Spencer Graves

Introduction to Functional Data Analysis

The main characteristics of functional data and of functional models are introduced. Data on the growth of girls illustrate samples of functional observations, and data on the US nondurable goods manufacturing index are an example of a single long multilayered functional observation. Data on the gait of children and handwriting are multivariate functional observations. Functional data analysis also involves estimating functional parameters describing data that are not themselves functional, and estimating a probability density function for rainfall data is an example. A theme in functional data analysis is the use of information in derivatives, and examples are drawn from growth and weather data. The chapter also introduces the important problem of registration: aligning functional features.

Registration: Aligning Features for Samples of Curves

This chapter presents two methods for separating phase variation from amplitude variation in functional data: landmark and continuous registration. We mentioned this problem in Section 1.1.1. We saw in the height acceleration curves in Figure 1.2 that the age of the pubertal growth spurt varies from girl to girl; this is phase variation. In addition, the intensity of the pubertal growth spurt also varies; this is amplitude variation. Landmark registration aligns features that are visible in all curves by estimating a strictly increasing nonlinear transformation of time that takes all the times of a given feature into a common value. Continuous registration uses the entire curve rather than specified features and can provide a more complete curve alignment. The chapter also describes a decomposition technique that permits the expression of the amount of phase variation in a sample of functional variation as a proportion of total variation.

How to Specify Basis Systems for Building Functions

This chapter is primarily about setting up a basis system. The next chapter will discuss the second step of bundling a set of coefficient values with the chosen basis system.

Smoothing: Computing Curves from Noisy Data

The previous two chapters have introduced the Matlab and R code needed to specify basis function systems and then to define curves by combining these coefficient arrays. For example, we saw how to construct a basis object such as heightbasis to define growth curves and how to combine it with a matrix of coefficients such as heightcoef so as to define growth functional data objects such as were plotted in Figure 1.1.

Exploring Variation: Functional Principal and Canonical Components Analysis

Now we look at how observations vary from one replication or sampled value to the next. There is, of course, also variation within observations, but we focused on that type of variation when considering data smoothing in Chapter 5.

Essential Comparisons of the Matlab and R Languages

We assume a working knowledge of either Matlab or R. For either language, there are many books that describe the basics for beginners. However, a brief comparison of the two languages might help someone familiar with one language read code written in the other.

Functional Linear Models for Scalar Responses

This is the first of two chapters on the functional linear model. Here we have a dependent or response variable whose value is to be predicted or approximated on the basis of a set of independent or covariate variables, and at least one of these is functional in nature. The focus here is on linear models, or functional analogues of linear regression analysis. This chapter is confined to considering the prediction of a scalar response on the basis of one or more functional covariates, as well as possible scalar covariates.

Functional Models and Dynamics

This chapter brings us to the study of continuous time dynamics, where functional data analysis has, perhaps, its greatest utility by providing direct access to relationships between derivatives that could otherwise be studied only indirectly. Although dynamic systems are the subject of a large mathematical literature, they are relatively uncommon in statistics. We have therefore devoted the first section of this chapter to reviewing them and their properties. Then we address how “principal differential analysis (PDA)” can contribute to their study from an empirical perspective.

Descriptions of Functional Data

This chapter and the next are the exploratory data analysis end of functional data analysis. Here we recast the concepts of mean, standard deviation, covariance and correlation into functional terms and provide R and Matlab functions for computing and viewing them.

Linear Models for Functional Responses

In this second chapter on the functional linear model, the dependent or response variable is functional. We first consider a situation in which all of the independent variables are scalar and in particular look at two functional analyses of variance.