Curve registration

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 x_i is a function. An essential preliminary to a functional data analysis is often the registration or alignment of salient curve features by suitable monotone transformations h_i of the argument t , so that the actual analyses are carried out on the values x_i { h_i ( t )}. This is referred to as dynamic time warping in the engineering literature. In effect, this conceptualizes variation among functions as being composed of two aspects: horizontal and vertical, or domain and range. A nonparametric function estimation technique is described for identifying the smooth monotone transformations h_i , and is illustrated by data analyses. A second-order linear stochastic differential equation is proposed to model these components of variation.