piero barone - Academia.edu (original) (raw)

Papers by piero barone

Research paper thumbnail of Prototipo di un sistema per l'elaborazione in linea delle elettromappe cardiache

Research paper thumbnail of Bayesian estimation of relaxation times T1 in MR images of irradiated Fricke-agarose gels

Magnetic Resonance Imaging, 2000

The authors present a novel method for processing T 1-weighted images acquired with Inversion-Rec... more The authors present a novel method for processing T 1-weighted images acquired with Inversion-Recovery (IR) sequence. The method, developed within the Bayesian framework, takes into account a priori knowledge about the spatial regularity of the parameters to be estimated. Inference is drawn by means of Markov Chains Monte Carlo algorithms. The method has been applied to the processing of IR images from irradiated Fricke-agarose gels, proposed in the past as relative dosimeter to verify radiotherapeutic treatment planning systems. Comparison with results obtained from a standard approach shows that signal-to noise ratio (SNR) is strongly enhanced when the estimation of the longitudinal relaxation rate (R1) is performed with the newly proposed statistical approach. Furthermore, the method allows the use of more complex models of the signal. Finally, an appreciable reduction of total acquisition time can be obtained due to the possibility of using a reduced number of images. The method can also be applied to T 1 mapping of other systems.

Research paper thumbnail of A high-field1 H nuclear magnetic resonance study of the minor components in virgin olive oils

Journal of the American Oil Chemists' Society, 1996

Research paper thumbnail of Prony methods in NMR spectroscopy

International Journal of Imaging Systems and Technology, 1997

ABSTRACT The Prony method and a modified Prony method (MPM), developed to improve the performance... more ABSTRACT The Prony method and a modified Prony method (MPM), developed to improve the performance of this technique at low signal-to-noise ratio, are described and applied to analysis of magnetic resonance spectroscopy (MRS) signals. Furthermore, the way in which results from MPM can be used as prior information in a Bayesian model is also described. First, analysis on simulated data is used to establish the methods' limits of reliability. Their performance with respect to peak identification and quantification of nuclear magnetic resonance parameters are then assayed on real data. Results of application of the methods to 1H-MRS signals from cultured cells are discussed and compared with those deriving from application of fast Fourier transform. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 565–571, 1997

Research paper thumbnail of A black box method for solving the complex exponentials approximation problem

A common problem, arising in many different applied contexts, consists in estimating the number o... more A common problem, arising in many different applied contexts, consists in estimating the number of exponentially damped sinusoids whose weighted sum best fits a finite set of noisy data and in estimating their parameters. Many different methods exist to this purpose. The best of them are based on approximate Maximum Likelihood estimators, assuming to know the number of damped sinusoids, which can then be estimated by an order selection procedure. As the problem can be severely ill posed, a stochastic perturbation method is proposed which provides better results than Maximum Likelihood based methods when the signal-to-noise ratio is low. The method depends on some hyperparameters which turn out to be essentially independent of the application. Therefore they can be fixed once and for all, giving rise to a black box method.

Research paper thumbnail of On the condensed density of the generalized eigenvalues of pencils of Hankel Gaussian random matrices and applications

Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean an... more Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean and not identical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided.

Research paper thumbnail of A diffusion equation for the density of the ratio of two jointly distributed Gaussian variables and the numerical inversion of Laplace transform

It is shown that the density of the ratio of two random variables with the same variance and join... more It is shown that the density of the ratio of two random variables with the same variance and joint Gaussian density satisfies a non stationary diffusion equation. Implications of this result for kernel density estimation of the condensed density of the generalized eigenvalues of a random matrix pencil useful for the numerical inversion of the Laplace transform is discussed.

Research paper thumbnail of Kernel density estimation via diffusion and the complex exponentials approximation problem, acceped for publication on Quart

A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of p... more A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when the complex exponentials approximation problem is considered in Gaussian noise. Several moments problems can be formulated in this framework and the estimation of the condensed density above is the main critical step for their solution. It is shown that the condensed density satisfies approximately a diffusion equation, which allows to estimate an optimal bandwidth. It is proved by simulation that good results can be obtained even when the signal-to-noise ratio is so small that other methods fail.

