## Theoretical foundations of functional data analysis, with an introduction to linear operators.(English)Zbl 1338.62009

Wiley Series in Probability and Statistics. Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-01691-6/hbk; 1-118-76254-1/ebook). xiii, 334 p. (2015).
Functional data analysis is concerned with the statistical analysis of data, each of which is a random function. In most situations it is assumed that these functions are independent replicates of a stochastic process. There are many options to describe the behavior of such processes. One popular approach, which in particular is investigated in the monograph under review, is based on covariances. These kernels give rise to classical integral operators which have been extensively studied in functional analysis. What is of special importance is the eigenvalue and eigenvector representation of these kernels (Mercer’s theorem). These comments show that functional data analysis makes heavy use of tools from functional analysis. What is required is an appropriate combination with probabilistic and statistical methodology. This is exactly what the monograph provides. In the first six of its eleven chapters, the text reviews classical facts and tools from functional analysis. In a brief introduction, the authors collect some basic facts from the multivariate (i.e., finite-dimensional) case. Chapter 2 deals with function spaces (e.g., $$L^2$$, Hilbert, RKHS, Sobolev) which are of central importance. Chapter 3 discusses some aspects of linear operators, while Chapter 4 focuses on compact operators. Since in later chapters these operators will depend on data and may be viewed as perturbed versions of theoretical operators of interest, a brief chapter on perturbation theory is included. Finally, since in applications empirical operators need to be inverted, some smoothing is required. Chapter 6 presents useful facts from regularization theory. These chapters are fairly self-contained and present all relevant proofs. In the second half of the monograph some probabilistic and statistical issues are discussed. Chapter 7 presents the aforementioned link between functional analysis and stochastic processes viewed as elements of an appropriate Hilbert space and characterized through its covariance kernel (Karhunen-Loève representation). Starting with Chapter 8, the monograph discusses estimation of relevant parameters, functions and kernels, which are unknown in practical situations, from a sequence of i.i.d. realizations. Applications to principal component analysis, canonical correlation analysis and regression follow in Chapters 9 to 11. Special emphasis is on the role of eigenvalue and eigenvector representations of relevant estimators.
Summarizing, the monograph presents a self-contained introduction to important aspects of functional data analysis based on covariance operators. The stochastic part of the book has a probabilistic flavour. Statistical issues are not studied in greater detail. Simulations or applications to real data are not included. Overall, the book presents a very good treatment of the subject as described by its title.

### MSC:

 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62-07 Data analysis (statistics) (MSC2010) 46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis 47A99 General theory of linear operators 62H25 Factor analysis and principal components; correspondence analysis
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