Total Least Squares (TLS)

A strategy for adaptive control and energetic optimization of aerobic fermentors was implemented, with both air flow and agitation speed as manipulated variables. This strategy is separable in its components: control, optimization, estimation. We optimized parameter’s estimation (from the usual KLa correlation) using sinusoidal excitation of air flow and agitation speed. We have implemented parameter’s estimation trough recursive least squares algorithm with forgetting factor. We carried separate essays on control, optimization and estimation algorithms. We carried our essays using an original computational simulation environment, with noise and delay generating facilities for data sampling and filtering.

Our results show the convergence and robustness of the estimation algorithm used, improved with use of both forgetting factor and KLa dead-band facilities. Control algorithm used in our work compares favorably with PID using the integrated area criteria for deviation between oxygen molarity and critical molarity (set point). Optimization algorithm clearly reduces energetic consumption, respecting critical molarity. Integration of control, optimization and adaptive algorithms was implemented, but future work is needed for stability. Methods were defined and implemented for stability improvement. We have implemented data acquisition and computer manipulation of air flow and agitation speed for actual fermentors.

This dissertation deals with a class of nonlinear adjustment problems that has a direct least squares solutionfor certain weighting cases.  In the literature of mathematical statistics these problems are expressed in anonlinear model called Errors-In-Variables (EIV) and their solution became popular as total least squares(TLS). The TLS solution is direct and involves the use of singular value decomposition (SVD), presented inmost cases for adjustment problems with equally weighted and uncorrelated measurements.  Additionally,several weighted total least squares (WTLS) algorithms have been published in the last years for derivingiterative solutions, when more general weighting cases have to be taken into account and without linearizingthe problem in any step of the solution process.This research provides  rstly a well de ned mathematical relationship between TLS and direct least squaressolutions.  As a by-product, a systematic approach for the direct solution of these adjustments is established,using a consistent and complete mathematical formalization.  By transforming the problem to the solutionof a quadratic or cubic algebraic equation, which is identical with those resulting from TLS, it will be shownthat TLS is an algorithmic approach already known to the geodetic community and not a new method.A second contribution of this work is the clear overview of weighted least squares solutions for the discussedclass of problems, i.e.  the WTLS solution in the terminology of the statistical community.  It will be shownthat for certain weighting cases a direct solution still exists, for which two new solution strategies will beproposed. Further, stochastic models with more general weight matrices are examined, including correlationsbetween the measurements or even singular cofactor matrices.  New algorithms are developed and presented,that provide iterative weighted least squares solutions without linearizing the original nonlinear problem.The aim of this work is the popularization of the TLS approach, by presenting a complete framework forobtaining a (weighted) least squares solution for the investigated class of nonlinear adjustment problems.The proposed approaches and the implemented algorithms can be employed for obtaining direct solutionsin engineering tasks for which efficiency is important, while iterative solutions can be derived for stochasticmodels with more general weights.

A method to resolve cyclic ambiguities and increase the accuracy and the resolution in the direction-of-arrival (DOA) estimation using the Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT)algorithm is proposed. It is based on rotating the array and sampling the received signal at multiple positions. Using this approach, the gain in accuracy and resolution is addressed as function of the mean and variance of the DOA. Simulations results are provided as a means of verifying this analysis.

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is per...

Many problems, not only in signal processing, image processing , digital imaging, computer vision and visualization, lead to the Least Square Error (LSE) problem or Total (Orthogonal) Least Square Error (TLSE) problem computation. Usually the standard least square error approximation method is used due to its simplicity, but it is not an optimal solution, as it does not optimize the orthogonal distances, but only the vertical distances. There are many problems for which the LSE is not convenient and the TLSE is to be used. Unfortunately, the TLSE is computationally much more expensive. This paper presents a new, simple, robust and fast algorithm for the total least square error computation in E 2 .

In the problem of least squares min || Ax-b || with A ∈ Rm × n, m ≥ n and b ∈ Rm,
It is often assumed that the matrix is accurate and the vector b is tainted by errors. It is
hypothesis is not always realistic because existen problems where the matrix is also subject to errors.
One way to deal with such problems is by method of Least Squares Total (TLS). In this paper we use the TLS method to solve the problem spectroscopy
magnetic resonance and present a comparison of results obtained with the method of Least Squares (LS).

Recently it was shown how necessary and sufficient information for solving  an orthogonally invariant linear approximation problem $AX\approx B$ with multiple right-hand sides can be revealed through the so-called core problem reduction; see [I. Hnětynková et al., SIAM J. Matrix Anal. Appl., 34, 2013, pp. 917--931]. The total least squares (TLS) serves as an important example of such approximation problem. Solvability of TLS was discussed in the full generality in [I. Hnětynková et al., SIAM J. Matrix Anal. Appl., 32, 2011,  pp. 748--770]. This theoretical study investigates solvability of core problems with multiple right-hand sides in the TLS sense. It is shown that contrary to the single right-hand side case, a core problem with multiple right-hand sides may not have a TLS solution.  Further possible internal structure of core problems is studied. Outputs of the classical TLS algorithm for the original problem $AX\approx B$ and for the core problem within $AX\approx B$ are compared.

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is per...

The total least squares (TLS) represents a popular data fitting approach for solving linear approximation problems $Ax\approx b$ (i.e., with a vector right-hand side) and $AX\approx B$ (i.e., with a matrix right-hand side) contaminated by errors. This paper introduces a generalization of TLS formulation to problems with structured right-hand sides. First, we focus on the case, where the right-hand side and consequently also the solution are tensors. We show that whereas the basic solvability result can be obtained directly by matricization of both tensors, generalization of the core problem reduction is more complicated. The core reduction allows to reduce mathematically the problem dimensions by removing all redundant and irrelevant data from the system matrix and the right-hand side. We prove that the core problems within the original tensor problem and its matricized counterpart are in general different. Then, we concentrate on problems with even more structured right-hand sides, where the same model A corresponds to a set of various tensor right-hand sides. Finally, relations between the matrix and tensor core problem are discussed.