Research paper thumbnail of On a Class of Parameters Estimators in Linear Models Dominating the Least Squares one, Based on Compressed Sensing Techniques

The estimation of parameters in a linear model is considered under the hypothesis that the noise,... more The estimation of parameters in a linear model is considered under the hypothesis that the noise, with finite second order statistics, can be represented in a given deterministic basis by random coefficients. An extended underdetermined design matrix is then considered and an estimator of the extended parameters is proposed with minimum l 1 norm. It is proved that if the noise variance is larger than a threshold, which depends on the unknown parameters and on the extended design matrix, then the proposed estimator of the original parameters dominates the least-squares estimator in the sense of the mean square error. A small simulation illustrates the behavior of the proposed estimator. Moreover it is shown experimentally that the proposed estimator can be convenient even if the design matrix is not known but only an estimate can be used. Furthermore the noise basis can eventually be used to introduce some prior information in the estimation process. These points are illustrated by simulation by using the proposed estimator for solving a difficult inverse ill-posed problem related to the complex moments of an atomic complex measure.

Research paper thumbnail of Computational aspects and applications of a new transform for solving the complex exponentials approximation problem

Many real life problems can be reduced to the solution of a complex exponentials approximation pr... more Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has been proposed in a theoretical framework. In this work some computational issues are addressed to make this new tool useful in practice. An algorithm is developed and used to solve a Nuclear Magnetic Resonance spectrometry problem, two time series interpolation and extrapolation problems and a shape from moments problem.

Research paper thumbnail of Toward an automatic method of modal analysis

A common problem, arising in many different applied contexts, consists in estimating the number o... more A common problem, arising in many different applied contexts, consists in estimating the number of exponentially damped sinusoids whose weighted sum best fits a finite set of noisy data and in estimating their parameters. Many different methods exist to this purpose. The best of them are based on approximate Maximum Likelihood estimators, assuming to know the number of damped sinusoids, which is then estimated by an order selection procedure. It turns out that Maximum Likelihood estimators are biased in this specific case. The idea pursued here is to cope with the bias, by a stochastic perturbation method, in order to get an estimator with smaller Mean Squared Error than the Maximum Likelihood one. Moreover the problem of estimating the number of damped sinusoids and the problem of estimating their parameters are solved jointly. The method is automatic, provided that a few hyperparameters have been chosen, and faster than standard best alternatives. MSC: 30E10, 65C60

Research paper thumbnail of Over-relaxation methods and Metropolis-Hastings coupledMarkov chains for Monte Carlo simulationPiero

This paper is concerned with improving the performance of Markov chain algorithms for Monte Carlo... more This paper is concerned with improving the performance of Markov chain algorithms for Monte Carlo simulation. We propose a new algorithm for simulating from multivariate Gaussian densities. This algorithm combines ideas from Metropolis-coupled Markov chain Monte Carlo methods and from an existing algorithm based only on over-relaxation. The speed of convergence of the proposed and existing algorithms can be measured by the spectral radius of certain matrices. We present examples in which the proposed algorithm converges faster than the existing algorithm and the Gibbs sampler. We also derive an expression for the asymptotic variance of any linear combination of the variables simulated by the proposed algorithm. From this expression it follows that the proposed algorithm ooers no asymptotic variance reduction compared with the existing algorithm. We extend the proposed algorithm to the non-Gaussian case and discuss its performance by means of examples from Bayesian image analysis. We...

