  Constructing generative models for functional observations is an important
task in statistical functional analysis. In general, functional data contains
both phase (or x or horizontal) and amplitude (or y or vertical) variability.
Tradi- tional methods often ignore the phase variability and focus solely on
the amplitude variation, using cross-sectional techniques such as fPCA for
dimensional reduction and data modeling. Ignoring phase variability leads to a
loss of structure in the data and inefficiency in data models. This paper
presents an approach that relies on separating the phase (x-axis) and amplitude
(y-axis), then modeling these components using joint distributions. This
separation, in turn, is performed using a technique called elastic shape
analysis of curves that involves a new mathematical representation of
functional data. Then, using individual fPCAs, one each for phase and amplitude
components, while respecting the nonlinear geometry of the phase representation
space; impose joint probability models on principal coefficients of these
components. These ideas are demonstrated using random sampling, for models
estimated from simulated and real datasets, and show their superiority over
models that ignore phase-amplitude separation. Furthermore, the generative
models are applied to classification of functional data and achieve high
performance in applications involv- ing SONAR signals of underwater objects,
handwritten signatures, and periodic body movements recorded by smart phones.
