Total Least-squares Algorithm Research Articles

A weighted total least-squares algorithm for any fitting model with correlated variables

An algorithm able to deal with any desired fitting model was developed for regression problems with uncertain and correlated variables.A typical application concerns the determination of calibration curves, especially (i) in those cases in which the uncertainties on the independent variables xi cannot be considered negligible with respect to those associated with the dependent variables yi, and (ii) when correlations exist among xi and yi. In the metrological field, several types of software have already been dedicated to the determination of calibration curves, some being focused just on problem (i) and a few others considering also problem (ii) but only for a straight-line fitting model. The proposed algorithm is able to deal with problems (i) and (ii) at the same time, for a generic fitting model. The tool was developed in the MATLAB® environment and validated on several benchmark data sets, fitted with linear and non-linear regression models.A review of the most commonly applied approximations to the parameter uncertainty is also presented, together with a Monte Carlo method proposed for comparison purposes with the results provided by the formula for the uncertainty evaluation which is implemented in the software.

Total least square method applied to rating curves

AbstractThe ordinary least square method (OLS) has been the most frequently used least square method in hydrological data analysis. Its computational algorithm is simple, and the error analysis is also simple and clear. However, the primary assumption of the OLS method, which states that the dependent variable is the only error‐contaminated variable and all other variables are error free, is often violated in hydrological data analyses. Recently, a matrix algorithm using the singular value decomposition for the total least square (TLS) method has been developed and used in data analyses as errors‐in‐variables model where several variables could be contaminated with observational errors. In our study, the algorithm of the TLS is introduced in the evaluation of rating curves between the flow discharge and the water level. Then, the TLS algorithm is applied to real data set for rating curves. The evaluated TLS rating curves are compared with the OLS rating curves, and the result indicates that the TLS rating curve and the OLS rating curve are in good agreement. The TLS and OLS rating curves are discussed about their algorithms and error terms in the study. Copyright © 2013 John Wiley & Sons, Ltd.

Computation and Uncertainty Evaluation of Offset Yield Strength

Abstract This paper presents computer procedures for the calculation of offset yield strength (Sy) and for the evaluation of the uncertainty in its computation. Offset yield strength is obtained from the plot of stress-strain data recorded in a tension test, as the stress that corresponds to the intersection between the stress-strain curve and a line parallel to its proportional region (offset by a prescribed strain). In the proposed method, the problem is reduced to finding the point of intersection between two straight lines, one that fits the curve in the neighborhood of the intersection and the offset line. For the fitting of each line, we propose the use of a weighted total least-squares algorithm that takes into account uncertainties in both ordinates and abscissas. The evaluation of the uncertainty associated with Sy, in accordance with the Guide to the Expression of Uncertainty in Measurement, considers the correlation between the parameters involved in its calculation. The implementation of these procedures motivated the development of dedicated software for the computation of tensile parameters from tension-test raw data and for the estimation of their associated uncertainties. To validate the program, developed in MATLAB as a standalone application, we used a set of ASCII data curves that have agreed values for the tensile parameters and which are publicly available at the web site of the National Physical Laboratory of the United Kingdom. Using these curves we demonstrate the validity of the proposed method for the computation of Sy; to validate the uncertainty-evaluation procedure, we use the law of propagation of probability distributions through Monte Carlo simulation. The computational tool, whose capabilities are presented in this work, is currently being used at the Laboratory of Mechanical Testing of the National Institute of Industrial Technology (INTI), in Córdoba, Argentina.

Constrained total least-squares localisation algorithm using time difference of arrival and frequency difference of arrival measurements with sensor location uncertainties

In this study, a constrained total least-squares (CTLS) algorithm for estimating the position and velocity of a moving source with sensor location uncertainties that uses the time difference of arrival and frequency difference of arrival measurements of a signal received at a number of sensors is proposed. The CTLS method, as a natural extension of LS when noise occurs in all the data and the noise components of the coefficients are linearly dependent, is more appropriate than the LS method for the above problem. By utilising the Lagrange multipliers technique, the known relation between the intermediate variables and the source localisation coordinates has been exploited to constrain the solution. In addition, the Lagrange multipliers can be obtained efficiently and robustly, which can allow real-time implementation as well as ensure global convergence. After a perturbation analysis, the bias and covariance of the proposed CTLS algorithm are also derived, indicating that the proposed CTLS algorithm is an unbiased estimator, and it could achieve the Cramer Rao lower bound (CRLB) when the measurement noise and the sensor location errors are sufficiently small. The simulation results show that the proposed estimator achieves remarkably better performance than the TLS and two-step weighted least squares approach, which makes it possible that the CRLB is attained at a sufficiently high noise level before the threshold effect occurs.