Research paper thumbnail of Recursive estimation with replications for multivariate covariance stationary processes

Research paper thumbnail of Random matrices in Magnetic Resonance signal processing

Many important problems in the processing of Magnetic Resonance data can be reduced to well known... more Many important problems in the processing of Magnetic Resonance data can be reduced to well known ill-posed inverse problems. MR relaxometry, spectroscopy and MR image formation problems can be formulated respectively as the numerical inversion of Laplace transform, the modal analysis problem and the truncated trigonometric moment problem. A unified framework to efficiently solve these problems is provided by considering random Pade’ approximants to the Z-transform s(z) of the measured signal. It turns out that the singularities of s(z) on the complex plane are the key quantities to make inference. Their location can be estimated from the logarithmic potential of the condensed density of the Pade’ poles or, equivalently, of the eigenvalues of a random pencil of Hankel matrices. Several alternatives to compute these quantities are discussed and compared by numerical examples.

Research paper thumbnail of Time structure of the rapid variations of BL Lacertae during the active phase of 1997 summer

Research paper thumbnail of A novel class of Padé based methods in spectral analysis

Journal of Computational Methods in Sciences and Engineering

Research paper thumbnail of Modified Prony Method to Resolve and Quantify in Vivo31P NMR Spectra of Tumors

Journal of Magnetic Resonance, Series B

ABSTRACT Prony′s method, successfully used in processing NMR signals, performs poorly at low sign... more ABSTRACT Prony′s method, successfully used in processing NMR signals, performs poorly at low signal-to-noise ratios. To overcome this problem, a statistical approach has been adopted by using Prony′s method as a sampling device from the distribution associated with the true spectrum. Specifically, Prony′s method is applied for each regression order p and number of data points n, both considered in a suitable range, and the estimates of frequencies, amplitudes, and decay factors are pooled separately. A histogram of the pooled frequencies is computed and, looking at the histogram, a lower and an upper frequency bound for each line of interest is determined. All frequency estimates in each of the determined intervals as well as associated decay factors and amplitudes are considered to be independent normal variates. A mean value and a corresponding 95% confidence interval are computed for each parameter. 31P NMR signals from MCF7 human breast cancer cells, inoculated into athymic mice and which developed into tumors, have been processed with traditional methods and with this modified Prony′s method. The main components of the phosphomonoester peak, namely those deriving from phosphorylcholine and phosphorylethanolamine, are always well resolved with this new approach and their relative amplitudes can be consequently evaluated. Peak intensities of these two signals show different behavior during treatment of tumors with the antiestrogenic drug tamoxifen. The results of this new approach are compared with those obtainable with traditional techniques.

Research paper thumbnail of Kernel density estimation via diffusion and the complex exponentials approximation problem

Quarterly of Applied Mathematics

A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of p... more A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when the complex exponentials approximation problem is considered in Gaussian noise. Several moments problems can be formulated in this framework and the estimation of the condensed density above is the main critical step for their solution. It is shown that the condensed density satisfies approximately a diffusion equation, which allows to estimate an optimal bandwidth. It is proved by simulation that good results can be obtained even when the signal-to-noise ratio is so small that other methods fail.

Research paper thumbnail of On the Universality of the Distribution of the Generalized Eigenvalues of a Pencil of Hankel Random Matrices

Random Matrices: Theory and Applications

Universality properties of the distribution of the generalized eigenvalues of a pencil of random ... more Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

Research paper thumbnail of On a class of parameters estimators in linear models dominating the least squares one

Digital Signal Processing

The estimation of parameters in a linear model is considered under the hypothesis that the noise,... more The estimation of parameters in a linear model is considered under the hypothesis that the noise, with finite second order statistics, can be represented in a given deterministic basis by random coefficients. An extended underdetermined design matrix is then considered and an estimator of the extended parameters is proposed with minimum l 1 norm. It is proved that if the noise variance is larger than a threshold, which depends on the unknown parameters and on the extended design matrix, then the proposed estimator of the original parameters dominates the least-squares estimator in the sense of the mean square error. A small simulation illustrates the behavior of the proposed estimator. Moreover it is shown experimentally that the proposed estimator can be convenient even if the design matrix is not known but only an estimate can be used. Furthermore the noise basis can eventually be used to introduce some prior information in the estimation process. These points are illustrated by simulation by using the proposed estimator for solving a difficult inverse ill-posed problem related to the complex moments of an atomic complex measure.

