performance than the conventional EMD and wavelet transform（WT） based denoising methods.

Joint DOA and polarization estimation for unequal power sources based on reconstructed noise subspace

In most literature about joint direction of arrival（DOA） and polarization estimation, the case that sources possess different power levels is seldom discussed. However, this case exists widely in practical applications, especially in passive radar systems. In this paper, we propose a joint DOA and polarization estimation method for unequal power sources based on the reconstructed noise subspace. The invariance property of noise subspace（IPNS） to power of sources has been proved an effective method to estimate DOA of unequal power sources. We develop the IPNS method for joint DOA and polarization estimation based on a dual polarized array. Moreover, we propose an improved IPNS method based on the reconstructed noise subspace, which has higher resolution probability than the IPNS method. It is theoretically proved that the IPNS to power of sources is still valid when the eigenvalues of the noise subspace are changed artificially. Simulation results show that the resolution probability of the proposed method is enhanced compared with the methods based on the IPNS and the polarimetric multiple signal classification（MUSIC） method. Meanwhile, the proposed method has approximately the same estimation accuracy as the IPNS method for the weak source.

Joint MAP channel estimation and data detection for OFDM in presence of phase noise from free running and phase locked loop oscillator

This paper addresses a computationally compact and statistically optimal joint Maximum a Posteriori(MAP)algorithm for channel estimation and data detection in the presence of Phase Noise(PHN)in iterative Orthogonal Frequency Division Multiplexing(OFDM)receivers used for high speed and high spectral efficient wireless communication systems.The MAP cost function for joint estimation and detection is derived and optimized further with the proposed cyclic gradient descent optimization algorithm.The proposed joint estimation and detection algorithm relaxes the restriction of small PHN assumptions and utilizes the prior statistical knowledge of PHN spectral components to produce a statistically optimal solution.The frequency-domain estimation of Channel Transfer Function(CTF)in frequency selective fading makes the method simpler,compared with the estimation of Channel Impulse Response(CIR)in the time domain.Two different time-varying PHN models,produced by Free Running Oscillator(FRO)and Phase-Locked Loop(PLL)oscillator,are presented and compared for performance difference with proposed OFDM receiver.Simulation results for joint MAP channel estimation are compared with Cramer-Rao Lower Bound(CRLB),and the simulation results for joint MAP data detection are compared with“NO PHN"performance to demonstrate that the proposed joint MAP estimation and detection algorithm achieve near-optimum performance even under multipath channel fading.

Joint Noise Reduction and lp-Norm Minimization for Enhancing Time Delay Estimation in Colored Noise

Time delay estimation (TDE) is an important issue in signal processing. Conventional TDE algorithms are usually efficient under white noise environments. In this paper, a joint noise reduction and -norm minimization method is presented to enhance TDE in colored noise. An improved subspace method for colored noise reduction is first performed. Then the time delay is estimated by using an -norm minimization method. Because the clean speech signal form the noisy signal is well extracted by noise reduction and the -norm minimization method is robust, the TDE accuracy can be enhanced. Experiment results confirm that the proposed joint estimation method obtains more accurate TDE than several conventional algorithms in colored noise, especially in the case of low signal-to-noise ratio.

Speech Signal Detection Based on Bayesian Estimation by Observing Air-Conducted Speech under Existence of Surrounding Noise with the Aid of Bone-Conducted Speech

In order to apply speech recognition systems to actual circumstances such as inspection and maintenance operations in industrial factories to recording and reporting routines at construction sites, etc. where hand-writing is difficult, some countermeasure methods for surrounding noise are indispensable. In this study, a signal detection method to remove the noise for actual speech signals is proposed by using Bayesian estimation with the aid of bone-conducted speech. More specifically, by introducing Bayes’ theorem based on the observation of air-conducted speech contaminated by surrounding background noise, a new type of algorithm for noise removal is theoretically derived. In the proposed speech detection method, bone-conducted speech is utilized in order to obtain precise estimation for speech signals. The effectiveness of the proposed method is experimentally confirmed by applying it to air- and bone-conducted speeches measured in real environment under the existence of surrounding background noise.

Adaptive Gaussian Noise Image Removal Algorithm Using Filtering-Based Noise Estimation

This paper proposes a spatially denoising algorithm using filtering-based noise estimation for an image corrupted by Gaussian noise.The proposed algorithm consists of two stages:estimation and elimination of noise density.To adaptively deal with variety of the noise amount,a noisy input image is firstly filtered by a lowpass filter.Standard deviation of the noise is computed from different images between the noisy input and its filtered image.In addition,a modified Gaussian noise removal filter based on the local statistics such as local weighted mean,local weighted activity and local maximum is used to control the degree of noise suppression.Experiments show the effectiveness of the proposed algorithm.

Direction of arrival estimation method based on quantum electromagnetic field optimization in the impulse noise

In order to resolve direction finding problems in the impulse noise,a direction of arrival(DOA)estimation method is proposed.The proposed DOA estimation method can restrain the impulse noise by using infinite norm exponential kernel covariance matrix and obtain excellent performance via the maximumlikelihood(ML)algorithm.In order to obtain the global optimal solutions of this method,a quantum electromagnetic field optimization(QEFO)algorithm is designed.In view of the QEFO algorithm,the proposed method can resolve the difficulties of DOA estimation in the impulse noise.Comparing with some traditional DOA estimation methods,the proposed DOA estimation method shows high superiority and robustness for determining the DOA of independent and coherent sources,which has been verified via the Monte-Carlo experiments of different schemes,especially in the case of snapshot deficiency,low generalized signal to noise ratio(GSNR)and strong impulse noise.Beyond that,the Cramer-Rao bound(CRB)of angle estimation in the impulse noise and the proof of the convergence of the QEFO algorithm are provided in this paper.

