Gradual Functional Changes

随着计算机设备和数据存储仓库的进步，越来越多的数据在一段时间内被连续记录或在过多的离散时间点上间歇性地记录。这两个都是函数数据的例子，而且包含随机场或流形。我们的注意力集中在主要由随机曲线或曲面表示的随机函数上，其中每个函数都被视为单一的观测。这些观察自然是关于时间的，而且可能会随着时间的推移而变化。感兴趣的不是每条曲线内的单个变化，而是曲线序列上的图案变化。几乎所有现有的变化检测方法都是为发现突然中断而设计的。人们很少关注平稳的结构性变化，这在实践中可能更现实。借助泛函分析和经验过程的优势，我们可以利用Bootstrap或Lasso等先进的统计工具来诊断随机曲线时间序列中函数形式的渐变。

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 within each 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.