Research paper thumbnail of Prototipo di un sistema per l'elaborazione in linea delle elettromappe cardiache

Research paper thumbnail of Bayesian estimation of relaxation times T1 in MR images of irradiated Fricke-agarose gels

Magnetic Resonance Imaging, 2000

The authors present a novel method for processing T 1-weighted images acquired with Inversion-Rec... more The authors present a novel method for processing T 1-weighted images acquired with Inversion-Recovery (IR) sequence. The method, developed within the Bayesian framework, takes into account a priori knowledge about the spatial regularity of the parameters to be estimated. Inference is drawn by means of Markov Chains Monte Carlo algorithms. The method has been applied to the processing of IR images from irradiated Fricke-agarose gels, proposed in the past as relative dosimeter to verify radiotherapeutic treatment planning systems. Comparison with results obtained from a standard approach shows that signal-to noise ratio (SNR) is strongly enhanced when the estimation of the longitudinal relaxation rate (R1) is performed with the newly proposed statistical approach. Furthermore, the method allows the use of more complex models of the signal. Finally, an appreciable reduction of total acquisition time can be obtained due to the possibility of using a reduced number of images. The method can also be applied to T 1 mapping of other systems.

Research paper thumbnail of A high-field1 H nuclear magnetic resonance study of the minor components in virgin olive oils

Journal of the American Oil Chemists' Society, 1996

Research paper thumbnail of Prony methods in NMR spectroscopy

International Journal of Imaging Systems and Technology, 1997

ABSTRACT The Prony method and a modified Prony method (MPM), developed to improve the performance... more ABSTRACT The Prony method and a modified Prony method (MPM), developed to improve the performance of this technique at low signal-to-noise ratio, are described and applied to analysis of magnetic resonance spectroscopy (MRS) signals. Furthermore, the way in which results from MPM can be used as prior information in a Bayesian model is also described. First, analysis on simulated data is used to establish the methods' limits of reliability. Their performance with respect to peak identification and quantification of nuclear magnetic resonance parameters are then assayed on real data. Results of application of the methods to 1H-MRS signals from cultured cells are discussed and compared with those deriving from application of fast Fourier transform. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 565–571, 1997

Research paper thumbnail of A black box method for solving the complex exponentials approximation problem

A common problem, arising in many different applied contexts, consists in estimating the number o... more A common problem, arising in many different applied contexts, consists in estimating the number of exponentially damped sinusoids whose weighted sum best fits a finite set of noisy data and in estimating their parameters. Many different methods exist to this purpose. The best of them are based on approximate Maximum Likelihood estimators, assuming to know the number of damped sinusoids, which can then be estimated by an order selection procedure. As the problem can be severely ill posed, a stochastic perturbation method is proposed which provides better results than Maximum Likelihood based methods when the signal-to-noise ratio is low. The method depends on some hyperparameters which turn out to be essentially independent of the application. Therefore they can be fixed once and for all, giving rise to a black box method.

Research paper thumbnail of On the condensed density of the generalized eigenvalues of pencils of Hankel Gaussian random matrices and applications

Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean an... more Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean and not identical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided.

Research paper thumbnail of A diffusion equation for the density of the ratio of two jointly distributed Gaussian variables and the numerical inversion of Laplace transform

It is shown that the density of the ratio of two random variables with the same variance and join... more It is shown that the density of the ratio of two random variables with the same variance and joint Gaussian density satisfies a non stationary diffusion equation. Implications of this result for kernel density estimation of the condensed density of the generalized eigenvalues of a random matrix pencil useful for the numerical inversion of the Laplace transform is discussed.

Research paper thumbnail of Kernel density estimation via diffusion and the complex exponentials approximation problem, acceped for publication on Quart

A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of p... more A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when the complex exponentials approximation problem is considered in Gaussian noise. Several moments problems can be formulated in this framework and the estimation of the condensed density above is the main critical step for their solution. It is shown that the condensed density satisfies approximately a diffusion equation, which allows to estimate an optimal bandwidth. It is proved by simulation that good results can be obtained even when the signal-to-noise ratio is so small that other methods fail.

