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**Nonlinear multivariate analysis.**
*(English)*
Zbl 0697.62048

Wiley Series in Probability and Mathematical Statistics. Chichester etc.: John Wiley & Sons. xx, 579 p. £34.95 (1990).

Gifi is the nom de plume of a group of Dutch statisticians and data analysts. They developed a set of methods for analyzing multivariate data of mixed type, i.e. of nominal, ordinal, and metric scale, along some guidelines which are called the Gifi philosophy. After a lot of separate publications and after extensive circulation of preliminary versions Gifi’s work has come out with the present book.

As a definition it is stated that Multivariate Analysis (MVA) studies systems of correlated random variables or random samples from such systems. As a consequence the point of view is adopted that, given some of the most common MVA questions, it is possible to start either from the model or from the data (without having any model in mind). MVA is called nonlinear if the results are invariant under all one-to-one nonlinear transformations. To achieve that, the techniques proposed here start from an indicator matrix which is one form to convert data into nominal scale.

The book is organized around various key techniques of MVA. They are identified, much more than is usually done, with computer programs and the type of output they produce. The techniques discussed here have a strong emphasis on geometry, in the sense that they make pictures of data.

Homogeneity analysis, nonlinear principal components analysis, nonlinear generalized canonical analysis, nonlinear canonical correlation analysis, multidimensional scaling, and correspondence analysis are the headers of the chapters proposing the techniques. There are some additional chapters dealing with the theoretical foundations, too. At the end, the proof of the pudding is given by discussion of some large-scaled examples.

As a definition it is stated that Multivariate Analysis (MVA) studies systems of correlated random variables or random samples from such systems. As a consequence the point of view is adopted that, given some of the most common MVA questions, it is possible to start either from the model or from the data (without having any model in mind). MVA is called nonlinear if the results are invariant under all one-to-one nonlinear transformations. To achieve that, the techniques proposed here start from an indicator matrix which is one form to convert data into nominal scale.

The book is organized around various key techniques of MVA. They are identified, much more than is usually done, with computer programs and the type of output they produce. The techniques discussed here have a strong emphasis on geometry, in the sense that they make pictures of data.

Homogeneity analysis, nonlinear principal components analysis, nonlinear generalized canonical analysis, nonlinear canonical correlation analysis, multidimensional scaling, and correspondence analysis are the headers of the chapters proposing the techniques. There are some additional chapters dealing with the theoretical foundations, too. At the end, the proof of the pudding is given by discussion of some large-scaled examples.

Reviewer: R.Schlittgen

### MSC:

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

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

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

62H99 | Multivariate analysis |

62-07 | Data analysis (statistics) (MSC2010) |

62H20 | Measures of association (correlation, canonical correlation, etc.) |