Linearly-Constrained Recursive Total Least-Squares Algorithm

We develop a new linearly-constrained recursive total least squares adaptive filtering algorithm by incorporating the linear constraints into the underlying total least squares problem using an approach similar to the method of weighting and searching for the solution (filter weights) along the input vector. The proposed algorithm outperforms the previously proposed constrained recursive least square (CRLS) algorithm when both input and output data are observed with noise. It also has a significantly smaller computational complexity than CRLS. Simulations demonstrate the efficacy of the proposed algorithm.

Channel Estimation and Interference Cancellation for OFDM Systems Based on Total Least Squares Solution

In this paper, we study the effects of interference to channel estimation and present an iterative total least-squares algorithm for channel estimation by using the conjugate gradient method in time-varying multipath fading channels. The Doppler shift of fast-fading channels will generate inter-carrier interference (ICI), hence, degrade the performance of orthogonal frequency-division multiplexing (OFDM) systems. Furthermore, when the length of channel impulse response (CIR) overstep the duration of cyclic prefix (CP), which will cause inter-symbol interference (ISI) and degrade system performance severely. Conventional channel estimation and equalization schemes, if applied to this case of insufficient CP, suffer significant performance degradation. By ISI and ICI analysis, an Interference elimination technique is proposed for channel estimation. After analyzing perturbation of least squares problems, we present total least-squares (TLS) scheme to eliminate the ICI, ISI and noise. We present a short theoretical overview of the TLS problem. In our estimation, residual ISI and ICI are not being handled as a noise. Since the introduction of interference, the operation of matrix becomes perplexed. We implement the Conjugate gradient total least-squares method. Simulation results confirm the performance of Conjugate gradient total least-squares method and our theoretical analysis.

The Total Least Squares Problem in AX≈B: A New Classification with the Relationship to the Classical Works

This paper revisits the analysis of the total least squares (TLS) problem AX≈B with multiple right-hand sides given by Van Huffel and Vandewalle in the monograph, The Total Least Squares Problem: Computational Aspects and Analysis, SIAM, Philadelphia, 1991. The newly proposed classification is based on properties of the singular value decomposition of the extended matrix [B|A]. It aims at identifying the cases when a TLS solution does or does not exist and when the output computed by the classical TLS algorithm, given by Van Huffel and Vandewalle, is actually a TLS solution. The presented results on existence and uniqueness of the TLS solution reveal subtleties that were not captured in the known literature.

Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling

Solving linear regression problems based on the total least-squares (TLS) criterion has well-documented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However, existing TLS approaches do not account for sparsity possibly present in the unknown vector of regression coefficients. On the other hand, sparsity is the key attribute exploited by modern compressive sampling and variable selection approaches to linear regression, which include noise in the data, but do not account for perturbations in the regression matrix. The present paper fills this gap by formulating and solving (regularized) TLS optimization problems under sparsity constraints. Near-optimum and reduced-complexity suboptimum sparse (S-) TLS algorithms are developed to address the perturbed compressive sampling (and the related dictionary learning) challenge, when there is a mismatch between the true and adopted bases over which the unknown vector is sparse. The novel S-TLS schemes also allow for perturbations in the regression matrix of the least-absolute selection and shrinkage selection operator (Lasso), and endow TLS approaches with ability to cope with sparse, under-determined “errors-in-variables” models. Interesting generalizations can further exploit prior knowledge on the perturbations to obtain novel weighted and structured S-TLS solvers. Analysis and simulations demonstrate the practical impact of S-TLS in calibrating the mismatch effects of contemporary grid-based approaches to cognitive radio sensing, and robust direction-of-arrival estimation using antenna arrays.

