Maximum-likelihood symmetric α-stable parameter estimation

Using the close relation between Fisher scoring and Newton maximization, and an efficient density function evaluation, we develop a fast maximum-likelihood parameter estimation method. Simulations show the algorithm to be superior in accuracy to McCulloch's (1986) method and to achieve the Cramer-Rao bound     

Maximum-likelihood based estimation of the Nakagami m parameter

Maximum-likelihood estimation of the Nakagami (1960) m parameter is considered. Two new estimators are proposed and examined. The sample mean and the sample variance of the new estimators are compared with the best reported estimator. The new estimators offer superior performance.     

Maximum-likelihood parameter estimation for unsupervised stochasticmodel-based image segmentation

An unsupervised stochastic model-based approach to image segmentation is described, and some of its properties investigated. In this approach, the problem of model parameter estimation is formulated as a problem of parameter estimation from incomplete data, and the expectation-maximization (EM) algorithm is used to determine a maximum-likelihood (ML) estimate. Previously, the use of the EM algorithm in this application has encountered difficulties since an analytical expression for the conditional expectations required in the EM procedure is generally unavailable, except for the simplest models. In this paper, two solutions are proposed to solve this problem: a Monte Carlo scheme and a scheme related to Besag's (1986) iterated conditional mode (ICM) method. Both schemes make use of Markov random-field modeling assumptions. Examples are provided to illustrate the implementation of the EM algorithm for several general classes of image models. Experimental results on both synthetic and real images are provided.     

Maximum-likelihood bearing estimation with partly calibrated arraysin spatially correlated noise fields

The problem of using a partly calibrated array for maximum likelihood (ML) bearing estimation of possibly coherent signals buried in unknown correlated noise fields is shown to admit a neat solution under fairly general conditions. More exactly, this paper assumes that the array contains some calibrated sensors, whose number is only required to be larger than the number of signals impinging on the array, and also that the noise in the calibrated sensors is uncorrelated with the noise in the other sensors. These two noise vectors, however, may have arbitrary spatial autocovariance matrices. Under these assumptions the many nuisance parameters (viz., the elements of the signal and noise covariance matrices and the transfer and location characteristics of the uncalibrated sensors) can be eliminated from the likelihood function, leaving a significantly simplified concentrated likelihood whose maximum yields the ML bearing estimates. The ML estimator introduced in this paper, and referred to as MLE, is shown to be asymptotically equivalent to a recently proposed subspace-based bearing estimator called UNCLE and rederived herein by a much simpler approach than in the original work. A statistical analysis derives the asymptotic distribution of the MLE and UNCLE estimates, and proves that they are asymptotically equivalent and statistically efficient. In a simulation study, the MLE and UNCLE methods are found to possess very similar finite-sample properties as well. As UNCLE is computationally more efficient, it may be the preferred technique in a given application     

Maximum-likelihood parameter estimation of the harmonic,evanescent, and purely indeterministic components of discretehomogeneous random fields

This paper presents a maximum-likelihood solution to the general problem of fitting a parametric model to observations from a single realization of a two-dimensional (2-D) homogeneous random field with mixed spectral distribution. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. The suggested algorithm involves a two-stage procedure. In the first stage, we obtain a suboptimal initial estimate for the parameters of the spectral support of the evanescent and harmonic components. In the second stage, we refine these initial estimates by iterative maximization of the conditional likelihood of the observed data, which is expressed as a function of only the parameters of the spectral supports of the evanescent and harmonic components. The solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the other unknown parameters of the field to a linear least squares. The Cramer-Rao lower bound on the accuracy of jointly estimating the parameters of the different components is derived, and it is shown that the bounds on the purely indeterministic and deterministic components are decoupled. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy. The performance of the maximum-likelihood algorithm is illustrated by Monte Carlo simulations and is compared with the Cramer-Rao bound     

The evanescent field transform for estimating the parameters ofhomogeneous random fields with mixed spectral distributions

Parametric modeling and estimation of complex valued homogeneous random fields with mixed spectral distributions is a fundamental problem in two-dimensional (2-D) signal processing. The parametric model under consideration results from the 2-D Wold-type decomposition of the random field. The same model naturally arises as the physical model in problems of space-time adaptive processing of airborne radar. A computationally efficient algorithm for estimating the parameters of the field components is presented. The algorithm is based on a nonlinear operator that uniquely maps each evanescent component to a single exponential. The exponential's spatial frequency is a function of the spectral support parameters of the evanescent component. Hence, employing this operator, the problem of estimating the spectral support parameters of an evanescent field is replaced by the simpler problem of estimating the spatial frequency of a 2-D exponential. The properties of the operator are analyzed. The algorithm performance is illustrated and investigated using Monte Carlo simulations     

