On the definition of phase and amplitude variability in functional data analysis

We introduce a mathematical framework in which a functional data reg-istration problem can be soundly and coherently set. In detail, we show that the introduction of a metric/semi-metric and of a group of warping function respect to which the metric/semi-metric is invariant is the key to a clear and not ambiguous definition of phase and amplitude variability. Moreover, an amplitude-to-total variability index is proposed. This index turns to be useful in practical situations to measure to what extent ampli-tude variability affects the data and to compare the effectiveness of different registration methods.