Gradual Functional Changes

With the advance of computer facilities and data storage warehouses, more and more data are being recorded continuously during a time period or intermittently at plethora of discrete time points. These are both examples of functional data, which furthermore embrace random fields or manifolds. Our attention is devoted to stochastic functions predominantly represented by random curves or surfaces, where every function is considered as a single observation. These observations are naturally ordered with respect to time and possibly changing over time. The interest is not in an individual change withineach curve, but in a change of the pattern across the sequence of curves. Almost all existing change detection methods are designed to discover abrupt breaks. Little attention has been paid to smooth structural changes, which may be more realistic in practice. With the

vantage of functional analysis and empirical processes, we can deploy advanced statistical tools like bootstrap or lasso to diagnose the gradual change of functional form in the time series of random curves.