Ill-conditioned Estimation Problem Research Articles

A Faster Procedure for Estimating SEMs Applying Minimum Distance Estimators With a Fixed Weight Matrix

This study presents a separable nonlinear least squares (SNLLS) implementation of the minimum distance (MD) estimator employing a fixed-weight matrix for estimating structural equation models (SEMs). In contrast to the standard implementation of the MD estimator, in which the complete set of parameters is estimated using nonlinear optimization, the SNLLS implementation allows a subset of parameters to be estimated using (linear) least squares (LS). The SNLLS implementation possesses a number of benefits, such as faster convergence, better performance in ill-conditioned estimation problems, and fewer required starting values. The present work demonstrates that SNLLS, when applied to SEM estimation problems, significantly reduces the estimation time. Reduced estimation time makes SNLLS particularly useful in applications involving some form of resampling, such as simulation and bootstrapping.

Online Distributed Learning Over Graphs With Multitask Graph-Filter Models

In this article, we are interested in adaptive and distributed estimation of graph filters from streaming data. We formulate this problem as a consensus estimation problem over graphs, which can be addressed with diffusion LMS strategies. Most popular graph-shift operators such as those based on the graph Laplacian matrix, or the adjacency matrix, are not energy preserving. This may result in an ill-conditioned estimation problem, and reduce the convergence speed of the distributed algorithms. To address this issue and improve the transient performance, we introduce a preconditioned graph diffusion LMS algorithm. We also propose a computationally efficient version of this algorithm by approximating the Hessian matrix with local information. Performance analyses in the mean and mean-square sense are provided. Finally, we consider a more general problem where the filter coefficients to estimate may vary over the graph. To avoid a large estimation bias, we introduce an unsupervised clustering method for splitting the global estimation problem into local ones. Numerical results show the effectiveness of the proposed algorithms and validate the theoretical results.

Parameter subset selection and biased estimation for a class of ill-conditioned estimation problems

In cases of ill-conditioned estimation problems, not all parameters can be estimated accurately and a selection of parameter subset composed of more influential and less correlated parameters may be needed to increase parameter estimability. This paper proposes to choose a subset for estimation from transformed parameters along the directions of the principal components of the parameter covariance matrix while retaining the initial guesses for the unselected transformed parameters. Since estimating a subset of the transformed parameters can adjust all the original parameter values and any constraint on the original parameter values can still be applied to the transformed parameters, the proposed regularization method can overcome the limitation of the existing methods, i.e., parameter subset selection and truncated singular value decomposition (also known as ‘principal component regression’). It is demonstrated that the proposed method can provide better parameter estimates with smaller variances than the existing parameter subset selection methods, first through statistical analysis, and then through case studies of linear and nonlinear regressions. In addition, based on the derived statistical properties, a criterion is suggested for selecting an optimal subset, which gives the smallest mean squared error of the estimates. Furthermore, with the advantage of a lower variance of the estimates, the proposed regularization method gives a more consistent choice of the number of parameters to estimate with the smallest mean squared error of the estimates even when significant errors in the initial guesses and high levels of measurement noise exist.

Three-Dimensional Imaging of Objects Concealed Below a Forest Canopy Using SAR Tomography at L-Band and Wavelet-Based Sparse Estimation

Despite its ability to characterize 3-D environments, synthetic aperture radar (SAR) tomographic imaging, when applied to the characterization of targets concealed beneath forest canopies, may appear as an ill-conditioned estimation problem, with a complex mixture of numerous scattering mechanisms measured from a few different positions. Among the set of tomographic estimators that may be used to characterize such complex scattering environments, nonparametric tomographic techniques are more robust to focus on artifacts but limited in resolution and, hence, may fail to discriminate objects, whereas parametric ones provide better vertical resolution but cannot adequately handle continuously distributed volumetric scattering densities, characteristic of forest canopies. This letter addresses a new wavelet-based sparse tomographic estimation method for the 3-D imaging and discrimination of underfoliage objects that overcomes these limitations. The effectiveness of this new approach is demonstrated using L-band airborne tomographic SAR data acquired by the German Aerospace Center over Dornstetten, Germany.

A Spectral Conjugate Gradient Method for Unconstrained Optimization

A family of scaled conjugate gradient algorithms for large-scale unconstrained minimization is defined. The Perry, the Polak—Ribiere and the Fletcher—Reeves formulae are compared using a spectral scaling derived from Raydan's spectral gradient optimization method. The best combination of formula, scaling and initial choice of step-length is compared against well known algorithms using a classical set of problems. An additional comparison involving an ill-conditioned estimation problem in Optics is presented.

Ballistic missile track initiation from satellite observations

An algorithm is presented to initiate tracks of a ballistic missile in the initial exoatmospheric phase, using line of sight (LOS) measurements from one or more moving platforms (typically satellites). The major feature of this problem is the poor target motion observability which results in a very ill-conditioned estimation problem. The Gauss-Newton iterative least squares minimization algorithm for estimating the state of a nonlinear deterministic system with nonlinear noisy measurements has been previously applied to the problem of angles-only orbit determination using more than three observations. A major shortcoming of this approach is that convergence of the algorithm depends strongly on the initial guess. By using the more sophisticated Levenberg-Marquardt method in place of the simpler Gauss-Newton algorithm and by developing robust new methods for obtaining the initial guess in both single and multiple satellite scenarios, the above mentioned difficulties have been overcome. In addition, an expression for the Cramer-Rao lower bound (CRLB) on the error covariance matrix of the estimate is derived. We also incorporate additional partial information as an extra pseudomeasurement and determine a modified maximum likelihood (ML) estimate of the target state and the associated bound on the covariance matrix. In most practical situations, probabilistic models of the target altitude and/or speed at the initial point constitute the most useful additional information. Monte Carlo simulation studies on some typical scenarios were performed, and the results indicate that the estimation errors are commensurate with the theoretical lower bounds, thus illustrating that the proposed estimators are efficient.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Sensitivity of ridge-type estimation methods to observation accuracy and sampling rate

Ridge-type estimation methods have been shown to provide increased solution accuracy over minimum variance methods in ill-conditioned estimation problems. The accuracy gained by the use of such methods is a function of the level of ill conditioning present in the problem, which is dependent on the accuracy of the observations and the sampling rate (over a given observation span). This study presents results of a simulation designed to demonstrate the sensitivity of ridge-type estimation methods to the observation accuracy and the sampling rate used in the estimation process. The results are analyzed, and conclusions are drawn concerning the effectiveness of ridge-type estimation methods in the type of ill-conditioned problems addressed here.