Total least-squares reconstruction with wavelets for optical tomography.

Summary In this study, we show significant improvements over conventional pressure-transient analysis on the basis of nonlinear regression by using total least squares (TLS), which minimizes errors in both pressure and time simultaneously. To our knowledge, TLS has not been applied to pressure-transient analysis before in this sense. TLS regression is not an easy problem to solve mathematically, especially for nonlinear pressure-transient-model functions. In this work, we compare four different versions of TLS. We formulate a robust approximation of the TLS solution, which can handle a variety of tests and reservoir models yet does not compromise the performance of the analysis. We show that our technique reduces ambiguity in the estimation of parameters to a large extent, especially in the presence of noise in time. Using our TLS algorithm, we obtain much narrower confidence intervals on the parameter estimates of a variety of real data sets, compared to the conventional least squares (LS) approach. For synthetic data sets, we observe that the TLS estimates are often closer to the true values than estimates made with LS, especially for poorly determined problems. When the deviation is in pressure only, TLS and LS results are comparable. However, in the presence of deviations in time in addition to pressure, the performance of TLS algorithms is substantially better. We, therefore, expect that our technique will provide more accurate estimation of reservoir parameters, allowing for better forecasting of reservoir performance.

On the accuracy of total least squares and least squares techniques in the presence of errors on all data

State-of-charge (SOC) estimation of lithium-ion batteries (LIBs) is the basis of other state estimations. However, its accuracy can be affected by many factors, such as temperature and ageing. To handle this bottleneck issue, we here propose a joint SOC-SOH estimation method considering the influence of the temperature. It combines the Forgetting Factor Recursive Least Squares (FFRLS) algorithm, Total Least Squares (TLS) algorithm, and Unscented Kalman Filter (UKF) algorithm. First, the FFRLS algorithm is used to identify and update the parameters of the equivalent circuit model in real time under different battery ageing degrees. Then, the TLS algorithm is used to estimate the battery SOH to improve the prior estimation accuracy of SOC. Next, the SOC is calculated by the UKF algorithm, and finally, a more accurate SOH can be obtained according to the UKF-based SOC trajectory. The battery-in-the-loop experiments are utilized to verify the proposed algorithm. For the cases of temperature change up to 35 °C and capacity decay up to 10%, our joint estimator can achieve ultra-low errors, bounded by 2%, respectively, for SOH and SOC. The proposed method paves the way for the advancement of battery use in applications, such as electric vehicles and microgrid applications.

Observation systems known as errors-in-variables (EIV) models with model parameters estimated by total least squares (TLS) have been discussed for more than a century, though the terms EIV and TLS were coined much more recently. So far, it has only been shown that the inequality-constrained TLS (ICTLS) solution can be obtained by the combinatorial methods, assuming that the weight matrices of observations involved in the data vector and the data matrix are identity matrices. Although the previous works test all combinations of active sets or solution schemes in a clear way, some aspects have received little or no attention such as admissible weights, solution characteristics and numerical efficiency. Therefore, the aim of this study was to adjust the EIV model, subject to linear inequality constraints. In particular, (1) This work deals with a symmetrical positive-definite cofactor matrix that could otherwise be quite arbitrary. It also considers cross-correlations between cofactor matrices for the random coefficient matrix and the random observation vector. (2) From a theoretical perspective, we present first-order Karush–Kuhn–Tucker (KKT) necessary conditions and the second-order sufficient conditions of the inequality-constrained weighted TLS (ICWTLS) solution by analytical formulation. (3) From a numerical perspective, an active set method without combinatorial tests as well as a method based on sequential quadratic programming (SQP) is established. By way of applications, computational costs of the proposed algorithms are shown to be significantly lower than the currently existing ICTLS methods. It is also shown that the proposed methods can treat the ICWTLS problem in the case of more general weight matrices. Finally, we study the ICWTLS solution in terms of non-convex weighted TLS contours from a geometrical perspective.

A graph subspace approach to system identification based on errors-in-variables system models

The classical Least-Squares (LS) adjustment has been widely used in processing and analysing observations from Global Satellite Navigation System (GNSS). However, in detecting temporal correlations of GNSS observations, which can be described by means of autoregressive (AR) process, the LS method may not provide reliable estimates of process coefficients, since the Yule-Walker (YW) equations refer to structured Errors-In-Variables (EIV) equations. In this contribution, we proposed a Total Least-Squares (TLS) solution with the singular cofactor matrix to solve the YW equations. The proposed TLS solution is obtained based on the fact that random errors belong to column space of its cofactor matrix. In addition the proposed solution does not need any substitution of the squared true parameter vector as done by the current publications. Finally, we simulate the AR process to prove that our solution is more reliable than the existing methods.

A unified treatment of several least squares (LS) algorithms is presented for bearings-only tracking of a target moving at constant acceleration. The close link between the maximum likelihood (ML) estimator and other nonlinear and "linearized" LS algorithms is explored under the assumption of Gaussian bearing noise. In this context, a new asymptotically unbiased closed-form instrumental variables (IV) algorithm is derived. Reduced-bias total least squares (TLS) and constrained TLS (CTLS) algorithms are developed. The equivalence of the ML algorithm to the structured TLS (STLS) algorithm is established. Simulation examples are provided to demonstrate the improved performance of the IV and TLS estimators vis-a-vis the pseudolinear estimator.

