Classical Linear Estimation Research Articles

A review on principles, performance and complexity of linear estimation and detection techniques for MIMO systems

The advent of the fifth generation (5G) of mobile networks has introduced several new use cases that are pushing mobile networks in environments that are typically supported by wired technologies. The initial discussions around the sixth generation (6G) of mobile networks signalizes that different approaches are needed to address all contrasting requirements, where multiple-input multiple-output (MIMO) technique stands as a key technology for most future wireless systems. In this review, we present an introduction on classical linear estimators and coherent detectors along with an innovative and accurate complexity formulation within a common framework, allowing a fair comparison and providing an initial guideline for researchers that are looking for a general view of the main techniques available for spatial multiplexing (SM)-MIMO detection and estimation.

Fault Estimation for Discrete-Time Systems With Lipschitz Perturbation and Time-Variant Coefficients

This brief investigates the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fault estimation problem for a class of Lipschitz nonlinear systems with timevariant coefficient matrices in discrete-time settings. By introducing an auxiliary unknown input based on the nonlinear term, a quasi-linear model and its corresponding indefinite quadratic performance function for fault estimation are respectively given in lieu of the original nonlinear dynamics and the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance metric, such that the estimation problem is converted as an indefinite optimization problem. By artificially constructing a Krein-space based dynamic model, the classical linear estimation technique in H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> sense is employed to seek a suitable choice of the estimation of the fault. A condition that ensures the existence of the estimator is derived analytically. A Kalman-filter-like estimator recursion is proposed simultaneously.

Early Detection of Sudden Cardiac Death by Using Ensemble Empirical Mode Decomposition-Based Entropy and Classical Linear Features From Heart Rate Variability Signals.

Sudden cardiac death (SCD), which can deprive a person of life within minutes, is a destructive heart abnormality. Thus, providing early warning information for patients at risk of SCD, especially those outside hospitals, is essential. In this study, we investigated the performances of ensemble empirical mode decomposition (EEMD)-based entropy features on SCD identification. EEMD-based entropy features were obtained by using the following technology: (1) EEMD was performed on HRV beats to decompose them into intrinsic mode functions (IMFs), (2) five entropy parameters, namely Rényi entropy (RenEn), fuzzy entropy (FuEn), dispersion Entropy (DisEn), improved multiscale permutation entropy (IMPE), and Renyi distribution entropy(RdisEn), were computed from the first four IMFs obtained, which were named EEMD-based entropy features. Additionally, an automated scheme combining EEMD-based entropy and classical linear (time and frequency domains) features was proposed with the intention of detecting SCD early by analyzing 14 min (at seven successive intervals of 2 min) heart rate variability (HRV) in signals from a normal population and subjects at risk of SCD. Firstly, EEMD-based entropy and classical linear measurements were extracted from HRV beats, and then the integrated measurements were ranked by various methodologies, i.e., t-test, entropy, receiver-operating characteristics (ROC), Wilcoxon, and Bhattacharyya. Finally, these ranked features were fed into a k-Nearest Neighbor algorithm for classification. Compared with several state-of-the-art methods, the proposed scheme firstly predicted subjects at risk of SCD up to 14 min earlier with an accuracy of 96.1%, a sensitivity of 97.5%, and a specificity of 94.4% 14 min before SCD onset. The simulation results exhibited that EEMD-based entropy estimators showed significant difference between SCD patients and normal individuals and outperformed the classical linear estimators in SCD detection, the EEMD-based FuEn and IMPE indexes were particularly useful assessments for identification of patients at risk of SCD and can be used as novel indices to reveal the disorders of rhythm variations of the autonomic nervous system when affected by SCD.

Non-Diagonal-Type Estimator in Linear Regression

This short article mainly aims to introduce the notion of the non-diagonal-type estimator (NDTE) by means of the singular value decomposition theorem in the linear regression model to improve some classical linear estimators that can be called the diagonal-type estimators. We derive the optimal NDTE under the mean squared error criterion and its iterative version through matrix techniques. A simulation study is finally conducted to illustrate the theoretical results.