Research paper thumbnail of On a Class of Parameters Estimators in Linear Models Dominating the Least Squares one, Based on Compressed Sensing Techniques

The estimation of parameters in a linear model is considered under the hypothesis that the noise,... more The estimation of parameters in a linear model is considered under the hypothesis that the noise, with finite second order statistics, can be represented in a given deterministic basis by random coefficients. An extended underdetermined design matrix is then considered and an estimator of the extended parameters is proposed with minimum l 1 norm. It is proved that if the noise variance is larger than a threshold, which depends on the unknown parameters and on the extended design matrix, then the proposed estimator of the original parameters dominates the least-squares estimator in the sense of the mean square error. A small simulation illustrates the behavior of the proposed estimator. Moreover it is shown experimentally that the proposed estimator can be convenient even if the design matrix is not known but only an estimate can be used. Furthermore the noise basis can eventually be used to introduce some prior information in the estimation process. These points are illustrated by simulation by using the proposed estimator for solving a difficult inverse ill-posed problem related to the complex moments of an atomic complex measure.

Research paper thumbnail of Computational aspects and applications of a new transform for solving the complex exponentials approximation problem

Many real life problems can be reduced to the solution of a complex exponentials approximation pr... more Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has been proposed in a theoretical framework. In this work some computational issues are addressed to make this new tool useful in practice. An algorithm is developed and used to solve a Nuclear Magnetic Resonance spectrometry problem, two time series interpolation and extrapolation problems and a shape from moments problem.

Research paper thumbnail of Toward an automatic method of modal analysis

A common problem, arising in many different applied contexts, consists in estimating the number o... more A common problem, arising in many different applied contexts, consists in estimating the number of exponentially damped sinusoids whose weighted sum best fits a finite set of noisy data and in estimating their parameters. Many different methods exist to this purpose. The best of them are based on approximate Maximum Likelihood estimators, assuming to know the number of damped sinusoids, which is then estimated by an order selection procedure. It turns out that Maximum Likelihood estimators are biased in this specific case. The idea pursued here is to cope with the bias, by a stochastic perturbation method, in order to get an estimator with smaller Mean Squared Error than the Maximum Likelihood one. Moreover the problem of estimating the number of damped sinusoids and the problem of estimating their parameters are solved jointly. The method is automatic, provided that a few hyperparameters have been chosen, and faster than standard best alternatives. MSC: 30E10, 65C60

Research paper thumbnail of Over-relaxation methods and Metropolis-Hastings coupledMarkov chains for Monte Carlo simulationPiero

This paper is concerned with improving the performance of Markov chain algorithms for Monte Carlo... more This paper is concerned with improving the performance of Markov chain algorithms for Monte Carlo simulation. We propose a new algorithm for simulating from multivariate Gaussian densities. This algorithm combines ideas from Metropolis-coupled Markov chain Monte Carlo methods and from an existing algorithm based only on over-relaxation. The speed of convergence of the proposed and existing algorithms can be measured by the spectral radius of certain matrices. We present examples in which the proposed algorithm converges faster than the existing algorithm and the Gibbs sampler. We also derive an expression for the asymptotic variance of any linear combination of the variables simulated by the proposed algorithm. From this expression it follows that the proposed algorithm ooers no asymptotic variance reduction compared with the existing algorithm. We extend the proposed algorithm to the non-Gaussian case and discuss its performance by means of examples from Bayesian image analysis. We...

