Abstract
The problem of aligning time series arises naturally in a number of fields, but our investigation is motivated by the problem of extracting a paleoclimatic signal from glacial varve chronologies. Glacial varves are laminated sediments that record annual depositional cycles in certain lakes that are fed by glaciers. Varve thickness is determined by temperature and to a lesser extent by precipitation so that an accurate reconstruction of varve chronologies can be regarded as providing a potential long term proxy for paleoclimate. An important aspect of the varve problem is to align series from different locations. What makes the alignment problem particularly interesting is that in either series, one or more varves may be missing or a single varve might have been mistaken for a pair of varves.
Traditionally, alignment problems have been treated using linear methods and concepts such as coherency and cross correlation. Since one of the pairs of varve series we look at is not well described by a linear model, in this paper we also employ nonlinear criteria that are derived from recent work on nonlinear identification and dependence modeling. More precisely, we use nonlinear least squares and measures of dependence such as the Kullback—Leibler measure and the Hellinger distance. Both local and global alignment are considered, and in particular we treat the situation where one or more observations may be missing from either series. Properties of the criteria are given and illustrated by simulation. We display examples where linear methods fail, but nonlinear criteria work. For the real data set of varves at our disposal, it will be seen that the nonlinear criteria perform as well as (or better than) the linear ones for the linearly related series, and that only the nonlinear criteria work for the two series that are not linearly related. Copyright © 2007 John Wiley & Sons, Ltd.