Optimal estimation of the amplitude of signal with known frequency in the presence of thermal noise

In the torsion pendulum experiments, the thermal noise sets the most fundamental limit to the accurate estimation of the amplitude of the signal with known frequency. The variance of the conventional method can meet the limit only when the measurement time is much longer than the relaxation time of the pendulum. By using the maximum likelihood estimation and the equation-of-motion filter operator, we propose an optimal(minimum variance, unbiased) amplitude estimation method without limitation of the measurement time, where thermal fluctuation is the leading noise. While processing the experimental data tests of the Newtonian gravitational inverse square law, the variance of our method has been improved than before and the measurement time of determining the amplitude with this method has been reduced about half than before for the same uncertainty. These results are significant for the torsion experiment when the measurement time is limited.

ADAPTIVE ESTIMATION OF THE MEANS AND COVARIANCES OF SYSTEM NOISES

This paper presents a simple and feasible method of estimating the means andcovariances of system noises.The information about the means and covariances is derivedfrom a suboptimal Kalman filter formed via the approximate statistics.An efficientalgorithm is obtained through WLS(weighted-least-square)estimation by using themeasurement residuals of the suboptimal Kalman filter.Simulations are given to show theefficiency of the method.

Automatic estimation and removal of noise on digital image

An spatially adaptive noise detection and removal algorithm is proposed.Under the assumption that an observed image and its additive noise have Gaussian distribution,the noise parameters are estimated with local statistics from an observed degraded image,and the parameters are used to define the constraints on the noise detection process.In addition,an adaptive low-pass filter having a variable filter window defined by the constraints on noise detection is used to control the degree of smoothness of the reconstructed image.Experimental results demonstrate the capability of the proposed algorithm.

Robust Frequency Estimation Under Additive Mixture Noise

In many applications such as multiuser radar communications and astrophysical imaging processing,the encountered noise is usually described by the finite sum ofα-stable(1≤α<2)variables.In this paper,a new parameter estimator is developed,in the presence of this new heavy-tailed noise.Since the closed-formPDF of theα-stable variable does not exist exceptα=1 andα=2,we take the sum of the Cauchy(α=1)and Gaussian(α=2)noise as an example,namely,additive Cauchy-Gaussian(ACG)noise.The probability density function(PDF)of the mixed random variable,can be calculated by the convolution of the Cauchy’s PDF and Gaussian’s PDF.Because of the complicated integral in the PDF expression of the ACG noise,traditional estimators,e.g.,maximum likelihood,are analytically not tractable.To obtain the optimal estimates,a new robust frequency estimator is devised by employing the Metropolis-Hastings(M-H)algorithm.Meanwhile,to guarantee the fast convergence of the M-H chain,a new proposal covariance criterion is also devised,where the batch of previous samples are utilized to iteratively update the proposal covariance in each sampling process.Computer simulations are carried out to indicate the superiority of the developed scheme,when compared with several conventional estimators and the Cramér-Rao lower bound.

A NOVEL APPROACH FOR DOA ESTIMATION IN UNKNOWN CORRELATED NOISE FIELDS

The key of the subspace-based Direction Of Arrival (DOA) estimation lies in the estimation of signal subspace with high quality. In the case of uncorrelated signals while the signals are temporally correlated, a novel approach for the estimation of DOA in unknown correlated noise fields is proposed in this paper. The approach is based on the biorthogonality between a matrix and its Moore-Penrose pseudo inverse, and made no assumption on the spatial covariance matrix of the noise. The approach exploits the structural information of a set of spatio-temporal correlation matrices, and it can give a robust and precise estimation of signal subspace, so a precise estimation of DOA is obtained. Its performances are confirmed by computer simulation results.

Wavelet-Based Density Estimation in Presence of Additive Noise under Various Dependence Structures

We study the following model: . The aim is to estimate the distribution of X when only are observed. In the classical model, the distribution of is assumed to be known, and this is often considered as an important drawback of this simple model. Indeed, in most practical applications, the distribution of the errors cannot be perfectly known. In this paper, the author will construct wavelet estimators and analyze their asymptotic mean integrated squared error for additive noise models under certain dependent conditions, the strong mixing case, the β-mixing case and the ρ-mixing case. Under mild conditions on the family of wavelets, the estimator is shown to be -consistent and fast rates of convergence have been established.

State Estimation for Sound and Vibration Emitted from a Machine Based on Sound Measurement Under Existenceof Background Noise

Many studies on the diagnosis for machines have become important recently because of increased use of various complex industrial systems.The correlation information between sound and vibration is very important for machine diagnosis.Usually,vibration pickups are attached directly to the machine in order to measure vibration data.However,in some cases,the sensors can not be attached directly on highly precise devices.In this study,a method to estimate the fluctuation of sound and vibration is proposed based on the measurement data of sound emitted from the machine under existence of background noise.The effectiveness of the proposed theory is experimentally confirmed by applying it to the observed data emitted from a rotational machine driven by an electric motor.