Moments Based Functional Synchronization
Gareth M. James · 2005
A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. A common “synchronization” approach involves equating “landmarks ” such as peaks or troughs. The landmarks method can work well but will fail if marker events can not be identified or are missing from some curves. It may also involve a manual identification of marker events. We develop automated alignment methods based on equating the “moments ” of a given set of curves. These moments do not depend on the identification of markers. For example, the first moment is a measure of the average value of a curve in the x, or time, axis while the second moment measures its spread. We explore both linear and non-linear synchronization procedures. Finally, we discuss the advantages of utilizing, not only the “amplitude ” information, which measures the general shape of the curves, but also the “warping ” information, which measures the way the curves have been distorted in time. Illustrations are provided on functional analyses involving principal components, clustering, classification and regression.