Consensus-Based Distributed Total Least Squares Estimation in Ad Hoc Wireless Sensor Networks

Total least squares (TLS) is a popular solution technique for overdetermined systems of linear equations, where both the right-hand side and the input data matrix are assumed to be noisy. We consider a TLS problem in an ad hoc wireless sensor network, where each node collects observations that yield a node-specific subset of linear equations. The goal is to compute the TLS solution of the full set of equations in a distributed fashion, without gathering all these equations in a fusion center. To facilitate the use of the dual-based subgradient algorithm (DBSA), we transform the TLS problem to an equivalent convex semidefinite program (SDP), based on semidefinite relaxation (SDR). This allows us to derive a distributed TLS (D-TLS) algorithm, that satisfies the conditions for convergence of the DBSA, and obtains the same solution as the original (unrelaxed) TLS problem. Even though we make a detour through SDR and SDP theory, the resulting D-TLS algorithm relies on solving local TLS-like problems at each node, rather than computationally expensive SDP optimization techniques. The algorithm is flexible and fully distributed, i.e., it does not make any assumptions on the network topology and nodes only share data with their neighbors through local broadcasts. Due to the flexibility and the uniformity of the network, there is no single point of failure, which makes the algorithm robust to sensor failures. Monte Carlo simulation results are provided to demonstrate the effectiveness of the method.

A Novel Power-Bearing Approach and Asymptotically Optimum Estimator for Target Motion Analysis

The problem of target motion analysis (TMA) has been extensively investigated using bearing-only (BO), Doppler-bearing (DB), and other measurement data. For radio frequency (RF) emitters, signal powers follow the well-known path loss law that can be utilized to aid localization and target tracking in BO-TMA, leading to the novel power-bearing (PB) approach as proposed in this paper. Our main contribution lies in the proof of the asymptotic optimality of the total least-squares (TLS) algorithm for PB-TMA under some mild condition. The Cramer-Rao lower bound (CRLB) is employed to demonstrate the performance improvements of the PB-TMA over the BO-TMA. Our work shows the promise of the PB-TMA approach for RF signals.

Total Least-Squares Algorithm for Time of Arrival Based Wireless Sensor Networks Location

Location of a source is of considerable interest in wireless sensor networks, and it can be estimated from passive measurements of the arrival times. A novel algorithm for source location by utilizing the time of arrival (TOA) measurements of a signal received at spatially separated sensors is proposed. The algorithm is based on total least-squares (TLS) method, which is a generalized least-squares method to solve an overdetermined set of equations whose coefficients are noisy, and gives an explicit solution. Comparisons of performance with standard least-squares method are made, and Monte Carlo simulations are performed. Simulation results indicate that the proposed TLS algorithm gives better results than LS algorithm.

Closed-Form Uniform Asymptotic Expansions of Green's Functions in Layered Media

Far-field closed-form expressions are derived for spatial domain multilayered Green's functions (GF). For the derivation of these far-field expressions, the spectral domain multilayered GF are approximated by means of the total least square algorithm (TLSA) in terms of the spectral variable u0=(kρ2-k02)1/2, and uniform asymptotic expansions are determined for the Sommerfeld integrals of the resulting TLSA approximations. Numerical results show that the far-field asymptotic expressions are accurate within 0.1% for distances between source and field points larger than one free-space wavelength. Also, it is shown that the hybrid use of the discrete complex image method and the TLSA in terms of the spectral variable kρ leads to near-field closed-form expressions of multilayered GF that are typically accurate within 0.1% for distances between source and field points smaller than one free-space wavelength. Therefore, the combination of the novel far-field asymptotic expressions and the well known near-field expressions makes it possible to compute multilayered GF with a great accuracy in the whole range of distances between source and field points, and in a wide range of frequencies.

Significant Improvement in the Accuracy of Pressure-Transient Analysis Using Total Least Squares

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.

Optimal selection of weights for the linear total least‐squares solution to impulsive source localization.

At a recent ASA meeting, Monte Carlo‐based noise performance analysis of several formulations and solution algorithms for the time‐of‐arrival (TOA) based impulsive source localization problem were presented. That analysis included two algorithms for parameter estimation using the (exact) linear formulation for the localization problem. Those algorithms were standard linear least‐squares (LLS) and linear total least‐squares (TLS). Using systematically derived, but in no way optimized, choice of weights, the TLS algorithm clearly demonstrated the ability to overcome the bias associated with LLS under conditions associated with low signal‐to‐noise ratio. Unfortunately, the bias reduction was obtained at a cost of high variance as compared to solutions associated with nonlinear formulations. This presentation presents results from investigations into performance of optimally chosen (variance reducing) weighting schemes for TLS and compares them to the performance of nonlinear formulations.

Efficient Determination of the Poles and Residues of Spectral Domain Multilayered Green's Functions That are Relevant in Far-Field Calculations

In this work the total least squares algorithm (TLSA) is applied to the determination of the proper and improper poles of spectral domain multilayered Green's functions that are closer to the branch point, and to the determination of the residues at these poles. The introduction of an adequate transformation in the spectral domain permits that the TLSA provides accurate values of the poles and residues, regardless of the proximity of the poles to the branch point. It is shown that the poles and residues supplied by the TLSA can be used to write the far field of the spatial domain Green's functions in terms of closed-form expressions that are reliable in a wide variety of scenarios.

