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**Nonparametric functional data analysis. Theory and practice.**
*(English)*
Zbl 1119.62046

Springer Series in Statistics. New York, NY: Springer (ISBN 0-387-30369-3/hbk). xx, 258 p. (2006).

High-dimensional responses arise naturally in many fields of application, such as spectroscopy and speech recognition, to name a few. In functional data analysis, high-dimensional responses are viewed as (discretised) realizations of a continuous function. Viewing data from this perspective and developing models for functional responses leads to new insights and new statistical methodologies.

In this monograph, the authors discuss models for functional data analysis and estimators for such models based on nonparametric kernel smoothing techniques. Taking a kernel smoothing approach allows the authors to derive the properties of the estimators and all derivations and proofs are given in the text. By way of contrasts, the books by J. O. Ramsay and B. W. Silverman [Functional data analysis. (1997; Zbl 0882.62002), Applied functional data analysis. Methods and Case Studies. (2002; Zbl 1011.62002); Functional data analysis. 2nd ed. (2005; Zbl 1079.62006)] employ a penalized smoothing approach which typically leads to spline smoothers.

Example data sets that motivate the development of the models are also provided. These data sets are available from a web-site (http://www.lsp.ups-tlse.fr/staph/npfda/) associated with this book. This web-site also provides R and S-Plus implementations of the estimators discussed in the theoretical parts of the monograph.

The monograph is split into five parts. The first part, “Statistical Background for Nonparametric Statistics and Functional Data”, contains four chapters; “Introduction to Nonparametric Statistics”, “Some Functional Datasets and Associated Statistical Problematics”, “What is a Well–Adapted Space for Functional Data” and “Local Weighting of Functional Variables”. The following part, “Nonparametric Prediction from Functional Data”, contains three chapters; “Functional Nonparametric Prediction Methodologies”, “Some Selected Asymptotics” and “Computational Issues”. The third part, “Nonparametric Classification of Functional Data”, contains a chapter on “Functional Nonparametric Supervised Classification” and on “Functional Nonparametric Unsupervised Classification”. The fourth part , “Nonparametric Methods for Dependent Functional Data”, has three chapters; “Mixing, Nonparametric and Functional Statistics”, “Some Selected Asymptotics” and “Application to Continuous Time Processes Prediction”. The final part, “Conclusions”, contains a chapter on “Small Ball Probabilities and Semi–metrics” and on “Some Perspectives”. An appendix provides “Some Probabilistic Tools”.

The references section lists the references (25 pages) in alphabetical order sorted by names of the author(s). However, the labels are derived from the first letter(s) of the surnames of the author(s) and the publication year and, hence, are not sorted. Within the monograph, the authors use these labels for references, which makes it on occasions a bit time consuming to look up a reference. The index provided seems to be fairly complete and is helpful in looking up topics discussed in this monograph.

Several chapters end in a section in which the authors provide additional comments, discussions and pose some open problems in this area, which should be appealing for researchers in this field. But this book is by no means focussed on researchers and practitioners will also find plenty of stimulating ideas in this book. This book should be useful for all people interested in the area of functional data analysis.

In this monograph, the authors discuss models for functional data analysis and estimators for such models based on nonparametric kernel smoothing techniques. Taking a kernel smoothing approach allows the authors to derive the properties of the estimators and all derivations and proofs are given in the text. By way of contrasts, the books by J. O. Ramsay and B. W. Silverman [Functional data analysis. (1997; Zbl 0882.62002), Applied functional data analysis. Methods and Case Studies. (2002; Zbl 1011.62002); Functional data analysis. 2nd ed. (2005; Zbl 1079.62006)] employ a penalized smoothing approach which typically leads to spline smoothers.

Example data sets that motivate the development of the models are also provided. These data sets are available from a web-site (http://www.lsp.ups-tlse.fr/staph/npfda/) associated with this book. This web-site also provides R and S-Plus implementations of the estimators discussed in the theoretical parts of the monograph.

The monograph is split into five parts. The first part, “Statistical Background for Nonparametric Statistics and Functional Data”, contains four chapters; “Introduction to Nonparametric Statistics”, “Some Functional Datasets and Associated Statistical Problematics”, “What is a Well–Adapted Space for Functional Data” and “Local Weighting of Functional Variables”. The following part, “Nonparametric Prediction from Functional Data”, contains three chapters; “Functional Nonparametric Prediction Methodologies”, “Some Selected Asymptotics” and “Computational Issues”. The third part, “Nonparametric Classification of Functional Data”, contains a chapter on “Functional Nonparametric Supervised Classification” and on “Functional Nonparametric Unsupervised Classification”. The fourth part , “Nonparametric Methods for Dependent Functional Data”, has three chapters; “Mixing, Nonparametric and Functional Statistics”, “Some Selected Asymptotics” and “Application to Continuous Time Processes Prediction”. The final part, “Conclusions”, contains a chapter on “Small Ball Probabilities and Semi–metrics” and on “Some Perspectives”. An appendix provides “Some Probabilistic Tools”.

The references section lists the references (25 pages) in alphabetical order sorted by names of the author(s). However, the labels are derived from the first letter(s) of the surnames of the author(s) and the publication year and, hence, are not sorted. Within the monograph, the authors use these labels for references, which makes it on occasions a bit time consuming to look up a reference. The index provided seems to be fairly complete and is helpful in looking up topics discussed in this monograph.

Several chapters end in a section in which the authors provide additional comments, discussions and pose some open problems in this area, which should be appealing for researchers in this field. But this book is by no means focussed on researchers and practitioners will also find plenty of stimulating ideas in this book. This book should be useful for all people interested in the area of functional data analysis.

Reviewer: Berwin A. Turlach (Singapore)