  When functional data manifest amplitude and phase variations, a
commonly-employed framework for analyzing them is to take away the phase
variation through a function alignment and then to apply standard tools to the
aligned functions. A downside of this approach is that the important variations
contained in the phases are completely ignored. To combine both of amplitude
and phase variations, we propose a variant of principal component analysis
(PCA) that captures non-linear components representing the amplitude, phase and
their associations simultaneously. The proposed method, which we call
functional combined PCA, is aimed to provide more efficient dimension reduction
with interpretable components, in particular when the amplitudes and phases are
clearly associated. We model principal components by non-linearly combining
time-warping functions and aligned functions. A data-adaptive weighting
procedure helps our dimension reduction to attain a maximal explaining power of
observed functions. We also discuss an application of functional canonical
correlation analysis in investigation of the correlation structure between the
two variations. We show that for two sets of real data the proposed method
provides interpretable major non-linear components, which are not typically
found in the usual functional PCA.