In order to determine the transformation parameters between two reference frames empirically, a sufficient number of point coordinates (or possibly higher dimensional features such as, e.g., straight lines or conics) need to be observed in both systems. A proper adjustment of the observed data must take the different variances and covariances into account.The resulting adjustment model is of type Errors-in-Variables rather than of type Gauss-Markov because both sets of coordinates are associated with errors. In the homoscedastic case, the least-squares principle thus leads to the Total Least-Squares Solution (TLSS) rather than the standard Least-Squares Solution (LESS).Here, we generalize the TLSS by allowing the individual variances to be different and the covariances to be non-zero, while still maintaining a certain reasonable variance-covariance structure. This leads to the Weighted TLSS. We show experimentally that the Weighted TLSS yields slightly better estimates (in terms of precision) than LESS or TLSS, but significantly more accurate dispersion measures.KeywordsTotal Least-Squares adjustmentweightsreference framesempirical coordinate transformations

Orthogonal frequency-division multiplexing (OFDM) combines the advantages of high performance and relatively low implementation complexity. However, for reliable coherent detection of the input signal, the OFDM receiver needs accurate channel information. The Doppler shift of fast-fading channels will generate intercarrier interference (ICI) and, hence, degrade the performance of orthogonal frequency-division multiplexing (OFDM) systems. In this paper, the ICI is considered disturbance of channel information.We present total least-squares (TLS) scheme to eliminate the ICI and noise.

We propose a new distributed algorithm to solve the total least-squares (TLS) problem when data are distributed over a multi-agent network. To develop the proposed algorithm, named distributed ADMM TLS (DA-TLS), we reformulate the TLS problem as a parametric semidefinite program and solve it using the alternating direction method of multipliers (ADMM). Unlike the existing consensus-based approaches to distributed TLS estimation, DA-TLS does not require careful tuning of any design parameter. Numerical experiments demonstrate that the DA-TLS converges to the centralized solution significantly faster than the existing consensus-based TLS algorithms.

The Total Least Squares (TLS) method is a generalized least square technique to solve an overdetermined system of equations Ax\simeqb . The TLS solution differs from the usual Least Square (LS) in that it tries to compensate for arbitrary present in both A and b . In certain problems the perturbations of A and b are linear functions of a common noise source vector. In this case we obtain a generalization of the TLS criterion called the Constrained Total Least Squares (CTLS) method by taking into account the linear dependence of the terms in A and b . If the columns of A and b are linearly related then the CTLS solution is obtained in terms of the largest eigenvalue and corresponding eigenvector of a certain matrix. The CTLS technique can be applied to problems like Maximum Likelihood Signal Parameter Estimation, Frequency Estimation of Sinusoids in white or colored by Linear Prediction and others.

Modified Yule-Walker (MYW) equations are often used to estimate autoregressive parameters of an autoregressive moving average (ARMA) model. Commonly used algorithms, i.e., the least square (LS) algorithm, the total least square (TLS) algorithm, cannot give an optimal estimate because they do not exploit the Toeplitz property and covariance of the perturbation matrix. In this paper, a constrained total least squares (CTLS) algorithm is applied to solve modified Yule-Walker equations. The perturbation covariance matrix of the autocorrelation functions required by the CTLS algorithm is estimated by introducing the bootstrap method. By utilizing the Toeplitz property and the covariance of perturbation matrix, a Newton method based CTLS algorithm is shown to outperform TLS and LS solutions.

Algebraic relationships between classical regression and total least-squares estimation

This paper presents a total least squares (TLS) method in an iterative way when the observations are linear with applications in two-dimensional linear regression and three-dimensional coordinate transformation. The second order smaller terms are preserved and the unbiased solution and the variance component estimate are both obtained rigorously from traditional non-linear least squares theory. Compared with the traditional TLS algorithm dealing with the so called errors-in-variables (EIV) model, this algorithm can be used to analyse all the observations involved in the observation vector and the design matrix coequally; the non-linear adjustment with constraints or partial EIV model can also be solved using the same method. In addition, all the errors of observations can be considered in a heteroscedastic or correlated case, and the calculation of the solution and the variance component estimate are much simpler than the traditional TLS and its related improved algorithms. Experiments using statistical methods show the deviations between the designed true value of the variables and the estimated ones using this algorithm and the traditional least squares algorithm respectively, and the mean value of the posteriori variance in 1000 simulations of coordinate transformation is computed as well to test and verify the efficiency and unbiased estimation of this algorithm.

In this paper, a new Hebbian-type learning algorithm for the total least-squares parameter estimation is presented. The algorithm is derived from the classical Hebbian rule. An asymptotic analysis is carried out to show that the algorithm allows the weight vector of a linear neuron unit to converge to the eigenvector associated with the smallest eigenvalue of the correlation matrix of the input signal. When the algorithm is applied to solve parameter estimation problems, the converged weights directly yield the total least-squares solution. Since the process of obtaining the estimate is optimal in the total least-squares sense, its noise rejection capability is superior to those of the least-squares-based algorithms. It is shown that the implementations of the proposed algorithm have the simplicity of those of the LMS algorithm. The applicability and performance of the algorithm are demonstrated through computer simulations of adaptive FIR and IIR parameter estimation problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>