The total least squares problem: computational aspects and analysis.

*(English)*Zbl 0789.62054
Frontiers in Applied Mathematics. 9. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xiii, 300 p. (1991).

Total least squares (TLS) is one of the several linear parameter estimation techniques that have been devised to compensate for data errors. It is also known as the errors-in-variables model. The renewed interest in the TLS method is mainly due to the development of computationally efficient and numerically reliable TLS algorithms. Much attention is paid in this book to the computational aspects of TLS and new algorithms are presented.

The book is divided into 10 chapters. The first chapter is an introduction and describes the techniques of TLS and its applications. The remaining chapters give different aspects of the TLS problem.

Chapter 2 surveys the main principles of the basic TLS problem \(Ax\approx b\) and shows how to compute its solution in a reliable way by means of the singular value decomposition (SVD). The word ‘basic’ means that only one right-hand side of the vector \(b\) is considered and that a solution of the TLS problem exists and is unique. A geometric comparison between TLS and LS problems enlightens the main differences between both principles.

Extensions of the basic TLS problems are investigated in Chapter 3. Here are discussed multidimensional TLS problems \(AX\approx B\) having more than one right-hand side vector, problems in which the TLS solution is no longer unique, TLS problems that fail to have a solution altogether, and mixed LS-TLS problems that assume some of the columns of the data matrix \(A\) to be error free. At the end, the TLS computations are summarized in one algorithm that takes into account all the extensions given above.

In Chapter 4, the TLS computations are sped up in a direct way by modifying appropriately the SVD computations. These modifications are summarized in a computationally improved algorithm PTLS. An analysis of its operation counts, as well as computational results, show its relative efficiency with respect to the classical TLS computations. Chapter 5 describes how the TLS computations can be sped up in an iterative way if a priori information about the TLS is available. Different algorithms are presented and their convergence properties are analyzed.

Chapters 6 to 9 analyze the properties of the TLS to delineate its domain of applicability and to evaluate its practical significance. The last chapter summarizes the conclusions of this book and surveys some recent extensions of the classical TLS problem currently under investigation. Suggestions for further research are also made.

To understand the book some basic knowledge of linear algebra, matrix computations and statistics is required. The book will be useful to researchers, practising engineers and scientists, and graduate students. Graduate students can use it as a course on linear equations and least squares problems (Chapters 2, 3, 6 and 7), or as a course on errors-in- variables regression (Chapters 2, 3, 8 and 9).

The book is divided into 10 chapters. The first chapter is an introduction and describes the techniques of TLS and its applications. The remaining chapters give different aspects of the TLS problem.

Chapter 2 surveys the main principles of the basic TLS problem \(Ax\approx b\) and shows how to compute its solution in a reliable way by means of the singular value decomposition (SVD). The word ‘basic’ means that only one right-hand side of the vector \(b\) is considered and that a solution of the TLS problem exists and is unique. A geometric comparison between TLS and LS problems enlightens the main differences between both principles.

Extensions of the basic TLS problems are investigated in Chapter 3. Here are discussed multidimensional TLS problems \(AX\approx B\) having more than one right-hand side vector, problems in which the TLS solution is no longer unique, TLS problems that fail to have a solution altogether, and mixed LS-TLS problems that assume some of the columns of the data matrix \(A\) to be error free. At the end, the TLS computations are summarized in one algorithm that takes into account all the extensions given above.

In Chapter 4, the TLS computations are sped up in a direct way by modifying appropriately the SVD computations. These modifications are summarized in a computationally improved algorithm PTLS. An analysis of its operation counts, as well as computational results, show its relative efficiency with respect to the classical TLS computations. Chapter 5 describes how the TLS computations can be sped up in an iterative way if a priori information about the TLS is available. Different algorithms are presented and their convergence properties are analyzed.

Chapters 6 to 9 analyze the properties of the TLS to delineate its domain of applicability and to evaluate its practical significance. The last chapter summarizes the conclusions of this book and surveys some recent extensions of the classical TLS problem currently under investigation. Suggestions for further research are also made.

To understand the book some basic knowledge of linear algebra, matrix computations and statistics is required. The book will be useful to researchers, practising engineers and scientists, and graduate students. Graduate students can use it as a course on linear equations and least squares problems (Chapters 2, 3, 6 and 7), or as a course on errors-in- variables regression (Chapters 2, 3, 8 and 9).

Reviewer: V.P.Gupta (Jaipur)

##### MSC:

62J05 | Linear regression; mixed models |

65C99 | Probabilistic methods, stochastic differential equations |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62J99 | Linear inference, regression |