The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

We propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data. The method aims at irregularly spaced longitudinal data, where the number of repeated measurements available per subject is small. In contrast, classical functional data analysis requires a large number of regularly spaced measurements per subject. We assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random (subject-specific) trajectory plus measurement errors. Basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories, and the estimation of the variance of the measurement errors. The eigenfunction basis is estimated from the data, and functional principal components score estimates are obtained by a conditioning step. This conditional estimation method is conceptually simple and straightforward to implement...

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Functional Data Analysis for Sparse Longitudinal Data

/pdf/remote-sensing-of-the-environment-4yfxjaaswe.pdf

Remote sensing of the environment

With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. These are both examples of functional data, which has become a commonly encountered type of data. Functional data analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is functional principal component analysis (FPCA). FPCA is an important dimension reduction tool, and in sparse data situations it can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional d...

/pdf/functional-data-analysis-2yrey8q6kx.pdf

Functional Data Analysis

Shifts in the timing of spring phenology are a central feature of global change research. Long-term observations of plant phenology have been used to track vegetation responses to climate variability but are often limited to particular species and locations and may not represent synoptic patterns. Satellite remote sensing is instead used for continental to global monitoring. Although numerous methods exist to extract phenological timing, in particular start-of-spring (SOS), from time series of reflectance data, a comprehensive intercomparison and interpretation of SOS methods has not been conducted. Here, we assess 10 SOS methods for North America between 1982 and 2006. The techniques include consistent inputs from the 8km Global Inventory Modeling and Mapping Studies Advanced Very High Resolution Radiometer NDVIg dataset, independent data for snow cover, soil thaw, lake ice dynamics, spring streamflow timing, over 16000 individual measurements of ground-based phenology, and two temperature-driven models of spring phenology. Compared with an ensemble of the 10 SOS methods, we found that individual methods differed in average day-of-year estimates by ! 60 days and in standard deviation by ! 20 days. The ability of the satellite methods to retrieve SOS estimates was highest in northern latitudes and lowest in arid, tropical, and Mediterranean ecoregions. The ordinal rank of SOS methods varied geographically, as did the relationships between SOS estimates and the cryospheric/hydrologic metrics. Compared with ground observations, SOS estimates were more related to the first leaf and first flowers expanding phenological stages. We found no evidence for time trends in spring arrival from ground- or model-based data; using an ensemble estimate from two methods that were more closely related to ground observations than other methods, SOS

Intercomparison, interpretation, and assessment of spring phenology in North America estimated from remote sensing for 1982-2006 M I C H A E L A. W H I T E*, K I R S T E N M. DE BEURS w , K A M E L D I D A Nz, D AV I D W. I N O U Y E § ,

Functional linear model

Many variations such as the annual cycle in sea surface temperatures can be considered to be smooth functions and are appropriately described using methods from functional data analysis. This study defines a class of functional autoregressive (FAR) models which can be used as robust predictors for making forecasts of entire smooth functions in the future. The methods are illustrated and compared with pointwise predictors such as SARIMA by applying them to forecasting the entire annual cycle of climatological El Nino- Southern Oscillation (ENSO) time series one year ahead. Forecasts for the period 1987-1996 suggest that the FAR functional predictors show some promising skill, compared to traditional scalar SARIMA forecasts which perform poorly.

http://empslocal.ex.ac.uk/people/staff/dbs202/publications/2000/besse.pdf

Autoregressive Forecasting of Some Functional Climatic Variations

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of R and the response is scalar. The response is modelled as Y=ψ(X)+e, where ψ is some linear continuous operator defined on the space of square integrable functions and valued in R. The random input X is independent from the noise e. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of ψ restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross-covariance operator of (X, Y). The first test statistic relies on a X 2 approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

http://cardot.perso.math.cnrs.fr/SJS2003.pdf

Testing hypotheses in the functional linear model

Le climat est un element important de la vie des territoires car il conditionne le comportement et les decisions des individus et des groupes sociaux comme celui de l’ensemble des especes vivantes et des ecosystemes. A ce titre, la differenciation de l’espace selon les climats et les aptitudes qui en resultent est un domaine qui merite d’etre reinvesti par la recherche en mettant a profit des moyens de traitement modernes de l’information. Avec cet objectif en vue, les auteurs proposent une approche spatiale de definition des climats. Partant des mesures stationnelles de precipitation et de temperature mises a disposition par Meteo-France, un jeu de 14 variables integrant une serie temporelle de 30 ans (1971-2000) est defini pour caracteriser les climats et leurs modalites distinctives de variation. Une methode originale dite d’interpolation locale permet de reconstituer les champs spatiaux continus des variables en question et de les exprimer sous forme de couches d’information gerables par SIG. Ces donnees sont ensuite soumises a un traitement associant analyse factorielle des correspondances et classification hierarchique ascendante pour en obtenir une typologie ou huit climats sont identifies et cartographies sur le territoire metropolitain. Un traitement complementaire faisant appel aux probabilites permet de restituer l’espace de distribution potentielle de chacun des types et de nuancer la partition stricte tiree de la classification. Une synthese met en perspective les resultats obtenus pour les reinterpreter dans un schema general de comprehension du climat. Deux annexes permettent de telecharger les bases cartographiques et les donnees associees a cet article.

Les types de climats en France, une construction spatiale

http://cardot.perso.math.cnrs.fr/JMVACardotSarda2005.pdf

Hervé Cardot

Papers

Functional linear model

Autoregressive Forecasting of Some Functional Climatic Variations

Testing hypotheses in the functional linear model

Les types de climats en France, une construction spatiale

Estimation in generalized linear models for functional data via penalized likelihood