Measurement, regression, and calibration.

*(English)*Zbl 0829.62064
Oxford Statistical Science Series. 12. Oxford: Clarendon Press. ix, 201 p. (1993).

According to the author this book is a research monograph for a range of regression problems in which one set of variables is predicted from another. Coverage includes traditional least squares topics as well as less standard topics which the author collectively calls regularized regression. The book has eight chapters and five appendices.

The first chapter is introductory, where a roadmap of the coverage is provided. Six different data sets are presented to motivate the range of applications discussed in the book. These data sets are analyzed in later chapters after introducing the relevant techniques.

Simple linear regression is the topic of chapter 2. Standard least squares analysis, analysis of variance, residual plotting and prediction intervals are some of the topics included. Generalized and weighted least squares regression are discussed. Two classes of calibration problems are introduced – natural (or random) calibration and controlled (or fixed) calibration. Some of the problems associated with construction of confidence sets for the unknown “calibrand” are pointed out. Chapter 3 is concerned with multiple linear regression. Standard hypothesis testing, prediction, selection of variables and calibration are the main topics included in this chapter.

Chapter 4 is perhaps the heart of the text. This is where the author introduces some of the less-standard methods in multiple regression under a single umbrella of “regularized multiple regression”. Methods described in detail include ridge regression, principal components regression, partial least squares regression and continuum regression. The cross-validation technique and its role in choosing the “correct” number of “components” for modeling the response are explained.

Natural calibration as well as controlled calibration in the multiple regression framework form the core of chapter 5. The “classical” estimator and the “inverse” estimator are compared. Bayesian approaches for prediction and calibration are also presented. Chapter 6 is devoted to Bayesian regression methods specifically developed in the context of spectroscopy although applications in other fields are not that uncommon. The discussion naturally leads to Bayesian alternatives to kriging in spatial modeling and data analysis. Nonlinear regression and calibration problems are covered in chapter 7. Discussion includes diagnostics for detecting nonlinearity, variable selection and a brief overview of the nonlinear theory.

Chapter 8 is titled “pattern recognition” but the main problem addressed is that of discrimination among \(p\) multivariate normal populations with possibly unequal variance structures. The treatment is standard with the exception of the use of proper prior distributions to make it possible to discriminate when there are more variables than observations.

There is a useful collection of appendices. Appendix A is an overview of the following distributions – multivariate normal, matrix normal, Wishart, Inverse Wishart, Matrix Student-T and Matrix Fisher-F. Appendix B gives some results for Bayesian prior to posterior analysis in the context of multivariate and matrix-variate normals. Appendix C derives sufficient conditions for a member of a particular class of “regularized estimators” to dominate the least squares estimator. A short list of some useful matrix results is presented in Appendix D. Appendix E gives a listing of an S-Plus code for partial least squares. A bibliography and an index conclude the book.

The writing is elegant and the presentation is clear. An enormous amount of information is packed in less than two hundred pages. This is bound to be a very valuable reference book for practitioners who have a strong background in standard regression theory.

The first chapter is introductory, where a roadmap of the coverage is provided. Six different data sets are presented to motivate the range of applications discussed in the book. These data sets are analyzed in later chapters after introducing the relevant techniques.

Simple linear regression is the topic of chapter 2. Standard least squares analysis, analysis of variance, residual plotting and prediction intervals are some of the topics included. Generalized and weighted least squares regression are discussed. Two classes of calibration problems are introduced – natural (or random) calibration and controlled (or fixed) calibration. Some of the problems associated with construction of confidence sets for the unknown “calibrand” are pointed out. Chapter 3 is concerned with multiple linear regression. Standard hypothesis testing, prediction, selection of variables and calibration are the main topics included in this chapter.

Chapter 4 is perhaps the heart of the text. This is where the author introduces some of the less-standard methods in multiple regression under a single umbrella of “regularized multiple regression”. Methods described in detail include ridge regression, principal components regression, partial least squares regression and continuum regression. The cross-validation technique and its role in choosing the “correct” number of “components” for modeling the response are explained.

Natural calibration as well as controlled calibration in the multiple regression framework form the core of chapter 5. The “classical” estimator and the “inverse” estimator are compared. Bayesian approaches for prediction and calibration are also presented. Chapter 6 is devoted to Bayesian regression methods specifically developed in the context of spectroscopy although applications in other fields are not that uncommon. The discussion naturally leads to Bayesian alternatives to kriging in spatial modeling and data analysis. Nonlinear regression and calibration problems are covered in chapter 7. Discussion includes diagnostics for detecting nonlinearity, variable selection and a brief overview of the nonlinear theory.

Chapter 8 is titled “pattern recognition” but the main problem addressed is that of discrimination among \(p\) multivariate normal populations with possibly unequal variance structures. The treatment is standard with the exception of the use of proper prior distributions to make it possible to discriminate when there are more variables than observations.

There is a useful collection of appendices. Appendix A is an overview of the following distributions – multivariate normal, matrix normal, Wishart, Inverse Wishart, Matrix Student-T and Matrix Fisher-F. Appendix B gives some results for Bayesian prior to posterior analysis in the context of multivariate and matrix-variate normals. Appendix C derives sufficient conditions for a member of a particular class of “regularized estimators” to dominate the least squares estimator. A short list of some useful matrix results is presented in Appendix D. Appendix E gives a listing of an S-Plus code for partial least squares. A bibliography and an index conclude the book.

The writing is elegant and the presentation is clear. An enormous amount of information is packed in less than two hundred pages. This is bound to be a very valuable reference book for practitioners who have a strong background in standard regression theory.

Reviewer: H.Iyer (Fort Collins)

##### MSC:

62Jxx | Linear inference, regression |

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

62J05 | Linear regression; mixed models |

62H25 | Factor analysis and principal components; correspondence analysis |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62N99 | Survival analysis and censored data |