Research paper thumbnail of Recursive estimation with replications for multivariate covariance stationary processes

Research paper thumbnail of Random matrices in Magnetic Resonance signal processing

Many important problems in the processing of Magnetic Resonance data can be reduced to well known... more Many important problems in the processing of Magnetic Resonance data can be reduced to well known ill-posed inverse problems. MR relaxometry, spectroscopy and MR image formation problems can be formulated respectively as the numerical inversion of Laplace transform, the modal analysis problem and the truncated trigonometric moment problem. A unified framework to efficiently solve these problems is provided by considering random Pade’ approximants to the Z-transform s(z) of the measured signal. It turns out that the singularities of s(z) on the complex plane are the key quantities to make inference. Their location can be estimated from the logarithmic potential of the condensed density of the Pade’ poles or, equivalently, of the eigenvalues of a random pencil of Hankel matrices. Several alternatives to compute these quantities are discussed and compared by numerical examples.

Research paper thumbnail of Time structure of the rapid variations of BL Lacertae during the active phase of 1997 summer

Research paper thumbnail of A novel class of Padé based methods in spectral analysis

Journal of Computational Methods in Sciences and Engineering

Research paper thumbnail of Modified Prony Method to Resolve and Quantify in Vivo31P NMR Spectra of Tumors

Journal of Magnetic Resonance, Series B

ABSTRACT Prony′s method, successfully used in processing NMR signals, performs poorly at low sign... more ABSTRACT Prony′s method, successfully used in processing NMR signals, performs poorly at low signal-to-noise ratios. To overcome this problem, a statistical approach has been adopted by using Prony′s method as a sampling device from the distribution associated with the true spectrum. Specifically, Prony′s method is applied for each regression order p and number of data points n, both considered in a suitable range, and the estimates of frequencies, amplitudes, and decay factors are pooled separately. A histogram of the pooled frequencies is computed and, looking at the histogram, a lower and an upper frequency bound for each line of interest is determined. All frequency estimates in each of the determined intervals as well as associated decay factors and amplitudes are considered to be independent normal variates. A mean value and a corresponding 95% confidence interval are computed for each parameter. 31P NMR signals from MCF7 human breast cancer cells, inoculated into athymic mice and which developed into tumors, have been processed with traditional methods and with this modified Prony′s method. The main components of the phosphomonoester peak, namely those deriving from phosphorylcholine and phosphorylethanolamine, are always well resolved with this new approach and their relative amplitudes can be consequently evaluated. Peak intensities of these two signals show different behavior during treatment of tumors with the antiestrogenic drug tamoxifen. The results of this new approach are compared with those obtainable with traditional techniques.

Research paper thumbnail of Kernel density estimation via diffusion and the complex exponentials approximation problem

Quarterly of Applied Mathematics

A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of p... more A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when the complex exponentials approximation problem is considered in Gaussian noise. Several moments problems can be formulated in this framework and the estimation of the condensed density above is the main critical step for their solution. It is shown that the condensed density satisfies approximately a diffusion equation, which allows to estimate an optimal bandwidth. It is proved by simulation that good results can be obtained even when the signal-to-noise ratio is so small that other methods fail.

Research paper thumbnail of On the Universality of the Distribution of the Generalized Eigenvalues of a Pencil of Hankel Random Matrices

Random Matrices: Theory and Applications

Universality properties of the distribution of the generalized eigenvalues of a pencil of random ... more Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

Research paper thumbnail of On a class of parameters estimators in linear models dominating the least squares one

Digital Signal Processing

The estimation of parameters in a linear model is considered under the hypothesis that the noise,... more The estimation of parameters in a linear model is considered under the hypothesis that the noise, with finite second order statistics, can be represented in a given deterministic basis by random coefficients. An extended underdetermined design matrix is then considered and an estimator of the extended parameters is proposed with minimum l 1 norm. It is proved that if the noise variance is larger than a threshold, which depends on the unknown parameters and on the extended design matrix, then the proposed estimator of the original parameters dominates the least-squares estimator in the sense of the mean square error. A small simulation illustrates the behavior of the proposed estimator. Moreover it is shown experimentally that the proposed estimator can be convenient even if the design matrix is not known but only an estimate can be used. Furthermore the noise basis can eventually be used to introduce some prior information in the estimation process. These points are illustrated by simulation by using the proposed estimator for solving a difficult inverse ill-posed problem related to the complex moments of an atomic complex measure.