Demonstration of Super-Resolution for Tomographic SAR Imaging in Urban Environment

Tomographic synthetic aperture radar (SAR) inversion, including SAR tomography and differential SAR tomography, is essentially a spectral analysis problem. The resolution in the elevation direction depends on the elevation aperture size, i.e., on the spread of orbit tracks. Since the orbits of modern meter-resolution spaceborne SAR systems, such as TerraSAR-X, are tightly controlled, the tomographic elevation resolution is at least an order of magnitude lower than in range and azimuth. Hence, super-resolution (SR) reconstruction algorithms are desired. Considering the sparsity of the signal in elevation, a compressive sensing based super-resolving algorithm, named “Scale-down by L1 norm Minimization, Model selection, and Estimation Reconstruction” (SL1MMER, pronounced “slimmer”), was proposed by the authors in a previous paper. The ultimate bounds of the technique on localization accuracy and SR power were investigated. In this paper, the essential role of SR for layover separation in urban infrastructure monitoring is indicated by geometric and statistical analysis. It is shown that double scatterers with small elevation distances are more frequent than those with large elevation distances. Furthermore, the SR capability of SL1MMER is demonstrated using TerraSAR-X real data examples. For a high rise building complex, the percentage of detected double scatterers is almost doubled compared to classical linear estimators. Among them, half of the detected double scatterer pairs have elevation distances smaller than the Rayleigh elevation resolution. This confirms the importance of SR for this type of applications.

General linear estimators under the prediction error sum of squares criterion in a linear regression model

In this paper, the notion of the general linear estimator and its modified version are introduced using the singular value decomposition theorem in the linear regression model y=X β+e to improve some classical linear estimators. The optimal selections of the biasing parameters involved are theoretically given under the prediction error sum of squares criterion. A numerical example and a simulation study are finally conducted to illustrate the superiority of the proposed estimators.

Sequential kriging and cokriging: Two powerful geostatistical approaches

A sequential linear estimator is developed in this study to progressively incorporate new or different spatial data sets into the estimation. It begins with a classical linear estimator (i.e., kriging or cokriging) to estimate means conditioned to a given observed data set. When an additional data set becomes available, the sequential estimator improves the previous estimate by using linearly weighted sums of differences between the new data set and previous estimates at sample locations. Like the classical linear estimator, the weights used in the sequential linear estimator are derived from a system of equations that contains covariances and cross-covariances between sample locations and the location where the estimate is to be made. However, the covariances and cross- covariances are conditioned upon the previous data sets. The sequential estimator is shown to produce the best, unbiased linear esti- mate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used. It also provides a new way to incorporate additional information into a previous estimation.

Evaluation of a Maximum Likelihood System Identification Technique on Flutter Testing Data

The paper deals with the application of a frequency domain maximum likelihood estimation method for linear system identification in the field of flutter data analysis. Unlike methods based on least squares error minimization, the proposed method takes into account the disturbing process or measurement noise on as well the input as the output of the device under test. This enables the optimal and unbiased identification of the parameters in the linear system model. A few examples of the identification of the system parameters of a mechanical single input single output system from flutter test data are discussed and compared to more classical linear estimation methods.

A multiplication-free solution for linear minimum mean-square estimation and equalization using the branch-and-bound principle

An optimal linear mean-square estimation algorithm is derived under the constraint that the algorithm be multiplication-free. A classical linear estimation problem with block length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> generally requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N^{2}</tex> multiplications. For many on-line signal processing situations a large number of multiplications is objectionable. This class of estimation problems includes the classical linear filtering of a random signal in random noise, as well as the linear equalization of digital data over a dispersive channel with additive noise. Here we consider the linear estimation problem on a binary computer where the estimation parameters are constrained to be powers of two and thus all multiplications are replaced by shifts. Then the optimal constrained linear estimation problem resembles an integer-programming problem except that the allowable discrete points are nonintegers. The branch-and-bound principle is used to convert this minimization problem to a series of convex programming problems. An algorithm is given for the solution as well as numerical results for filtering and data equalization. These examples show that the multiplication-free constraint does not generally increase the mean-square error significantly compared with the classical optimal solution. Furthermore, the intuitive "round to the nearest power of two" procedure for the estimation parameters can be inferior to the optimal brunch-and-bound solution.

