Abstract

Advances in data collection and storage have tremendously increased the presence of functional data, whose graphical representations are curves, images or shapes. As a new area of statistics, functional data analysis extends existing methodologies and theories from the realms of functional analysis, generalized linear model, multivariate data analysis, nonparametric statistics, regression models and many others. From both methodological and practical viewpoints, this paper provides a review of functional principal component analysis, and its use in explanatory analysis, modeling and forecasting, and classification of functional data.

Highlights

  • In probability theory, random functions have being studied for quite a long time

  • I revisit two different bootstrap techniques given by Gonzalez-Manteiga & Martınez-Calvo (2011) and Poskitt & Sengarapillai (2009) that are applicable to re-sample functional time series

  • Illustrated by the Australian fertility data, this paper has broadly revisited some functional principal component techniques for analyzing increasingly high-dimensional data, with the main emphasis being on three popular areas, namely functional principal component analysis (FPCA), functional principal component regression (FPCR), and bootstrap in FPCR

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Summary

Introduction

Random functions have being studied for quite a long time. Due to recent advances in computing and the opportunity to collect and store high-dimensional data, statisticians can study models for “infinite-dimensional functional data”. In 2004, Statistica Sinica published a special issue (vol 14, issue 3) based on that conference, which dealt exclusively with the close connection between longitudinal data and functional data, and contained two review articles by Rice (2004) and Davidian et al (2004). In 2007, Computational Statistics & Data Analysis published a special issue (vol 51, issue 10) on functional data analysis, along with a review article by Gonzalez-Manteiga & Vieu (2007). Computational Statistics published a special issue (vol 22, issue 3) on modeling functional data, along with a review article by Valderrama (2007). This paper contains six sections, and reviews the research on functional data analysis undertaken in both the statistics and probabilistic fields.

Functional data analysis
Functional principal component analysis
Multicollinearity in multivariate linear regression
Multivariate PCR
Conclusion
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