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The Oxford handbook of functional data analysis
\"As technology progresses, we are able to handle larger and larger datasets. At the same time, monitoring devices such as electronic equipment and sensors (for registering images, temperature, etc.) have become more and more sophisticated. This high-tech revolution offers the opportunity to observe phenomena in an increasingly accurate way by producing statistical units sampled over a finer and finer grid, with the measurement points so close that the data can be considered as observations varying over a continuum. Such continuous (or functional) data may occur in biomechanics (e.g. human movements), chemometrics (e.g. spectrometric curves), econometrics (e.g. the stock market index), geophysics (e.g. spatio-temporal events such as El Nino or time series of satellite images), or medicine (electro-cardiograms/electro-encephalograms). It is well known that standard multivariate statistical analyses fail with functional data. However, the great potential for applications has encouraged new methodologies able to extract relevant information from functional datasets. This Handbook aims to present a state of the art exploration of this high-tech field, by gathering together most of major advances in this area. Leading international experts have contributed to this volume with each chapter giving the key original ideas and comprehensive bibliographical information. The main statistical topics (classification, inference, factor-based analysis, regression modelling, resampling methods, time series, random processes) are covered in the setting of functional data. The twin challenges of the subject are the practical issues of implementing new methodologies and the theoretical techniques needed to expand the mathematical foundations and toolbox. The volume therefore mixes practical, methodological and theoretical aspects of the subject, sometimes within the same chapter. As a consequence, this book should appeal to a wide audience of engineers, practitioners and graduate students, as well as academic researchers, not only in statistics and probability but also in the numerous related application areas\"-- Provided by publisher.
Book
DISTRIBUTION AND QUANTILE FUNCTIONS, RANKS AND SIGNS IN DIMENSION d
Unlike the real line, the real space ℝ
d
, for d ≥ 2, is not canonically ordered. As a consequence, such fundamental univariate concepts as quantile and distribution functions and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has been an open problem for more than half a century, generating an abundant literature and motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions proposed in the literature, the measure transportation-based ranks and signs introduced in Chernozhukov, Galichon, Hallin and Henry (Ann. Statist. 45 (2017) 223–256) enjoy all the properties that make univariate ranks a successful tool for semiparametric inference. Related with those ranks, we propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko–Cantelli result. Our approach is based on McCann (Duke Math. J. 80 (1995) 309–323) and our results do not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free and essentially maximal ancillary in the sense of Basu (Sankhyā 21 (1959) 247–256) which, in semiparametric models involving noise with unspecified density, can be interpreted as a finite-sample form of semiparametric efficiency. Although constituting a sufficient summary of the sample, empirical center-outward distribution functions are defined at observed values only. A continuous extension to the entire d-dimensional space, yielding smooth empirical quantile contours and sign curves while preserving the essential monotonicity and Glivenko–Cantelli features of the concept, is provided. A numerical study of the resulting empirical quantile contours is conducted.
Journal Article
Structural Equation Modeling with Mplus
[This book] reviews the basic concepts and applications of SEM using Mplus Version 6. ... The first two chapters introduce the fundamental concepts of SEM and important basics of the Mplus program. The remaining chapters focus on SEM applications and include a variety of SEM models presented within the context of three sections: Single-group analyses, Multiple-group analyses, and other important topics, the latter of which includes the multitrait-multimethod, latent growth curve, and multilevel models. (DIPF/Orig.).
eBook
Modeling and analysis of compositional data
Modeling and Analysis of Compositional Data presents a practical and comprehensive introduction to the analysis of compositional data along with numerous examples to illustrate both theory and application of each method. Based upon short courses delivered by the authors, it provides a complete and current compendium of fundamental to advanced methodologies along with exercises at the end of each chapter to improve understanding, as well as data and a solutions manual which is available on an accompanying website. Complementing Pawlowsky-Glahn's earlier collective text that provides an overview of the state-of-the-art in this field, Modeling and Analysis of Compositional Data fills a gap in the literature for a much-needed manual for teaching, self learning or consulting.
eBook
Deconstructing multivariate decoding for the study of brain function
Multivariate decoding methods were developed originally as tools to enable accurate predictions in real-world applications. The realization that these methods can also be employed to study brain function has led to their widespread adoption in the neurosciences. However, prior to the rise of multivariate decoding, the study of brain function was firmly embedded in a statistical philosophy grounded on univariate methods of data analysis. In this way, multivariate decoding for brain interpretation grew out of two established frameworks: multivariate decoding for predictions in real-world applications, and classical univariate analysis based on the study and interpretation of brain activation. We argue that this led to two confusions, one reflecting a mixture of multivariate decoding for prediction or interpretation, and the other a mixture of the conceptual and statistical philosophies underlying multivariate decoding and classical univariate analysis. Here we attempt to systematically disambiguate multivariate decoding for the study of brain function from the frameworks it grew out of. After elaborating these confusions and their consequences, we describe six, often unappreciated, differences between classical univariate analysis and multivariate decoding. We then focus on how the common interpretation of what is signal and noise changes in multivariate decoding. Finally, we use four examples to illustrate where these confusions may impact the interpretation of neuroimaging data. We conclude with a discussion of potential strategies to help resolve these confusions in interpreting multivariate decoding results, including the potential departure from multivariate decoding methods for the study of brain function.
•We highlight two sources of confusion that affect the interpretation of multivariate decoding results.•Confusion 1: The dual use of multivariate decoding for real-world predictions and interpretation in terms of brain function.•Confusion 2: A mixture of statistical and conceptual frameworks of classical univariate analysis and multivariate decoding.•We show six differences between univariate analysis and multivariate decoding and a different meaning of signal and noise.•We use four illustrative examples to reveal these confusions and the assumptions of multivariate decoding for interpretation.
Journal Article
Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains
Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, such as functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error. A flexible R implementation is available on CRAN. The new method is shown to be competitive to existing approaches for data observed on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. Supplementary material, including detailed proofs, additional simulation results, and software is available online.
Journal Article