Constrained Total Least-Squares Algorithm for Hyperbolic Location

A novel algorithm for source location by utilizing the time-difference-of-arrival (TDOA) of a signal received at spatially separated sensors is proposed. The algorithm is based on the constrained total least-squares (CTLS) technique and gives an explicit solution. Simulation results demonstrate that the proposed algorithm has high location accuracy and its performance is close to the Cramer-Rao lower bound (CRLB).

A Quadratic Constraint Total Least-squares Algorithm for Hyperbolic Location

A novel algorithm for source location by utilizing the time difference of arrival (TDOA) measurements of a signal received at spatially separated sensors is proposed. The algorithm is based on quadratic constraint total least-squares (QC-TLS) method and gives an explicit solution. The total least-squares method is a generalized data fitting method that is appropriate for cases when the system model contains error or is not known exactly, and quadratic constraint, which could be realized via Lagrange multipliers technique, could constrain the solution to the location equations to improve location accuracy. Comparisons of performance with ordinary least-squares are made, and Monte Carlo simulations are performed. Simulation results indicate that the proposed algorithm has high location accuracy and achieves accuracy close to the Cramer-Rao lower bound (CRLB) near the small TDOA measurement error region.

A weighted total least-squares algorithm for fitting a straight line

The well-known problem of fitting a straight line to data with uncertainties in both coordinates is revisited. An algorithm is developed which treats x- and y-data in a symmetrical way. The problem is reduced to a one-dimensional search for a minimum. Global convergence and stability are assured by determining the angle of the straight line with respect to the abscissa instead of the slope. As opposed to previous publications on the subject, the complete uncertainty matrix is calculated, i.e. variances and covariance of the fitting parameters. The algorithm is tested using Pearson's data with York's weights. Although the algorithm is implemented in MATLAB, implementation in a different programming language is straightforward using the formulae presented. An application example is given, a calibration line for dosimetry based on electron spin resonance of alanine is investigated.

A Maximum Likelihood Approach to Parallel Imaging With Coil Sensitivity Noise

Parallel imaging is a powerful technique to speed up magnetic resonance (MR) image acquisition via multiple coils. Both the received signal of each coil and its sensitivity map, which describes its spatial response, are needed during reconstruction. Widely used schemes such as SENSE assume that sensitivity maps of the coils are noiseless while the only errors are in coil outputs. In practice, however, sensitivity maps are subject to a wide variety of errors. At first glance, sensitivity noise appears to result in an errors-in-variables problem of the kind that is typically solved using total least squares (TLSs). However, existing TLS algorithms are in general inappropriate for the specific type of block structure that arises in parallel imaging. In this paper, we take a maximum likelihood approach to the problem of parallel imaging in the presence of independent Gaussian sensitivity noise. This results in a quasi-quadratic objective function, which can be efficiently minimized. Experimental evidence suggests substantial gains over conventional SENSE, especially in nonideal imaging conditions like low signal-to-noise ratio (SNR), high g-factors and large acceleration, using sensitivity maps suffering from misalignment, ringing, and random noise.

Computerized ionospheric tomography with the IRI model

Computerized ionospheric tomography (CIT) is a method to estimate ionospheric electron density distribution by using the global positioning system (GPS) signals recorded by the GPS receivers. Ionospheric electron density is a function of latitude, longitude, height and time. A general approach in CIT is to represent the ionosphere as a linear combination of basis functions. In this study, the model of the ionosphere is obtained from the IRI in latitude and height only. The goal is to determine the best representing basis function from the set of Squeezed Legendre polynomials, truncated Legendre polynomials, Haar Wavelets and singular value decomposition (SVD). The reconstruction algorithms used in this study can be listed as total least squares (TLS), regularized least squares, algebraic reconstruction technique (ART) and a hybrid algorithm where the reconstruction from the TLS algorithm is used as the initial estimate for the ART. The error performance of the reconstruction algorithms are compared with respect to the electron density generated by the IRI-2001 model. In the investigated scenario, the measurements are obtained from the IRI-2001 as the line integral of the electron density profiles, imitating the total electron content estimated from GPS measurements. It has been observed that the minimum error between the reconstructed and model ionospheres depends on both the reconstruction algorithm and the basis functions where the best results have been obtained for the basis functions from the model itself through SVD.