Some Tools for Functional Data Analysis

TLDR

Functional data analysis extends multivariate techniques to infinite‑dimensional processes, using spline smoothing to separate structural and residual components of observed functions. The paper demonstrates that L‑splines enable generalizations of linear modelling and principal components analysis to samples drawn from random functions. This is achieved by partitioning function space with a linear differential operator and applying polynomial spline theory with arbitrary operators and boundary constraints. The methods are illustrated through an analysis of temperature–precipitation variation across Canadian weather stations.

Abstract

SUMMARY Multivariate data analysis permits the study of observations which are finite sets of numbers, but modern data collection situations can involve data, or the processes giving rise to them, which are functions. Functional data analysis involves infinite dimensional processes and/or data. The paper shows how the theory of L-splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions. Spline smoothing rests on a partition of a function space into two orthogonal subspaces, one of which contains the obvious or structural components of variation among a set of observed functions, and the other of which contains residual components. This partitioning is achieved through the use of a linear differential operator, and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints. These data analysis tools are illustrated by a study of variation in temperature–precipitation patterns among some Canadian weather-stations.