More and more often, the outcome of a study is not a random
variable but a noisy function for each experimental unit, resulting in a sample
of curves. Typically, the individual curves vary not only in amplitude or
intensity, but also with respect to the time axis: different subjects
experience certain events sooner or later. Analyzing such data involves finding
out the time changes (or curve registration) among curves. Following our
previous work where modified dynamic time warping is applied to align two
curves, we formulate a global minimization problem to align all curves in a
sample and to compute the aligned average curve. Algorithms for solving the
minimization problem are presented and tested with simulated and real data. The
test results are promising. The method, which involves kernel smoothing of
regression functions, estimates the time changes and the average of the aligned
curves from noisy data. Large sample asymptotics is derived.