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Functional Data Analysis (functional + data_analysis)

Distribution by Scientific Domains

Mathematics and Statistics	100%

Selected Abstracts

Bayesian Hierarchical Functional Data Analysis Via Contaminated Informative Priors

BIOMETRICS, Issue 3 2009
Bruno Scarpa
Summary A variety of flexible approaches have been proposed for functional data analysis, allowing both the mean curve and the distribution about the mean to be unknown. Such methods are most useful when there is limited prior information. Motivated by applications to modeling of temperature curves in the menstrual cycle, this article proposes a flexible approach for incorporating prior information in semiparametric Bayesian analyses of hierarchical functional data. The proposed approach is based on specifying the distribution of functions as a mixture of a parametric hierarchical model and a nonparametric contamination. The parametric component is chosen based on prior knowledge, while the contamination is characterized as a functional Dirichlet process. In the motivating application, the contamination component allows unanticipated curve shapes in unhealthy menstrual cycles. Methods are developed for posterior computation, and the approach is applied to data from a European fecundability study. [source]

Statistics for spatial functional data: some recent contributions

ENVIRONMETRICS, Issue 3-4 2010
P. Delicado
Abstract Functional data analysis (FDA) is a relatively new branch in statistics. Experiments where a complete function is observed for each individual give rise to functional data. In this work we focus on the case of functional data presenting spatial dependence. The three classic types of spatial data structures (geostatistical data, point patterns, and areal data) can be combined with functional data as it is shown in the examples of each situation provided here. We also review some contributions in the literature on spatial functional data. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Curve registration by local regression

THE CANADIAN JOURNAL OF STATISTICS, Issue 1 2000
A. Kneip
Abstract Functional data analysis involves the extension of familiar statistical procedures such as principal-components analysis, linear modelling and canonical correlation analysis to data where the raw observation is a function x, (t). An essential preliminary to a functional data analysis is often the registration or alignment of salient curve features by suitable monotone transformations hi(t). In effect, this conceptualizes variation among functions as being composed of two aspects: phase and amplitude. Registration aims to remove phase variation as a preliminary to statistical analyses of amplitude variation. A local nonlinear regression technique is described for identifying the smooth monotone transformations hi, and is illustrated by analyses of simulated and actual data. L'analyse de données se présentant sous la forme de fonctions x,(t) repose sur la généralisation d'outils statistiques familiers tels que l'analyse en composantes principales, les modèles linéaires et l'analyse des corrélations canoniques. L'étalonnage des caractéristiques saillantes des courbes à l'aide de transformations monotones hi(t) constitue souvent un préiequis essentiel au traitement statistique de telles données. II découle d'une décomposition en deux parties de la variation entre les fonctions observées: une phase et une amplitude. L'étalonnage vise à éliminer la première de ces deux sources de variation, ce qui permet de concentrer ensuite l'analyse sur la seconde. Les auteurs décrivent ici une technique de régression non linéaire locale facilitant l'identification de transformations monotones lisses hi appropriées. Leur propos est illustré à l'aide de données réelles et simulées. [source]

Detecting changes in the mean of functional observations

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 5 2009
István Berkes
Summary., Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as Xi(t)=,(t)+,1,l<,,i, l+ vl(t), where , is the common mean, vl are the eigenfunctions of the covariance operator and the ,i, l are the scores. Inferential procedures assume that the mean function ,(t) is the same for all values of i. If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England. [source]

Hybrid Dirichlet mixture models for functional data

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 4 2009
Sonia Petrone
Summary., In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individual-specific regions that provide heterogeneous behaviour (e.g. ,damaged' areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: non-homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with k -levels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to , and their connection with Dirichlet process mixtures. [source]

Modelling price paths in on-line auctions: smoothing sparse and unevenly sampled curves by using semiparametric mixed models

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES C (APPLIED STATISTICS), Issue 2 2008
Florian Reithinger
Summary., On-line auctions pose many challenges for the empirical researcher, one of which is the effective and reliable modelling of price paths. We propose a novel way of modelling price paths in eBay's on-line auctions by using functional data analysis. One of the practical challenges is that the functional objects are sampled only very sparsely and unevenly. Most approaches rely on smoothing to recover the underlying functional object from the data, which can be difficult if the data are irregularly distributed. We present a new approach that can overcome this challenge. The approach is based on the ideas of mixed models. Specifically, we propose a semiparametric mixed model with boosting to recover the functional object. As well as being able to handle sparse and unevenly distributed data, the model also results in conceptually more meaningful functional objects. In particular, we motivate our method within the framework of eBay's on-line auctions. On-line auctions produce monotonic increasing price curves that are often correlated across auctions. The semiparametric mixed model accounts for this correlation in a parsimonious way. It also manages to capture the underlying monotonic trend in the data without imposing model constraints. Our application shows that the resulting functional objects are conceptually more appealing. Moreover, when used to forecast the outcome of an on-line auction, our approach also results in more accurate price predictions compared with standard approaches. We illustrate our model on a set of 183 closed auctions for Palm M515 personal digital assistants. [source]

Curve registration by local regression

Bayesian Hierarchical Functional Data Analysis Via Contaminated Informative Priors

Bayesian Adaptive Regression Splines for Hierarchical Data

BIOMETRICS, Issue 3 2007
Jamie L. Bigelow
Summary This article considers methodology for hierarchical functional data analysis, motivated by studies of reproductive hormone profiles in the menstrual cycle. Current methods standardize the cycle lengths and ignore the timing of ovulation within the cycle, both of which are biologically informative. Methods are needed that avoid standardization, while flexibly incorporating information on covariates and the timing of reference events, such as ovulation and onset of menses. In addition, it is necessary to account for within-woman dependency when data are collected for multiple cycles. We propose an approach based on a hierarchical generalization of Bayesian multivariate adaptive regression splines. Our formulation allows for an unknown set of basis functions characterizing the population-averaged and woman-specific trajectories in relation to covariates. A reversible jump Markov chain Monte Carlo algorithm is developed for posterior computation. Applying the methods to data from the North Carolina Early Pregnancy Study, we investigate differences in urinary progesterone profiles between conception and nonconception cycles. [source]