Best Linear Unbiased Estimation by Recursive Methods

Introduction. The classical linear estimation problem for a finite number of parameters using least squares dates back to Gauss [1]. In a paper by Aitken [2], the method of parameter estimation was generalized. Instead of obtaining the set of parameters which minimize the sum of squares of the residuals (difference between the observed and expected values), a quadratic form of the residuals was minimized. The quadratic form consisted of the product of the residual vectors times the inverse covariance matrix of the observational errors. The estimator so obtained is known as the Markov estimator, minimum variance estimator, or the best linear unbiased estimator. The estimator is optimal in the following sense. If the random errors of observation are jointly normally distributed then the Markov estimator is also the maximum likelihood estimator. Further, even if the random errors are not jointly normally distributed, the covariance matrix of the best linear unbiased estimator (B.L.U.) is equal to or less than the covariance matrix of any other linear unbiased estimator [3]. A closely related theory of linear smoothing and prediction has been developed beginning with the work of Wiener [4] and Kolmogorov [5]. In recent years application of the classical linear smoothing and estimation problems has led to computational difficulties because of the need to handle large amounts of data or perform on line or near real-time parameter estimations. Attempts to lighten the computational load has led to the development of recursive solutions whereby it is not necessary to store all previous data, but only the previous parameter estiinates and some recent data. Blum [6] considers a problem equivalent to estimating the coefficients of a least squares polynomial curve fit by recursive means which is suitable for real-time smoothing and prediction. The solution permits a substantial reduction in the storage and arithmetic operations when the number of data points is much greater than the degree of the polynomial. In [7] Blum considers the design of a smoothing filter which is not derived by least squares or minimum variance criteria. Again the solution is equivalent to a polynomial curve fitting procedure but the output of the filter may be biased. The advantage gained is the simple difference equation between the present output and the present input and past output values which is computationally simple to implement.

A Tchebychev Inequality for the Convergence of a Generalized Classical Linear Estimator, Sample Size Being Fixed

A Tchebychev Inequality for the Convergence of a Generalized Classical Linear Estimator, Sample Size Being Fixed

A Note on the Exact Finite Sample Frequency Functions of Generalized Classical Linear Estimators in a Leading Three-Equation Case

A Note on the Exact Finite Sample Frequency Functions of Generalized Classical Linear Estimators in a Leading Three-Equation Case

A Note on the Exact Finite Sample Frequency Functions of Generalized Classical Linear Estimators in a Leading Three-Equation Case

Abstract In this article some leading results of a mathematical investigation of exact finite sample distributions of GCL estimators are given. These results pertain to the distributions of estimators of coefficients appearing in econometric systems of three structural equations. The results show that the general forms of exact distributions derived for coefficient estimators in two-equation systems re-appear in the corresponding cases of three equations systems.

A Note on the Exact Finite Sample Frequency Functions of Generalized Classical Linear Estimators in Two Leading Over-Identified Cases

Abstract The asymptotic unbiasedness and normality of alternative statistical estimators θ, θ,1 ··· of a given parameter θ* are generally proved without reference to any explicit knowledge of the exact finite sample distribution functions, F n (x), G n (x) ···. (Here n denotes sample size.) Within the class of asymptotically unbiased and normally distributed estimators of a given parameter it is sometimes possible to demonstrate that one estimator possesses a smaller asymptotic variance than another, or that one estimator possesses the smallest asymptotic variance within a particular subclass. Asymptotic theory obviously does not predict anything about finite sample distribution functions Fn(x), Gn(x) ···. In particular we cannot deduce from asymptotic theorems that the estimator with the smallest asymptotic variance will continue to exhibit the smallest dispersion in finite samples. Consequently it remains an essential task in positive estimation theory to derive the exact finite sample distribution functions of the alternative estimators that appear to be promising on the basis of asymptotic considerations. In econometric statistics the alternative estimators of structural coefficients in systems of simultaneous equations appear to possess rather complicated finite sample distribution functions. Monte Carlo experiments are resorted to for the purpose of providing useful leads in mathematical research into the nature of these distributions. In this article are presented some leading results of a mathematical investigation of the exact finite sample distribution functions of generalized classical linear estimators. The results presented here exhibit some particularly important implications for the conduct and evaluation of Monte Carlo experiments in econometric statistics.

A Note on the Exact Finite Sample Frequency Functions of Generalized Classical Linear Estimators in Two Leading Over-Identified Cases

Abstract The asymptotic unbiasedness and normality of alternative statistical estimators θ, θ,1 ··· of a given parameter θ* are generally proved without reference to any explicit knowledge of the exact finite sample distribution functions, F n (x), G n (x) ···. (Here n denotes sample size.) Within the class of asymptotically unbiased and normally distributed estimators of a given parameter it is sometimes possible to demonstrate that one estimator possesses a smaller asymptotic variance than another, or that one estimator possesses the smallest asymptotic variance within a particular subclass. Asymptotic theory obviously does not predict anything about finite sample distribution functions Fn(x), Gn(x) ···. In particular we cannot deduce from asymptotic theorems that the estimator with the smallest asymptotic variance will continue to exhibit the smallest dispersion in finite samples. Consequently it remains an essential task in positive estimation theory to derive the exact finite sample distribution func...