Total least squares methods

Abstract

Recent advances in total least squares approaches for solving various errorsin-variables modeling problems are reviewed, with emphasis on the following generalizations:1. the use of weighted norms as a measure of the data perturbation size, capturing prior knowledge about uncertainty in the data;2. the addition of constraints on the perturbation to preserve the structure of the data matrix, motivated by structured data matrices occurring in signal and image processing, systems and control, and computer algebra;3. the use of regularization in the problem formulation, aiming at stabilizing the solution by decreasing the effect because of intrinsic ill-conditioning of certain problems. The focus of this paper is on computational algorithms for solving the generalized TLS problems. The reader is referred to the errors-in-variables literature for the statistical properties of the corresponding estimators, as well as for a wider range of applications. WEIGHTED AND STRUCTURED TOTAL LEAST SQUARES PROBLEMSThe TLS solution where the vector of perturbations vec ([ A b]) is zero mean and has covariance matrix that is equal to the identity up to a scaling factor, i.e.,The noise assumption (3) implies that all elements of the data matrix are measured with equal precision, an assumption that may not be satisfied in practice. A natural generalization of the EIV model (Eq. (2,3)) is to allow the covariance matrix of the vectorized noise to be of the form σ 2 V, where V is a given positive definite matrix. The corresponding estimation problem is the TLS problem (1) with the Frobenius norm · F replaced by the weighted matrix norm

Two empirical regimes of the planetary mass-radius relation

“…In cases where the transformation led to asymmetric uncertainties, we still assigned symmetric errors by taking the more conservative (larger error) of the two error estimates. In our quest for the best-fit piecewise-linear function, we chose what is probably the most intuitive approach to EIV: a total least-squares approach (TLS, e.g., Markovsky et al 2010). Similarly to standard regression, in TLS the problem is represented as a minimization problem of a sum of squares.…”

Consistency of the total least squares estimator in the linear errors-in-variables regression

“…Then every ∆ that minimizes (7) also minimizes (12), and every ∆ that minimizes (12) also minimizes (11). If M U is the Frobenius norm, then optimization problems (7) and (12) coincide, and if M U is the spectral norm, then optimization problems (11) and (12) coincide. Remark 2.2.…”

Force calibration of KUKA LWR-like robots including embedded joint torque sensors and robot structure

“…To identify the torque sensor gains, it is necessary to introduce them into the robot base parameters and to use a new method developed recently for identify the robot drive gains [5], which is based on a Total Least Squares Identification (IDIM-TLS) procedure (see [20][21][22][23][24] for a global overview of the TLS techniques). This procedure is detailed below.…”