Bounds on the accuracy of estimating the parameters of discretehomogeneous random fields with mixed spectral distributions

This paper considers the achievable accuracy in jointly estimating the parameters of a real-valued two-dimensional (2-D) homogeneous random field with mixed spectral distribution, from a single observed realization of it. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. An exact form of the Cramer-Rao lower bound on the error variance in jointly estimating the parameters of the different components is derived. It is shown that the estimation of the harmonic component is decoupled from that of the purely indeterministic and the evanescent components. Moreover, the bound on the parameters of the purely indeterministic and the evanescent components is independent of the harmonic component. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy     

Higher order spectral estimation for random fields

This paper is concerned with the nonparametric estimation of the higher order cumulant spectra of vector-valued stationary random fields onZ ^d by smoothing the periodograms, whereZ is the space of integers and the dimensiond1. We derive the asymptotic cumulant properties of the spectral estimates, and consider an application to multidimensional nonlinear systems identification. Numerical examples with simulated data are provided.     

Density parameter estimation of skewed α-stable distributions

Over the last few years, there has been a great interest in α-stable distributions for modeling impulsive data. As a critical step in modeling with α-stable distributions, the problem of estimating the parameters of stable distributions have been addressed by several works in the literature. However, many of these works consider only the special case of symmetric stable random variables. This is an important restriction, however, since most real-life signals are skewed. The existing techniques on estimating skewed distribution parameters are either computationally too expensive, require lookup tables, or have poor convergence properties. We introduce three novel classes of estimators for the parameters of general stable distributions, which are generalizations of the methods previously suggested for parameter estimation of symmetric stable distributions. These estimators exploit expressions we develop for fractional lower order, negative order, and logarithmic moments and tail statistics. We also introduce simple transformations that allow one to use existing symmetric stable parameter estimation techniques. Techniques suggested in this paper provide the only closed-form solutions we are aware of for parameters that may be efficiently computed. Simulation results show that at least one of our new estimators has better performance than the existing techniques over most of the parameter space. Furthermore, our techniques require substantially less computation     

Maximum-likelihood estimation of Rician distribution parameters

The problem of parameter estimation from Rician distributed data (e.g., magnitude magnetic resonance images) is addressed. The properties of conventional estimation methods are discussed and compared to maximum-likelihood (ML) estimation which is known to yield optimal results asymptotically. In contrast to previously proposed methods, ML estimation is demonstrated to be unbiased for high signal-to-noise ratio (SNR) and to yield physical relevant results for low SNR     

ML parameter estimation for Markov random fields with applicationsto Bayesian tomography

Markov random fields (MRFs) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: (i) direct parameter estimation for MRFs is known to be mathematically and numerically challenging; (ii) parameters can not be directly estimated because the true image cross section is unavailable. We propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as generalized Gaussian MRF (GGMRF). We derive methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, sigma, has a simple closed-form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an off-line numerical computation of the dependence of the partition function on p. We present a fast algorithm for computing ML parameter estimates when the true image is unavailable. To do this, we use the expectation maximization (EM) algorithm. We develop a fast simulation method to replace the E-step, and a method to improve the parameter estimates when the simulations are terminated prior to convergence. Experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets.     

Multi-dimensional Capon spectral estimation using discrete Zhang neural networks

The minimum variance spectral estimator, also known as the Capon spectral estimator, is a high resolution spectral estimator used extensively in practice. In this paper, we derive a novel implementation of a very computationally demanding matched filter-bank based a spectral estimator, namely the multi-dimensional Capon spectral estimator. To avoid the direct computation of the inverse covariance matrix used to estimate the Capon spectrum which can be computationally very expensive, particularly when the dimension of the matrix is large, we propose to use the discrete Zhang neural network for the online covariance matrix inversion. The computational complexity of the proposed algorithm for one-dimensional (1-D), as well as for two-dimensional (2-D) and three-dimensional (3-D) data sequences is lower when a parallel implementation is used.     

