  We develop a method for constructing tolerance bounds for functional data
with random warping variability. In particular, we define a generative,
probabilistic model for the amplitude and phase components of such
observations, which parsimoniously characterizes variability in the baseline
data. Based on the proposed model, we define two different types of tolerance
bounds that are able to measure both types of variability, and as a result,
identify when the data has gone beyond the bounds of amplitude and/or phase.
The first functional tolerance bounds are computed via a bootstrap procedure on
the geometric space of amplitude and phase functions. The second functional
tolerance bounds utilize functional Principal Component Analysis to construct a
tolerance factor. This work is motivated by two main applications: process
control and disease monitoring. The problem of statistical analysis and
modeling of functional data in process control is important in determining when
a production has moved beyond a baseline. Similarly, in biomedical
applications, doctors use long, approximately periodic signals (such as the
electrocardiogram) to diagnose and monitor diseases. In this context, it is
desirable to identify abnormalities in these signals. We additionally consider
a simulated example to assess our approach and compare it to two existing
methods.
