ABSTRACT
Incorporating information about derivatives is a distinctive feature of functional data analysis (FDA), as compared to traditional multivariate analysis. This chapter deals with a very popular approach to smoothing which imposes a penalty on functions that are too "wiggly". It discusses the variability in a sample curves which can be decomposed into two sources. The first is the random curve-to-curve variation; this is called amplitude variation. The second, as just discussed, constitutes possible shifting of the curves with respect to the domain; this is called phase variation. In most situations, phase variability in the sample goes together with amplitude variability. The water level in a mountain stream measured over the spring months provides one curve per year. Curve alignment and registration have been subject of continued research. An important research area which has some relation to FDA has been the application of smoothing methodology to differential equations; Ramsay et al. is a frequently cited work in this area.