Robust parameter estimation of intensity distributions for brainmagnetic resonance images

Presents two new methods for robust parameter estimation of mixtures in the context of magnetic resonance (MR) data segmentation. The head is constituted of different types of tissue that can be modeled by a finite mixture of multivariate Gaussian distributions. The authors' goal is to estimate accurately the statistics of desired tissues in presence of other ones of lesser interest. These latter can be considered as outliers and can severly bias the estimates of the former. For this purpose, the authors introduce a first method, which is an extension of the expectation-maximization (EM) algorithm, that estimates parameters of Gaussian mixtures but incorporates an outlier rejection scheme which allows to compute the properties of the desired tissues in presence of atypical data. The second method is based on genetic algorithms and is well suited for estimating the parameters of mixtures of different kind of distributions. The authors use this property by adding a uniform distribution to the Gaussian mixture for modeling the outliers. The proposed genetic algorithm can efficiently estimate the parameters of this extended mixture for various initial settings. Also, by changing the minimization criterion, estimates of the parameters can be obtained by histogram fitting which considerably reduces the computational cost. Experiments on synthetic and real MR data show that accurate estimates of the gray and white matters parameters are computed     

基于谱相关的CBOC信号参数盲估计

针对低信噪比下组合二进制偏移载波(CBOC)调制信号的参数盲估计问题,提出了利用谱相关对CBOC信号进行参数估计方法。首先给出了CBOC信号模型,然后根据CBOC信号的数据通道和导频通道之间有良好的正交性特点,详细推导出其谱相关函数可以化简为两个BOC信号谱相关函数的叠加,最后根据CBOC信号循环频率截面的特点进行峰值检索后,实现对伪码速率、载频速率和副载波速率的盲估计。推导结果和计算机仿真分析表明,该方法可以实现在低信噪比下对伪码速率、载频速率和副载波速率的有效估计。     

Maximum-likelihood estimation of a class of chaotic signals

The chaotic sequences corresponding to tent map dynamics are potentially attractive in a range of engineering applications. Optimal estimation algorithms for signal filtering, prediction, and smoothing in the presence of white Gaussian noise are derived for this class of sequences based on the method of maximum likelihood. The resulting algorithms are highly nonlinear but have convenient recursive implementations that are efficient both in terms of computation and storage. Performance evaluations are also included and compared with the associated Cramer-Rao bounds     

Maximum likelihood estimation of parameters of several continuous and discrete failure distributions

A comprehensive quantitative treatment is presented for maximum-likelihood estimation of parameters of the following continuous and discrete failure distributions: (1) Exponential, (2) Gamma, (3) Weibull, (4) Normal, (5) Lognormal, (6) Extreme value, (7) Poisson, (8) Binomial and (9) Geometric.     

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)     

Maximum-likelihood versus maximum a posteriori parameter estimation of physiological system models: the C-peptide impulse response case study

Maximum-likelihood (ML), also given its connection to least-squares (LS), is widely adopted in parameter estimation of physiological system models, i.e., assigning numerical values to the unknown model parameters from the experimental data. A more sophisticated but less used approach is maximum a posteriori (MAP) estimation. Conceptually, while ML adopts a Fisherian approach, i.e., only experimental measurements are supplied to the estimator, MAP estimation is a Bayesian approach, i.e., a priori available statistical information on the unknown parameters is also exploited for their estimation. In this paper, after a brief review of the theory behind ML and MAP estimators, we compare their performance in the solution of a case study concerning the determination of the parameters of a sum of exponential model which describes the impulse response of C-peptide (CP), a key substance for reconstructing insulin secretion. The results show that MAP estimation always leads to parameter estimates with a precision (sometimes significantly) higher than that obtained through ML, at the cost of only a slightly worse fit. Thus, a three exponential model can be adopted to describe the CP impulse response model in place of the two exponential model usually identified in the literature by the ML/LS approach. Simulated case studies are also reported to evidence the importance of taking into account a priori information in a data poor situation, e.g., when a few or too noisy measurements are available. In conclusion, our results show that, when a priori information on the unknown model parameters is available, Bayes estimation can be of relevant interest, since it can significantly improve the precision of parameter estimates with respect to Fisher estimation. This may also allow the adoption of more complex models than those determinable by a Fisherian approach.     

Maximum-likelihood estimation of parameters of exponentially damped sinusoids

Maximum-likelihood estimates of the parameters of exponentially damped sinusoidal signals in noise can be found by using an iterative procedure based on Newton's method. The initial estimate is obtained by an improved linear-prediction method. The properties of the Newton estimates obtained from short data records is experimentally studied.